Abstract
We compute the elliptic genus of abelian 2d (0, 2) gauge theories corresponding to brane brick models. These theories are worldvolume theories on a single D1-brane probing a toric Calabi-Yau 4-fold singularity. We identify a match with the elliptic genus of the non-linear sigma model on the same Calabi-Yau background, which is computed using a new localization formula. The matching implies that the quantum effects do not drastically alter the correspondence between the geometry and the 2d (0, 2) gauge theory. In theories whose matter sector suffers from abelian gauge anomaly, we propose an ansatz for an anomaly cancelling term in the integral formula for the elliptic genus. We provide an example in which two brane brick models related to each other by Gadde-Gukov-Putrov triality give the same elliptic genus.
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References
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
K. Hori and D. Tong, Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N = (2,2) Theories, JHEP 05 (2007) 079 [hep-th/0609032] [INSPIRE].
J. Distler and S. Kachru, (0, 2) Landau-Ginzburg theory, Nucl. Phys. B 413 (1994) 213 [hep-th/9309110] [INSPIRE].
E. Silverstein and E. Witten, Global U(1) R symmetry and conformal invariance of (0, 2) models, Phys. Lett. B 328 (1994) 307 [hep-th/9403054] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, (0, 2) trialities, JHEP 03 (2014) 076 [arXiv:1310.0818] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Exact Solutions of 2d Supersymmetric Gauge Theories, arXiv:1404.5314 [INSPIRE].
K. Mohri, D-branes and quotient singularities of Calabi-Yau fourfolds, Nucl. Phys. B 521 (1998) 161 [hep-th/9707012] [INSPIRE].
H. Garcia-Compean and A.M. Uranga, Brane box realization of chiral gauge theories in two-dimensions, Nucl. Phys. B 539 (1999) 329 [hep-th/9806177] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
S. Franco, D. Ghim, S. Lee, R.-K. Seong and D. Yokoyama, 2d (0, 2) Quiver Gauge Theories and D-branes, JHEP 09 (2015) 072 [arXiv:1506.03818] [INSPIRE].
S. Franco, S. Lee and R.-K. Seong, Brane Brick Models, Toric Calabi-Yau 4-Folds and 2d (0, 2) Quivers, JHEP 02 (2016) 047 [arXiv:1510.01744] [INSPIRE].
S. Franco, S. Lee and R.-K. Seong, Brane brick models and 2d (0, 2) triality, JHEP 05 (2016) 020 [arXiv:1602.01834] [INSPIRE].
S. Franco, S. Lee, R.-K. Seong and C. Vafa, Brane Brick Models in the Mirror, JHEP 02 (2017) 106 [arXiv:1609.01723] [INSPIRE].
S. Franco, S. Lee and R.-K. Seong, Orbifold Reduction and 2d (0, 2) Gauge Theories, JHEP 03 (2017) 016 [arXiv:1609.07144] [INSPIRE].
F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [INSPIRE].
F. Benini, N. Bobev and P.M. Crichigno, Two-dimensional SCFTs from D3-branes, JHEP 07 (2016) 020 [arXiv:1511.09462] [INSPIRE].
R. Tatar, Geometric Constructions of Two Dimensional (0, 2) SUSY Theories, Phys. Rev. D 92 (2015) 045006 [arXiv:1506.05372] [INSPIRE].
S. Schäfer-Nameki and T. Weigand, F-theory and 2d (0, 2) theories, JHEP 05 (2016) 059 [arXiv:1601.02015] [INSPIRE].
F. Apruzzi, F. Hassler, J.J. Heckman and I.V. Melnikov, UV Completions for Non-Critical Strings, JHEP 07 (2016) 045 [arXiv:1602.04221] [INSPIRE].
F. Apruzzi, F. Hassler, J.J. Heckman and I.V. Melnikov, From 6D SCFTs to Dynamic GLSMs, arXiv:1610.00718 [INSPIRE].
A. Gadde and S. Gukov, 2d Index and Surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic Genera of 2d N = 2 Gauge Theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
F. Benini and B. Le Floch, Supersymmetric localization in two dimensions, arXiv:1608.02955 [INSPIRE].
L.C. Jeffrey and F.C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995) 291 [alg-geom/9307001].
W. Lerche, Elliptic Index and Superstring Effective Actions, Nucl. Phys. B 308 (1988) 102 [INSPIRE].
D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Commun. Math. Phys. 280 (2008) 611 [hep-th/0603021] [INSPIRE].
F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].
S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, A new supersymmetric index, Nucl. Phys. B 386 (1992) 405 [hep-th/9204102] [INSPIRE].
M. Koloğlu, Quantum Vacua of 2d Maximally Supersymmetric Yang-Mills Theory, arXiv:1609.08232 [INSPIRE].
S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].
J.J. Duistermaat and G.J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259.
D. Ghim, J.-W. Kim and S. Lee, Elliptic Genus of 2d (0, 2) Non-linear Sigma Models, to appear.
T. Kawai and K. Mohri, Geometry of (0, 2) Landau-Ginzburg orbifolds, Nucl. Phys. B 425 (1994) 191 [hep-th/9402148] [INSPIRE].
S. Lee and S. Lee, Klebanov-Witten Flows in M-theory, JHEP 09 (2012) 098 [arXiv:1207.7169] [INSPIRE].
J. Davey, A. Hanany and R.-K. Seong, Counting Orbifolds, JHEP 06 (2010) 010 [arXiv:1002.3609] [INSPIRE].
R. Dijkgraaf, G.W. Moore, E.P. Verlinde and H.L. Verlinde, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys. 185 (1997) 197 [hep-th/9608096] [INSPIRE].
C. Closset, W. Gu, B. Jia and E. Sharpe, Localization of twisted \( \mathcal{N} \) = (0, 2) gauged linear σ-models in two dimensions, JHEP 03 (2016) 070 [arXiv:1512.08058] [INSPIRE].
K. Hori et al., Mirror Symmetry, Clay Mathematics Monographs, American Mathematical Society (2003).
R. Blumenhagen and E. Plauschinn, Introduction to Conformal Field Theory: With Applications to String Theory, Lecture Notes in Physics, Springer Berlin Heidelberg (2009).
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Franco, S., Ghim, D., Lee, S. et al. Elliptic genera of 2d (0,2) gauge theories from brane brick models. J. High Energ. Phys. 2017, 68 (2017). https://doi.org/10.1007/JHEP06(2017)068
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DOI: https://doi.org/10.1007/JHEP06(2017)068