Abstract
We study a class of two-dimensional \({\mathcal{N}=(2,2)}\) sigma models called squashed toric sigma models, using their Gauged Linear Sigma Models (GLSM) description. These models are obtained by gauging the global \({U(1)}\) symmetries of toric GLSMs and introducing a set of corresponding compensator superfields. The geometry of the resulting vacuum manifold is a deformation of the corresponding toric manifold in which the torus fibration maintains a constant size in the interior of the manifold, thus producing a neck-like region. We compute the elliptic genus of these models, using localization, in the case when the unsquashed vacuum manifolds obey the Calabi–Yau condition. The elliptic genera have a non-holomorphic dependence on the modular parameter \({\tau}\) coming from the continuum produced by the neck. In the simplest case corresponding to squashed \({\mathbb{C} / \mathbb{Z}_{2}}\) the elliptic genus is a mixed mock Jacobi form which coincides with the elliptic genus of the \({\mathcal{N}=(2,2)}\) \({SL(2,\mathbb{R}) / U(1)}\) cigar coset.
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Schellekens A.N., Warner N.P.: Anomalies and modular invariance in string theory. Phys. Lett. B 177, 317–323 (1986)
Schellekens A.N., Warner N.P.: Anomalies, characters and strings. Nucl. Phys. B 287, 317 (1987)
Pilch K., Schellekens A.N., Warner N.P.: Path integral calculation of string anomalies. Nucl. Phys. B 287, 362–380 (1987)
Witten E.: Elliptic genera and quantum field theory. Commun. Math. Phys., 109, 525 (1987)
Witten, E.: The index of the Dirac operator in loop space. http://alice.cern.ch/format/showfull?sysnb=0088339 (1987)
Alvarez O., Killingback T.P., Mangano M.L., Windey P.: String theory and loop space index theorems. Commun. Math. Phys. 111, 1 (1987)
Alvarez, O., Killingback, T.P., Mangano, M.L., Windey, P.: The Dirac–Ramond operator in string theory and loop space index theorems, Nucl. Phys. Proc. Suppl., 1A(1), 189–215 (1987)
Troost, J.: The non-compact elliptic genus: mock or modular, JHEP 06, 104 (2010). [arXiv:1004.3649].
Eguchi, T., Sugawara, Y.: Non-holomorphic Modular Forms and SL(2,R)/U(1) Superconformal Field Theory, JHEP 03, 107 (2011). [arXiv:1012.5721].
Ashok S.K., Troost J.: A twisted non-compact elliptic genus. JHEP 03, 067 (2011) arXiv:1101.1059
Gawedzki, K.: Noncompact WZW conformal field theories, New symmetry principles in quantum field theory. In: Proceedings, NATO Advanced Study Institute, Cargese, France, July 16–27, 1991, pp. 0247–274 (1991). arXiv:hep-th/9110076
Harvey J.A., Murthy S.:: Moonshine in Fivebrane Spacetimes. JHEP. 01, 146 (2014) arXiv:1307.7717
Harvey, J.A., Murthy, S., Nazaroglu, C.: ADE double scaled little string theories, mock modular forms and umbral moonshine. JHEP 05, 126, (2015). arXiv:1410.6174
Cheng M.C.N., Harrison S.: Umbral moonshine and K3 surfaces. Commun. Math. Phys. 339(1), 221–261 arXiv:1406.0619 (2015)
Murthy, S.: A holomorphic anomaly in the elliptic genus. JHEP, 06, 165 (2014). arXiv:1311.0918
Ashok S.K., Doroud N., Troost J.: Localization and real Jacobi forms. JHEP. 04, 119 arXiv:1311.1110 (2014)
Zwegers S.P.: . Mock theta functions, Thesis, Utrecht (2002)
Zagier, D.: Ramanujan’s mock theta functions and their applications [d’aprè s Zwegers and Bringmann-Ono], Séminaire BOURBAKI, 60 ème année, 2006–2007 986 (2007)
Dabholkar, A., Murthy, S., Zagier, D.: Quantum black holes, wall crossing, and mock modular forms. arXiv:1208.4074
Eguchi T., Ooguri H., Tachikawa Y.: Notes on the K3 Surface and the Mathieu group M 24. Exper. Math., 20, 91–96 arXiv:1004.0956 (2011)
Cheng M.C.N., Duncan J.F.R., Harvey J.A.: Umbral moonshine. Commun. Num. Theor. Phys. 08, 101–242 arXiv:1204.2779 (2014)
Cheng M.C.N, Duncan J.F.R, Harvey J.A.: Umbral Moonshine and the Niemeier Lattices. Math. Sci., 1, 3 (2014). arXiv:1307.5793
Hori K., Kapustin A.: Duality of the fermionic 2-D black hole and N = 2 liouville theory as mirror symmetry. JHEP. 08, 045 (2001) arXiv:hep-th/0104202
Morrison, D.R., Plesser, M.R.: Summing the instantons: quantum cohomology and mirror symmetry in toric varieties. Nucl. Phys. B440, 279–354 (1995). arXiv:hep-th/9412236
Harvey J.A., Lee S., Murthy S.: Elliptic genera of ALE and ALF manifolds from gauged linear sigma models.. JHEP 02, 110 (2015) arXiv:1406.6342
Dabholkar, A., Gomes, J., Murthy, S., Sen, A.: Supersymmetric index from black hole entropy. JHEP 04(2011), 034 (2015). arXiv:1009.3226
Haghighat B, Murthy S., Vafa C., Vandoren S.: F-theory, spinning black holes and multi-string branches. JHEP. 01, 009 (2016) arXiv:1509.0045
Gromov, N., Gupta, R., Murthy, S.: Work in progress.
Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror Symmetry, vol. 1 of Clay Mathematics Monographs. AMS, Providence, USA (2003)
da Silva, A.C.: Symplectic Toric Manifolds. Birkhäuser series Advanced Courses in Mathematics, CRM Barcelona, Birkhauser (Springer) (2003)
Aharony O., Razamat S.S., Seiberg N., Willett B.: The long flow to freedom. JHEP 02, 056 (2003) arXiv:1611.0276
Fateev, V.A., Onofri, E., Zamolodchikov, A.B.: The sausage model (integrable deformations of O(3) sigma model). Nucl. Phys. B406, 521–565 (1993)
Fendley, P., Intriligator, K.A.: Scattering and thermodynamics in integrable N=2 theories. Nucl. Phys. B380, 265–290 (1992). arXiv:hep-th/9202011
Hori K., Kapustin A.: World sheet descriptions of wrapped NS five-branes. JHEP 11, 038 (2002) arXiv:hep-th/0203147
Benini, F., Eager, R., Hori, K., Tachikawa, Y.: Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups. Lett. Math. Phys. 104, 465–493 (2014) arXiv:1305.0533
Benini, F., Eager, R., Hori, K., Tachikawa Y.: Elliptic genera of 2d \({\mathcal{N} = 2}\) Gauge Theories. Commun. Math. Phys. 333(3) 1241–1286 (2015) arXiv:1308.4896
Jeffrey L., Kirwan F.: Localization for nonabelian group actions. Topology. 34(2), 291–327 (1995) arXiv:alg-geom/9307001
Eguchi T., Hanson A.J.: Selfdual solutions to euclidean gravity. Ann. Phys. 120, 82 (1979)
Eguchi T., Sugawara Y.: Conifold type singularities, N = 2 Liouville and SL(2:R)/U(1) theories. JHEP 0501, 027 (2005) arXiv:hep-th/0411041
Ashok, S.K., Benichou, R., Troost, J.: Non-compact Gepner Models, Landau–Ginzburg Orbifolds and Mirror Symmetry. JHEP 0801, 050 (2008). arXiv:0710.1990
Ashok, S.K., Nampuri, S., Troost, J.: Counting strings, wound and bound. JHEP, 04, 096 (2013). arXiv:1302.1045
Giveon A., Harvey J., Kutasov D., Lee S.: Three-charge black holes and quarter BPS states in little string theory. JHEP. 12, 145 (2015) arXiv:1508.0443
Eichler M., Zagier D.: The Theory of Jacobi Forms. Birkhäuser, Basel (1985)
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Gupta, R.K., Murthy, S. Squashed Toric Sigma Models and Mock Modular Forms. Commun. Math. Phys. 360, 405–437 (2018). https://doi.org/10.1007/s00220-017-3069-5
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DOI: https://doi.org/10.1007/s00220-017-3069-5