We evaluate the ideas of -stability at the Landau-Ginzburg (LG) point in moduli space of compact Calabi-Yau manifolds, using matrix factorizations to -model the topological -brane category. The standard requirement of unitarity at the IR fixed point is argued to lead to a notion of “-stability” for matrix factorizations of quasihomogeneous LG potentials. The -brane on the quintic at the Landau-Ginzburg point is not obviously unstable. Aiming to relate -stability to a moduli space problem, we then study the action of the gauge group of similarity transformations on matrix factorizations. We define a naive moment maplike flow on the gauge orbits and use it to study boundary flows in several examples. Gauge transformations of nonzero degree play an interesting role for brane-antibrane annihilation. We also give a careful exposition of the grading of the Landau-Ginzburg category of -branes, and prove an index theorem for matrix factorizations.
REFERENCES
1.
2.
M. R.
Douglas
, J. Math. Phys.
42
, 2818
(2001
).3.
4.
5.
T.
Bridgeland
, math.ag/0212237.6.
P. S.
Aspinwall
, “D-branes on Calabi-Yau manifolds
,” Talk given at Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2003): Recent Trends in String Theory, Boulder, Colorado, 1–27 June 2003
.7.
W.
Lerche
, C.
Vafa
, and N. P.
Warner
, Nucl. Phys. B
324
, 427
(1989
).8.
J.
Fuchs
, C.
Schweigert
, and J.
Walcher
, Nucl. Phys. B
588
, 110
(2000
).9.
M.
Kontsevich
, “Homological algebra of mirror symmetry
,” Proceedings of ICM, Zürich, 1994
(Birkhäuser
, Boston, 1995
), Vols. 1, 2
, pp. 120
–139
.10.
P. S.
Aspinwall
, “D-branes, Pi-stability and Theta-stability
,” Lectures given at AMS-IMS-SIAM Summer Research Conference on String Geometry, Snowbird, Utah, 5–11 Jun (2004
).11.
12.
K.
Hori
and J.
Walcher
, “F-term equations near Gepner points
,” J. High Energy Phys.
0501
, 008
(2005
).13.
K.
Hori
and J.
Walcher
, “D-branes from matrix factorizations
,” C. R. Phys.
5
, 1061
(2004
).14.
15.
Y.
Yoshino
, Cohen-Macaulay Modules over Cohen-Macaulay Rings
(Cambridge University Press
, Cambridge, 1990
).16.
W.
Bruns
and J.
Herzog
, Cohen-Macaulay Rings
(Cambridge University Press
, Cambridge, 1993
).17.
M.
Kontsevich
(unpublished), as cited in Ref. 30.18.
A. I.
Bondal
and M. M.
Kapranov
, translation in Math. USSR. Sb.
70
, 93
(1991
).19.
D. O.
Orlov
Algebr. Geom. Metody
, Svyazi i Prilozh., 240
–262
;20.
21.
B. R.
Greene
, C.
Vafa
, and N. P.
Warner
, Nucl. Phys. B
324
, 371
(1989
).22.
E.
Witten
, Nucl. Phys. B
403
, 159
(1993
).23.
K.
Hori
et al., Mirror Symmetry, Clay Mathematics Monographs
(American Mathematical Society
, Providence, RI, 2003
), Vol. 1
.24.
N. P.
Warner
, Nucl. Phys. B
450
, 663
(1995
).25.
S.
Govindarajan
, T.
Jayaraman
, and T.
Sarkar
, Nucl. Phys. B
580
, 519
(2000
).26.
27.
K.
Hori
, hep-th∕0012179.28.
S.
Hellerman
, S.
Kachru
, A. E.
Lawrence
, and J.
McGreevy
, J. High Energy Phys.
0207
, 002
(2002
).29.
K.
Hori
, hep-th∕0207068.30.
31.
32.
A.
Kapustin
and Y.
Li
, “Topological correlators in Landau-Ginzburg models with boundaries
,” Adv. Theor. Math. Phys.
7
, 727
(2004
).33.
A.
Kapustin
and Y.
Li
, “D-branes in topological minimal models: The Landau-Ginzburg approach
,” J. High Energy Phys.
0407
, 045
(2004
).34.
C. I.
Lazaroiu
, “On the boundary coupling of topological Landau-Ginzburg models
,” J. High Energy Phys.
0505
, 037
(2005
).35.
36.
K.
Hori
, hep-th∕0401139.37.
M.
Herbst
, C. I.
Lazaroiu
, and W.
Lerche
, “Superpotentials, A(infinity) relations and WDVV equations for open topological strings
,” J. High Energy Phys.
0502
, 071
(2005
).38.
39.
M.
Herbst
and C. I.
Lazaroiu
, “Localization and traces in open-closed toplogical Landau-Ginzburg models
,” J. High Energy Phys.
0505
, 044
(2005
).40.
41.
M.
Herbst
, C. I.
Lazaroiu
, and W.
Lerche
, “D-brane effective action and tachyon condensation in topological minimal models
,” J. High Energy Phys.
0503
, 078
(2005
).42.
43.
44.
45.
K.
Intriligator
and B.
Wecht
, Nucl. Phys. B
667
, 183
(2003
).46.
47.
I.
Brunner
, M. R.
Douglas
, A. E.
Lawrence
, and C.
Romelsberger
, J. High Energy Phys.
0008
, 015
(2000
).48.
49.
C. I.
Lazaroiu
, Nucl. Phys. B
603
, 497
(2001
).50.
G.
Moore
and G.
Segal
(unpublished); see also lectures by51.
D.
Gepner
, Nucl. Phys. B
296
, 757
(1988
).52.
A.
Recknagel
and V.
Schomerus
, Nucl. Phys. B
531
, 185
(1998
).53.
54.
J.
Fuchs
, P.
Kaste
, W.
Lerche
, C. A.
Lutken
, C.
Schweigert
, and J.
Walcher
, Nucl. Phys. B
598
, 57
(2001
).55.
56.
R. P.
Thomas
, Symplectic Geometry and Mirror Symmetry (Seoul, 2000)
(World Scientific
, River Edge, NJ, 2001
), pp. 467
–498
.57.
58.
D.
Mumford
, J.
Fogarty
, and F.
Kirwan
, Geometric Invariant Theory
, 3rd ed. (Springer
, Berlin, 1994
).59.
60.
A.
Schappert
, “A characterisation of strictly unimodular plane curve singularities
,” Singularities, Representation of Algebras, and Vector Bundles (Lambrecht, 1985)
, Lect. Notes Math. 1273
(Springer
, Berlin, 1987
), pp. 168
–177
.61.
V. I.
Arnold
, S. M.
Gusein-Zade
, and A. N.
Varchenko
, Singularities of Differentiable Maps
(Birkäuser
, Boston, 1988
), Vol. II
.© 2005 American Institute of Physics.
2005
American Institute of Physics
You do not currently have access to this content.
