N=(4,4) vector multiplets on curved two-manifolds

Lawrence, Albion; Soroush, Masoud

doi:10.1063/1.4945585

We study the necessary conditions for preserving $N = (4, 4)$ supersymmetry on curved 2d backgrounds, following the strategy of Dumitrescu, Festuccia, and Seiberg. We derive the transformation laws and invariant action for off-shell Abelian vector multiplets. An explicit solution of the supersymmetry conditions is found on the round two-sphere.

I. INTRODUCTION

Localization techniques for supersymmetric quantum field theories on curved Euclidean backgrounds have yielded a remarkable set of exact results. Pestun’s pioneering work computed partition functions and Wilson line expectation values for supersymmetric gauge theories on S⁴.¹ There has since been a host of results for three-dimensional quantum field theories.^2–9 (For a review see Ref. 10.)

In two dimensions, these techniques were used to compute the exact partition function on S² for $N = (2, 2)$ gauged linear sigma models (GLSMs).^11,12 For GLSMs which flow to nonlinear sigma models with Calabi-Yau target spaces, these results were shown to yield the exact Kähler potential on the moduli space of Kähler deformations of the Calabi-Yau.^13–16 They have also been used to study 2d theories on spaces with boundaries^17,18 and on nonorientable surfaces.¹⁹

The starting point of these calculations is the realization of supersymmetry on curved backgrounds, which involves constraints on the supersymmetry parameters—for example, they might be Killing or conformal Killing spinors^1,11,12—and a deformation of the flat space algebra. A systematic approach to finding these constraints and deformations was initiated for four-dimensional theories in Refs. 20–22. In this approach, one considers off-shell supergravity multiplets in the limit that the Planck scale goes to infinity. Demanding that supersymmetry transformations do not transform the background metric and gravitino constrains the spinors; allowed background auxiliary fields parametrize the deformations of the flat-space superalgebra.^23,24,54 These constraints have been carefully analyzed in various cases in four^{20–22,25–27} and two dimensions,²⁸ the latter for $N = (2, 2)$ theories. In four dimensions, localization techniques have been applied to study the dependence of the partition functions of these theories on the geometry of the spacetime manifold.^25,26

$N = (4, 4)$ backgrounds retain a great deal of interest—see, for example, Refs. 29–38. In this article, we study manifestly $N = (4, 4)$ supersymmetric quantum field theories on curved two-dimensional backgrounds, following the procedure in Refs. 20 and 21.⁵⁵

We will focus on theories of Abelian vector multiplets only. These have an off-shell realization with a finite number of auxiliary fields. Our eventual goal is to consider $N = (4, 4)$ gauged linear sigma models which might, for example, realize A_n asymptotically locally euclidean spaces via hyper-Kähler quotients. The completely off-shell hypermultiplets which transform linearly under gauge symmetries cannot be realized via a finite number of auxiliary fields; partially off-shell multiplets (which close on an off-shell central charge) with finite set of auxiliary fields do exist,³⁹ as well as hypermultiplets which cannot be separately linearly gauged without breaking the SU(2) R-symmetry.^39–41 We found the story for vector multiplets to be sufficiently rich to merit its own treatment.

This paper is organized as follows. In Section II, we review the “large” $N = (4, 4)$ superalgebra, as described in Ref. 42. In Section III, we study the supersymmetry transformations of the field components of the gravity multiplet and derive the necessary conditions for preserving $N = 4$ rigid supersymmetry on a curved background. In Section IV, we study the resulting constraints on 2d manifolds which admit $N = (4, 4)$ supersymmetry. In Section V, we derive the supersymmetry transformations of the vector multiplet propagating on curved backgrounds and show that they transform under the large $N = (4, 4)$ superalgebra. In Section VI, we explicitly solve the supersymmetry equations derived in Section III on the two-sphere. Section VII is devoted to our concluding remarks and possible directions for future research. Appendix A summarizes our notations and conventions and presents a list of useful identities used throughout the paper; Appendix B reviews the isomorphism of SO(4) and SU(2) × SU(2).

II. $N = (4, 4)$ SUPERALGEBRA

$N = (4, 4)$ superconformal algebras come in a one-parameter family^43,42 of algebras which generically contain the Kac-Moody algebra 𝔰𝔲(2) ⊕ 𝔰𝔲(2) ⊕ 𝔲(1), of which the 𝔰𝔲(2) components correspond to an R symmetry. Two particular values of this parameter correspond to well-known versions of $N = (4, 4)$ algebras as follows:

The “large” $N = 4$ superconformal algebra with Kac-Moody subalgebra 𝔰𝔬(4) ≡ 𝔰𝔲(2)⁽⁺⁾ ⊕ 𝔰𝔲(2)⁽⁻⁾ and finite subalgebra 𝔬𝔰𝔭(4|2). This corresponds to $γ = \frac{1}{2}$ in Ref. 42. The labels (±) merely distinguish the two 𝔰𝔲(2) factors.
The “small” superconformal algebra with Kac-Moody subalgebra 𝔰𝔲(2) and finite subalgebra 𝔰𝔲(2|1, 1). This corresponds to γ → 1 or γ → 0 in Ref. 42. This is a subalgebra of the large algebra. In the limit γ → 1, the R-symmetry maps to 𝔰𝔲(2)⁽⁺⁾; in the limit γ → 0, the R-symmetry maps to 𝔰𝔲(2)⁽⁻⁾.

In all of the theories in Ref. 42, the supercharges G transform as a vector under 𝔰𝔬(4), with index a = 1, …, 4. If we write 𝔰𝔬(4) = 𝔰𝔲(2)⁽⁺⁾ ⊕ 𝔰𝔲(2)⁽⁻⁾, the generators of 𝔰𝔬(4) can be written in terms of generators $α_{a b}^{(\pm) I}$ of the two 𝔰𝔲(2) subalgebras, where I = 1, …3,

[) α^{(\pm) I}, α^{(\pm) J} (] = - ε^{I J K} α^{(\pm) K}, [) α^{(+) I}, α^{(-) J} (] = 0, \{) α^{(\pm) I}, α^{(\pm) J} (\} = - \frac{1}{2} δ^{I J} .

(2.1)

A convenient representation for $α_{a b}^{(\pm) I}$ is⁴²

α_{a b}^{(\pm) I} = \pm \frac{1}{2} () δ_{a}^{I} δ_{b}^{4} - δ_{b}^{I} δ_{a}^{4} () + \frac{1}{2} ε_{I a b},

(2.2)

with the understanding that the ε_Iab symbol vanishes if any of its indices is equal to 4. Appendix B describes the decomposition of the vector of 𝔰𝔬(4) into a (2, 2) of 𝔰𝔲(2) ⊕ 𝔰𝔲(2).

To set notation, we quote the chiral (left-moving) part of the “large” $N = (4, 4)$ superconformal algebra,⁴²

\begin{aligned} [) L_{m}, L_{n} (] = (m - n) L_{m + n} + \frac{1}{12} c (m^{3} - m) δ_{m + n}, \\ [) L_{m}, ϕ_{n} (] = () (d_{ϕ} - 1) m - n () ϕ_{m + n}, \\ \{) G_{m}^{a}, G_{n}^{b} (\} = 2 δ^{a b} L_{m + n} + \frac{1}{3} c δ^{a b} δ_{m + n} () m^{2} - \frac{1}{4} () + 2 (n - m) () α_{a b}^{(+) I} A_{n - m}^{(+) I} + α_{a b}^{(-) I} A_{n + m}^{(-) I} (), \\ [) A_{m}^{(\pm) I}, A_{n}^{(\pm) J} (] = ε^{I J K} A_{m + n}^{(\pm) K} - \frac{1}{6} c m δ^{I J} δ_{m + n}, \\ [) A_{m}^{(+) I}, A_{n}^{(-) J} (] = 0, \end{aligned}

(2.3)

\begin{aligned} [) A_{m}^{(\pm) I}, G_{n}^{a} (] = α_{a b}^{(\pm) I} () G_{m + n}^{b} \mp m Q_{m + n}^{b} (), [) A_{m}^{(\pm) I}, Q_{n}^{a} (] = α_{a b}^{(\pm) I} Q_{m + n}^{b}, \\ \{) Q_{m}^{a}, G_{n}^{b} (\} = 2 () α_{a b}^{(+) I} A_{m + n}^{(+) I} - α_{a b}^{(-) I} A_{m + n}^{(-) I} () + δ^{a b} U_{m + n}, \\ [) U_{m}, Q_{n}^{a} (] = 0, [) U_{m}, G_{n}^{a} (] = m Q_{m + n}^{a}, [) U_{m}, A_{n}^{(\pm) I} (] = 0, \\ [) U_{m}, U_{n} (] = - \frac{1}{3} c m δ_{m + n}, \{) Q_{m}^{a}, Q_{n}^{b} (\} = - \frac{1}{3} c δ^{a b} δ_{m + n} . \end{aligned}

(2.4)

Here c is the central charge of the superconformal algebra. In (2.3), ${L_{m}, G_{m}^{a}, A_{m}^{(\pm) I}, Q_{m}^{a}, U_{m}}$ are generators of the left-moving chiral Virasoro algebra, chiral supersymmetry generators of dimension 3/2, chiral operators of dimension 1, chiral operators of dimension 1/2, and finally a single chiral operator of dimension 2, respectively. The lower alphabetical indices m are mode numbers for the currents; in the usual fashion, for a chiral operator of dimension d_ϕ,

O (z) = \sum_{n} O_{n} z^{- d_{ϕ} - n} .

(2.5)

The bosonic operators $A_{n}^{(\pm) I}$ are generators of the 𝔰𝔲(2)⁽⁺⁾ ⊕ 𝔰𝔲(2)⁽⁻⁾ ≡ 𝔰𝔬(4) Kac-Moody R-symmetry. The indices (±) refer to the two 𝔰𝔲(2) factors; index I runs over {1, 2, 3}, labeling 𝔰𝔲(2) generators. The fermionic generators $G_{n}^{a}$ ⁠, and $Q_{n}^{a}$ transform as vectors of the global part of the 𝔰𝔬(4) R-symmetry, where a ∈ {1, 2, 3, 4}. ϕ_n is a collective notation for the fields $G_{n}^{a}$ ⁠, $A_{m}^{(\pm) I}$ ⁠, $Q_{n}^{a}$ ⁠, and U_n, with the corresponding dimension d_ϕ.

Finally, in the full $N = (4, 4)$ algebra, there is an equivalent antichiral (right-moving) algebra with operators ${{\tilde{L}}_{m}, {\tilde{G}}_{m}^{a}, {\tilde{A}}_{m}^{(\pm) I}, {\tilde{Q}}_{m}^{a}, {\tilde{U}}_{m}}$ ⁠.

In this work, the large $N = (4, 4)$ theory will play an important role. Following Refs. 11–13, we imagine studying gauged linear sigma models which flow to nontrivial superconformal theories in the IR; only the finite subalgebra is realized at intermediate points of the renormalization group flow, and this is all that are required for localization. The chiral part of this algebra is generated by operators ${L_{0}, L_{\pm 1}, G_{\pm 1 / 2}^{a}, A_{0}^{(\pm), I}}$ and has the commutation relations,

\begin{aligned} [) L_{0}, L_{\pm 1} (] = \mp L_{\pm 1}, & [) L_{1}, L_{- 1} (] = 2 L_{0}, \\ [) L_{0}, G_{\pm 1 / 2}^{a} (] = \mp \frac{1}{2} G_{\pm 1 / 2}^{a}, & [) L_{1}, G_{+ 1 / 2}^{a} (] = 0, \\ [) L_{1}, G_{- 1 / 2}^{a} (] = G_{+ 1 / 2}^{a}, & [) L_{- 1}, G_{+ 1 / 2}^{a} (] = - G_{- 1 / 2}^{a}, \\ [) L_{- 1}, G_{- 1 / 2}^{a} (] = 0, & [) L_{0}, A_{0}^{(\pm) I} (] = 0, \\ [) L_{\pm 1}, A_{0}^{(\pm) I} (] = 0, & [) A_{0}^{(+) I}, A_{0}^{(-) J} (] = 0, \\ [) A_{0}^{(\pm) I}, A_{0}^{(\pm) J} (] = ε^{I J K} A_{0}^{(\pm) K}, & [) A_{0}^{(\pm) I}, G_{\pm 1 / 2}^{a} (] = α_{a b}^{(\pm) I} G_{\pm 1 / 2}^{b}, \\ \{) G_{\pm 1 / 2}^{a}, G_{\pm 1 / 2}^{b} (\} = 2 δ^{a b} L_{\pm 1}, & \{) G_{+ 1 / 2}^{a}, G_{- 1 / 2}^{b} (\} = 2 δ^{a b} L_{0} - 2 () α_{a b}^{(+) I} A_{0}^{(+) I} + α_{a b}^{(-) I} A_{0}^{(-) I} () . \end{aligned}

(2.6)

Note that in the global supersymmetry algebra realized away from the conformal point, only the vector-like 𝔰𝔲(2)⁽⁺⁾ is generally realized (see, for example, Ref. 44).

The off-shell formalisms we will work with make the vector-like combination of the 𝔰𝔲(2)⁽⁺⁾ R-symmetry manifest. As we will review in Appendix B, the following complex linear combinations of supersymmetry generators will transform as doublets under 𝔰𝔲(2)⁽⁺⁾:

G_{\pm 1 / 2}^{-} = G_{\pm 1 / 2}^{1} + i G_{\pm 1 / 2}^{2},

(2.7)

G_{\pm 1 / 2}^{+} = G_{\pm 1 / 2}^{3} - i G_{\pm 1 / 2}^{4} .

(2.8)

We denote the complex conjugates of these as ${\bar{G}}_{α}^{\pm}$ ⁠. In this basis, supersymmetry algebra (2.6) becomes

\begin{aligned} \{) G_{+ 1 / 2}^{\pm}, G_{- 1 / 2}^{\pm} (\} = 0, & \{) {\bar{G}}_{+ 1 / 2}^{\pm}, {\bar{G}}_{- 1 / 2}^{\pm} (\} = 0, \\ \{) G_{+ 1 / 2}^{\pm}, G_{- 1 / 2}^{\mp} (\} = \pm 2 i A_{0}^{(-) +}, & \{) {\bar{G}}_{+ 1 / 2}^{\pm}, {\bar{G}}_{- 1 / 2}^{\mp} (\} = \mp 2 i A_{0}^{(-) -}, \\ \{) G_{\pm 1 / 2}^{+}, {\bar{G}}_{\pm 1 / 2}^{+} (\} = 4 L_{\pm 1}, & \{) G_{\pm 1 / 2}^{-}, {\bar{G}}_{\pm 1 / 2}^{-} (\} = 4 L_{\pm 1}, \\ \{) G_{+ 1 / 2}^{\pm}, {\bar{G}}_{- 1 / 2}^{\mp} (\} = - 2 i A_{0}^{(+) \mp}, & \{) G_{- 1 / 2}^{\pm}, {\bar{G}}_{+ 1 / 2}^{\mp} (\} = 2 i A_{0}^{(+) \mp}, \\ \{) G_{+ 1 / 2}^{\pm}, {\bar{G}}_{- 1 / 2}^{\pm} (\} = 4 L_{0} \mp 2 i (A_{0}^{(+) 3} \mp A_{0}^{(-) 3}), & \{) G_{- 1 / 2}^{\pm}, {\bar{G}}_{+ 1 / 2}^{\pm} (\} = 4 L_{0} \pm 2 i (A_{0}^{(+) 3} \mp A_{0}^{(-) 3}), \end{aligned}

(2.9)

where we have defined the raising and lowering operators of the 𝔰𝔲(2) part as

A_{0}^{(\pm) +} = A_{0}^{(\pm) 1} + i A_{0}^{(\pm) 2}, A_{0}^{(\pm) -} = A_{0}^{(\pm) 1} - i A_{0}^{(\pm) 2} .

(2.10)

III. $N = (4, 4)$ GRAVITY MULTIPLET

The off-shell $N = 4$ Poincaré supergravity in two dimensions has the following field content:^40,45

(e_{μ}^{a}, ψ_{μ, i}, A_{μ}^{I}, g, h, b), i = 1, 2, I = 1, 2, 3,

(3.1)

where $e_{μ}^{a}$ ⁠, ψ_μ,i, $A_{μ}^{I}$ ⁠, g, h, and b are the vielbein, the two Dirac gravitinos, an SU(2)-triplet one-form gauge field, an auxiliary real scalar, an auxiliary real scalar, and an auxiliary complex scalar field, respectively. This can be derived by various routes⁴⁵; one route is to start with conformal supergravity, add a conformal scalar multiplet, and fix the gauge freedom of local conformal transformations. Note that this multiplet makes a single SU(2) R-symmetry explicit; the corresponding Lie algebra is the vector-like combination of the 𝔰𝔲(2)⁽⁺⁾ algebra discussed in Sec. II. It would be interesting to consider the case that the R-symmetry of the full large $N = 4$ algebra is explicit after fixing gauge to arrive at Poincaré supergravity.

The transformation laws of the graviton and the two gravitinos are given by⁵⁶

\begin{aligned} δ e_{μ}^{a} = - \frac{1}{2} {\bar{\bar{ϵ}}}^{i} γ^{a} ψ_{μ, i} + \frac{1}{2} {\bar{\bar{ψ}}}_{μ}^{i} γ^{a} ϵ_{i}, \\ δ ψ_{μ, i} = D_{μ} ϵ_{i} - γ_{μ} η_{i}, \end{aligned}

(3.2)

where

\begin{aligned} D_{μ} ϵ_{i} = \partial_{μ} ϵ_{i} + \frac{1}{4} ω_{μ}^{a b} γ_{a b} ϵ_{i} - i A_{μ}^{I} {(σ^{I})}_{i}^{j} ϵ_{j} \equiv \nabla_{μ} ϵ_{i} - i A_{μ}^{I} {(σ^{I})}_{i}^{j} ϵ_{j}, \\ η_{i} = \frac{i}{4} h γ_{3} ϵ_{i} - \frac{1}{4} g ϵ_{i} - \frac{i}{2} b γ_{3} ϵ_{i}^{*}, \end{aligned}

(3.3)

where σ^I are the Pauli matrices. The gauge covariant derivative acting on the conjugate spinor is

D_{μ} {\bar{\bar{ϵ}}}^{i} = \partial_{μ} {\bar{\bar{ϵ}}}^{i} - \frac{1}{4} ω_{μ}^{a b} {\bar{\bar{ϵ}}}^{i} γ_{a b} + i A_{μ}^{I} {(σ^{I})}^{i}_{j} {\bar{\bar{ϵ}}}^{j} \equiv \nabla_{μ} {\bar{\bar{ϵ}}}^{i} + i A_{μ}^{I} {(σ^{I})}^{i}_{j} {\bar{\bar{ϵ}}}^{j} .

(3.4)

The transformations and field contents listed above are for the theory in Lorentzian signature. Once we have this picture, the Euclidean version proceeds by the following:

Promoting real fields to complex fields.
Promoting complex conjugate fields such as b, b^∗ to independent complex fields. This means that we also take $\bar{\bar{ϵ}}$ to be independent of ϵ.

Upon coupling the background supergravity fields to matter, the resulting action will not in general be real.

Following Refs. 20 and 21, in order to decouple gravity from the theory and to obtain a consistent rigid supersymmetric theory in a curved background, we set the gravitinos and their supersymmetry variations to zero: this ensures that the 2d metric and the background superfields are invariant under supersymmetry. When we ultimately couple the supergravity multiplet to matter, the metric and the auxiliary fields in the gravity multiplet will appear in the supersymmetry variations of the matter fields, leading to a consistent realization of supersymmetry on a curved background.

Setting δψ_μ,i = 0 in (3.2), we arrive at the following constraint on the Dirac SU(2)-doublet spinors of the curved background:

D_{μ} ϵ_{i} = γ_{μ} η_{i} .

(3.5)

Rewriting (3.5) in its explicit form, we find the following conditions on the ϵⁱ:

\begin{aligned} \nabla_{μ} ϵ_{i} = \frac{i}{4} h γ_{μ} γ_{3} ϵ_{i} - \frac{1}{4} g γ_{μ} ϵ_{i} - \frac{i}{2} b γ_{μ} γ_{3} ϵ_{i}^{*} + i A_{μ}^{I} {(σ^{I})}_{i}^{j} ϵ_{j}, \\ \nabla_{μ} {\bar{\bar{ϵ}}}^{i} = - \frac{i}{4} h {\bar{\bar{ϵ}}}^{i} γ_{3} γ_{μ} + \frac{1}{4} g {\bar{\bar{ϵ}}}^{i} γ_{μ} + \frac{i}{2} b^{*} {\bar{ϵ}}^{i} γ_{3} γ_{μ} - i A_{μ}^{I} {(σ^{I})}_{j}^{i} {\bar{\bar{ϵ}}}^{j}, \end{aligned}

(3.6)

where the auxiliary fields h, g, and b acquire complex values. One difference from the $N = (2, 2)$ case is that when b ≠ 0, the right hand side of the first (second) line contains a term proportional to ϵ^∗ (ϵ). A second difference is the appearance of the auxiliary SU(2) gauge field.

IV. CONSISTENCY CONDITIONS FOR $N = (4, 4)$ SUPERSYMMETRY

As in Refs. 20–22 and 28, the existence of spinors satisfying (3.6) places constraints on the 2d background Σ. We will focus on the case that maximal, global $N = (4, 4)$ supersymmetry is preserved. This means that there are two complex solutions {ϵⁱ, λⁱ} to (3.6) for each i. In flat space with no background fields, we could take

ϵ^{i} = (\begin{array}{l} ϵ_{+}^{i} \\ 0 \end{array}), λ^{i} = (\begin{array}{l} 0 \\ λ_{-}^{i} \end{array}) .

(4.1)

In curved space, the solutions will not generally have definite chirality, but we will assume that at each point on Σ, ϵ, λ are linearly independent spinors.

For oriented two-dimensional surfaces, the spacetime is automatically complex and Kähler. Following Refs. 20 and 21, we can build Kähler forms from bilinears of the spinors ϵ_i, λ_j, but these will simply reproduce the canonical Kähler form (the 2d volume form).

Next, we can build a set of complex Killing vectors ξ^A,μ using ϵ_i, λ_i,

ξ^{1, μ} = {\bar{\bar{ϵ}}}^{i} γ^{μ} ϵ_{i},

ξ^{2, μ} = {\bar{\bar{ϵ}}}^{i} γ^{μ} λ_{i},

ξ^{3, μ} = {\bar{\bar{λ}}}^{i} γ^{μ} ϵ_{i},

ξ^{4, μ} = {\bar{\bar{λ}}}^{i} γ^{μ} λ_{i} .

(4.2)

There is no guarantee that these are independent. The supersymmetry conditions imply the Killing vector equation

\nabla^{μ} ξ^{A, ν} + \nabla^{ν} ξ^{A, μ} - g^{μ ν} \nabla_{σ} ξ^{A, σ} = 0 .

(4.3)

Finally, we get constraints on the auxiliary fields by studying the commutator of covariant derivatives acting on the spinors

[\nabla_{μ}, \nabla_{ν}] ϵ_{i} = \frac{1}{4} R_{μ ν}^{a b} γ_{a b} ϵ_{i} = \frac{1}{4} R (x) ε_{μ ν} γ_{3} ϵ_{i} = \frac{i}{4} ε_{μ ν} γ^{ρ} (\partial_{ρ} h + i \partial_{ρ} g γ_{3}) ϵ_{i} - \frac{i}{2} ε_{μ ν} γ^{ρ} \partial_{ρ} b ϵ_{i}^{*}

- \frac{1}{8} ε_{μ ν} (h^{2} + g^{2} + 4 b b^{*}) γ_{3} ϵ_{i} + i F_{μ ν}^{I} σ^{I}_{i}^{j} ϵ_{j},

(4.4)

where

F_{μ ν}^{I} = \partial_{μ} A_{ν}^{I} - \partial_{ν} A_{μ}^{I} + 2 ε^{I J K} A_{μ}^{J} A_{ν}^{K}

(4.5)

is the SU(2)-covariant field strength for $A$ ⁠, and

[\nabla_{μ}, \nabla_{ν}] {\bar{\bar{ϵ}}}^{i} = - \frac{1}{4} {\bar{\bar{ϵ}}}^{i} γ_{3} R (x) ε_{μ ν} = \frac{i}{4} ε_{μ ν} {\bar{\bar{ϵ}}}^{i} γ^{ρ} (\partial_{ρ} h - i \partial_{ρ} g γ_{3}) - \frac{i}{2} ε_{μ ν} {\bar{ϵ}}^{i} γ^{ρ} \partial_{ρ} b^{*}

+ \frac{1}{8} ε_{μ ν} {\bar{\bar{ϵ}}}^{i} γ_{3} (h^{2} + g^{2} + 4 b b^{*}) - i {\bar{\bar{ϵ}}}^{j} σ_{j}^{I}^{i} F_{μ ν}^{I} .

(4.6)

We have the same equations if we substitute ϵⁱ → λⁱ. Now if ϵⁱ, λⁱ are linearly independent spinors at each point on Σ, and we are working in Euclidean space so that ${\bar{\bar{ϵ}}}^{i}$ is independent of ϵ_i (that is, not built from the complex conjugate spinor), the integrability conditions are as follows:

h, g, and b are all constant,
$F_{μ ν}^{I} = 0$ ⁠,
$R (x) = - \frac{1}{2} () h^{2} + g^{2} + 4 b b^{*} ()$ ⁠.

On a compact two-dimensional surface, the flat connection corresponds to Wilson lines of the manifest SU(2) R-symmetry. As in Ref. 28, nontrivial Wilson lines will break supersymmetry, so we demand that $A_{μ}^{I}$ is pure gauge. Note that the first and third conditions require constant curvature, and for spaces of constant positive curvature, we must take h or g to be imaginary, or let b, b^∗ be independent.

For our later use of constructing an invariant action on a curved space, we need to calculate the Laplacian of the Dirac spinor ϵ_i satisfying supersymmetry condition (3.6). Using (3.6), it is straightforward to calculate the action of the Laplacian operator on Dirac spinor doublet ϵ_i,

D_{μ} D^{μ} ϵ_{i} = \frac{1}{2} () \frac{h^{2}}{4} + \frac{g^{2}}{4} + b b^{*} () ϵ_{i} .

(4.7)

V. $N = (4, 4)$ VECTOR MULTIPLET

In this section, we will construct the supersymmetry transformations and invariant action for an Abelian vector multiplet propagating on a curved two-manifold. We will find that these transformations in general realize the “large” $N = (4, 4)$ supersymmetry algebra acting on vector multiplets.

Before considering the case of the curved space, we will briefly summarize the $N = (4, 4)$ vector multiplet story in flat space, following Ref. 45.

A. Flat space

$N = 4$ vector multiplets in two dimensions have an off-shell formulation with a finite number of auxiliary fields; the multiplets and action can be obtained via dimensional reduction from $N = 2$ theories in four dimensions.^45,46 The off-shell multiplet has 16° of freedom, consisting of the following fields:

(A, B, C, C^{*}, V_{μ}, d^{I}, χ_{i}, {\bar{\bar{χ}}}^{i}), i = 1, 2, I = 1, 2, 3,

(5.1)

in which A and B are real propagating scalar fields, C is a complex propagating scalar field, V_μ is a U(1) gauge field, d^I are an SU(2) triplet of auxiliary scalar fields, and χ_i are SU(2)-doublet Dirac spinors. We will find it useful to combine A, B into a complex field U = A + iB.

The rigid supersymmetry transformation laws of the fields of the vector multiplet are given by⁴⁵

\begin{aligned} δ C = {\bar{ϵ}}^{i} χ_{i}, δ C^{*} = - {\bar{\bar{χ}}}^{i} ϵ_{i}^{*}, δ U = {\bar{\bar{ϵ}}}^{i} P_{-} χ_{i} + {\bar{\bar{χ}}}^{i} P_{-} ϵ_{i}, δ U^{*} = {\bar{\bar{ϵ}}}^{i} P_{+} χ_{i} + {\bar{\bar{χ}}}^{i} P_{+} ϵ_{i}, \\ δ χ_{i} = \frac{1}{2} \partial_{μ} C γ^{μ} ϵ_{i}^{*} - \frac{1}{2} \partial_{μ} U P_{-} γ^{μ} ϵ_{i} - \frac{1}{2} \partial_{μ} U^{*} P_{+} γ^{μ} ϵ_{i} - \frac{i}{4} F γ_{3} ϵ_{i} - \frac{1}{2} d^{I} {(σ^{I})}_{i}^{j} ϵ_{j}, \\ δ {\bar{\bar{χ}}}^{i} = \frac{1}{2} \partial_{μ} C^{*} {\bar{ϵ}}^{i} γ^{μ} + \frac{1}{2} \partial_{μ} U {\bar{\bar{ϵ}}}^{i} γ^{μ} P_{-} + \frac{1}{2} \partial_{μ} U^{*} {\bar{\bar{ϵ}}}^{i} γ^{μ} P_{+} - \frac{i}{4} F {\bar{\bar{ϵ}}}^{i} γ_{3} - \frac{1}{2} d^{I} {(σ^{I})}_{j}^{i} {\bar{\bar{ϵ}}}^{j}, \\ δ V_{μ} = \frac{i}{2} ({\bar{\bar{ϵ}}}^{i} γ_{μ} χ_{i} + {\bar{\bar{χ}}}^{i} γ_{μ} ϵ_{i}), \\ δ d^{I} = \frac{1}{2} {(σ^{I})}_{j}^{i} {\bar{\bar{ϵ}}}^{j} γ^{μ} \partial_{μ} χ_{i} - \frac{1}{2} {(σ^{I})}_{j}^{i} \partial_{μ} {\bar{\bar{χ}}}^{j} γ^{μ} ϵ_{i}, \end{aligned}

(5.2)

where $P_{\pm} = \frac{1}{2} (1 \pm γ_{3})$ ⁠. In (5.2), F is defined as F = ε^μνF_μν where F_μν is the field strength associated with the Abelian vector field V_μ. The invariant Lagrangian which is preserved by supersymmetry transformations (5.2) is⁴⁵

L = \frac{1}{2} \partial_{μ} U \partial^{μ} U^{*} + \frac{1}{2} \partial_{μ} C \partial^{μ} C^{*} + \frac{1}{4} F_{μ ν} F^{μ ν} - \frac{1}{2} d^{I} d^{I} - {\bar{\bar{χ}}}^{i} γ^{μ} \partial_{μ} χ_{i}

(5.3)

and is invariant up to total derivatives.

B. Curved space

To find the transformation laws in the curved space, we replace ordinary derivatives with the appropriate covariant derivatives. We define the two following SU(2)-doublet covariant derivatives:

{(D_{μ})}_{i}^{j} = \nabla_{μ} δ_{i}^{j} - i A_{μ}^{I} {(σ^{I})}_{i}^{j}, {({\bar{D}}_{μ})}_{i}^{j} = \nabla_{μ} δ_{i}^{j} + i A_{μ}^{I} {(σ^{I})}_{i}^{j},

(5.4)

where $δ_{i}^{j}$ is the Kronecker delta symbol. In analogy with (5.2), we take the supersymmetry transformation rules for the components of the vector multiplet to be

\begin{aligned} δ C = {\bar{ϵ}}^{i} χ_{i}, δ C^{*} = - {\bar{\bar{χ}}}^{i} ϵ_{i}^{*}, δ U = {\bar{\bar{ϵ}}}^{i} P_{-} χ_{i} + {\bar{\bar{χ}}}^{i} P_{-} ϵ_{i}, δ U^{*} = {\bar{\bar{ϵ}}}^{i} P_{+} χ_{i} + {\bar{\bar{χ}}}^{i} P_{+} ϵ_{i}, \\ δ χ_{i} = \frac{1}{2} {(D_{μ})}_{i}^{j} (C γ^{μ} ϵ_{j}^{*}) - \frac{1}{2} {(D_{μ})}_{i}^{j} (U P_{-} γ^{μ} ϵ_{j}) - \frac{1}{2} {(D_{μ})}_{i}^{j} (U^{*} P_{+} γ^{μ} ϵ_{j}) - \frac{i}{4} F γ_{3} ϵ_{i} - \frac{1}{2} d^{I} {(σ^{I})}_{i}^{j} ϵ_{j}, \\ δ {\bar{\bar{χ}}}^{i} = \frac{1}{2} {({\bar{D}}_{μ})}_{j}^{i} (C^{*} {\bar{ϵ}}^{j} γ^{μ}) + \frac{1}{2} {({\bar{D}}_{μ})}_{j}^{i} (U {\bar{\bar{ϵ}}}^{j} γ^{μ} P_{-}) + \frac{1}{2} {({\bar{D}}_{μ})}_{j}^{i} (U^{*} {\bar{\bar{ϵ}}}^{j} γ^{μ} P_{+}) \\ - \frac{i}{4} F {\bar{\bar{ϵ}}}^{i} γ_{3} - \frac{1}{2} d^{I} {(σ^{I})}_{j}^{i} {\bar{\bar{ϵ}}}^{j}, \\ δ V_{μ} = \frac{i}{2} ({\bar{\bar{ϵ}}}^{i} γ_{μ} χ_{i} + {\bar{\bar{χ}}}^{i} γ_{μ} ϵ_{i}), \\ δ d^{I} = \frac{1}{2} {(σ^{I})}_{j}^{i} D_{μ} ({\bar{\bar{ϵ}}}^{j} γ^{μ} χ_{i}) - \frac{1}{2} {(σ^{I})}_{j}^{i} D_{μ} ({\bar{\bar{χ}}}^{j} γ^{μ} ϵ_{i}) . \end{aligned}

(5.5)

The following action is then invariant under (5.5):

\begin{aligned} ℒ = & \frac{1}{2} \nabla_{μ} U \nabla^{μ} U^{*} + \frac{1}{2} \nabla_{μ} C \nabla^{μ} C^{*} - {\bar{\bar{χ}}}^{i} γ^{μ} {(D_{μ})}_{i}^{j} χ_{j} + \frac{i}{2} h {\bar{\bar{χ}}}^{i} γ_{3} χ_{i} - \frac{1}{2} g {\bar{\bar{χ}}}^{i} χ_{i} \\ + \frac{1}{2} d^{I} d^{I} + \frac{1}{4} () \frac{h^{2}}{4} + \frac{g^{2}}{4} + b b^{*} () U U^{*} + \frac{1}{4} () \frac{h^{2}}{4} + \frac{g^{2}}{4} + b b^{*} () C C^{*} \\ - \frac{1}{4} () F + \frac{1}{2} (h - i g) U + \frac{1}{2} (h + i g) U^{*} + b^{*} C + b C^{*} {()}^{2}, \end{aligned}

(5.6)

if in addition we impose the constraint $F_{μ ν}^{I} = 0$ which follows from the integrability conditions in Section IV.

Given the above supersymmetry transformations, we can show that the commutator of these transformations closes on the large $N = (4, 4)$ algebra. A direct computation gives the first set as

\begin{aligned} [) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] U = L_{ξ} U + ρ U - β_{+} U + β_{-} U, \\ [) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] U^{*} = L_{ξ} U^{*} + ρ U^{*} + β_{+} U^{*} - β_{-} U^{*}, \\ [) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] C = L_{ξ} C + ρ C - β_{+} C - β_{-} C, \\ [) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] C^{*} = L_{ξ} C^{*} + ρ C^{*} + β_{+} C^{*} + β_{-} C^{*}, \\ [) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] V^{μ} = {(L_{ξ}^{A} V)}^{μ} + \nabla^{μ} Λ \\ [) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] χ_{i} = L_{ξ}^{A} χ_{i} + \frac{3}{2} ρ χ_{i} - β_{-} χ_{i, L} - β_{+} χ_{i, R} \\ - \frac{1}{4} ({\bar{\bar{λ}}}^{j} γ^{μ} γ^{ν} D_{μ} ϵ_{i}) γ_{ν} χ_{j} + \frac{1}{4} ({\bar{\bar{λ}}}^{j} γ^{μ} γ^{ν} D_{μ} ϵ_{j}) γ_{ν} χ_{i}, \\ [) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] {\bar{\bar{χ}}}^{i} = L_{ξ}^{A} {\bar{\bar{χ}}}^{i} + \frac{3}{2} ρ {\bar{\bar{χ}}}^{i} + β_{-} {\bar{\bar{χ}}}_{L}^{i} + β_{+} {\bar{\bar{χ}}}_{R}^{i} \\ - \frac{1}{4} (D_{μ} {\bar{\bar{λ}}}^{i} γ^{ν} γ^{μ} ϵ_{j}) {\bar{\bar{χ}}}^{j} γ_{ν} + \frac{1}{4} (D_{μ} {\bar{\bar{λ}}}^{j} γ^{ν} γ^{μ} ϵ_{j}) {\bar{\bar{χ}}}^{i} γ_{ν}, \\ [) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] d^{I} = L_{ξ} d^{I} + 2 ρ d^{I} + \frac{1}{4} {(σ^{I})}_{j}^{i} D_{μ} () D_{ν} {\bar{\bar{λ}}}^{j} γ^{μ} γ^{ν} P_{+} ϵ_{i} U + D_{ν} {\bar{\bar{λ}}}^{j} γ^{μ} γ^{ν} P_{-} ϵ_{i} U^{*} \\ - {\bar{\bar{λ}}}^{j} P_{+} γ^{ν} γ^{μ} D_{ν} ϵ_{i} U - {\bar{\bar{λ}}}^{j} P_{-} γ^{ν} γ^{μ} D_{ν} ϵ_{i} U^{*} (), \end{aligned}

(5.7)

where

\begin{aligned} χ_{i, L} = P_{-} χ_{i}, & χ_{i, R} = P_{+} χ_{i}, \\ ξ^{μ} = - \frac{1}{2} {\bar{\bar{λ}}}^{i} γ^{μ} ϵ_{i}, & ρ = - \frac{1}{4} \nabla_{μ} () {\bar{\bar{λ}}}^{i} γ^{μ} ϵ_{i} () = \frac{1}{2} \nabla_{μ} ξ^{μ}, \\ β_{\pm} = \frac{1}{4} () D_{μ} {\bar{\bar{λ}}}^{i} P_{\pm} γ^{μ} ϵ_{i} - {\bar{\bar{λ}}}^{i} P_{\pm} γ^{μ} D_{μ} ϵ_{i} (), & Λ = - \frac{i}{2} (U {\bar{\bar{λ}}}^{i} P_{+} ϵ_{i} + U^{*} {\bar{\bar{λ}}}^{i} P_{-} ϵ_{i}) \end{aligned}

(5.8)

and the action of the Lie derivatives acting on the field components is

\begin{aligned} L_{ξ} (C, C^{*}, U, U^{*}) = ξ^{μ} \partial_{μ} (C, C^{*}, U, U^{*}), \\ L_{ξ}^{A} V = L_{ξ} V - d^{A} (ι_{ξ} A) = ξ^{μ} F_{μ ν} d x^{ν}, \\ L_{ξ}^{A} χ_{i} = ξ^{μ} D_{μ} χ_{i} + \frac{1}{4} (\nabla_{μ} ξ_{ν}) γ^{μ ν} χ_{i}, \\ L_{ξ}^{A} {\bar{\bar{χ}}}^{i} = ξ^{μ} D_{μ} {\bar{\bar{χ}}}^{i} - \frac{1}{4} (\nabla_{μ} ξ_{ν}) {\bar{\bar{χ}}}^{i} γ^{μ ν}, \\ L_{ξ} d^{I} = ξ^{μ} (\partial_{μ} d^{I} - ε^{I J K} A_{μ}^{J} d^{K}) . \end{aligned}

(5.9)

Furthermore, the second lines of $[) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] V^{μ}$ ⁠, $[) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] χ_{i}$ ⁠, $[) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] {\bar{\bar{χ}}}^{i}$ ⁠, and $[) δ_{ϵ}, δ_{\bar{\bar{λ}}} (] d^{I}$ vanish, using supersymmetry conditions (3.6) and the fact that γ^μγ^νγ_μ = 0.

The next set of commutators are

\begin{aligned} [) δ_{ϵ}, δ_{λ} (] U = κ_{+} C^{*}, \\ [) δ_{ϵ}, δ_{λ} (] U^{*} = κ_{-} C^{*}, \\ [) δ_{ϵ}, δ_{λ} (] C = κ_{-} U + κ_{+} U^{*}, \\ [) δ_{ϵ}, δ_{λ} (] C^{*} = 0, \\ [) δ_{ϵ}, δ_{λ} (] V^{μ} = \nabla^{μ} () υ C^{*} (), \\ [) δ_{ϵ}, δ_{λ} (] χ_{i} = - κ_{-} χ_{i, L}^{*} - κ_{+} χ_{i, R}^{*} \\ - \frac{1}{4} () D_{μ} {\bar{λ}}^{j} γ^{ν} γ^{μ} ϵ_{i} () γ_{ν} χ_{j}^{*} + \frac{1}{4} () D_{μ} {\bar{ϵ}}^{j} γ^{ν} γ^{μ} λ_{i} () γ_{ν} χ_{j}^{*}, \\ [) δ_{ϵ}, δ_{λ} (] {\bar{\bar{χ}}}^{i} = 0, \\ [) δ_{ϵ}, δ_{λ} (] d^{I} = - \frac{1}{4} {(σ^{I})}_{j}^{i} \nabla_{ν} C^{*} \nabla_{μ} () {\bar{ϵ}}^{j} γ^{ν} γ^{μ} λ_{i} - {\bar{λ}}^{j} γ^{ν} γ^{μ} ϵ_{i} () \\ - \frac{1}{4} {(σ^{I})}_{j}^{i} \nabla_{μ} () () D_{ν} {\bar{ϵ}}^{j} γ^{ν} γ^{μ} λ_{i} - D_{ν} {\bar{λ}}^{j} γ^{ν} γ^{μ} ϵ_{i} () C^{*} (), \end{aligned}

(5.10)

where

\begin{aligned} κ_{\pm} = - \frac{1}{2} () D_{μ} {\bar{ϵ}}^{i} P_{\pm} γ^{μ} λ_{i} - {\bar{ϵ}}^{i} P_{\pm} γ^{μ} D_{μ} λ_{i} (), \\ υ = \frac{i}{2} {\bar{λ}}^{i} ϵ_{i} . \end{aligned}

(5.11)

Equation (3.6) guarantees that the second line of the commutator acting on χ vanishes. The commutator acting on d^I can be reduced to

[) δ_{ϵ}, δ_{λ} (] d^{I} = - \frac{i}{2} {(σ^{I} σ^{J})}_{j}^{i} ε^{μ ν} F_{μ ν}^{J} ({\bar{ϵ}}^{j} γ_{3} λ_{i}) C^{*}

(5.12)

which vanishes upon imposing the integrability condition $F_{μ ν}^{I} = 0$ ⁠.

Finally, commutators of the conjugate spinors are

\begin{aligned} [) δ_{\bar{\bar{ϵ}}}, δ_{\bar{\bar{λ}}} (] U = κ_{-}^{*} C, \\ [) δ_{\bar{\bar{ϵ}}}, δ_{\bar{\bar{λ}}} (] U^{*} = κ_{+}^{*} C, \\ [) δ_{\bar{\bar{ϵ}}}, δ_{\bar{\bar{λ}}} (] C = 0, \\ [) δ_{\bar{\bar{ϵ}}}, δ_{\bar{\bar{λ}}} (] C^{*} = κ_{-}^{*} U^{*} + κ_{+}^{*} U, \\ [) δ_{\bar{\bar{ϵ}}}, δ_{\bar{\bar{λ}}} (] V^{μ} = \nabla^{μ} () υ^{*} C (), \\ [) δ_{\bar{\bar{ϵ}}}, δ_{\bar{\bar{λ}}} (] χ_{i} = 0, \\ [) δ_{\bar{\bar{ϵ}}}, δ_{\bar{\bar{λ}}} (] {\bar{\bar{χ}}}^{i} = - κ_{-}^{*} {\bar{χ}}_{L}^{i} - κ_{+}^{*} {\bar{χ}}_{R}^{i}, \\ [) δ_{\bar{\bar{ϵ}}}, δ_{\bar{\bar{λ}}} (] d^{I} = 0 . \end{aligned}

(5.13)

While our formalism only makes a single vector 𝔰𝔲(2) R-symmetry manifest, we claim that the supersymmetry algebra here realizes the “large” $N = (4, 4)$ superalgebra, with R-symmetry 𝔰𝔲(2)⁽⁺⁾ ⊕ 𝔰𝔲(2)⁽⁻⁾, and 𝔰𝔲(2)⁽⁺⁾ comprising the manifest R-symmetry. More precisely we find specific actions of the R-symmetry, translations, and dilatations, depending on the background auxiliary fields: the parameters ρ, β_±, and κ_± depend on the auxiliary fields via supersymmetry conditions (3.6).

In the case of the fermions, 𝔰𝔲(2)⁽⁺⁾ acts on the doublet index i, while 𝔰𝔲(2)⁽⁻⁾ acts on the doublets

(\begin{array}{l} {(χ_{1})}^{*} \\ χ_{2} \end{array}) (\begin{array}{l} - {(χ_{2})}^{*} \\ χ_{1} \end{array}) .

(5.14)

This can be deduced from Appendix B.

In the case of the scalars, the left-moving 𝔰𝔲(2)⁽⁻⁾ acts on the doublet

(\begin{array}{l} U \\ C \end{array})

(5.15)

while the right-moving 𝔰𝔲(2)⁽⁻⁾ acts on the doublet

(\begin{array}{l} U^{*} \\ C \end{array}) .

(5.16)

Given these identifications, Eq. (2.9) is consistent with Eqs. (5.7), (5.10), and (5.13), we see that as follows:

ξ corresponds to translations.
ρ corresponds to dilatations.
The vectorlike combination of 𝔰𝔲(2)⁺ acts on $χ^{i}, {\bar{\bar{χ}}}^{i}$ via the term iξ^μA_μ,i^j in $L_{ξ}^{A}$ ⁠.
β₋, β₊ corresponds to the left- and right-moving actions of $A_{0}^{(-) 3}$ ⁠.
κ₋ corresponds to the left-moving part of A⁽⁻⁾⁺ and κ₊ to the right-moving part of A⁽⁻⁾⁺.
$κ_{-}^{*}$ corresponds to the left-moving part of A⁽⁻⁾⁻ and $κ_{+}^{*}$ to the right-moving action of A⁽⁻⁾⁻.
Λ, vC^∗, and v^∗C are gauge transformations under the complexified gauge group of the Abelian vector multiplet.

We can view the 2-dimensional $N = (4, 4)$ supersymmetry as the dimensional reduction of the 6-dimensional $N = (1, 0)$ theory on a two-manifold.⁴⁴ The R-symmetry of the 6-dimensional theory reduces to the 2-dimensional 𝔰𝔲(2)⁽⁺⁾ R-symmetry, whereas the 𝔰𝔬(4) rotational symmetry in the 4 extra dimensions becomes the product of the two 𝔰𝔲(2)⁽⁻⁾ symmetries in two dimensions.⁵⁷ The additional components of the 6d gauge field become a quartet of real scalars, transforming as a vector of 𝔰𝔬(4) = 𝔰𝔲(2) ⊕ 𝔰𝔲(2). The results of Appendix B show how such a vector can be organized into complex doublets of the two 𝔰𝔲(2) factors. The scalars U, U^∗, C, and C^∗ transform in precisely this fashion.

VI. $N = (4, 4)$ ON S²

A standard test case is to derive the supersymmetry conditions for S².^11,12,28 Here we consider solutions to the spinor equations for general h, g, b, b^∗ satisfying the integrability conditions in Section IV.

We consider S² with the metric

d s^{2} = r^{2} (d θ^{2} + {sin}^{2} θ d ϕ^{2}),

(6.1)

with zweibeins

e_{θ}^{1} = r, e_{ϕ}^{2} = r sin θ,

(6.2)

spin connection

ω_{ϕ}^{1}_{2} = - cos θ,

(6.3)

and Ricci scalar $R = \frac{2}{r^{2}}$ ⁠. In the Euclidean continuation, the gamma matrices become

γ_{1} = σ_{2}, γ_{2} = σ_{1}, γ_{12} = - i σ_{3} = i γ_{3} .

(6.4)

For simplicity, we define

\begin{aligned} H = \frac{r}{4} (h + i g), \\ \tilde{H} = \frac{r}{4} (h - i g), \\ B = \frac{r}{2} b, & \tilde{B} = \frac{r}{2} b^{*} . \end{aligned}

(6.5)

The curvature constraint from Section IV becomes $H \tilde{H} + B \tilde{B} = - \frac{1}{4}$ ⁠. In essence, $H, \tilde{H}$ are the same as $H, \tilde{H}$ in Ref. 28.

We define the spinors Ψ_i,α=± (where α is the spinor index) as

Ψ_{i} = (\begin{array}{c} Ψ_{i, -} \\ Ψ_{i, +} \end{array}) = (\begin{array}{c} c_{i} ϵ_{i, -} - d_{i} ϵ^{*}_{i, -} \\ - 2 i (c_{i} H + d_{i} \tilde{B}) ϵ_{i, +} + 2 i (c_{i} B - d_{i} \tilde{H}) ϵ^{*}_{i, +} \end{array}) .

(6.6)

With these definitions, and the curvature constraint, the equations for χ are the standard Killing spinor equations,

\nabla_{μ} Ψ_{i} = \frac{i}{2} γ_{μ} γ_{3} Ψ_{i},

(6.7)

with solutions

Ψ_{i} = C_{i} e^{i ϕ / 2} (\begin{array}{c} cos \frac{θ}{2} \\ i sin \frac{θ}{2} \end{array}) + D_{i} e^{- i ϕ / 2} (\begin{array}{c} sin \frac{θ}{2} \\ - i cos \frac{θ}{2} \end{array}),

(6.8)

where C_i and D_i are constants of integration. Note that Ψ_i has the same form as the solutions in Ref. 11, for each element of the doublet; however, it is a complicated linear combination of the spinors $ϵ_{i}, ϵ_{i}^{*}$ ⁠.

VII. CONCLUSIONS

There is obvious further technical work to be done. One is to study non-Abelian vector multiplets. Another is to study hypermultiplets on curved backgrounds. As stated above, hypermultiplets which can be linearly gauged in a straightforward way (that is, the scalars are written as an SU(2) doublet of auxiliary fields) do not have completely off-shell, manifestly SU(2)_R covariant representations with a finite number of auxiliary fields. There is, however, a partially off-shell multiplet in $N = 2, d = 4$ theories—see, for example, Chap. 12 of Ref. 39—which reduces to an $N = (4, 4)$ hypermultiplet upon dimensional reduction, for which the supersymmetries close on an off-shell central charge. There are also other completely off-shell hypermultiplets whose supersymmetry transformations close on a finite number of auxiliary fields.^40,41 For instance, in one case, the scalars form an SU(2) singlet and SU(2) triplet and so a single multiplet will not transform linearly under gauge transformations. Nonetheless, these are also worth studying. One could also study off-shell multiplets with infinite numbers of auxiliary fields coupled to supergravity, via harmonic^47–51 or projective^52,53 superspace.

An interesting extension of this work would be to study cases analogous to Ref. 28, with fewer unbroken supercharges, and nonvanishing A^I, F^I.

Finally, it would be interesting to study various localization calculations in this framework; this should be possible at least for noncompact theories, after the fashion of Ref. 36.

Acknowledgments

We would like to thank Thomas Dumitrescu, S. James Gates, Jim Halverson, Shamit Kachru, Martin Rocek, and David Tong for inspirational and useful discussions and correspondences. Part of this work was performed while A.L. was at the Aspen Center for Physics, which is supported by National Science Foundation Grant No. PHY-1066293. Part of this work was performed while A.L. was visiting the KITP at UC Santa Barbara, supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. A.L. would like to thank both institutions for providing stimulating environments conducive to productive work. The research of A.L. is supported by DOE Grant No. DE-SC0009987. The research of M.S. is supported by NSF FRG Grant No. DMS 1159049 and NSF Grant No. PHY 1053842.

APPENDIX A: NOTATIONS AND CONVENTIONS

In this section, we provide more details on our notation and conventions. In $N = (4, 4)$ theory, the parameters of supersymmetry transformation are Dirac spinors which carry an SU(2) index of the R-symmetry. The parameter of supersymmetry, ϵ_i, is an SU(2)-doublet Dirac spinor with two spinorial components. The SU(2) R-symmetry indices are raised and lowered by the totally antisymmetric tensor ϵ_ij in the following way:

ϵ^{i} = ε^{i j} ϵ_{j}, ϵ_{i} = ϵ^{j} ε_{j i} = - ε_{i j} ϵ^{j} .

(A1)

The SU(2) totally antisymmetric tensor, ϵ_ij, fulfills the following properties:

ε_{i j} = ε^{i j}, ε_{j i} ε^{i k} = - δ_{j}^{k} .

(A2)

Moreover, one can relate the product of two SU(2) Kronecker delta to Pauli matrices. The following identity turns out to be very useful:

2 δ_{l}^{i} δ_{k}^{j} = δ_{k}^{i} δ_{l}^{j} + {(σ^{I})}_{k}^{i} {(σ^{I})}_{l}^{j} .

(A3)

In the above equation, one performs a sum over the index I = 1, 2, 3. Also, one can easily show that

A_{i} B_{j} - A_{j} B_{i} = - ε_{i j} A^{k} B_{k}, A^{i} B^{j} - A^{j} B^{i} = ε^{i j} A^{k} B_{k},

(A4)

where A_i and B_i could be both complex numbers, as well as Grassmanians. Complex conjugation raises and lowers the SU(2) R-symmetry indices as follows:

ϵ^{* i} \equiv {(ϵ_{i})}^{*}, {(ϵ^{i})}^{*} = {(ε^{i j} ϵ_{j})}^{*} = ε_{i j} ϵ^{* j} = - ϵ_{i}^{*}, {(ϵ^{i})}^{* *} = - ϵ^{i} .

(A5)

In particular, we notice in (A5) that after two successive complex conjugations, a minus sign is produced. For spinors, we introduce two types of conjugation: namely, the Majorana and the Dirac conjugations. These two spinors are, respectively, defined as

{\bar{ϵ}}^{i} = - ε^{i j} ϵ_{j}^{T} C, {\bar{\bar{ϵ}}}^{i} = ε^{i j} {(ϵ_{j})}^{†} C,

(A6)

where ϵ^iT is the transpose spinor without complex conjugation, whereas (ϵ_i)^† includes complex conjugation operation in addition to transposition. In (A6), 𝒞 is the two dimensional charge conjugation matrix which happens to coincide with the Pauli matrix σ²,

C = σ^{2} = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) .

(A7)

In our conventions, the γ-matrices which are two dimensional representations of the Clifford algebra are matrices with real entries. The two γ-matrices γ₀ and γ₁ together with the helicity matrix γ₃ are defined by

γ_{0} = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}), γ_{1} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), γ_{3} = γ_{0} γ_{1} = (\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}) .

(A8)

Using the Clifford algebra, and the explicit representation of γ-matrices in (A8), it is straightforward to derive the following useful properties among product of γ-matrices:

\begin{aligned} γ_{μ}^{T} C = - C γ_{μ}, \\ {C, γ_{3}} = 0, \\ γ^{μ} γ^{ν} = g^{μ ν} + γ^{μ ν} = g^{μ ν} + ε^{μ ν} γ_{3}, \\ γ^{μ} γ_{ν} γ_{μ} = 0, \\ ε^{μ ν} γ_{ν} = γ_{3} γ^{μ} . \end{aligned}

(A9)

Fierz identity for both the Majorana and Dirac conjugations takes the same form and in our notation is expressed in the following form:

\begin{aligned} (\bar{ϵ} λ) χ = + \frac{1}{2} () (\bar{ϵ} χ) λ + (\bar{ϵ} γ_{3} χ) γ_{3} λ + (\bar{ϵ} γ^{μ} χ) γ_{μ} λ (), \\ (\bar{\bar{ϵ}} λ) χ = - \frac{1}{2} () (\bar{\bar{ϵ}} χ) λ + (\bar{\bar{ϵ}} γ_{3} χ) γ_{3} λ + (\bar{\bar{ϵ}} γ^{μ} χ) γ_{μ} λ (), \end{aligned}

(A10)

in which ϵ, λ, and χ are all Dirac spinors in two dimensions. From Fierz identity (A10), we can derive the following useful relations between the product of spinor bilinears:

\begin{aligned} (\bar{λ} γ^{μ} λ) (\bar{ϵ} γ_{μ} ϵ) + (\bar{λ} γ^{μ} γ_{3} λ) (\bar{ϵ} γ_{μ} γ_{3} ϵ) = 0, \\ (\bar{ϵ} γ^{μ} ϵ) (\bar{ϵ} γ_{μ} ϵ) + (\bar{ϵ} γ^{μ} γ_{3} ϵ) (\bar{ϵ} γ_{μ} γ_{3} ϵ) = 0, \\ (\bar{ϵ} ϵ) (\bar{ϵ} ϵ) + (\bar{ϵ} γ_{3} ϵ) (\bar{ϵ} γ_{3} ϵ) = 0, \\ (\bar{\bar{λ}} γ^{μ} λ) (\bar{\bar{ϵ}} γ_{μ} ϵ) + (\bar{\bar{λ}} γ^{μ} γ_{3} λ) (\bar{\bar{ϵ}} γ_{μ} γ_{3} ϵ) = 0, \\ (\bar{\bar{ϵ}} γ^{μ} ϵ) (\bar{\bar{ϵ}} γ_{μ} ϵ) + (\bar{\bar{ϵ}} γ^{μ} γ_{3} ϵ) (\bar{\bar{ϵ}} γ_{μ} γ_{3} ϵ) = 0, \\ (\bar{\bar{ϵ}} ϵ) (\bar{\bar{ϵ}} ϵ) + (\bar{\bar{ϵ}} γ_{3} ϵ) (\bar{\bar{ϵ}} γ_{3} ϵ) = 0 . \end{aligned}

(A11)

APPENDIX B: 𝔰𝔬(4) AND 𝔰𝔲(2) × 𝔰𝔲(2)

In this work, we move between the SO(4) and SU(2) × SU(2) representations of the R-symmetry of the“large” $N = 4$ supersymmetry algebra. We provide a few details of this equivalence; here in case it is useful to the reader.

The defining representation of the 𝔰𝔬(4) algebra consists of real 4 × 4 antisymmetric matrices, which we label as

M_{a b}^{α β} = \frac{1}{2} () δ_{a}^{α} δ_{b}^{β} - δ_{a}^{β} δ_{b}^{α} () .

(B1)

Here (a, b) = 1, …, 4 label the matrix indices, and α, β = (1, …, 4) label elements of the algebra. M^αβ generates an infinitesimal rotation of a four-vector in the α − β plane.

We can split the orthogonal axes of ℝ⁴ in three different ways, corresponding to orthogonal pairs of two-planes. The two 𝔰𝔲(2) subalgebras correspond to simultaneous rotations along both pairs of planes, with differing relative signs of the rotation. More specifically, we can rewrite the basis elements $α_{a b}^{(\pm)}$ as

\begin{aligned} α^{(\pm) 1} = M^{23} \pm M^{14}, \\ α^{(\pm) 2} = M^{31} \pm M^{24}, \\ α^{(\pm) 3} = M^{12} \pm M^{34} . \end{aligned}

(B2)

Given a vector V^a in ℝ⁴, the linear combination

(\begin{array}{l} V^{(+) +} \\ V^{(+) -} \end{array}) = (\begin{array}{l} V^{1} + i V^{2} \\ V^{3} - i V^{4} \end{array})

(B3)

is a doublet under 𝔰𝔲(2)⁽⁺⁾, as is the complex conjugate. Similarly, the linear combination

(\begin{array}{l} V^{(-) +} \\ V^{(-) -} \end{array}) = (\begin{array}{l} V^{1} + i V^{2} \\ V^{3} + i V^{4} \end{array})

(B4)

is a doublet under 𝔰𝔲(2)⁽⁻⁾, as is its complex conjugate.

REFERENCES

1.

V.

Pestun

, “

Localization of gauge theory on a four-sphere and supersymmetric Wilson loops

,”

Commun. Math. Phys.

313

,

71

–

129

(

2012

); e-print arXiv:0712.2824 [hep-th].

https://doi.org/10.1007/s00220-012-1485-0

Google Scholar

Crossref

2.

A.

Kapustin

,

B.

Willett

, and

I.

Yaakov

, “

Exact results for Wilson loops in superconformal Chern-Simons theories with matter

,”

J. High Energy Phys.

2010

(

03

),

089

; e-print arXiv:0909.4559 [hep-th].

https://doi.org/10.1007/JHEP03(2010)089

Crossref

3.

A.

Kapustin

,

B.

Willett

, and

I.

Yaakov

, “

Nonperturbative tests of three-dimensional dualities

,”

J. High Energy Phys.

2010

(

10

),

013

; e-print arXiv:1003.5694 [hep-th].

https://doi.org/10.1007/JHEP10(2010)013

Crossref

4.

N.

Drukker

,

M.

Marino

, and

P.

Putrov

, “

From weak to strong coupling in ABJM theory

,”

Commun. Math. Phys.

306

,

511

–

563

(

2011

); e-print arXiv:1007.3837 [hep-th].

https://doi.org/10.1007/s00220-011-1253-6

Google Scholar

Crossref

5.

N.

Drukker

,

M.

Marino

, and

P.

Putrov

, “

Nonperturbative aspects of ABJM theory

,”

J. High Energy Phys.

2011

(

11

),

141

; e-print arXiv:1103.4844 [hep-th].

https://doi.org/10.1007/JHEP11(2011)141

Crossref

6.

M.

Marino

and

P.

Putrov

, “

ABJM theory as a Fermi gas

,”

J. Stat. Mech.

2012

(

1203

),

P03001

; e-print arXiv:1110.4066 [hep-th].

https://doi.org/10.1088/1742-5468/2012/03/p03001

Crossref

7.

A.

Klemm

,

M.

Marino

,

M.

Schiereck

, and

M.

Soroush

, “

Aharony-Bergman-Jafferis–Maldacena Wilson loops in the Fermi gas approach

,”

Z. Naturforsch. A

68

,

178

–

209

(

2013

); e-print arXiv:1207.0611 [hep-th].

https://doi.org/10.5560/zna.2012-0118

Google Scholar

Crossref

8.

H.

Awata

,

S.

Hirano

, and

M.

Shigemori

, “

The partition function of ABJ theory

,”

Prog. Theor. Exp. Phys.

2013

,

053B04

; e-print arXiv:1212.2966.

https://doi.org/10.1093/ptep/ptt014

Crossref

9.

S.

Hirano

,

K.

Nii

, and

M.

Shigemori

, “

ABJ Wilson loops and Seiberg duality

,”

Prog. Theor. Exp. Phys.

2014

(

11

),

113B04

; e-print arXiv:1406.4141 [hep-th].

https://doi.org/10.1093/ptep/ptu156

Crossref

10.

K.

Hosomichi

, “

A review on SUSY gauge theories on S³

,” e-print arXiv:1412.7128 [hep-th].

11.

F.

Benini

and

S.

Cremonesi

, “

Partition functions of

N = (2, 2)

gauge theories on S² and vortices

,”

Commun. Math. Phys.

334

(

3

),

1483

–

1527

(

2015

); e-print arXiv:1206.2356 [hep-th].

https://doi.org/10.1007/s00220-014-2112-z

Google Scholar

Crossref

12.

N.

Doroud

,

J.

Gomis

,

B.

Le Floch

, and

S.

Lee

, “

Exact results in D=2 supersymmetric gauge theories

,”

J. High Energy Phys.

2013

(

05

),

093

; e-print arXiv:1206.2606 [hep-th].

https://doi.org/10.1007/JHEP05(2013)093

Crossref

13.

H.

Jockers

,

V.

Kumar

,

J. M.

Lapan

,

D. R.

Morrison

, and

M.

Romo

, “

Two-sphere partition functions and Gromov-Witten invariants

,”

Commun. Math. Phys.

325

,

1139

–

1170

(

2014

); e-print arXiv:1208.6244 [hep-th].

https://doi.org/10.1007/s00220-013-1874-z

Google Scholar

Crossref

14.

J.

Gomis

and

S.

Lee

, “

Exact Kahler potential from gauge theory and mirror symmetry

,”

J. High Energy Phys.

2013

(

04

),

019

; e-print arXiv:1210.6022 [hep-th].

https://doi.org/10.1007/JHEP04(2013)019

Crossref

15.

Y.

Honma

and

M.

Manabe

, “

Exact Kahler potential for Calabi-Yau fourfolds

,”

J. High Energy Phys.

2013

(

1305

),

102

; e-print arXiv:1302.3760 [hep-th].

https://doi.org/10.1007/JHEP05(2013)102

Crossref

16.

B.

Jia

and

E.

Sharpe

, “

Curvature couplings in

N = (2, 2)

nonlinear sigma models on S²

,”

J. High Energy Phys.

2013

(

09

),

031

; e-print arXiv:1306.2398 [hep-th].

https://doi.org/10.1007/JHEP09(2013)031

Crossref

17.

K.

Hori

and

M.

Romo

, “

Exact results in two-dimensional (2,2) supersymmetric gauge theories with boundary

,” e-print arXiv:1308.2438 [hep-th].

18.

L.

Di Pietro

,

N.

Klinghoffer

, and

I.

Shamir

, “

On supersymmetry, boundary actions and brane charges

,”

J. High Energy Phys.

2016

(

1602

),

163

.

https://doi.org/10.1007/JHEP02(2016)163

Crossref

19.

H.

Kim

,

S.

Lee

, and

P.

Yi

, “

Exact partition functions on RP² and orientifolds

,”

J. High Energy Phys.

2014

(

02

),

103

; e-print arXiv:1310.4505 [hep-th].

https://doi.org/10.1007/JHEP02(2014)103

Crossref

20.

G.

Festuccia

and

N.

Seiberg

, “

Rigid supersymmetric theories in curved superspace

,”

J. High Energy Phys.

2011

(

06

),

114

; e-print arXiv:1105.0689 [hep-th].

https://doi.org/10.1007/JHEP06(2011)114

Crossref

21.

T. T.

Dumitrescu

,

G.

Festuccia

, and

N.

Seiberg

, “

Exploring curved superspace

,”

J. High Energy Phys.

2012

(

08

),

141

; e-print arXiv:1205.1115 [hep-th].

https://doi.org/10.1007/JHEP08(2012)141

Crossref

22.

C.

Klare

,

A.

Tomasiello

, and

A.

Zaffaroni

, “

Supersymmetry on curved spaces and holography

,”

J. High Energy Phys.

2012

(

08

),

061

; e-print arXiv:1205.1062 [hep-th].

https://doi.org/10.1007/JHEP08(2012)061

Crossref

23.

A.

Adams

,

H.

Jockers

,

V.

Kumar

, and

J. M.

Lapan

, “

N=1 sigma models in AdS_4

,”

J. High Energy Phys.

2011

(

12

),

042

; e-print arXiv:1104.3155 [hep-th].

https://doi.org/10.1007/JHEP12(2011)042

Crossref

24.

B.

Jia

and

E.

Sharpe

, “

Rigidly supersymmetric gauge theories on curved superspace

,”

J. High Energy Phys.

2012

(

04

),

139

; e-print arXiv:1109.5421 [hep-th].

https://doi.org/10.1007/JHEP04(2012)139

Crossref

25.

C.

Closset

,

T. T.

Dumitrescu

,

G.

Festuccia

, and

Z.

Komargodski

, “

The geometry of supersymmetric partition functions

,”

J. High Energy Phys.

2014

(

01

),

124

; e-print arXiv:1309.5876 [hep-th].

https://doi.org/10.1007/JHEP01(2014)124

Crossref

26.

C.

Closset

,

T. T.

Dumitrescu

,

G.

Festuccia

, and

Z.

Komargodski

, “

From rigid supersymmetry to twisted holomorphic theories

,”

Phys. Rev. D

90

(

8

),

085006

(

2014

); e-print arXiv:1407.2598 [hep-th].

https://doi.org/10.1103/PhysRevD.90.085006

Google Scholar

Crossref

27.

C.

Closset

and

I.

Shamir

, “

The

N = 1

chiral multiplet on T² × S² and supersymmetric localization

,”

J. High Energy Phys.

2014

(

03

),

040

; e-print arXiv:1311.2430 [hep-th].

https://doi.org/10.1007/JHEP03(2014)040

Crossref

28.

C.

Closset

and

S.

Cremonesi

, “

Comments on

N = (2, 2)

supersymmetry on two-manifolds

,”

J. High Energy Phys.

2014

(

07

),

075

; e-print arXiv:1404.2636 [hep-th].

https://doi.org/10.1007/JHEP07(2014)075

Crossref

29.

S. H.

Katz

,

A.

Klemm

, and

C.

Vafa

, “

M theory, topological strings and spinning black holes

,”

Adv. Theor. Math. Phys.

3

,

1445

–

1537

(

1999

); e-print arXiv:hep-th/9910181 [hep-th].

https://doi.org/10.4310/ATMP.1999.v3.n5.a6

Google Scholar

Crossref

30.

T.

Eguchi

,

H.

Ooguri

, and

Y.

Tachikawa

, “

Notes on the K3 surface and the Mathieu group M₂₄

,”

Exp. Math.

20

,

91

–

96

(

2011

); e-print arXiv:1004.0956 [hep-th].

https://doi.org/10.1080/10586458.2011.544585

Google Scholar

Crossref

31.

M. C. N.

Cheng

,

J. F. R.

Duncan

, and

J. A.

Harvey

, “

Umbral moonshine

,”

Commun. Num. Theor. Phys.

08

,

101

–

242

(

2014

); e-print arXiv:1204.2779 [math.RT].

https://doi.org/10.4310/CNTP.2014.v8.n2.a1

Google Scholar

Crossref

32.

J. A.

Harvey

and

S.

Murthy

, “

Moonshine in fivebrane spacetimes

,”

J. High Energy Phys.

2014

(

01

),

146

; e-print arXiv:1307.7717 [hep-th].

https://doi.org/10.1007/JHEP01(2014)146

Crossref

33.

R.

Pandharipande

and

R. P.

Thomas

, “

The Katz-Klemm-Vafa conjecture for K3 surfaces

,” e-print arXiv:1404.6698 [math.AG].

34.

J. A.

Harvey

,

S.

Murthy

, and

C.

Nazaroglu

, “

ADE double scaled little string theories, mock modular forms and umbral moonshine

,”

J. High Energy Phys.

2015

(

05

),

126

; e-print arXiv:1410.6174 [hep-th].

https://doi.org/10.1007/JHEP05(2015)126

Crossref

35.

M. C. N.

Cheng

and

S.

Harrison

, “

Umbral moonshine and K3 surfaces

,”

Commun. Math. Phys.

339

(

1

),

221

–

261

(

2015

); e-print arXiv:1406.0619 [hep-th].

https://doi.org/10.1007/s00220-015-2398-5

Google Scholar

Crossref

36.

J. A.

Harvey

,

S.

Lee

, and

S.

Murthy

, “

Elliptic genera of ALE and ALF manifolds from gauged linear sigma models

,”

J. High Energy Phys.

2015

(

02

),

110

; e-print arXiv:1406.6342 [hep-th].

https://doi.org/10.1007/JHEP02(2015)110

Crossref

37.

S.

Katz

,

A.

Klemm

, and

R.

Pandharipande

, “

On the motivic stable pairs invariants of K3 surfaces

,” e-print arXiv:1407.3181 [math.AG].

38.

M. C. N.

Cheng

,

J. F. R.

Duncan

,

S. M.

Harrison

, and

S.

Kachru

, “

Equivariant K3 invariants

,” e-print arXiv:1508.02047 [hep-th].

39.

P. C.

West

,

Introduction to Supersymmetry and Supergravity

(

World Scientific Pub Co Inc.

,

1990

).

Google Scholar

Crossref

40.

S. J.

Gates

, Jr.,

L.

Lu

, and

R. N.

Oerter

, “

Simplified SU(2) spinning string superspace supergravity

,”

Phys. Lett.B

218

,

33

(

1989

).

https://doi.org/10.1016/0370-2693(89)90470-X

Google Scholar

Crossref

41.

S. J.

Gates

, Jr. and

S. V.

Ketov

, “

2-D (4,4) hypermultiplets

,”

Phys. Lett. B

418

,

111

–

118

(

1998

); e-print arXiv:hep-th/9504077 [hep-th].

https://doi.org/10.1016/S0370-2693(97)01504-9

Google Scholar

Crossref

42.

A.

Sevrin

,

W.

Troost

, and

A.

Van Proeyen

, “

Superconformal algebras in two-dimensions with N=4

,”

Phys. Lett. B

208

,

447

(

1988

).

https://doi.org/10.1016/0370-2693(88)90645-4

Google Scholar

Crossref

43.

R.

Gastmans

,

A.

Sevrin

,

W.

Troost

, and

A.

Van Proeyen

, “

Infinite dimensional extended superconformal lie algebras

,”

Int. J. Mod. Phys. A

2

,

195

–

216

(

1987

).

https://doi.org/10.1142/S0217751X87000077

Google Scholar

Crossref

44.

O.

Aharony

and

M.

Berkooz

, “

IR dynamics of D = 2, N=(4,4) gauge theories and DLCQ of ‘little string theories

,’”

J. High Energy Phys.

1999

(

10

),

030

; e-print arXiv:hep-th/9909101 [hep-th].

https://doi.org/10.1088/1126-6708/1999/10/030

Crossref

45.

S. J.

Gates

, Jr.,

Y.

Hassoun

, and

P.

van Nieuwenhuizen

, “

Auxiliary fields for d = 2, N = 4 supergravity

,”

Nucl. Phys. B

317

,

302

(

1989

).

https://doi.org/10.1016/0550-3213(89)90071-0

Google Scholar

Crossref

46.

R.

Grimm

,

M.

Sohnius

, and

J.

Wess

, “

Extended supersymmetry and gauge theories

,”

Nucl. Phys. B

133

,

275

(

1978

).

https://doi.org/10.1016/0550-3213(78)90303-6

Google Scholar

Crossref

47.

A.

Galperin

,

E.

Ivanov

,

S.

Kalitsyn

,

V.

Ogievetsky

, and

E.

Sokatchev

, “

Unconstrained N = 2 matter, Yang-Mills and supergravity theories in harmonic superspace

,”

Classical Quantum Gravity

1

,

469

–

498

(

1984

)

https://doi.org/10.1088/0264-9381/1/5/004

;

Crossref

Google Scholar

Erratum

A.

Galperin

,

E.

Ivanov

,

S.

Kalitsyn

,

V.

Ogievetsky

, and

E.

Sokatchev

,

Classical Quantum Gravity

2

,

127

(

1985

).

https://doi.org/10.1088/0264-9381/2/1/512

Crossref

Google Scholar

48.

A. S.

Galperin

,

N. A.

Ky

, and

E.

Sokatchev

, “

N = 2 supergravity in superspace: Solution to the constraints

,”

Classical Quantum Gravity

4

,

1235

(

1987

).

https://doi.org/10.1088/0264-9381/4/5/022

Google Scholar

Crossref

49.

A. S.

Galperin

,

E. A.

Ivanov

,

V. I.

Ogievetsky

, and

E.

Sokatchev

, “

N = 2 supergravity in superspace: Different versions and matter couplings

,”

Classical Quantum Gravity

4

,

1255

(

1987

).

https://doi.org/10.1088/0264-9381/4/5/023

Google Scholar

Crossref

50.

S.

Bellucci

and

E.

Ivanov

, “

N=(4,4), 2-D supergravity in SU(2) x SU(2) harmonic superspace

,”

Nucl. Phys. B

587

,

445

–

480

(

2000

); e-print arXiv:hep-th/0003154 [hep-th].

https://doi.org/10.1016/S0550-3213(00)00347-3

Google Scholar

Crossref

51.

G.

Tartaglino-Mazzucchelli

, “

On 2D N=(4,4) superspace supergravity

,”

Phys. Part. Nucl. Lett.

8

,

251

–

261

(

2011

); e-print arXiv:0912.5300 [hep-th].

https://doi.org/10.1134/S1547477111030204

Google Scholar

Crossref

52.

S. M.

Kuzenko

,

U.

Lindstrom

,

M.

Rocek

, and

G.

Tartaglino-Mazzucchelli

, “

4D N = 2 supergravity and projective superspace

,”

J. High Energy Phys.

2008

(

09

),

051

; e-print arXiv:0805.4683 [hep-th].

https://doi.org/10.1088/1126-6708/2008/09/051

Crossref

53.

G.

Tartaglino-Mazzucchelli

, “

2D N = (4,4) superspace supergravity and bi-projective superfields

,”

J. High Energy Phys.

2010

(

04

),

034

; e-print arXiv:0911.2546 [hep-th].

https://doi.org/10.1007/JHEP04(2010)034

Crossref

54.

For earlier work applied to AdS₄ backgrounds, see Refs. 23 and 24.

55.

Equivariant elliptic genera for $N = (4, 4)$ theories have been computed via localization when the worldsheet is a flat torus, by writing the theory in $N = (2, 2)$ language.³⁶

56.

We are using the notation in Ref. 45; a summary of our conventions and a list of useful identities used throughout this paper can be found in Appendix A.

57.

We would like to thank our anonymous referee for reminding us of this point.

2016

AIP Publishing LLC

$N = (4, 4)$ vector multiplets on curved two-manifolds

I. INTRODUCTION

II. $N = (4, 4)$ SUPERALGEBRA

III. $N = (4, 4)$ GRAVITY MULTIPLET

IV. CONSISTENCY CONDITIONS FOR $N = (4, 4)$ SUPERSYMMETRY

V. $N = (4, 4)$ VECTOR MULTIPLET

A. Flat space

B. Curved space

VI. $N = (4, 4)$ ON S²

VII. CONCLUSIONS

Acknowledgments

APPENDIX A: NOTATIONS AND CONVENTIONS

APPENDIX B: 𝔰𝔬(4) AND 𝔰𝔲(2) × 𝔰𝔲(2)

REFERENCES

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N=(4,4) vector multiplets on curved two-manifolds Free

I. INTRODUCTION

II. N=(4,4) SUPERALGEBRA

III. N=(4,4) GRAVITY MULTIPLET

IV. CONSISTENCY CONDITIONS FOR N=(4,4) SUPERSYMMETRY

V. N=(4,4) VECTOR MULTIPLET

A. Flat space

B. Curved space

VI. N=(4,4) ON S2

VII. CONCLUSIONS

Acknowledgments

APPENDIX A: NOTATIONS AND CONVENTIONS

APPENDIX B: 𝔰𝔬(4) AND 𝔰𝔲(2) × 𝔰𝔲(2)

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$N = (4, 4)$ vector multiplets on curved two-manifolds

II. $N = (4, 4)$ SUPERALGEBRA

III. $N = (4, 4)$ GRAVITY MULTIPLET

IV. CONSISTENCY CONDITIONS FOR $N = (4, 4)$ SUPERSYMMETRY

V. $N = (4, 4)$ VECTOR MULTIPLET

VI. $N = (4, 4)$ ON S²