A familiar anomaly affects SU(2) gauge theory in four dimensions: a theory with an odd number of fermion multiplets in the spin 1/2 representation of the gauge group, and more generally in representations of spin 2r + 1/2, is inconsistent. We describe here a more subtle anomaly that can affect SU(2) gauge theory in four dimensions under the condition that fermions transform with half-integer spin under SU(2) and bosons transform with integer spin. Such a theory, formulated in a way that requires no choice of spin structure, and with an odd number of fermion multiplets in representations of spin 4r + 3/2, is inconsistent. The theory is consistent if one picks a spin or spinc structure. Under Higgsing to U(1), the new SU(2) anomaly reduces to a known anomaly of “all-fermion electrodynamics.” Like that theory, an SU(2) theory with an odd number of fermion multiplets in representations of spin 4r + 3/2 can provide a boundary state for a five-dimensional gapped theory whose partition function on a closed five-manifold Y is (1)Yw2w3. All statements have analogs with SU(2) replaced by Sp(2N). There is also an analog in five dimensions.

An SU(2) gauge theory with an odd number of fermion multiplets that transform in the spin 1/2 representation of the gauge group is inconsistent.1 More generally, fermion multiplets that transform under SU(2) in the spin 2r + 1/2 representation—for brevity, we will call these multiplets of isospin 2r + 1/2—contribute to this anomaly: an SU(2) gauge theory with an odd number of fermion multiplets of isospin 2r + 1/2 is inconsistent.

This anomaly is most systematically described, as in Ref. 1, using the fact that π4(SU(2))=Z2 and the relation of this to the mod 2 index of the Dirac operator in five dimensions. However, as we will explain in Sec. II, there is a more elementary way to see the anomaly using the fact that in SU(2) gauge theory, the number of fermion zero-modes in an instanton field can be odd (Note 1 in the  Appendix).

This familiar SU(2) anomaly receives contributions from fermion multiplets of isospin 2r + 1/2 but not from multiplets in other representations. However, in the present paper, we will describe a similar but more subtle anomaly in SU(2) gauge theory that receives contributions from (and only from) fermion multiplets of isospin 4r + 3/2. The new anomaly depends on a refinement of what one might mean by SU(2) gauge theory. We consider an SU(2) gauge theory in which all fermion fields have half-integral isospin and all boson fields have integral isospin. Such a theory has no gauge-invariant local fields that are fermionic, so as in Ref. 2, one can view it as a possible critical point in a theory that microscopically is constructed from bosons only. In fact, such a theory can be formulated without a choice of spin structure, using only what we will call a spin-SU(2) structure, which is the SU(2) analog of a spinc structure. (For discussion of such structures, see, for example, Refs. 3–5.)

The new anomaly only arises when the theory is formulated in this generality. If one formulates an SU(2) gauge theory with fermions only on spin manifolds, one will see only the usual anomaly associated with π4(SU(2))=Z2. Thus, there is a fundamental difference between an SU(2) gauge theory that has the “old” anomaly and one that only has the “new” anomaly. An SU(2) gauge theory with the old anomaly—for example, a theory with a single fermion multiplet of isospin 1/2—is simply inconsistent, even locally, and it does not make sense to ask about its dynamics. An SU(2) gauge theory with only the new anomaly—for example, a theory with an odd number of fermion multiplets of isospin 3/2—is perfectly consistent as a conventional quantum field theory in Minkowski spacetime, so it makes sense to analyze its dynamics (see Ref. 6 for an attempt). The anomaly means that this theory cannot be formulated consistently in a general spacetime without some additional structure and cannot arise as an effective low energy description of a theory that microscopically has bosons only. Examples of the additional structure that do make it possible to formulate this theory consistently are a choice of a spin or spinc structure.

Roughly, a theory with the new anomaly is invariant under diffeomorphisms plus gauge transformations that preserve a spin structure (or some other suitable structure such as a spinc structure) but not under the larger class of diffeomorphisms plus gauge transformations that preserve a spin-SU(2) structure.

In Sec. II, we review the familiar SU(2) anomaly and describe the new one. In Sec. III, we describe the effects of gauge symmetry breaking. A theory with the new SU(2) anomaly can be Higgsed to a U(1) gauge theory that has a similar anomaly. This is a theory of “all-fermion electrodynamics”7 in which electrons, monopoles, and dyons of odd charges are all fermions. Further Higgsing leads to a theory that locally is a Z2 gauge theory but more globally can be described as a theory with a dynamical spin structure. In Sec. IV, we take a different perspective. A theory with the new SU(2) anomaly can arise as a boundary state of a five-dimensional gapped theory with SU(2) global symmetry. More interestingly and more surprisingly, it can also arise as a boundary state for a five-dimensional theory with no particular symmetry that is gapped but topologically nontrivial; its partition function on a closed five-manifold Y is (Note 2 in the  Appendix) (1)Yw2(TY)w3(TY). The anomalous SU(2) gauge theory gives a novel boundary state for that theory. Its Higgsing to U(1) gives a boundary state that is already known;8 further Higgsing to Z2 gives a gapped boundary state with a dynamical spin structure.

In most of this paper, we assume for simplicity that all four-manifolds and five-manifolds are orientable, and we consider only the gauge group SU(2). In Sec. V, we consider other gauge groups, including Sp(2N), SO(3) × SU(2), and Spin(10). For Sp(2N), one gets a very close analog of what happens for SU(2). For SO(3) × SU(2), we make contact with a recent analysis.5 For Spin(10), we explain that the grand unified theory with that gauge group can be formulated without a choice of spin structure. A generalization to unorientable manifolds and a similar anomaly in five dimensions are also briefly discussed in Sec. V.

The Lorentz group SO(1,3) or more precisely its spin double cover Spin(1, 3) has two different irreducible spinor representations, each of dimension two. They correspond to fermions of left-handed or right-handed helicity. These representations are complex conjugates of each other, so a left-handed fermion field ψα, α = 1, 2 in Lorentz signature is the hermitian adjoint of a right-handed fermion field ψ̃α̇, α̇=1,2.

Now, consider an SU(2) gauge theory in four dimensions coupled to a single multiplet of fermions of isospin 1/2, that is, to a single SU(2) doublet of left-handed fermions ψαi, where α = 1, 2 is a Lorentz index and i = 1, 2 is an SU(2) index, and a corresponding doublet of right-handed fermions ψ̃α̇j. In the Lorentz signature, since the two spinor representations are complex conjugates, ψ̃α̇j are simply the hermitian adjoints of ψαi. The theory can hence be viewed as a theory of eight hermitian Fermi fields (the hermitian parts and i times the antihermitian parts of ψαi).

As in this example, fermion fields in the Lorentz signature always carry a real structure; if a fermion field appears in the Lagrangian, so does its hermitian adjoint. In the Euclidean signature, nothing like that is true in general; what happens depends on the spacetime dimension and the gauge or global symmetry group assumed. But four-dimensional fermions in a pseudoreal representation of the gauge group are real in the Euclidean signature. That is because the two spinor representations of Spin(4) are each pseudoreal, as is the isospin 1/2 representation of SU(2). In general, the tensor product of two pseudoreal representations is real, so fermion fields ψαi or ψ̃α̇i valued in the isospin 1/2 representation of SU(2)—or more generally in a pseudoreal representation of any gauge and/or global symmetry—carry a real structure. Note that ψαi and ψ̃α̇j are each separately real in the Euclidean signature, while in the Lorentz signature they are complex conjugates of each other.

Because the fermion fields ψα,i and ψ̃α̇,j carry a natural real structure, the Dirac operator that acts on them is also naturally real. Let us spell out in detail what this assertion means. In doing this, we consider a combined 8 component Dirac operator D4 that acts on the four components of ψαi and the four components of ψ̃α̇j together (Note 3 in the  Appendix). This Dirac operator is

D4=μ=14γμxμ+a=13Aμata,
(2.1)

where Aμa is the SU(2) gauge field, γμ, μ = 1, …, 4, are gamma matrices, and ta, a = 1, …, 3, are antihermitian generators of the SU(2) Lie algebra. It is not possible to find 4 × 4 real gamma matrices obeying the Clifford algebra {γμ, γν} = 2δμν. Similarly, it is not possible to find 2 × 2 real matrices ta obeying the SU(2) commutation relations [ta, tb] = εabctc (where ε denotes the antisymmetric tensor). The nonexistence of such matrices is more or less equivalent to the assertion that the spinor representations of Spin(4) and the isospin 1/2 representation of SU(2) are pseudoreal, not real. However, to write a real Dirac equation D4 for an eight-component fermion field that combines ψai and ψ̃ȧj, we want not 4 × 4 real gamma matrices but 8 × 8 ones. There is no difficulty with this, and moreover, a set of four real 8 × 8 gamma matrices γμ will commute with a set of three real ta’s that satisfy the SU(2) commutation relations. This statement is more or less equivalent to the fact that the tensor product of the two representations is real.

We will also be interested in what happens in five dimensions. In the Lorentz signature, the group Spin(1, 4) has a unique spinor representation, which is a pseudoreal representation of dimension 4. The tensor product of this representation with the isospin 1/2 representation of SU(2) is real, so it is possible in 4 + 1 dimensions to have a theory with a single SU(2) doublet of fermions ψαi, where now α = 1, …, 4 is a Lorentz spinor index and i = 1, 2 is an SU(2) index. This field has eight hermitian components, just as in four dimensions.

Upon going to the Euclidean signature, we observe that the spinor representation of Spin(5) is pseudoreal, so the field ψαi remains real in the Euclidean signature. The Euclidean signature Dirac operator for this field is therefore also real. Concretely, if γ1, …, γ4 are four real 8 × 8 gamma matrices that commute with real SU(2) generators ta, then γ5 = γ1γ2γ3γ4 is a fifth real 8 × 8 gamma matrix that commutes with the same SU(2) generators. So once one has a real four-dimensional Dirac operator (2.1), there is no additional difficulty to construct a real five-dimensional Dirac operator

D5=μ=15γμxμ+a=13Aμata.
(2.2)

The only property of the isospin 1/2 representation of SU(2) that was important in any of this is that it is pseudoreal. So all statements carry over immediately to the representation of isospin j, for any half-integer j.

Now, let us return in four dimensions to the theory of a single multiplet of fermions of isospin 1/2, that is, the theory of a left-handed fermion field ψαi and its hermitian conjugate. Such a field cannot have a bare mass. Because the spinor representation of the Lorentz group and the j = 1/2 representation of SU(2) are both pseudoreal, the Lorentz and SU(2) symmetry would force a mass term to be εαβεijψαiψβj + h.c. This expression vanishes by Fermi statistics as it is antisymmetric in both Lorentz and gauge indices.

The impossibility of a bare mass means that such a fermion can potentially contribute an anomaly, and this is what actually happens, as we discuss starting in Sec. II B. More generally, the same holds if the isospin 1/2 representation of SU(2) is replaced by any irreducible, pseudoreal representation of an arbitrary gauge and/or global symmetry group G. In such a representation, a G-invariant bilinear form is antisymmetric, so a mass term would have to be εαβκijψαiψβj + h.c., with κij antisymmetric. This again vanishes by Fermi statistics. In this paper (except in Sec. V A), the pseudoreal representations we will consider will be the isospin j representations of SU(2), with half-integer j. The absence of bare masses again means that such fermion fields can contribute anomalies.

There is no problem in giving a bare mass to a pair of fermion multiplets in the isospin j representation of SU(2), for half-integer j. (If the two multiplets are ψ and χ, then obviously Fermi statistics do not constrain a mass term ψαiχβjεαβκij.) More generally, one can give bare masses to any even number of isospin j multiplets. So a possible anomaly can only be a mod 2 effect, sensitive to whether the number of multiplets of a given kind is even or odd.

By contrast, for integer j, the spin j representation of SU(2) is real and a single multiplet of fermions transforming in that representation could have a bare mass and hence could not contribute a gauge anomaly. For this reason, the interesting SU(2) representations for our purposes in this paper are the ones of half-integer j.

The following is the simplest illustration of the usual SU(2) anomaly for a single fermion multiplet of isospin 1/2. Consider an SU(2) gauge field of instanton number 1 on a four-sphere M = S4. Let nL and nR be the number of zero-modes of ψL and ψR, and let n = nL + nR be the total number of fermion zero-modes.

The Atiyah-Singer index theorem says in this situation that

nLnR=1.
(2.3)

We note that n = nL + nR is congruent to nLnR mod 2, so

n1  mod2.
(2.4)

Having an odd number of fermion zero-modes in an instanton field is a kind of anomaly. It means that the path integral measure, for integration over the fermions in the instanton background, is odd under the operator (−1)F that counts fermions mod 2. This operator can be viewed as a diffeomorphism—a trivial diffeomorphism taken to act on fermions as −1—or as a gauge transformation by the central element −1 of SU(2). The second interpretation shows that an anomaly in (−1)F can be interpreted as a breakdown of SU(2) gauge-invariance.

Obviously, having an odd number of zero-modes, each of which is odd under (−1)F, means that the path integral measure for the zero-modes is odd under (−1)F. On the other hand, the path integral measure for the nonzero fermion modes—that is, for all fermion modes other than zero-modes—is invariant under all symmetries of the classical action, including (−1)F.

This statement is fairly obvious in the present instance because nonzero modes come in pairs (Note 4 in the  Appendix) and a pair of fermion modes is invariant under (−1)F. Let us, however, pause to put the point in a more general perspective. Fermion anomalies are elementary to analyze when one considers anomalies in symmetries that leave fixed a given bosonic background. To understand anomalies in a more general situation requires more sophisticated arguments, as in Ref. 1 and later papers. But in the present paper, it will suffice to consider the easy case of symmetries that leave fixed the bosonic background. What makes this case easy is that if φ is any global and/or gauge symmetry that leaves fixed a given bosonic background, then φ-invariance of the action that describes fermions propagating in that background

I=Md4xgψ¯iDψ+
(2.5)

(where the ellipses refer to possible mass terms or additional couplings) means that the nonzero fermion modes are paired up in an φ-invariant way, ensuring invariance of the path integral measure for those modes. So a possible anomaly can be computed just by asking whether the measure for the fermion zero-modes is φ-invariant.9,10

Now, let us explain how this anomaly is related to a topological invariant in five dimensions, as described in Ref. 1. In general, let M be a four-manifold with some SU(2) gauge field and some choice of spin structure. Let φ be a symmetry whose potential anomaly we want to consider. Here, φ might be a diffeomorphism, a gauge transformation, a global symmetry, or a combination thereof. We use φ to construct a five-manifold Y known as the mapping torus of φ. The construction is as follows. Begin with the five-manifold Y0 = M × I, where I is the unit interval 0 ≤ t ≤ 1. If φ is a symmetry of the metric g and gauge field A of M, we can take the metric on Y0 to be a product, the gauge field on Y0 to be a pullback from M (meaning that it is independent of t and has no component in the t direction), and the spin structure of Y0 to be a pullback from M. In general, if g and A are not φ-invariant, one picks a metric and gauge field on Y0 that interpolate smoothly between g, A at t = 0 and gφ, Aφ at t = 1 (where gφ and Aφ are the transforms of g and A by φ). Finally, to construct Y with its gauge bundle, we glue together the two ends M × 0 and M × 1 of Y0, applying the symmetry φ in the process: thus, for xM, the point x × 0 is identified with φx × 1, and similarly, any bosonic or fermionic field χ(x × 0) is set equal to χφ(φx × 1) (where χφ is the transform of χ by φ).

The main result of Ref. 1 is that the fermion path integral for a fermion field on M in a given representation of SU(2) is φ-invariant if and only if the mod 2 index of the Dirac operator on Y—in that representation—vanishes. Here, the mod 2 index is defined as the number of zero modes of the five-dimensional Dirac operator, reduced mod 2. The mod 2 index is governed by a more subtle variant of the usual Atiyah-Singer index theorem.11 For an introduction to the mod 2 index, see Sec. III B of Ref. 12. Briefly, whenever one can write a fermion action ψ¯(D+)ψ, the number of fermion zero-modes is always a topological invariant mod 2 (provided one counts all fermion modes, not distinguishing fermions from antifermions in theories in which such a distinction could be made). That is because Fermi statistics means that the fermion kinetic operator D + ⋯ is antisymmetric. The canonical form of an antisymmetric matrix is

0aa00bb000
(2.6)

with nonzero modes that come in pairs and zero-modes that are not necessarily paired. When background fields are varied, the number of zero-modes can only change when one of the skew eigenvalues a, b, ⋯ becomes zero or nonzero. When this happens, the number of zero-modes changes by 2, so the number of zero-modes mod 2 is always a topological invariant. It is known as the mod 2 index. The mod 2 index is not in general the mod 2 reduction of an integer-valued invariant such as an index; indeed, there is no integer-valued index for a Dirac operator in five dimensions.

The mod 2 index is difficult to calculate in general; the mod 2 version of the index theorem11 provides a somewhat abstract description, not what one would usually regard as a formula. However, the mod 2 index is relatively easy to calculate if one considers a mapping torus built using a symmetry φ that leaves invariant the classical background. Let us consider our example with φ = (−1)F. This symmetry acts trivially as a diffeomorphism, and it also acts trivially on the gauge field A. The mapping torus in this case is just an ordinary product Y = M × S1, with a gauge field that is a pullback from M. The meaning of the twist by (−1)F is just that we should use on Y a spin structure in which the fermions are periodic in going around the S1. Now, we can easily compute the mod 2 index. To get a fermion zero-mode on M × S1, we have to take a fermion field that has zero momentum in the S1 direction and that satisfies the four-dimensional Dirac equation on M. So in this case, the fermion zero-modes in five dimensions are the same as the fermion zero-modes in four dimensions, and the mod 2 index is simply the number n of fermion zero-modes in four dimensions, reduced mod 2. Thus, the general statement that the fermion anomaly in four dimensions is given by the mod 2 index in five dimensions reduces in this situation to the more obvious statement that we began with: there is an anomaly if the number of fermion zero-modes in the original four-dimensional problem is odd.

More generally, the mod 2 index in five dimensions can be computed in four-dimensional terms whenever we consider a symmetry φ that leaves fixed the bosonic background. Before explaining this, we should explain the following. As described in Sec. II A, in the Euclidean signature, a five-dimensional fermion field in the spin 1/2 representation of SU(2) can be given a real structure because the spinor representation of Spin(5) is pseudoreal, as is the spin 1/2 representation of SU(2). Any diffeomorphism and/or gauge transformation φ preserves this real structure, and this constrains the possible action of φ. A complex eigenvalue eiα, α ≠ 0, π of φ will always be paired with a complex conjugate eigenvalue e−iα, with the complex conjugate wavefunction. In addition, φ might have eigenvalues ±1, and these are not necessarily paired. The anomaly of the fermion measure under φ is just the determinant of φ, regarded as a linear operator on the space of fermion zero-modes. Since a complex conjugate pair of eigenvalues e±iα does not contribute to this determinant, the determinant and therefore the anomaly are just (1)m1, where m−1 is the dimension of the space of fermion zero-modes with φ = −1. As a check on this, suppose we consider the symmetry (−1)Fφ instead of φ. Now, we are interested in eigenvalues of (−1)Fφ with eigenvalue −1. These are the same as eigenvalues of φ with eigenvalue +1, so now the anomaly is (1)m1, where m1 is the dimension of the space of fermion zero-modes with φ = +1. Note that the total number n of fermion zero-modes is

nm1+m1   mod2
(2.7)

because the paired modes with eigenvalues e±iα do not contribute to n mod 2. So

(1)m1=(1)n(1)m1,
(2.8)

saying, as one would expect, that the anomaly (1)m1 of the fermion path integral under φ(−1)F is the product of the anomaly (−1)n under (−1)F and the anomaly (1)m1 under φ. In particular, if there is no anomaly under (−1)F, then n is even, m1m−1 mod 2, and it does not matter whether we consider the anomaly under φ or under (−1)Fφ.

If φ acts nontrivially as a diffeomorphism (but is still a symmetry of the bosonic background), then the mapping torus is not a simple product M × S1 but a sort of “semidirect product” MS1. The spin structure in the S1 direction depends on whether the symmetry we want to consider is φ or (−1)Fφ. If the spin structure in the S1 direction is periodic, then a fermion zero-mode on MS1 corresponds to a φ-invariant zero-mode on M, so the five-dimensional mod 2 index is m1. If the spin structure is antiperiodic, a fermion zero-mode on MS1 corresponds to a mode on M with eigenvalue −1 of φ, so the mod 2 index is m−1. Let us for simplicity consider the case that there is no anomaly in (−1)F so that m1 and m−1 are equal. Then, (1)m1=(1)m1, and as we have seen, each of these corresponds to the transformation under φ of the fermion path integral measure in four dimensions. This confirms, in this situation, that the mod 2 index in five dimensions governs the anomaly in four dimensions. [To extend this to the case that there is an anomaly under (−1)F requires some care to decide which is which of φ and (−1)Fφ; one has an anomaly and one does not.]

For our purposes in the present paper, it will suffice to consider anomalies under symmetries φ that leave fixed a bosonic background so that the anomaly can be computed directly in terms of fermion zero-modes in four dimensions. This will suffice because every anomaly of interest can be illustrated by a special case with that property. The general relation between the anomaly in four dimensions and the mod 2 index in five dimensions can be deduced by an argument using spectral flow of a family of four-dimensional Dirac operators.1 

Now, let us consider the generalization of the familiar SU(2) anomaly that we have just reviewed to the case of fermions in some other representation of SU(2).

As we have seen, there is an anomaly when the number of fermion zero-modes in an instanton field is odd. Moreover, the number of zero-modes mod 2 is given by the Atiyah-Singer index theorem. The contribution to the index that depends on the gauge fields is a multiple of ∫MTr FF, where F is the Yang-Mills curvature and the trace is taken in the fermion representation. In a representation of isospin j, the quantity FF is a multiple of j(j + 1) and its trace is a multiple of j(j + 1)(2j + 1). The index is 1 for j = 1/2, so in general, it is 23j(j+1)(2j+1). The number nj of zero-modes of a fermion multiplet of isospin j in a gauge field on M = S4 of instanton number 1 is congruent to this mod 2

nj23j(j+1)(2j+1)   mod2.
(2.9)

This number is odd if j is of the form 2r + 1/2, rZ, and otherwise, it is even. So the general statement of the usual SU(2) anomaly is that the total number of fermion multiplets of isospin 2r + 1/2 must be even. It is possible to have, for example, a consistent theory with a single fermion multiplet of isospin 1/2 and another of isospin 5/2.

The usual SU(2) anomaly does not receive a contribution from a fermion multiplet of isospin 2r + 3/2. But we will see that there is a more subtle anomaly for a fermion multiplet of isospin 4r + 3/2, rZ. (The case 4r + 7/2 remains anomaly-free.)

If we consider SU(2) gauge theory on a spin manifold, there is no further anomaly beyond what we have described. But there is something more subtle that we can do.

Consider an SU(2) gauge theory in which fermions are in half-integer spin representations of SU(2) and bosons are in integer spin representations. In this case, we can have the option to formulate the theory on a four-manifold M without a choice of spin structure, choosing a weaker structure that we will call a spin-SU(2) structure (Note 5 in the  Appendix). A spin-SU(2) structure is the SU(2) analog of a spinc structure in the case of U(1). A spinc structure is appropriate for a theory that, locally, has gauge group U(1) but with the property that fermions have odd U(1) charge and bosons have even U(1) charge. This means (in four spacetime dimensions) that not all representations of Spin(4) × U(1) occur in the theory but only those representations that are actually representations of

Spinc(4)=(Spin(4)×U(1))/Z2.
(2.10)

Here, Z2 is embedded in Spin(4) × U(1) as the product of the element (−1)F ∈ Spin(4) and the element −1 ∈ U(1). A spinc connection is a connection with structure group Spinc(4) and with the property that if one forgets U(1) and projects Spinc(4)=(Spin(4)×U(1))/Z2 to Spin(4)/Z2=SO(4), the SO(4) connection is the Riemannian connection for some Riemannian metric on M.

A theory in which all fields provide representations of Spinc(4) [as opposed to more general representations of Spin(4) × U(1)] can be formulated on a four-manifold M that is endowed with a spinc structure [as opposed to a spin structure and a U(1) gauge field]. Concretely, a Riemannian metric plus a spinc structure gives the information one needs to define parallel transport for fermions of odd charge and for bosons of even charge. One does not have the additional structure that would be needed to define parallel transport for a fermion of even charge (for example, a neutral fermion) or a boson of odd charge.

Locally, a spinc connection is equivalent to a U(1) gauge field A, but globally this is not the case. The difference is most striking if M is such as to not admit a spin structure. The obstruction to a spin structure is the Stiefel-Whitney class w2(TM), where TM is the tangent bundle of M. Suppose that SM is a closed oriented two-manifold with ∫Sw2(TM) ≠ 0. This is an element of Z2 because w2 is a Z2-valued cohomology class. Then, the magnetic flux F = dA for a spinc connection does not obey conventional Dirac quantization, but rather

SF2π=12Sw2(TM)  modZ.
(2.11)

This implies that 2F does obey conventional Dirac quantization; 2F is the curvature of the gauge connection 2A, which is a conventional U(1) gauge field that can couple a charge 2 boson. Such a boson is a section of a complex line bundle L^M, and 2A is a connection on this line bundle.

A simple example of a four-manifold M with w2(TM) ≠ 0 is CP2, which can be parameterized by homogeneous complex coordinates z1, z2, z3, obeying ∑i|zi|2 = 1 and with the equivalence relation zieiαzi for real α. A CP1 subspace of CP2 is defined by setting, for example, z3 = 0. One has

CP1w2(TM)=1   mod2.
(2.12)

So to define a spinc structure, we need

CP1F2π=12+n,    nZ.
(2.13)

If we ask for a spinc structure to be invariant under the SU(3) [actually SU(3)/Z3 symmetries] of CP2, then it is uniquely determined by the flux condition (2.13). The basic case is to set n = 0, that is, one-half a unit of flux. In this case, the complex line bundle L^M is the fundamental line bundle over CP2; a charge 2 boson couples to a single flux quantum on CP1. The Atiyah-Singer index theorem can be used to compute the index I of the Dirac operator for a fermion coupled to a spinc structure with flux f = n + 1/2

I=4f218=n(n+1)2.
(2.14)

So, for example, for f = 1/2, one has I=0; for f = 3/2, one has I=1. It can be shown, using the positive curvature of CP2, or by relating the question to algebraic geometry, that this Dirac equation has zero-modes only of one chirality so that I is the total number of zero-modes for this Dirac equation.

Now, let us return to the SU(2) case. Here, we consider a theory in which all fields are representations of what we will call

SpinSU(2)(4)=(Spin(4)×SU(2))/Z2,
(2.15)

rather than more general representations of Spin(4) × SU(2). By a spin-SU(2) structure, we mean a connection with structure group SpinSU(2)(4), which lets one define parallel transport for representations of this group but not for general representations of Spin(4) × SU(2). [Again the SO(4) part comes from the Riemannian geometry.] Any example of a spinc structure, say one with flux f, gives an example of a spin-SU(2) structure, simply by embedding U(1) in SU(2). In other words, if A is any spinc connection, then we view

A^=A00A
(2.16)

as the SU(2) part of a spin-SU(2) connection. For example, the spin 1/2 representation of SU(2) decomposes under U(1) as the direct sum of representations of charges 1 and charge −1, so a fermion of isospin 1/2 has two components, which, respectively, see background spinc connections with flux f and −f, where f is the flux of A. For another example, the spin 3/2 representation of SU(2) decomposes under U(1) as the direct sum of representations of charges 3, 1, −1, and −3, so a fermion of isospin 3/2 has four components, which, respectively, see spinc connections of fluxes 3f, f, −f, and −3f.

Using (2.14), we can determine the index Ij of the Dirac operator for a fermion of spin 1/2 and isospin j from the case that A^ is constructed from a spinc connection A of flux f. For our purposes, it will suffice to consider the basic case f = 1/2. We can get Ij by simply summing the abelian answer (2.14) over the various components of an isospin j fermion field. This gives

I1/2=0,   I3/2=2,   Ij=f{j,j1,,j}4f218=124(4j21)(2j+3).
(2.17)

Just as in the spinc case, the zero-modes of such a fermion are all of the same chirality, and the number of such zero-modes is given by Ij.

For j = k + 1/2, we have

Ij=13k(k+1)(k+2).
(2.18)

Ij is always even, so an example of this kind will never give an anomaly in (−1)F. (For even k, a more simple example gives such an anomaly, as reviewed in Sec. II B.) However, we will see that a more subtle anomaly receives a contribution whenever Ij is not divisible by 4. From Eq. (2.18), the condition for this is that k ≅ 1 mod 4 or in other words that j is of the form 4r + 3/2, rZ.

To find a new anomaly, we consider the diffeomorphism φ of CP2 that acts by complex conjugation, that is, by ziz¯i, where zi are the homogeneous coordinates of CP2. This diffeomorphism reverses the orientation of any subspace CP1CP2, so it reverses the sign of the flux f of any spinc structure. Since f is a half-integer, it cannot vanish, and hence, there is no φ-invariant spinc structure on CP2.

However, there is no difficulty to find a spin-SU(2) structure that is invariant under φ combined with a suitable gauge transformation. We just take the spinc structure that is obtained by embedding U(1) in SU(2) diagonally as in Eq. (2.16), and we combine φ with the gauge transformation

W=0110,
(2.19)

which also reverses the sign of the flux. We write φ^ for the combination of the diffeomorphism φ with this gauge transformation.

Starting with any value of the flux of the underlying spinc connection, this construction gives a φ^-invariant spin-SU(2) structure on CP2. For our purposes, we will consider the basic case that the spinc connection has flux f = 1/2. We will consider a spin 1/2 fermion of half-integral isospin j coupled to this spin-SU(2) structure. We want to know if there is a quantum anomaly in the action of φ^.

Since φ^ is a symmetry of the classical background, all we have to do, as explained in Sec. II B, is to compute the determinant of φ^ in its action on the space of fermion zero-modes. The following fact will simplify the analysis. For any half-integer j, the space of zero-modes of the Dirac operator acting on a fermion field of spin 1/2 and isospin j is real. That is simply because the spin 1/2 representation of Spin(4) is pseudoreal, and the spin j representations of SU(2) are also pseudoreal, so their tensor products are real representations of the group SpinSU(2)(4)=(Spin(4)×SU(2))/Z2.

The space of zero-modes has some additional crucial properties. It has a U(1) symmetry that comes simply from constant diagonal gauge transformations in SU(2). These are symmetries of the spin-SU(2) structure, which was constructed in Eq. (2.16) using a diagonal embedding of U(1) in SU(2). Moreover, all zero-modes have odd U(1) charges, and in particular, none of them are U(1)-invariant. That is so simply because the spin j representation of SU(2) decomposes under U(1) as a sum of components of charges 2j, 2j − 2, …, −2j, all of which are odd (for half-integer j). Finally, φ^ commutes with the diagonal U(1) generator. That is because the gauge transformation W in (2.19) that is used in defining φ^ anticommutes with the generator of the diagonal U(1) symmetry in Eq. (2.16). Of course, the reason for this is that W was chosen to reverse the sign of the U(1) flux, to compensate for the fact that this flux is odd under ziz¯i.

The symmetry φ^ acting on the isospin j representation of (Spin(4)×SU(2))/Z2 obeys φ^2=1. This fairly subtle fact can be understood as follows. As a symmetry of CP2, φ has a codimension 2 fixed point set on which zi are all real. Since the fixed point set is of codimension 2, near a fixed point, we can pick real coordinates x1, ⋯, x4 such that x3, x4 are odd under φ and x1, x2 are even. Accordingly, φ acts near the fixed point set by (x1, x2, x3, x4) → (x1, x2, −x3, −x4). One can think of this as a π rotation of the x3x4 plane; its square is a 2π rotation and acts on fermions as −1. But φ^ is defined by combining the geometrical transformation φ with the gauge transformation in (2.19), which also squares to −1. So φ^2=+1. Note that there is no freedom to modify W to change this conclusion. Since W has to anticommute with a diagonal U(1) generator to make the construction possible, and since we want det W = 1 so that W is valued in SU(2), there is no way to avoid the fact that W2 = −1.

Since φ^2=1 and φ^ anticommutes with the U(1) generator, φ^ and the unbroken U(1) together generate a group of symmetries of the space of fermion zero-modes that is isomorphic to O(2). The group O(2) has irreducible representations of dimension 1 in which U(1) acts trivially and irreducible representations of dimension 2 in which U(1) acts nontrivially.

In a two-dimensional real representation of O(2) in which U(1) acts with charges ±n, the U(1) generator is, in a suitable basis,

0nn0.
(2.20)

Also, φ^ is conjugate by an element of U(1) to

1001.
(2.21)

In particular, in any such representation of O(2), φ^ has one eigenvalue +1 and one eigenvalue −1; its determinant in this two-dimensional space is −1. A quicker way to explain this is that since φ^2=1, its possible eigenvalues are 1 and −1; since φ^ anticommutes with the U(1) generator, it has one eigenvalue of each type.

The dimension of the space of zero-modes is Ij, defined in Eqs. (2.17) and (2.18); since there are no 1-dimensional representations of O(2) among the zero-modes, the space of zero-modes consists of Ij/2 representations each of which is of dimension 2. The determinant of φ^ in each such 2-dimensional representation is −1. So its determinant in the full space of zero-modes is

detφ^=(1)Ij/2.
(2.22)

When this determinant is −1, the path integral measure is odd under the combined diffeomorphism plus gauge transformation φ^. This is our anomaly. An anomaly-free theory has detφ^=+1. In view of Eq. (2.18), in a theory that is free of this anomaly, the total number of fermion multiplets of isospin 4r + 3/2 is even. This is in addition to requiring that the total number of fermion multiplets of isospin 2r + 1/2 should be even, to avoid the usual SU(2) anomaly.

The new SU(2) anomaly has a five-dimensional interpretation that is quite analogous to what we explained for the old one.

We use the diffeomorphism plus gauge transformation φ^ to build a mapping torus, which in this case we may describe as a “semidirect product” (Note 6 in the  Appendix) CP2S1, with a certain spin-SU(2) structure. We recall that this is constructed starting with CP2×I, where I is a unit interval. The metric of CP2×I is taken to be a product metric, and the spin-SU(2) structure is pulled back from CP2. Then, one glues together the two ends of CP2×I, via the diffeomorphism plus gauge transformation φ^, to make a closed five-manifold CP2S1 with a spin-SU(2) structure. It is fibered over S1 with fiber CP2; the monodromy when one goes around S1 is φ^.

A fermion zero-mode on CP2×I is a φ^-invariant fermion zero-mode on CP2. So in view of the analysis in Sec. II E, the five-dimensional mod 2 index is I/2, where I is the dimension of the space of zero-modes in four dimensions. Thus, the mod 2 index in five dimensions is nonzero precisely when I/2 is odd, that is, precisely when the theory is anomalous.

The following is one systematic way to understand the fact that the old and new SU(2) anomalies both have five-dimensional interpretations in terms of a mod 2 index. In general, anomalies for fermions in D dimensions are always governed by a corresponding η-invariant in D +1 dimensions. This was originally shown in Ref. 21, with an important later refinement in Ref. 22. For a theory with no perturbative anomaly, the η-invariant in question is a cobordism invariant, so, in particular, for D-dimensional fermions with no perturbative anomaly, a possible global anomaly is governed by a cobordism invariant in D + 1 dimensions. When the relevant D + 1-dimensional Dirac operator is real—and in certain other situations—the η-invariant reduces to a mod 2 index. (This is explained in Ref. 12, for example.) We are in that situation for the old and new SU(2) anomalies in four dimensions.

We found the new SU(2) anomaly as an anomaly in a combined gauge transformation plus diffeomorphism in a particular example. The new SU(2) anomaly is never an anomaly in a gauge transformation only. A partial explanation of this is that to test for an anomaly under a gauge transformation φ of some spin-SU(2) bundle on a four-manifold M, we would have to compute a mod 2 index on M × S1, with a spin-SU(2) structure that depends on φ. But in Sec. IV B, by relating the new anomaly to the five-dimensional invariant ∫w2w3, we will show that it always vanishes on M × S1, regardless of the spin-SU(2) structure. For a more complete explanation, we show in Sec. IV G how to define the partition function of a theory with the new SU(2) anomaly on a four-manifold M endowed with a suitable additional structure (a spin or spinc structure). This shows that the theory is invariant under SU(2) gauge transformations, though not in general under diffeomorphisms that do not preserve the additional structure used in defining it.

We have identified two types of anomalies in SU(2) gauge theory in four dimensions. The simplest example of the usual SU(2) anomaly is a theory with a single multiplet of fermions in the isospin 1/2 representation of SU(2). The simplest example of the new anomaly is a theory with a single multiplet of fermions in the isospin 3/2 representation of SU(2).

In each case, these are discrete, mod 2 anomalies. A mod 2 anomaly in four dimensions should be related to a mod 2 topological invariant in five dimensions. We have identified the relevant invariants. Let j be a half-integer (Note 7 in the  Appendix), and let us denote as Ij the mod 2 index in five dimensions for a fermion in the isospin j representation. The five-dimensional invariants associated to the usual SU(2) anomaly and the new one are I1/2 and I3/2.

We have shown that I1/2 and I3/2 are different from each other and are each nontrivial by giving examples. An example with I1/20, I3/2=0 is provided by Y1 = S4 × S1, with an SU(2) bundle of instanton number 1 on S4. An example with I1/2=0, I3/20 is provided by Y2=CP2S1, as discussed in Sec. II F.

What about Ij with j > 3/2? Are these new invariants? The answer to this question is “no.” In any dimension, the mod 2 index of the Dirac operator is a cobordism invariant whenever it can be defined. Thus, in particular, for any half-integer j, Ij is a cobordism invariant of a five-manifold Y with spin-SU(2) structure. [This statement means that if Y is the boundary of a six-manifold Z and the spin-SU(2) structure of Y extends over Z, then Ij(Y)=0 for all j.] On the other hand, as shown in Refs. 13 and 14, slightly extending computations in Ref. 4, the group of cobordism invariants for five-manifolds with spin-SU(2) structure is Z2×Z2. So any set of two independent Z2-valued cobordism invariants for such manifolds is a complete set. In particular, I1/2 and I3/2 are a complete set of invariants, and any Ij can be expressed in terms of them. In view of the computations in Sec. II for the special cases of Y1 and Y2, we have in general I2r+1/2=I1/2, I4r+3/2=I3/2, and I4r+7/2=0.

Among our examples, Y1 is a spin manifold, but Y2 has only a spin-SU(2) structure. Thus, our examples show that I1/2 can be nonzero on a spin manifold but do not demonstrate this for I3/2. In fact, I3/2 is identically zero on a spin manifold. This follows from the fact that the group of cobordism invariants for a five-dimensional spin manifold endowed with an SU(2) bundle is Z2.4,13,14I1/2 is a nontrivial invariant for this cobordism problem, so any other invariant is either trivial or equal to I1/2. To decide which is the case, it suffices to evaluate the invariant for Y1. Since I3/2 vanishes for Y1, it is identically zero for spin manifolds. For a manifold that admits only a spin-SU(2) structure and not a spin structure, I3/2 can be nonzero; in Sec. IV B, we will show that on such a manifold, I3/2 is actually independent of the choice of spin-SU(2) structure.

Now, let us consider an SU(2) gauge theory which is consistent—because it lacks the original SU(2) anomaly—but is affected by the more subtle anomaly that we found in Sec. II E. Such a theory is consistent if formulated on a spin manifold, so it can be studied sensibly. But it cannot be consistently formulated on a manifold with only a spin-SU(2) structure. A simple example is an SU(2) gauge theory with a single fermion multiplet of isospin 3/2.

We can introduce a Higgs field ϕ in the spin 1 representation of SU(2). Its expectation value can spontaneously break SU(2) to U(1). The resulting U(1) theory must somehow embody the same anomaly: it must be consistent if formulated on a spin manifold but not if formulated on a bare four-manifold with no additional structure. We would like to see how this comes about.

Symmetry breaking from SU(2) to U(1) changes the nature of the problem, as follows. A single isospin 3/2 fermion multiplet cannot receive an SU(2)-invariant bare mass. Hence, it is possible for the path integral of such a field to be anomalous, and this is what we have found in Sec. II E. However, once SU(2) is spontaneously broken to U(1), the isospin 3/2 fermion can gain a bare mass by coupling to ϕ. Hence, we can integrate out the fermions, and it must be possible to see the anomaly in terms of effective couplings of a U(1) gauge theory only. In fact, what is relevant here is an anomalous theory that is known as “all-fermion electrodynamics.” This anomaly was first demonstrated in Ref. 7; we will give an explanation of it in Sec. III B.

The low energy theory that we will get after symmetry breaking from SU(2) to U(1) has a full set of electric and magnetic charges allowed by Dirac quantization; the ’t Hooft-Polyakov monopole of the theory has the minimum possible magnetic charge relative to the component of the elementary fermion field of electric charge 1.

In general, a U(1) gauge theory might have fermions that carry neither electric nor magnetic charge. But that is not the case here; because we are considering a theory in which all elementary fields are in representations of (Spin(4)×SU(2))/Z2, all magnetically neutral states (at least in weakly coupled perturbation theory) are fermions of odd electric charge or bosons of even electric charge.

In general, in a theory that does not have any neutral fermions, one can ask whether an electrically charged particle with electric and magnetic charges (1, 0) is a boson or a fermion. (The answer will be the same for each such particle since otherwise a particle-antiparticle pair would be a neutral fermion.) Likewise, one can ask the same question for a monopole with charges (0, 1) or a dyon with charges (1, 1). Duality in the low energy effective U(1) gauge theory permutes these three particle types, so any general constraints are invariant under such permutations.

If two of these basic particles are bosons or two are fermions, then after taking into account the angular momentum in the electromagnetic field, we can deduce that the third is a fermion; likewise, if one is a boson and another is a fermion, then the third is a boson. Thus, on this ground alone, there are two possibilities: two of the three basic charges are bosons and one is a fermion, or all three are fermions. The first case is anomaly-free, as we will discuss in a moment, but the second case, which is “all-fermion electrodynamics,” is anomalous.

To show that the first case is anomaly-free, it suffices to give a manifestly anomaly-free realization. For this, we simply consider an SU(2) gauge theory with the Higgs triplet field ϕ that breaks SU(2) to U(1) and add two multiplets of spin 1/2 fermions in the isospin 1/2 representation of the gauge group. This is actually a much-studied theory. It is anomaly-free (and fermion bare masses are possible) because we included an even number of identical fermion multiplets. The elementary fermions have electric and magnetic charges (±1, 0) and are, of course, fermions. The ’t Hooft-Polyakov monopole is rotationally invariant at the classical level (up to a gauge transformation), so in the absence of fermions, it is a boson. Fermion zero-modes can give the monopole unusual quantum numbers under global symmetries (Note 8 in the  Appendix), but the zero-modes of isospin 1/2 fermions do not carry angular momentum and the monopole is a spinless boson. The (1,1) dyon is therefore also a boson. Consistency of this theory makes clear that in general a U(1) gauge theory in which two of the basic charges are bosons and one is a fermion can be consistent.

However, we will now see that the anomalous SU(2) gauge theory with a single fermion multiplet of isospin 3/2 reduces, after symmetry breaking, to the all-fermion case, which is anomalous. Since the particles in this theory with minimum electric charge are elementary fermions, we just have to show that the ’t Hooft-Polyakov monopole is also a fermion. The Callias index theorem16 implies that a single isospin 3/2 fermion multiplet in the field of an ’t Hooft-Polyakov monopole of magnetic charge 1 has a four-dimensional space of zero-modes (Note 9 in the  Appendix). The ’t Hooft-Polyakov monopole is spherically symmetric (up to a gauge transformation), so the four zero-modes furnish a representation of the rotation group. Allowing for the angular momentum in the electromagnetic field, an isospin 3/2 fermion in the field of the monopole carries integer angular momentum. A four-dimensional representation of the rotation group with only integer spin appearing is the sum of either four copies of spin 0 or one copy of spin 0 and one copy of spin 1. As we will explain in a moment, in this particular case, the four zero-modes transform as spin 0 ⊕ 1.

Having described the zero-modes, we follow the logic of Ref. 15 for quantization of the monopole. Quantization turns the classical zero-modes into quantum operators that obey the anticommutation relations of Dirac gamma matrices. Thus, in the present case, we have a single gamma matrix γ0 that commutes with the rotation group and a triplet of gamma matrices γ of spin 1. To represent γ, we need a two-dimensional Hilbert space that transforms as spin 1/2 under rotations; the operators γ act in this space like the Pauli angular momentum matrices σ. The fourth gamma matrix γ0, because it anticommutes with γ but carries no angular momentum, forces a doubling of the spectrum but without changing the fact that the states have spin 1/2. Thus the ’t Hooft-Polyakov monopole in this theory has angular momentum 1/2 and is a fermion. Accordingly, the dyon is also a fermion and this theory gives a realization of all-fermion electrodynamics.

The statement that the four zero-modes have spin 0 ⊕ 1 rather than 0 ⊕ 0 ⊕ 0 ⊕ 0 can be deduced from Secs. III A and III B of Ref. 17. A less technical explanation is as follows. A spin 1/2 fermion has components of helicity ±1/2. In the absence of the monopole, the minimum angular momentum of a particle of helicity h is |h|. However, if the particle has electric charge q and interacts with a monopole of charge g, then its minimum angular momentum is shifted from |h| to |h + qg/4π|. To get angular momentum zero, we need h + qg/4π = 0. For an elementary fermion field that transforms under the SU(2) gauge symmetry with isospin 3/2, the possible values of q are 3, 1, −1, and −3, and the corresponding values of qg/4π (in the field of a minimum charge monopole) are 3/2, 1/2, −1/2, and −3/2. To get h + qg/4π = 0, we take h = 1/2, q = −1, or h = −1/2, q = 1. Thus, in the partial wave decomposition of the elementary fermion field of this theory, angular momentum 0 occurs precisely twice. Since the Dirac equation is a first order differential equation, a given partial wave can lead to at most 1 zero-mode, and actually since only one linear combination of the two partial waves of spin 0 obeys the boundary condition at the origin, those two partial waves together cannot possibly give more than 1 zero-mode. So there is no way to make four zero-modes of spin 0, and the zero-modes in this problem carry angular momentum 0 ⊕ 1, as assumed in the derivation of all-fermion electrodynamics.

More generally, consider an elementary fermion multiplet that couples to the SU(2) gauge field with any half-integral isospin j. The Callias index theorem says that in the field of a monopole, such a fermion has (j + 1/2)2 zero-modes. These modes transform under rotations with spin 0 ⊕ 1 ⊕ ⋯ ⊕ (j − 1/2). Quantizing these (j + 1/2)2 zero-modes gives states of half-integer angular momentum if and only if j is of the form 4r + 3/2, rZ. If and only if the total number of fermion multiplets of isospin 4r + 3/2 is odd, the ’t Hooft-Polyakov monopole turns out to be a fermion and we get a model of all-fermion electrodynamics.

The minimal example that illustrates the new anomaly is an SU(2) gauge theory with a single multiplet of fermions of isospin 3/2. This theory is asymptotically free (just barely, that is, the coefficient of the one-loop beta function is just slightly negative), so it gives an ultraviolet completion of all-fermion electrodynamics.

Here, we will give a direct explanation of the anomaly in all-fermion electrodynamics (Note 10 in the  Appendix).

We work on a four-manifold M without a chosen spin structure. The choice of a spin structure would trivialize the discussion of all-fermion electrodynamics [just as it trivializes the new SU(2) anomaly discussed in the present paper] because it means that a neutral fermion is possible and therefore that a quasiparticle of any given electric and magnetic charges can be either a boson or a fermion. (Less obviously, as explained at the end of the present section, the choice of a background spinc structure on M also trivializes the discussion.)

The most obvious form of electrodynamics is the ordinary U(1) gauge theory of an ordinary U(1) gauge field. In the absence of a spin structure, a particle with an electric charge but no magnetic charge then has to be a boson.

We already discussed in Sec. II D how to modify this if we want a field of electric charge 1 to be fermionic rather than bosonic: the gauge field must be a spinc connection rather than an ordinary U(1) gauge field.

Let us, however, go back to ordinary U(1) gauge theory and ask what one has to do in order to make a charge 1 magnetic monopole a fermion. The answer is that this will happen if the path integral measure of the low energy theory has a factor that roughly speaking is

(1)Mw2(TM)F/2π.
(3.1)

This formula needs some improvement since F/2π is a differential form and w2(TM) is a Z2-valued cohomology class. A more accurate description is as follows. The gauge field A is a connection on a complex line bundle L over spacetime. This line bundle has a first Chern class c1(L)H2(M,Z). Reducing it mod 2, we get a class in H2(M,Z2), which we also denote as c1(L). Then, taking a cup product in Z2-valued cohomology, we can define the mod 2 invariant ∫Mw2(TM) · c1(L), and the factor that we want in the path integral is

(1)Mw2(TM)c1(L).
(3.2)

(Another version of this formula is described in Sec. III B 1.) To be more exact, (3.2) is a topological invariant assuming that M is compact and without boundary. We make this assumption to simplify the analysis. Otherwise, it is necessary to specify the behavior along the boundary of M or at infinity.

Now, we want to include a monopole. Since we have assumed that M is a compact four-manifold without boundary, it is natural to take the worldline of the monopole to be a circle. In the low energy U(1) gauge theory, the worldline of a charge 1 monopole is represented by a charge 1 ’t Hooft operator. Concretely, to compute in the presence of this ’t Hooft operator, we do U(1) gauge theory on the complement of , but along , the U(1) gauge field has a singularity that is characterized by

dF=2πδ,
(3.3)

where δ is a three-form delta function that is Poincaré dual to .

The line bundle L and the corresponding cohomology class c1(L) are only defined on the complement of , not on all of M. Therefore, the formula (3.2) does not make sense in the presence of the ’t Hooft operator.

To try to remedy the situation, we can remove from M a small tubular neighborhood of , replacing M with a manifold M′ with boundary on which L is defined. The boundary of M′ is a copy of S2 × . The fact that the ’t Hooft operator carries magnetic charge 1 means that

S2F2π=1.
(3.4)

We now replace (3.2) with

(1)Mw2(TM)c1(L).
(3.5)

We have made some progress because w2(TM) and c1(L) both make sense as elements of H2(M,Z2). However, we still have a problem because M′ is a manifold with boundary. To integrate a cohomology class—in this case w2(TM) · c1(L)—on a manifold with boundary, we need a trivialization of the cohomology class on the boundary.

In the present case, since w2(TM′) · c1(L) is the product of two factors, we could a priori trivialize it along ∂M′ = S2 × by trivializing either of the two factors. However, we cannot trivialize c1(L) along ∂M′ = S2 × because Eq. (3.4) says that the restriction of c1(L) to ∂M′ is nontrivial. The remaining option is to trivialize w2(TM′) along ∂M′.

A trivialization of w2(TM′) is a spin structure. We have learned that to define the factor (3.5), we need a spin structure along ∂M′ = S2 × . Since S2 is simply connected, there are precisely two spin structures along S2 × —differing by the sign that arises in parallel transport of a fermion all the way around . Since S2 plays no role in classifying spin structures on ∂M′ = S2 × , the choices of a spin structure along S2 × are directly in correspondence with the choices of a spin structure along M (or equivalently in a tubular neighborhood of M).

We have learned that if the integrand of the path integral contains the factor (3.1), then to define the propagation of a charge 1 monopole around the worldline , we need a spin structure along . In other words, the charge 1 monopole is a fermion.

So to make the elementary electric charge a fermion, we take the gauge field to be a spinc connection rather than a U(1) gauge field. To make the elementary monopole a fermion, we need to include in the integrand of the path integral the factor (3.2).

To get all-fermion electrodynamics, we mostly do both of these. But here there is a problem. If the gauge field A is really a spinc connection, then there is no “charge 1” line bundle L whose curvature would be F/2π. Likewise, there is no integral cohomology class c1(L). So in the spinc case, we cannot define the factor (3.2). If A is a spinc connection, then 2A is an ordinary U(1) gauge field, and there exists a complex line bundle L^ with connection 2A, which would reduce to L2 in the case of U(1) gauge theory. One can then define an integral cohomology class c1(L^), which would correspond to 2c1(L) in U(1) gauge theory. But in the spinc case, there is no way to eliminate the factor of 2 and thus no way to define the interaction (3.2). This is a topological explanation of why all-fermion electrodynamics is anomalous, that is, not well-defined. The anomaly is trivialized by a choice of spin structure because such a choice eliminates the distinction between a U(1) gauge field [for which Eq. (3.2) can be defined] and a spinc connection (for which it cannot be).

The choice of a background, nondynamical spinc connection A0 would trivialize the discussion of all-fermion electrodynamics for the following reason. By subtracting A0 from the dynamical spinc connection A, we would get an ordinary U(1) gauge field A′ = AA0. Therefore, a charge 1 bosonic quasiparticle is possible. Once such a quasiparticle is possible, it follows that a quasiparticle of any electric and magnetic charges can be either a boson or a fermion.

1. Some more detail on the amplitude for monopole propagation

To complete this explanation, we would like to show that if the spin structure of is changed; this changes the sign of the amplitude for a monopole to propagate around . We will give two versions of the explanation. For an abstract explanation, two trivializations β1 and β2 of a class w2H2(M,Z2) differ by an element aH1(M,Z2). In the present case, the trivializations (or spin structures) β1 and β2 are defined only near ∂M′, so likewise a is defined only near ∂M′. For i = 1, 2, let M;βiw2(TM)c1(L) be the integral in question defined using the trivialization βi along ∂M′. The general topological formula in this situation is

M;β1w2(TM)c1(L)M;β2w2(TM)c1(L)=Mac1(L)=×S2ac1(L).
(3.6)

If the two spin structures along ∂M′ are different, this means that ∫a = 1, and from Eq. (3.4), we have (Note 11 in the  Appendix) S2c1(L)=1. So the right hand side of Eq. (3.6) is 1. Hence, flipping the spin structure along the monopole worldline shifts what we mean by ∫Mw2 · c1(L) by 1, and therefore, it changes the sign of the factor (3.2) in the path integral measure. This is the expected effect of flipping the spin structure if the monopole is a fermion.

For a somewhat more concrete version of this explanation, we will first explain the notion of an integral lift of w2(TM), which will also be useful in Sec. IV. We look at the short exact sequence of abelian groups

0Z2ZrZ20,
(3.7)

where the first map is multiplication by 2 and the second is reduction mod 2. This leads to a long exact sequence of cohomology groups that reads in part

H2(M,Z)rH2(M,Z2)βH3(M,Z).
(3.8)

A class xH2(M,Z) is called an integral lift of w2(TM)H2(M,Z2) if w2(TM) = r(x). From the exactness of the sequence (3.8), such an x exists if and only if W3(TM) = 0, where

W3(TM)=β(w2(TM)).
(3.9)

The mod 2 reduction of W3(TM) is the Stiefel-Whitney class w3(TM). This fact will be important in Sec. IV.

For the moment, however, we are interested in the case that W3(TM) = 0. In fact, this is always true in four dimensions. Given W3(TM) = 0, we pick any integral lift x of w2(TM). There will then be a complex line bundle RM with x=c1(R). We write B for a connection on R and G = dB for the corresponding field strength. The fact that x is congruent mod 2 to w2(TM) means that for any closed oriented two-surface SM,

SG2π=Sw2(TM)   mod2.
(3.10)

Comparing to Eq. (2.11), we see that C=12B can be regarded as a spinc connection, and thus, the existence of a spinc structure on M is equivalent to the existence of an integral lift of w2(TM).

For now, though, we continue the discussion with the ordinary U(1) gauge field B and its field strength G. Equation (3.10) means that if M has no boundary (and no monopole singularity) we can write

Mw2(TM)c1(L)=MG2πF2π.
(3.11)

Now, let us include the monopole worldline and replace M with the manifold with boundary M′. We consider the same integral as in Eq. (3.11), but now on M′,

MG2πF2π.
(3.12)

Let us try to define this, as before, by trivializing w2(TM′) along ∂M′. The advantage of the formalism we are using here is that we can give a more direct explanation of what this means. Trivializing w2(TM′) along ∂M′ corresponds, in the language we are using now, to picking B to be a pure gauge along ∂M′. With this restriction, what freedom is there in picking B? Equation (3.11) would allow us to change the flux of G in the interior of M only by an even number of flux quanta. This corresponds to replacing the integral lift x with a different integral lift of w2(TM), and it does not affect the integral in Eq. (3.12) mod 2. However, we are free to change the flux of G near ∂M′ by an odd number of flux quanta. To be precise, factorize M′ near its boundary as (×R+)×S2, where R+ is a half-line (whose endpoint is on ∂M′). Without affecting Eq. (3.10) or the fact that B is pure gauge along ∂M′, we can change G to G + G′ where

×R+G2π=1.
(3.13)

This is the operation that corresponds to changing the trivialization of w2(TM′) near ∂M′. Using Eq. (3.4) again, we see that shifting G to G + G′ shifts the integral (3.12) by 1. This is the effect that corresponds to the monopole being a fermion.

In Sec. III A, we used a Higgs triplet to break SU(2) to U(1), thereby reducing an anomalous (Spin(4)×SU(2))/Z2 theory to an anomalous Spinc(4)=(Spin(4)×U(1))/Z2 theory. We can go further and add a second Higgs triplet, making it possible to break U(1) to Z2, the center of SU(2). This gives a theory that at low energies can be viewed locally as a Z2 gauge theory. But globally, it is much more accurate to say that the effect of the Higgsing from SU(2) to Z2 is to reduce (Spin(4)×SU(2))/Z2 to (Spin(4)×Z2)/Z2=Spin(4).

So the low energy theory can have fields in all possible representations of Spin(4). In other words, the low energy theory has dynamically generated a spin structure, which is determined by the Higgs expectation values. Although the low energy theory can be described locally as a Z2 gauge theory, globally, it is more accurately described as a theory in which one has to sum over the possible choices of a dynamically generated spin structure.

Since the underlying anomaly in the original SU(2) description is trivialized once a spin structure is chosen, there is no mystery about the consistency of this low energy phase: the dynamical generation of a spin structure eliminates the anomaly.

It is interesting to consider the vortex line of the low energy Z2 gauge theory. Its worldvolume is a codimension two submanifold CM, with the property that the spin structure of M does not extend over C; rather, a fermion parallel transported on a tiny loop that is linked with C will change sign (Fig. 1). In the spinc description, it is clear how to describe the vortex line: although the spin structure of M does not extend over C as a spin structure, one can at least locally extend it over C as a spinc structure. Physically, this means that in the U(1) description, there is a half-integral flux in the normal plane to C. An interesting phenomenon occurs if the normal bundle to C in M is unorientable (which is possible even if M itself is orientable). In that case, there is an obstruction to extending the spin structure of M over C as an everywhere defined spinc structure because the sign of the U(1) flux in the normal plane cannot be chosen continuously. There will then have to be a defect line on C along which the sign of the flux changes; this defect line is actually the worldline of a magnetic monopole. This monopole is a fermion. The spinc structure cannot be extended over the monopole worldline as a spinc structure, but it can be extended as a spin-SU(2) structure.

FIG. 1.

A codimension two vortex line L, around which a spin structure does not extend. A fermion that is parallel transported around a small loop γ that links with L will come back with the opposite sign. The worldvolume of L is a two-manifold C, discussed in the text.

FIG. 1.

A codimension two vortex line L, around which a spin structure does not extend. A fermion that is parallel transported around a small loop γ that links with L will come back with the opposite sign. The worldvolume of L is a two-manifold C, discussed in the text.

Close modal

As is by now familiar from many other examples, an anomalous fermion theory in four dimensions can be interpreted as the boundary state of a gapped theory in five dimensions. We will actually describe two rather different ways to do this. For the more obvious and standard approach, we consider a five-dimensional gapped theory with fermions and SU(2) global symmetry. Turn on background SU(2) gauge fields. There are two cases to consider: a theory that requires a spin structure, and in which the background gauge fields define an SU(2) bundle with connection, and a theory that can be formulated more generally with a background spin-SU(2) structure. The old SU(2) anomaly fits in either of these frameworks, but the new anomaly is relevant only if one considers a theory in which the background fields can define a general spin-SU(2) structure. (That is because the new anomaly is trivial if a spin structure is assumed, as explained in Sec. II G.)

We consider then a theory whose partition function in the presence of these background fields is (1)I1/2 or (1)I3/2 (or the product of these two). Now, suppose that Y is a manifold with boundary and with background fields of the same type [either a spin structure plus an SU(2) bundle with connection or a spin-SU(2) structure]. Then, there is no way to define I1/2 or I3/2 as a topological invariant. To make sense of the spectrum of the Dirac operator D5 on a manifold with boundary, one needs a boundary condition. Here, D5 is the five-dimensional Dirac operator coupled to a pseudoreal representation of SU(2); it is an antisymmetric operator, and this is important in being able to define its mod 2 index. There is no SU(2)-invariant local boundary condition on D5 that preserves its antisymmetry and would let us define its mod 2 index as a topological invariant. We can use global Atiyah-Patodi-Singer boundary conditions,18 which do preserve the antisymmetry, and with those boundary conditions, it is possible to define the mod 2 index. But the mod 2 index defined with Atiyah-Patodi-Singer (APS) boundary conditions is not a topological invariant. It jumps whenever an eigenvalue (more precisely, a pair of eigenvalues; see Ref. 12) of the four-dimensional Dirac operator on the boundary M = ∂Y of Y passes through zero. This is possible because APS boundary conditions depend explicitly on the spectrum of the Dirac operator on M.

This difficulty, however, mirrors the problem in defining a path integral for four-dimensional fermions in a pseudoreal representation of SU(2). Formally, the fermion path integral is Pf(D4), the Pfaffian of the four-dimensional Dirac operator D4 [acting on fermions in the appropriate representation of SU(2)]. There is no problem to define the absolute value |Pf(D4)| of the Pfaffian. However, there is no simple way to define its sign, and the SU(2) anomalies that we have discussed mean that in some theories, the sign has no satisfactory definition.

The two problems cancel if we combine them. The absolute value |Pf(D4)| is not satisfactory for the fermion path integral in the boundary theory because it gives unphysical behavior when a pair of eigenvalues of D4 passes through zero. Also, (1)I(D5) is not satisfactory as the partition function of a gapped theory on a five-manifold with boundary because it jumps in sign in an unphysical way when a pair of eigenvalues of D4 passes through zero. However, the product

Z=|Pf(D4)|(1)I(D5)
(4.1)

is physically sensible and describes a combined system consisting of a gapped but topologically nontrivial system in bulk and a boundary state of gapless fermions transforming in a suitable representation of SU(2).

This construction of the combined path integral of a bulk system plus a boundary state, which is described more fully in Ref. 12, has various applications in string/M-theory (see Sec. II of Ref. 19 and Sec. VII C of Ref. 20). It has a fairly direct analog (in certain spacetime dimensions) for fermions in a real representation of the gauge group. For fermions in a complex representation, the construction has a more subtle analog using the Atiyah-Patodi-Singer η-invariant18 and the Dai-Freed theorem,22 rather than the mod 2 index. See Ref. 12 for an introduction and Ref. 25 for an application to the worldsheet path integral of the heterotic string.

There is a more subtle way to use the new SU(2) anomaly in four dimensions to give a boundary state for a topologically nontrivial theory in five dimensions. In this case, we consider a five-dimensional theory with no symmetry assumed. Let J be the mod 2 invariant

J(Y)=Yw2(TY)w3(TY),
(4.2)

and consider a gapped theory23,24 whose partition function on a closed five-manifold Y is

(1)J(Y).
(4.3)

We want to find a boundary state for this theory. This is not straightforward because ∫Yw2(TY)w3(TY) is not a topological invariant on a manifold with boundary. To correct for this, there must be some additional structure or some additional degrees of freedom on the boundary.

The quantity J is not just a topological invariant but a cobordism invariant, meaning that if Y is the boundary of a six-manifold Z, then J = 0. This is true without any restriction on Y and Z beyond which they are manifolds. So certainly if Y is assumed to have a spin-SU(2) structure that extends over Z, still with Y = ∂Z, then it remains true—since this is true even without the spin-SU(2) structures—that J = 0. So we can view J as a cobordism invariant of a five-manifold with spin-SU(2) structure [a special one that actually does not depend on the spin-SU(2) structure]. But the group of cobordism invariants of a five-manifold with spin-SU(2) structure is Z2×Z2,4,13,14 and for generators, one can take I1/2 and I3/2, as we discussed in Sec. II G. So J must be a linear combination of I1/2 and I3/2. To determine the coefficients, we just need to evaluate J for the two examples Y1 = S4 × S1 and Y2=CP2S1 that were discussed in Sec. II G. In fact, J(S4 × S1) = 0 since all Stiefel-Whitney classes of S4 × S1 vanish. But J(CP2S1)=1.

To understand that last statement, let x be the nonzero element of H2(CP2,Z2)=Z2 and y be the nonzero element of H1(S1,Z2)=Z2. The cohomology ring of CP2S1 is generated by x and y with relations x3 = y2 = 0 and with CP2S1x2y=1. We have w2(CP2S1)=x, basically because w2(CP2)=x, and this is unaffected by the presence of S1. On the other hand, w3(CP2S1)=xy, leading to J(CP2S1)=1. The fact that w3(CP2S1)=xy is related to the following. As we learned in discussing Eq. (3.8), the obstruction to the existence of a spinc structure on a manifold Y is measured by a class W3H3(Y,Z). CP2S1 does not admit a spinc structure, as we know from Sec. II D, so it has W3 ≠ 0. The Stiefel-Whitney class w3 is defined as the mod 2 reduction of W3. In the particular case of Y=CP2S1, one has H3(Y,Z)=Z2=H3(Y,Z2), so nothing is lost in the mod 2 reduction and the fact that W3 is nonzero implies that w3 is also nonzero and is the unique nonzero element xyH3(CP2S1,Z2).

So it follows that for any five-manifold Y with spin-SU(2) structure, we have

J(Y)=I3/2(Y).
(4.4)

Since the left hand side of Eq. (4.4) does not depend on the spin-SU(2) structure of Y, this shows that I3/2(Y) likewise does not depend on the spin-SU(2) structure.

A corollary of this result is that the new SU(2) anomaly is never an anomaly in gauge invariance; it is always an anomaly in a combined diffeomorphism plus gauge transformation. For as remarked at the end of Sec. II F, an anomaly in a gauge transformation on a four-manifold M would correspond to a nonzero mod 2 index on M × S1, with some spin-SU(2) structure. But the Stiefel-Whitney classes of M × S1 are pulled back from M, so M×S1w2w3=0.

The relation (4.4) suggests that the bulk theory with partition function (−1)J could have a boundary state with an emergent SU(2) gauge symmetry coupled to a single multiplet of fermions of isospin 3/2 (or any other set of fermions that carries the same anomalies). This is true, but the explanation involves a few ideas beyond what we needed in Sec. IV A.

Suppose first that there is some manifold Ỹ such that the spin-SU(2) structure of M extends over Ỹ. We do not assume that Ỹ is isomorphic to Y, and it is not necessary to know if the spin-SU(2) structure of M extends over Y. We orient Ỹ so that it can be glued to Y along M to make a smooth manifold Y^. Then, denoting the spin-SU(2) structure of M as A, we define the following substitute for the product of (−1)J(Y) and the fermion path integral Pf(DM):

ZM,A,Y=(1)J(Y^)(1)Ij(DỸ)|Pf(DM)|.
(4.5)

Here, DỸ is the five-dimensional Dirac operator (Note 12 in the  Appendix) on Ỹ for isospin j, and Ij(DỸ) is its mod 2 index with APS boundary conditions (we assume that the isospin j is of the form 4r + 3/2 so that the fermions under discussion do carry the relevant anomaly). The point of this definition is twofold. First, it is physically sensible, for the same reason as Eq. (4.1): whenever a pair of eigenvalues of DM passes through 0 so that the fermion amplitude should change sign, such a sign change occurs because of a jump in Ij(DỸ). Second, although the factors (1)J(Y^) and (1)Ij(DỸ) on the right hand side both depend on the choice of Y^, this dependence cancels out in the product. Therefore, ZM,A,Y only depends on M, A, and Y, that is, on Y and the spin-SU(2) structure of M = ∂Y, as it should.

To prove the cancellation of the dependence on Y^, we proceed as follows. Let Ỹ be some other five-manifold of boundary M over which the spin-SU(2) structure of M extends. Glue Ỹ to Y along M to make a new five-manifold Y^, and glue Ỹ to Ỹ to make another new five-manifold Y*. The gluing law for the mod 2 index says that it is additive under gluing, in the sense that, if Y* is made by gluing Ỹ and Ỹ as in Fig. 2, then (Note 13 in the  Appendix)

(1)Ij(DỸ)(1)Ij(DỸ)=(1)Ij(DY*).
(4.6)

On the other hand, from Eq. (4.4), we have

(1)Ij(DY*)=(1)J(Y*).
(4.7)

We also have a gluing law for integrals in mod 2 cohomology that is analogous to Eq. (4.6)

Y^w2(TY^)w3(TY^)=Y^w2(TY^)w3(TY^)+Y*w2(TY*)w3(TY*),
(4.8)

or simply, J(Y^)=J(Y^)+J(Y*). This statement would follow from elementary calculus if w2 and w3 were differential forms; one would write the integral over Y^ as the sum of an integral over Y and an integral over Ỹ, and after making a similar decomposition of each term on the right hand side of Eq. (4.8), one would find that all terms cancel in pairs. This sort of formula is also valid for integration in mod 2 cohomology. One way to prove it is to observe (Fig. 3) that there is a fairly obvious six-manifold Z whose boundary is the union of Y^, Y^, and Y*. The existence of Z implies that 0 = ∫∂Zw2(T∂Z)w3(T∂Z), and writing this out in detail, we get Eq. (4.8). (The signs in that formula are not important as these quantities are all Z2-valued. They have been chosen to make the formulas look natural in ordinary calculus.) Using (4.6)–(4.8), it is straightforward to show that the right hand side of Eq. (4.5) does not change if Ỹ is replaced by Ỹ and correspondingly Y^ is replaced by Y^. Thus, the right hand side of this formula only depends on Y and A, as it should.

FIG. 2.

Y* is obtained by gluing Ỹ and Ỹ along their common boundary M. The mod 2 index is additive in such a gluing, leading to Eq. (4.6).

FIG. 2.

Y* is obtained by gluing Ỹ and Ỹ along their common boundary M. The mod 2 index is additive in such a gluing, leading to Eq. (4.6).

Close modal
FIG. 3.

The manifold Y^ is made by gluing together Y and Ỹ along their common boundary M, and similarly, Y^ is made by gluing together Y and Ỹ along M, and Y* is made by gluing Ỹ and Ỹ along M. The shaded region schematically represents a six-manifold Z whose boundary is the union of Y^, Y^, and Y*. The existence of this cobordism establishes Eq. (4.8).

FIG. 3.

The manifold Y^ is made by gluing together Y and Ỹ along their common boundary M, and similarly, Y^ is made by gluing together Y and Ỹ along M, and Y* is made by gluing Ỹ and Ỹ along M. The shaded region schematically represents a six-manifold Z whose boundary is the union of Y^, Y^, and Y*. The existence of this cobordism establishes Eq. (4.8).

Close modal

There is still a gap in this construction of a boundary state because we assumed that there exists some Y^ with boundary M over which the spin-SU(2) structure of M extends. In general, this is not the case. If M is an abstract four-manifold with spin-SU(2) structures, there are two obstructions to the existence of a manifold Y^ with boundary M over which the spin-SU(2) structure of M extends. First, the signature of M is an obstruction to the existence of Y^, even if one does not ask to extend the spin-SU(2) structure of M over Y^. In our case, this obstruction vanishes since M is given as the boundary of Y. The second obstruction is that if the SU(2) instanton number of M is nonzero, then, regardless of what Y^ we choose with boundary M, the spin-SU(2) structure of M will not extend over Y^. The SU(2) instanton number normalized to be integer-valued for an ordinary SU(2) gauge field is

Q=MTrFF8π2.
(4.9)

In gauge theory, one usually introduces an angular parameter θ and includes a factor exp(iθQ) in the integrand of the path integral. As Q is integer-valued, θ has period 2π. On a spin-SU(2) manifold M, in general, Q is not an integer, but if M has zero signature (as is the case if it is the boundary of some Y^), then Q is integer-valued (Note 14 in the  Appendix). Let us pick a particular example consisting of a five-manifold Y0, a four-manifold M0 = ∂Y0, and an SU(2) gauge field A0 on M0 with Q = 1. For instance, we can pick M0 to be a four-sphere S4 with some metric and some SU(2) gauge field A0 of instanton number 1, and pick Y0 to be a five-ball B5 whose boundary is M0. We do not have a natural way to define the sign of (1)J(Y)Pf(DM0,A0). We arbitrarily declare this object to be positive and so write

ZM0,A0,Y0=eiθ|Pf(DM0,A0)|.
(4.10)

We can regard this as the definition of what we mean by θ at the quantum level. If we took (1)J(Y)Pf(DM0,A0) to be negative, this would have the same effect in Eq. (4.10) and later as shifting θ by π.

Now, we want to define ZM,A,Y for an arbitrary triple M, A, Y. If the instanton number on M is a nonzero integer n, then we cannot directly use Eq. (4.5) to define ZM,A,Y because there is no Ỹ with boundary M over which the spin-SU(2) structure of M extends. But there is a Ỹ with boundary (Note 15 in the  Appendix) MnM0 over which the spin-SU(2) structures of M and M0 extend. Finally, by gluing together Y, Ỹ, and n copies of Y0 along their boundaries—which together consist of M and n copies of M0—we build a closed five-manifold Y¯, depicted in Fig. 4. The general definition of ZM,A,Y is then

ZM,A,Y=einθ(1)J(Y¯)(1)Ij(DỸ)|Pf(DM,A)|.
(4.11)

As usual, this definition is physically sensible in part because whenever an eigenvalue pair of DM,A passes through zero, the desired jump in sign of the amplitude comes from the behavior of the mod 2 index Ij(DỸ). Beyond this, the consistency of Eq. (4.11) amounts to two facts. First, the right hand side does not depend on the choice of Ỹ. This follows from the same reasoning that we used in discussing Eq. (4.5). Second, if we take M = M0, A = A0, Y = Y0, then Eq. (4.11) is compatible with Eq. (4.10). To show this, we observe that if M = M0, then n = 1 and we can take Ỹ to be M0 × I (where I is a unit interval) with a spin-SU(2) structure pulled back from M0. Then, J(Y¯)=Ij(DỸ)=0 and the two formulas for ZM0,A0,Y0 coincide.

FIG. 4.

M is cobordant to n copies of M0, for some n, via a five-manifold Ỹ over which the spin-SU(2) structures of M and M0 extend. This is depicted here for n = 2. Each copy of M0 is the boundary of some Y0, and M is the boundary of Y. So Y, Ỹ, and n copies of Y0 glue together naturally to a closed five-manifold Y¯, depicted here. This construction is used in the general definition of the amplitude ZM,A,Y.

FIG. 4.

M is cobordant to n copies of M0, for some n, via a five-manifold Ỹ over which the spin-SU(2) structures of M and M0 extend. This is depicted here for n = 2. Each copy of M0 is the boundary of some Y0, and M is the boundary of Y. So Y, Ỹ, and n copies of Y0 glue together naturally to a closed five-manifold Y¯, depicted here. This construction is used in the general definition of the amplitude ZM,A,Y.

Close modal

At this level of description, we have found a boundary state with an emergent SU(2) gauge field coupled to massless fermions. Next, we should discuss the dynamics of this SU(2) gauge theory. If the fermion representation is large enough so that the SU(2) gauge theory is infrared-free, then the SU(2) dynamics will not qualitatively change the picture. For example, if the anomaly is carried by a boundary fermion of isospin 4r + 3/2, r > 0, then the one-loop beta function of the SU(2) gauge theory is positive, and if the SU(2) coupling is small enough, it will flow to zero in the infrared. However, for a minimal set of boundary fermions—a single multiplet of isospin 3/2—the one-loop beta function of the SU(2) gauge theory is slightly negative. The SU(2) dynamics cannot be ignored in this case, and the outcome is not clear. It is difficult to decide, for example, if the theory flows to an infrared fixed point or to a confining state with a mass gap. An analysis leading to confinement was made in Ref. 6. A possibility, suggested by the most attractive channel (MAC) hypothesis,26 is that the theory might Higgs itself to U(1) or possibly to Z2. The logic here is that for this particular theory, the most attractive channel for fermion condensation (assuming Lorentz invariance is maintained) is the isospin 1 channel. A fermion condensate in this channel will break the gauge symmetry to U(1) or Z2, depending on whether the condensate can be made real by a chiral U(1) rotation of the fermions.

Once we have found a boundary state with emergent SU(2) gauge symmetry, it is clear that it is possible to get a boundary state with emergent U(1) gauge theory. We simply follow the same symmetry-breaking as in Sec. III A: we add a Higgs field ϕ of isospin 1, whose expectation value breaks SU(2) to U(1).

After symmetry breaking, the isospin 3/2 fermions that contribute the anomaly in the SU(2) description can gain mass, so the anomaly must be carried by effective couplings of the U(1) gauge field. How this happens was already analyzed in Secs. III A and III B: we get the anomalous “all-fermion electrodynamics” in which particles carrying a unit of electric or magnetic charge are fermions. In this theory, since there are no massless charged fermions, the anomaly is carried by effective couplings of the U(1) gauge fields only. From our previous analysis, we already know how this happens. To make the charge 1 monopole a fermion, the integrand of the path integral has a factor

(1)Mw2(TM)c1(L),
(4.12)

where L is the line bundle on which the U(1) gauge field is a connection. To make a particle of electric charge 1 a fermion, the “U(1) gauge field” is really a spinc connection. Together these two conditions are incompatible, leading to an anomaly.

Thus, it follows by Higgsing from our construction in Sec. IV B that an emergent U(1) theory of all-fermion electrodynamics gives a possible boundary state (Note 16 in the  Appendix) for the five-dimensional theory with partition function (−1)J. The boundary state with emergent all-fermion electrodynamics can be described in the following way, more in the spirit of Ref. 28. The problem we are facing, which is to define what we mean by J(Y) = ∫Yw2(TY)w3(TY) when Y has a boundary M, is analogous to the problem that we already discussed in Sec. III B 1, where we were trying to define the integral of a different cohomology class [namely, w2(TM′) · c1(L)] on a manifold M′ with boundary. As in that case, a trivialization of the cohomology class along the boundary gives a way to define the integral J(Y).

We can trivialize w2(TY)w3(TY) along the boundary of Y by trivializing either w2(TY) or w3(TY) along the boundary. For our present application, we want to trivialize w3(TY) along the boundary. Physically, if we assume the existence of an emergent spinc connection on M = ∂Y, this gives a natural mechanism to trivialize W3(Y) and hence w3(Y) along M. With this assumption, let us see how to describe the boundary state. Suppose first that there is some manifold Ỹ with boundary M such that the spinc connection A of M extends over Ỹ. We do not assume that Ỹ is isomorphic to Y, and in general, A does not extend over Y. The allowed choices of Ỹ depend on A because of the condition that A must extend over Ỹ. We orient Ỹ so that Ỹ and Y can be glued together along M to make a closed five-manifold Y^. Then, we define

JA(Y)=J(Y^).
(4.13)

The right hand side is not necessarily zero because Y^ may not admit a spinc connection (only Ỹ is guaranteed from the construction to admit one), and it may depend on the original spinc connection A on M because this affects the allowed choices of Ỹ. The point of the definition in Eq. (4.13) is, however, that the right hand side does not depend on the choice of Ỹ but only on the original five-manifold Y with the emergent spinc connection A on its boundary. To see this, let Ỹ be some other five-manifold with boundary M over which A extends, and let Y^ be made by gluing together Y and Ỹ. Also, let Y* be made by gluing together Ỹ and Ỹ along M, after reversing the orientation of one so that they fit smoothly. Then, we have the identity (4.8) that we used previously in a similar context. But now we have the further simplification that Y* is a spinc manifold since it was defined by gluing together Ỹ and Ỹ, over each of which the spinc structure of M extends. So w3(TY*) = 0, and therefore, the second term on the right hand side of Eq. (4.8) vanishes. The identity reduces to Y^w2(TY^)w3(TY^)=Y^w2(TY^)w3(TY^), showing that JA(Y) as defined in Eq. (4.13) does not depend on the choice of Ỹ.

This enables us to define the boundary state. As a substitute for (−1)J(Y), which is the partition function in the absence of a boundary, we include in the path integral over A a factor (1)JA(Y). This makes sense because as we have seen, this factor only depends on A and Y. In addition to this factor, the integrand of the path integral contains some factor exp(−I(A)), where I(A) is a conventional local action; for example, we certainly expect a Maxwell term 14e2Md4xgFijFij, where F = dA. (We also expect a topological term θMFF/(2π)2, about which more later.) An important detail, however, is that we do not add the factor (1)Mw2(TM)c1(L) that was important in Sec. III B. Rather, this term is automatically incorporated in the factor (1)JA(Y) that we have already included. How this happens is similar to what we explained in Sec. III B 1. The definition of JA(Y) used a spinc connection A to trivialize w3(Y) along M = ∂Y. If A and A′ are two spinc connections, then their difference AA′ is an ordinary U(1) gauge field, a connection on a complex line bundle L*M. Reasoning similar to that in Sec. III B leads to

(1)JA(Y)=(1)JA(Y)(1)Mw2(TM)c1(L*).
(4.14)

Informally, the last factor on the right hand side is (1)Mw2(TM)(F/2πF/2π). Keeping A′ fixed and letting A vary, we get, up to a constant multiple (that is, an additive constant in the exponent), the interaction (1)Mw2(TM)F/2π that was discussed in Sec. III B. The reason that we recover the previous result only up to a constant multiple is that in the present discussion, we are studying not just all-fermion electrodynamics on the boundary but its combination with a bulk topological field theory.

In the above discussion, we assumed that M is the boundary of some manifold Ỹ over which the spinc structure of M extends. In general, this is not the case. Just as in Sec. IV B, this means not that the boundary state does not exist, but that it is subtle to describe in this state what we mean by the topological angle θ. The obstruction to the existence of Ỹ (given that M, ignoring its spinc structure, is the boundary of Y) is that the spinc connection on M might have a nonzero instanton number. The integer-valued instanton number in U(1) gauge theory is

Q=MFF(2π)2.
(4.15)

In gauge theory, one usually includes in the path integral a factor exp(iθQ). As Q is an integer, it follows that in purely four-dimensional gauge theory, θ is an angular variable with period 2π. In the present context, matters are somewhat different. Given that M is the boundary of some Y, Q takes values in (Note 17 in the  Appendix) 2Z, so naively θ has period π. But there is a subtlety in defining what we mean by θ, related to the fact that if Q ≠ 0, then Ỹ does not exist. We proceed as in Sec. IV C. Pick an example with Q = 2, the smallest possible value; for instance, one can take M0 = S2 × S2, with a spinc connection A0, of field strength F0, such that F0/2π integrates to 1 over either S2 factor in M0. M0 is the boundary of, for example, Y0 = B3 × S2, where B3 is a three-ball with boundary S2. (The spinc structure of M0 does not extend over Y0 since this would contradict the fact that B3F0/2π=S2F0/2π=1.) Because we do not have a natural way to define (1)J(Y0) in this example, we arbitrarily declare it to be positive. This means that the phase of the amplitude for (Y0, M0, A0) is e2iθ, fixing what we mean by θ at the quantum level. Once this is specified, we proceed for any example (Y, M, A) in the same way as in Sec. IV C. We simply observe that any (M, A) is cobordant to a certain number of copies of (M0, A0), and then, we proceed as before. The upshot is that the arbitrarily chosen overall sign of the path integral amplitude for (Y0, M0, A0) enables us to define the path integral amplitude for any (Y, M, A). If the instanton number integrated on M is n, then the path integral amplitude is

ZM,A,Y=(1)J(Y¯)einθexp(I(A,M)),
(4.16)

where Y¯ is defined as in Fig. 4. In Eq. (4.16), I(A, M) is the action for the spinc connection on M, except that we omit the θ term in I and write it explicitly as a factor ei. The reason for doing this is that it is subtle to explain what we mean by θ. Making a different choice of the reference point (Y0, M0, A0) would change what we mean by Y¯ and thus would possibly shift J(Y¯). If so, this would have the same effect as shifting θ by π/2. Likewise, changing the sign of the amplitude at (Y0, M0, A0) would have the effect of shifting θ by π/2. So the parameter θ that appears in Eq. (4.16) is really not just a parameter from the classical action but is a sort of quantum-corrected parameter.

As in Sec. III C, we can go farther and, by giving an expectation value to a Higgs field of charge 2, we can spontaneously break the U(1) gauge symmetry to Z2. To describe the situation more geometrically, this reduces (Spin(4)×U(1))/Z2 to (Spin(4)×Z2)/Z2=Spin(4) and means that the emergent spinc connection is reduced to an emergent spin structure.

There is a fairly evident reason that such a boundary state is possible. An emergent spin structure on M = ∂Y trivializes w2(TY) near the boundary and therefore, topologically, makes it possible to define what we mean by J(Y) = ∫Yw2(TY)w3(TY).

At least one more boundary state of this theory can be described in a somewhat similar way. By Higgsing from SU(2) to U(1), or by directly postulating the existence on M = ∂Y of an emergent spinc connection, we described a boundary state in which W3(Y) is trivialized along the boundary. This implies a trivialization of w3(Y) and made it possible to define J(Y).

Existence of a spinc connection does not in general lead to an arbitrary trivialization of w3(Y), and of course, a spinc connection [since it looks locally like a U(1) gauge field] describes much more than a trivialization of w3(Y). A boundary state defined with an emergent spinc connection is gapless.

We could instead have a gapped boundary state in which the emergent variable along the boundary M is a trivialization of w3(M)—nothing more nor less. Two trivializations of w3(M) differ by an element of H2(M,Z2), so such a boundary state looks locally like we are summing over an element of H2(M,Z2), though the global structure is more subtle.

It is not completely clear if this boundary state is new or is a dual formulation of the one described in Sec. IV E.

We can also use these ideas to explain in a precise way the statement that a theory with the new SU(2) anomaly can be formulated in a consistent fashion on a four-manifold M that is endowed with a choice of a spin or spinc structure. Spin and spinc structures are relevant because they trivialize w2 and w3, respectively, and therefore trivialize the anomaly ∫Yw2w3.

As always, the problem is to explain what is meant by the Pfaffian Pf(DM) of the four-dimensional fermion operator DM. In the abstract, on a four-manifold M with a spin-SU(2) structure and no further structure, there is no way to determine the sign of this Pfaffian. However, suppose that M is endowed with a spin structure or a spinc structure, and suppose that M is the boundary of a five-manifold Y over which the spin or spinc structure of M extends. Then, as in Eq. (4.1), we define the fermion path integral on M as

|Pf(DM)|(1)I(DY).
(4.17)

By familiar arguments, using the fact that ∫Yw2w3 = 0 on a spin or spinc manifold, we deduce that in the case of a theory that has the new SU(2) anomaly but not the old one, this definition of the fermion path integral does not depend on the choice of Y. If a suitable Y does not exist, we deal with this as in Sec. IV C by assigning arbitrary phases to the generators of the appropriate cobordism group. The group that classifies four-manifolds M with spin-SU(2) structure up to cobordism is Z×Z, with the two invariants being the SU(2) instanton number and the signature of M (which one could call the gravitational instanton number). After assigning arbitrary phases to the generators of the cobordism group, one gets a definition of the fermion path integral on M along with a definition of the gauge theory and gravitational theta-angles at the quantum level. But all this depends on having picked on M a spin or spinc structure to trivialize the anomaly.

Much of what we have said has interesting analogs for certain topological phases of matter in four dimensions. The following discussion will be in the spirit of Ref. 28 (and part of the following was explained there).

In four dimensions, there is a gapped theory of bosons, not protected by any symmetry, whose partition function on a four-manifold M is (−1)K(M), where K(M) = ∫Mw2(TM) · w2(TM). From what we have said, it is clear how to construct a gapped boundary state for this theory. If M has a nonempty boundary W, then K(M) is not well-defined a priori, but it becomes well-defined if one is given a trivialization of w2(TM) near W. Such a trivialization corresponds naturally to a spin structure on W. So this gives a gapped boundary state that can locally be described in terms of an emergent Z2 gauge symmetry on W but is better described by saying that the theory has a dynamically generated spin structure. A spin structure makes it possible to describe the propagation of fermions, so such a theory will generically have fermionic quasiparticles on the boundary, though in bulk it has bosons only.

The theory with partition function (−1)K(M) makes sense on unorientable manifolds since K(M) is well-defined on a possibly unorientable closed four-manifold. Placing this theory on an unorientable manifold is only natural if one assumes time-reversal or reflection symmetry, and for brevity, we will refer to time-reversal symmetry (Note 18 in the  Appendix). There are two possible generalizations of a spin structure to an unorientable manifold, namely, a pin+ structure and a pin structure, where in the former case, a fermionic quasiparticle will transform as a Kramers doublet under time-reversal, and in the latter case, it will transform as a Kramers singlet. Since on an unorientable manifold w2(TM) is the obstruction to a pin+ structure (and so is trivialized by a choice of pin+ structure), the dynamical spin structure of the last paragraph becomes a dynamical pin+ structure once we allow spacetime to be unorientable. So once we assume time-reversal symmetry, the theory with partition function (−1)K(M) has a gapped boundary state with dynamical pin+ structure and with Kramers doublet fermions on the boundary.

Once we assume time-reversal symmetry, another invariant is possible, namely, L(M) = ∫Mw1(TM)4. There are consequently two more theories, namely, the ones whose partition functions on a closed manifold are (−1)L(M) and (−1)L(M)+K(M) = (−1)T(M), where T(M)=M(w2(TM)+w1(TM)2)2. Both of these theories admit gapped boundary states that can locally be described by an emergent Z2 gauge theory but globally have a more illuminating description.

We consider first the theory whose bulk partition function is (−1)T(M). Here, if M has boundary W, we need to trivialize (w2(TM)+w1(TM)2)2 near W. An obvious way to do this is to trivialize w2(TM) + w1(TM)2 near W. As w2(TM) + w1(TM)2 is the obstruction to a pin structure, it is trivialized by the choice of a pin structure. Thus, this theory has a gapped boundary state with a dynamical pin structure on the boundary. This boundary state will have Kramers singlet fermionic quasiparticles.

Finally, we consider the theory with bulk partition function (−1)L(M). To define ∫Mw1(TM)4 on a four-manifold M with boundary W, we need to trivialize w1(TM)4 near the boundary. One way to do this is to trivialize w1(TM)2. Let us discuss what sort of boundary state this will give. We follow a logic similar to what we discussed in Sec. III B 1. We start with the short exact sequence of abelian groups

0Z22Z4rZ20,
(4.18)

where the first map is multiplication by 2 and the second is reduction mod 2. This leads to a long exact sequence of cohomology groups that reads in part

H1(M,Z4)rH1(M,Z2)βH2(M,Z2).
(4.19)

It can be shown that for any class xH1(M,Z2), β(x) = x2. Thus, the exactness of the sequence tells us that x2 = 0 if and only if x = r(y) for some y and more to the point that the choice of such a y gives a trivialization of x2.

In our application, we want to think of the sequence as follows:

0Z2emergent2Z4totalrZ2T0.
(4.20)

Here, we take Z2T to be the Z2 symmetry generated by time-reversal T. Z2emergent will be an emergent Z2 gauge symmetry that exists only along the boundary. The combined symmetry group will be Z4total. We choose x = w1(T). x is the first Stiefel-Whitney class of a Z2 bundle over M that is known as the orientation bundle of M: it is a Z2 bundle ε whose holonomy around a loop is +1 or −1 depending on whether the orientation of M is preserved in going around . We have found that we can trivialize x2 by picking a “lift” of x to a mod 4 class y whose mod 2 reduction is x. A choice of y corresponds to a Z4 bundle whose holonomy around a loop is the square root of the holonomy of ε (so for an orientation-preserving loop, the holonomy is ±1, and for an orientation-reversing one, it is ±i). In a boundary state based on emergent trivializations of w1(TM)2, we sum over such structures. Two such structures differ by an element of H1(M,Z2), that is, a flat Z2-valued gauge field, so the boundary state locally looks like a theory with an emergent Z2 gauge symmetry, though that is not the best description globally.

Let us consider the quasiparticles in such a boundary state. The boundary state gives no choice of spin or pin± structure, so the quasiparticles are bosons. Consider a quasiparticle state |Ψ⟩ that is odd under Z2emergent. The embedding in Eq. (3.7) tells us that the nonzero element of this group corresponds to the element 2Z4total. So a quasiparticle state |Ψ⟩ that carries the emergent gauge charge is odd under the element 2 of Z4total, that is, this element acts by |Ψ⟩ → −|Ψ⟩. We can lift the time-reversal symmetry T to either generator 1 or 3 of Z4total. Whichever one we pick, T2 corresponds to element 2 of Z4total and, as we have just seen, acts by |Ψ⟩ → −|Ψ⟩. Since T2|Ψ⟩ = −|Ψ⟩, the quasiparticles that carry the emergent gauge charge are Kramers doublet bosons.

A Z2 theory in 2 + 1 dimensions has vortex particles as well as particles carrying the Z2 charge. By extending the reasoning, it is possible to show that in each of the three examples, the vortex has the same properties (boson or fermion, Kramers singlet or doublet) as the charge.

For the theory with partition function (−1)L(M), we can also construct a boundary state in which the emergent variable on the boundary W is a trivialization of w1(TW)3. This can be discussed analogously to what we said in Sec. IV F. One might wonder what is a boundary state with an emergent variable that trivializes w1(TW), rather than one of its powers. This corresponds to a state in which T is spontaneously broken since a field whose expectation value trivializes w1(TW), while preserving Poincaré symmetry, is a T-odd scalar field.

What we have described in this paper has some generalizations that are worth at least brief mention.

1. Sp(2N)

First of all, one can replace SU(2) with another gauge group. The closest analog is (Note 19 in the  Appendix) Sp(2N), which of course has pseudoreal representations just like SU(2). Following the logic of Sec. II, it is not difficult to see that the original SU(2) anomaly has a close analog for a single multiplet of four-dimensional fermions in the fundamental representation of Sp(2N). Moreover, the new SU(2) anomaly has a close analog for a single multiplet in the third symmetric tensor product of the fundamental representation of Sp(2N). Both of these statements can be generalized to arbitrary Sp(2N) representations by repeating what we have done for SU(2), but we will not explore that direction.

2. SO(3) × SU(2)

An SO(3) gauge theory with SU(2) global symmetry and a single multiplet of fermions transforming as (1, 1/2) under SO(3) × SU(2) [that is, isospin 1 under SO(3) and isospin 1/2 under SU(2)] was recently studied in Ref. 5. It was shown that interesting anomalies arise if one gauges the SU(2) global symmetry and then formulates the theory on a spin-SU(2) manifold. There are three possible cobordism invariants, which one can take to be I1/2, ∫Yw2(TY)w3(TY), and ∫Yw2(E)w3(TY), where E is the SO(3) bundle. It was found that the anomaly of this theory is I1/2+Yw2(E)w3(TY). The term ∫Yw2(E)w3(TY) was found by reinterpreting a computation29 in a topologically twisted supersymmetric gauge theory.

From the point of view of the present paper, one would say that since the (1, 1/2) representation of SO(3) × SU(2) is pseudoreal, the anomaly for this representation is given by the mod 2 index I1,1/2 of the Dirac operator with values in this representation. To show that I1,1/2=I1/2+Yw2(E)w3(TY), it suffices to calculate for a set of manifolds that detect the three invariants. For such a set, we can take Y1 = S4 × S1, with an SU(2) bundle of instanton number 1 on S4 and a trivial SO(3) bundle; Y2=CP2S1, with a spin-SU(2) structure of flux 1/2 and trivial SO(3) bundle; and Y2=CP2S1, with a spin-SU(2) structure of flux 1/2 and an SO(3) bundle whose restriction to CP2 has w2(E) ≠ 0. One can compute I1,1/2 for these three examples, using the facts described in Sec. II, and [using also the determination of w3(TY) in Sec. IV B] confirm that I1,1/2=I1/2+Yw2(E)w3(TY). The left and right hand sides are both nonzero for Y1 and Y2 and zero for Y2.

After suitable Higgsing to U(1), an SO(3) × SU(2) gauge theory with fermions in the (1, 1/2) ⊕ (0, 1/2) representation gives an ultraviolet completion of all-fermion electrodynamics. [The role of the (0, 1/2) fermions is to cancel the old SU(2) anomaly.] For this, one first breaks the symmetry from SO(3) × SU(2) to a diagonally embedded SU(2) subgroup. Under this subgroup, the fermions transform as the direct sum of isospin 3/2 and two copies of isospin 1/2, so further symmetry breaking to U(1) will give a model of all-fermion electrodynamics.

3. Grand unification without spin structure

An interesting case that does not lead to a new anomaly is the following. One of the standard grand unified theories of four-dimensional particle physics is a Spin(10) gauge theory in which fermions are in spinor representation of Spin(10) and bosons are in tensor representations. Thus, fermions are odd under a 2π rotation in Spin(10), while bosons are even. Hence, this theory can potentially be formulated on a manifold with (Spin(4)×Spin(10))/Z2 structure.

This theory is completely anomaly-free. By standard criteria, the model lacks perturbative anomalies. One way to show that it has no global anomaly is to first observe that the only cobordism invariant of a five-manifold Y with (Spin(5)×Spin(10))/Z2 structure is ∫Yw2(TY)w3(TY) [which does not really depend on the (Spin(5)×Spin(10))/Z2 structure]. To check that the Spin(10) grand unified theory has no anomaly associated to the invariant ∫Yw2(TY)w3(TY), it suffices to consider an example. In Sec. II, we constructed a (Spin(4)×SU(2))/Z2 structure on CP2 and found an anomaly because the path integral measure was not invariant under a certain diffeomorphism plus gauge transformation φ^. Since SU(2) is isomorphic to Spin(3), which embeds in Spin(10) in an obvious way, we can promote this example to a (Spin(4)×Spin(10))/Z2 structure on CP2. Now, recall that in this example, the (Spin(4)×SU(2))/Z2 Dirac operator acting in the isospin 1/2 representation of SU(2) has no zero-modes. The spinor representation 16 of Spin(10) (the usual representation for grand unification) decomposes under SU(2) ≅ Spin(3) as the direct sum of eight copies of the isospin 1/2 representation. So it too has no zero-modes, and hence, the path integral measure is invariant under any classical symmetry of the bosonic background, such as φ^.

One interesting consequence of this is that, since it can be formulated without any choice of spin structure, the Spin(10) grand unified theory can conceivably arise as a critical point in a purely bosonic theory. Upon symmetry breaking to a subgroup that does not contain the 2π rotation in Spin(10)—such as SU(5), another standard gauge group for grand unification in particle physics—a dynamical spin structure is generated.

Another generalization is to incorporate time-reversal symmetry. The basic issue here is to ask what time-reversal properties are possessed by a four-dimensional SU(2) gauge theory with fermions consisting of, for example, a single multiplet with some half-integral isospin j. Let us suppose that we want a time-reversal symmetry that commutes with SU(2). To construct a time-reversal invariant theory with a single isospin j multiplet, what we need is then a four-dimensional pseudoreal spinor representation of either Pin+(1, 3) or Pin(1, 3). If the four-dimensional spinor representation of one of these groups is pseudoreal, then upon taking the tensor product with the isospin j representation of SU(2), we will get a real representation of Pin±(1, 3) × SU(2) or of (Pin±(1,3)×SU(2))/Z2 (depending on which version of the theory one wants to consider). This representation will have dimension 4(2j + 1), as is appropriate to describe a single multiplet of fermions of isospin j.

To describe Pin+(1, 3), start with a Clifford algebra of signature − + + +, that is, a Clifford algebra with

{γμ,γν}=2ημν,    ημν=diag(1,1,1,1).
(5.1)

This Clifford algebra has a representation by 4 × 4 real matrices. So the four-dimensional spinor representation of Pin+(1, 3) is real. Upon taking the tensor product with the isospin j representation of SU(2), for half-integral j, we get a pseudoreal representation of (Pin+(1,3)×SU(2))/Z2 so that with Pin+(1,3)×SU(2))/Z2 symmetry, it is not possible to have a single fermion multiplet of isospin j.

On the other hand, the spinor representation of Pin(1, 3) is a four-dimensional space on which acts a Clifford algebra of signature + − − −, that is, a Clifford algebra with

{γμ,γν}=2ημν.
(5.2)

This algebra cannot be represented by 4 × 4 real matrices, so the four-dimensional spinor representation of Pin(1, 3) is not real. Rather, it is pseudoreal, so (Pin(1,3)×SU(2))/Z2 has a real representation of the desired type.

Once we generalize the anomalous SU(2) gauge theories studied in this paper to be time-reversal invariant, the anomalies that we have studied will have to generalize. After all, we are free to forget time-reversal symmetry and just look at these theories on an orientable manifold; the anomalies that we have found cannot disappear because we have learned that the theory actually has time-reversal symmetry.

So the discussion in this paper will generalize to pin manifolds with SU(2) gauge fields and more generally to manifolds with (Pin(1,3)×SU(2))/Z2 structure. The mod 2 index in five dimensions generalizes nicely in this situation. A few details in the presentation need some modification. The obstruction to a pin structure is w2(TM) + w1(TM)2, so this will play the role that was played by w2(TM) in some statements. In a theory that is required to be time-reversal invariant, the θ parameter is no longer an arbitrary angle. Rather, with the normalization that we used in Sec. IV, its allowed values are 0, π (or 0, π/2 in the spinc case).

Finally, we can consider an SU(2) gauge theory in five dimensions. The spinor representation of Spin(1, 4) is pseudoreal, so it is possible in five dimensions to consider a single multiplet of fermions in a pseudoreal representation, say, the isospin j representation of SU(2) for some half-integer j. (This was discussed in Sec. II A.) Upon dimensional reduction to four dimensions, this theory reduces to the four-dimensional theory with a single multiplet of fermions with isospin j. This class of four-dimensional theories has anomalies that were studied in Sec. II. As we will now explain, the five-dimensional version of the theories has very similar anomalies.

In fact, the examples that we considered in Sec. II to exhibit anomalies in four dimensions can be reinterpreted to exhibit anomalies in five dimensions. Consider the five-manifold Y1 = S4 × S1 with an SU(2) instanton bundle of instanton number 1 on the S4 and gauge fields and spin structure pulled back from S4. The mod 2 index of an isospin 1/2 fermion on this five-manifold is nonzero, as discussed in Sec. II B. We interpreted this nonzero index in our earlier discussion as a five-dimensional invariant that measures an anomaly in a four-dimensional theory. But the same calculation has a direct interpretation in five dimensions: having an odd number of fermion zero-modes in this example shows that the operator (−1)F can be anomalous in a five-dimensional theory with a single fermion multiplet of j = 1/2. Thus, this theory is anomalous. [This anomaly was pointed out in Ref. 30. A certain string theory construction with gauge symmetry Sp(2N) in five dimensions always gives an even number of fermion multiplets in the fundamental representation;31 this reflects the fact that the case with an odd number of such multiplets is anomalous.]

The new SU(2) anomaly similarly has an analog in five dimensions. We simply consider the five-manifold Y2=CP2S1 with the (Spin(5)×SU(2))/Z2 structure that was exploited in Sec. II. For this (Spin(5)×SU(2))/Z2 structure, the mod 2 index of the Dirac operator in the j = 3/2 representation of SU(2) is nonzero. In Sec. II F, we interpreted this as a five-dimensional invariant that diagnoses a new SU(2) anomaly in four dimensions. But the same nonzero mod 2 index represents an anomaly in (−1)F in a five-dimensional theory with a single fermion of isospin 3/2. Thus, also the new SU(2) anomaly has a five-dimensional analog.

The six-dimensional cobordism invariant that detects the old and new SU(2) anomalies in five dimensions is simply the mod 2 index of a chiral six-dimensional Dirac operator acting on fermions in the isospin 1/2 or isospin 3/2 representation of SU(2). (Here, by a chiral Dirac operator in six dimensions, we mean a Dirac operator acting on a spinor field of one definite chirality.) Following the logic of Sec. II, as an example of six-manifolds that detect the old and new SU(2) anomalies, we can just take products Y1 × S1 or Y2 × S1, with metrics, gauge fields, and spin structures all pulled back from Y1 or Y2.

We thank C. Cordova for helpful remarks. J.W. gratefully acknowledges support from a Corning Glass Works Foundation Fellowship and from NSF Grant No. PHY-1606531. X.G.W. is partially supported by NSF Grant Nos. DMR-1506475 and DMS-1664412. E.W. is partially supported by NSF Grant No. PHY-1606531.

  • Note 1.

    This important property of SU(2) gauge theory was pointed out by S. Coleman at Aspen in the summer of 1976.

  • Note 2.

    Here, wk(TY), kZ, which are often denoted simply as wk, are Stiefel-Whitney classes of the tangent bundle TY of Y.

  • Note 3.

    D4 maps ψ to ψ̃ and vice versa, so in a basis ψψ̃, one has D4=0**0. D4 cannot be block-diagonalized in a real basis.

  • Note 4.

    If this assertion is unfamiliar, the reader may wish to jump ahead to Eq. (2.6).

  • Note 5.

    Such structures have been discussed in Refs. 3–5. They are sometimes called Spinh structures.

  • Note 6.

    This manifold is a special case of the Dold manifold (CPn×Sm)/Z2, where Z2 acts as complex conjugation on CPn and as the antipodal map on Sm.

  • Note 7.

    This is the interesting case, as explained at the beginning of Sec. II B. For integer j, one can still formally define a mod 2 index of the five-dimensional Dirac operator, but it vanishes identically. This is clear from the fact that a bare mass is possible for such a field. A direct proof can be given by showing that the Euclidean Dirac operator for such a field has an antilinear symmetry whose square is −1, so by a Euclidean analog of Kramers doubling, the space of its zero-modes has even dimension. See Ref. 12 for such arguments.

  • Note 8.

    Concretely,15 after giving the fermions a mass by coupling to ϕ, one can be left with an SO(2) global symmetry that rotates them. Monopoles can have half-integral charge under this SO(2).

  • Note 9.

    This assertion, and its generalization for any isospin j, follows from the formula for index(L) on p. 231 of Ref. 16. In that formula, one has to take T = j. One also takes m = 0 (because we are interested in a case in which a bare mass m is not possible) and hence {m} = −1/2.

  • Note 10.

    This anomaly has been discussed in Refs. 24 and 27. The fact that the interaction (3.2) makes the monopole a fermion is explained in Ref. 24 and in Ref. 5, Sec. IV C.

  • Note 11.

    All statements are mod 2.

  • Note 12.

    Since there are now different five-manifolds in play, instead of just denoting a five-dimensional Dirac operator as D5, we write DỸ, etc., indicating the five-manifold. To specify the spin-SU(2) structure, we sometimes write DỸ,A, etc.

  • Note 13.

    The gluing theorem can be proved by picking on M a spin-SU(2) structure such that the Dirac operator DM has no zero-modes and then picking on Y* a metric that near M looks like a long tube M×R. In this limit, the low-lying modes of the Dirac operator on Y* come from what in the limit of an infinitely long tube would be zero-modes localized on either Ỹ or Ỹ, and this leads to the additivity of the mod 2 index.

  • Note 14.

    One way to prove this is to use the index theorem for the index of the spin-SU(2) Dirac operator on M in the spin 1/2 representation of SU(2). This theorem writes the index, which is an integer, as the sum of Q and a multiple of the signature of M. But the signature vanishes if M is a boundary, so Q is an integer, the index.

  • Note 15.

    Here, MnM0 is the disjoint union of M with |n| copies of M0; if n > 0, the minus sign in MnM0 means that one reverses the orientation of M0.

  • Note 16.

    This boundary state is already known.24,8 The underlying anomaly, viewed as a mixed anomaly between 1-form global symmetries in four-dimensional U(1) gauge theory, was described in Ref. 27, and the use of this anomaly to construct this boundary state was briefly indicated in Footnote 31 of Ref. 5.

  • Note 17.

    We will explain the proof somewhat informally. The flux condition (2.11) means that we can think of F/π as w2 + 2x, where x is an integral class, so Q=14M(w22+4w2x+4x2). Since Mw22 is a cobordism invariant and M = ∂Y is a boundary, this term vanishes and Q = M(w2x + x2). A standard result is that for any integral class x, Mw2x is congruent mod 2 to Mx2, and hence, Q is congruent mod 2 to 2Mx2 and is an even integer.

  • Note 18.

    In a relativistic theory, the CRT theorem (where CRT is the product of a spatial reflection and time-reversal; because time-reversal is antiunitary, the combination is traditionally called CRT rather than just RT) relates time-reversal symmetry to reflection symmetry, so either of these implies the other and there is no distinction. In condensed matter physics, the two cases are potentially different. They lead to the same physics at low energies when there is an emergent relativistic description (as we assume here). But in condensed matter physics, R and T can allow different relevant parameters that might potentially obstruct the emergence of a relativistic description at low energies, so the problem of finding an emergent relativistic description can be essentially different depending on whether a system has microscopic R symmetry or microscopic T symmetry.

  • Note 19.

    Our terminology is such that Sp(2) coincides with SU(2), and in general, the fundamental representation of Sp(2N) has complex dimension 2N.

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