Abstract
A large class of two-dimensional \( \mathcal{N}=\left(2,\ 2\right) \) superconformal field theories can be understood as IR fixed-points of Landau-Ginzburg models. In particular, there are rational conformal field theories that also have a Landau-Ginzburg description. To understand better the relation between the structures in the rational conformal field theory and in the Landau-Ginzburg theory, we investigate how rational B-type boundary conditions are realised as matrix factorisations in the SU(3)/U(2) Grassmannian Kazama-Suzuki model. As a tool to generate the matrix factorisations we make use of a particular interface between the Kazama-Suzuki model and products of minimal models, whose fusion can be realised as a simple functor on ring modules. This allows us to formulate a proposal for all matrix factorisations corresponding to rational boundary conditions in the SU(3)/U(2) model.
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References
B.R. Greene, String theory on Calabi-Yau manifolds, hep-th/9702155 [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
Y. Kazama and H. Suzuki, New N = 2 superconformal field theories and superstring compactification, Nucl. Phys. B 321 (1989) 232 [INSPIRE].
Y. Kazama and H. Suzuki, Characterization of N = 2 Superconformal Models Generated by Coset Space Method, Phys. Lett. B 216 (1989) 112 [INSPIRE].
W. Lerche, C. Vafa and N.P. Warner, Chiral Rings in N = 2 Superconformal Theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].
D. Gepner, Fusion rings and geometry, Commun. Math. Phys. 141 (1991) 381 [INSPIRE].
I. Brunner, M. Herbst, W. Lerche and B. Scheuner, Landau-Ginzburg realization of open string TFT, JHEP 11 (2006) 043 [hep-th/0305133] [INSPIRE].
A. Kapustin and Y. Li, D-branes in topological minimal models: the Landau-Ginzburg approach, JHEP 07 (2004) 045 [hep-th/0306001] [INSPIRE].
I. Brunner and M.R. Gaberdiel, The matrix factorisations of the D-model, J. Phys. A 38 (2005) 7901 [hep-th/0506208] [INSPIRE].
I. Brunner and M.R. Gaberdiel, Matrix factorisations and permutation branes, JHEP 07 (2005) 012 [hep-th/0503207] [INSPIRE].
H. Enger, A. Recknagel and D. Roggenkamp, Permutation branes and linear matrix factorisations, JHEP 01 (2006) 087 [hep-th/0508053] [INSPIRE].
C.A. Keller and S. Rossi, Boundary states, matrix factorisations and correlation functions for the E-models, JHEP 03 (2007) 038 [hep-th/0610175] [INSPIRE].
N. Behr and S. Fredenhagen, D-branes and matrix factorisations in supersymmetric coset models, JHEP 11 (2010) 136 [arXiv:1005.2117] [INSPIRE].
N. Behr and S. Fredenhagen, Variable transformation defects, Proc. Symp. Pure Math. 85 (2012) 303 [arXiv:1202.1678] [INSPIRE].
M. Kontsevich, unpublished.
A. Kapustin and Y. Li, D branes in Landau-Ginzburg models and algebraic geometry, JHEP 12 (2003) 005 [hep-th/0210296] [INSPIRE].
D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, math/0302304 [INSPIRE].
A. Kapustin and Y. Li, Topological correlators in Landau-Ginzburg models with boundaries, Adv. Theor. Math. Phys. 7 (2004) 727 [hep-th/0305136] [INSPIRE].
M. Herbst, C.-I. Lazaroiu and W. Lerche, D-brane effective action and tachyon condensation in topological minimal models, JHEP 03 (2005) 078 [hep-th/0405138] [INSPIRE].
S. Govindarajan, H. Jockers, W. Lerche and N.P. Warner, Tachyon condensation on the elliptic curve, Nucl. Phys. B 765 (2007) 240 [hep-th/0512208] [INSPIRE].
I. Brunner and D. Roggenkamp, B-type defects in Landau-Ginzburg models, JHEP 08 (2007) 093 [arXiv:0707.0922] [INSPIRE].
A. Kapustin and L. Rozansky, On the relation between open and closed topological strings, Commun. Math. Phys. 252 (2004) 393 [hep-th/0405232] [INSPIRE].
M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1 [math/0401268].
Y. Yoshino, Tensor products of matrix factorizations, Nagoya Math. J. 152 (1998) 39.
N. Carqueville and I. Runkel, On the monoidal structure of matrix bi-factorisations, J. Phys. A 43 (2010) 275401 [arXiv:0909.4381] [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354 [hep-th/0607247] [INSPIRE].
D. Gepner, Scalar field theory and string compactification, Nucl. Phys. B 322 (1989) 65 [INSPIRE].
H. Ooguri, Y. Oz and Z. Yin, D-branes on Calabi-Yau spaces and their mirrors, Nucl. Phys. B 477 (1996) 407 [hep-th/9606112] [INSPIRE].
J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
H. Ishikawa and T. Tani, Twisted boundary states in Kazama-Suzuki models, Nucl. Phys. B 678 (2004) 363 [hep-th/0306227] [INSPIRE].
M.R. Gaberdiel and T. Gannon, Boundary states for WZW models, Nucl. Phys. B 639 (2002) 471 [hep-th/0202067] [INSPIRE].
S. Fredenhagen and V. Schomerus, Brane dynamics in CFT backgrounds, hep-th/0104043 [INSPIRE].
S. Fredenhagen and V. Schomerus, D-branes in coset models, JHEP 02 (2002) 005 [hep-th/0111189] [INSPIRE].
S. Fredenhagen, D-brane dynamics in curved backgrounds, Ph.D. Thesis, Humboldt University, Berlin Germany (2002), http://edoc.hu-berlin.de/docviews/abstract.php?id=10498.
S. Fredenhagen and V. Schomerus, On boundary RG flows in coset conformal field theories, Phys. Rev. D 67 (2003) 085001 [hep-th/0205011] [INSPIRE].
S. Fredenhagen, Organizing boundary RG flows, Nucl. Phys. B 660 (2003) 436 [hep-th/0301229] [INSPIRE].
C. Bachas and S. Monnier, Defect loops in gauged Wess-Zumino-Witten models, JHEP 02 (2010) 003 [arXiv:0911.1562] [INSPIRE].
V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [INSPIRE].
M. Nozaki, Comments on D-branes in Kazama-Suzuki models and Landau-Ginzburg theories, JHEP 03 (2002) 027 [hep-th/0112221] [INSPIRE].
K. Graham and G.M.T. Watts, Defect lines and boundary flows, JHEP 04 (2004) 019 [hep-th/0306167] [INSPIRE].
S.K. Ashok, E. Dell’Aquila and D.-E. Diaconescu, Fractional branes in Landau-Ginzburg orbifolds, Adv. Theor. Math. Phys. 8 (2004) 461 [hep-th/0401135] [INSPIRE].
S.K. Ashok, E. Dell’Aquila, D.-E. Diaconescu and B. Florea, Obstructed D-branes in Landau-Ginzburg orbifolds, Adv. Theor. Math. Phys. 8 (2004) 427 [hep-th/0404167] [INSPIRE].
M. Herbst and C.-I. Lazaroiu, Localization and traces in open-closed topological Landau-Ginzburg models, JHEP 05 (2005) 044 [hep-th/0404184] [INSPIRE].
I. Brunner, N. Carqueville and D. Plencner, Discrete torsion defects, Commun. Math. Phys. 337 (2015) 429 [arXiv:1404.7497] [INSPIRE].
I. Brunner, N. Carqueville and D. Plencner, Orbifolds and topological defects, Commun. Math. Phys. 332 (2014) 669 [arXiv:1307.3141] [INSPIRE].
N. Carqueville and D. Murfet, Adjunctions and defects in Landau-Ginzburg models, arXiv:1208.1481 [INSPIRE].
N. Carqueville and I. Runkel, Rigidity and defect actions in Landau-Ginzburg models, Commun. Math. Phys. 310 (2012) 135 [arXiv:1006.5609] [INSPIRE].
N. Behr, S. Fredenhagen, Rational defects in Landau-Ginzburg models, in preparation.
N. Carqueville and D. Murfet, Computing Khovanov-Rozansky homology and defect fusion, Topology 14 (2014) 489 [arXiv:1108.1081] [INSPIRE].
N. Behr, S. Fredenhagen, Fusion functors, in preparation.
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Behr, N., Fredenhagen, S. Matrix factorisations for rational boundary conditions by defect fusion. J. High Energ. Phys. 2015, 55 (2015). https://doi.org/10.1007/JHEP05(2015)055
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DOI: https://doi.org/10.1007/JHEP05(2015)055