Abstract
We define and compute the dressed elliptic genus of \( \mathcal{N}=2 \) heterotic compactifications with torsion that are principal two-torus bundles over a K3 surface. We consider a large class of gauge bundles compatible with supersymmetry, consisting of a stable holomorphic vector bundle over the base together with an Abelian bundle over the total space, generalizing the computation previously done by the authors in the absence of the latter. Starting from a (0,2) gauged linear sigma-model with torsion we use supersymmetric localization to obtain the result. We provide also a mathematical definition of the dressed elliptic genus as a modified Euler characteristic and prove that both expressions agree for hypersurfaces in weighted projective spaces. Finally we show that it admits a natural decomposition in terms of \( \mathcal{N}=4 \) superconformal characters, that may be useful to investigate moonshine phenomena for this wide class of \( \mathcal{N}=2 \) vacua, that includes K3 × T 2 compactifications as special cases.
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References
C.M. Hull, Compactifications of the Heterotic Superstring, Phys. Lett. B 178 (1986) 357 [INSPIRE].
A. Strominger, Superstrings with Torsion, Nucl. Phys. B 274 (1986) 253 [INSPIRE].
K. Dasgupta, G. Rajesh and S. Sethi, M theory, orientifolds and G-flux, JHEP 08 (1999) 023 [hep-th/9908088] [INSPIRE].
E. Goldstein and S. Prokushkin, Geometric model for complex nonKähler manifolds with SU(3) structure, Commun. Math. Phys. 251 (2004) 65 [hep-th/0212307] [INSPIRE].
J.-X. Fu and S.-T. Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampere equation, J. Diff. Geom. 78 (2008) 369 [hep-th/0604063] [INSPIRE].
K. Becker and S. Sethi, Torsional Heterotic Geometries, Nucl. Phys. B 820 (2009) 1 [arXiv:0903.3769] [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
E. Silverstein and E. Witten, Criteria for conformal invariance of (0, 2) models, Nucl. Phys. B 444 (1995) 161 [hep-th/9503212] [INSPIRE].
C. Beasley and E. Witten, Residues and world sheet instantons, JHEP 10 (2003) 065 [hep-th/0304115] [INSPIRE].
M. Bertolini and M.R. Plesser, Worldsheet instantons and (0, 2) linear models, JHEP 08 (2015) 081 [arXiv:1410.4541] [INSPIRE].
A. Adams, M. Ernebjerg and J.M. Lapan, Linear models for flux vacua, Adv. Theor. Math. Phys. 12 (2008) 817 [hep-th/0611084] [INSPIRE].
A. Adams and D. Guarrera, Heterotic Flux Vacua from Hybrid Linear Models, arXiv:0902.4440 [INSPIRE].
A. Adams and J.M. Lapan, Computing the Spectrum of a Heterotic Flux Vacuum, JHEP 03 (2011) 045 [arXiv:0908.4294] [INSPIRE].
M. Blaszczyk, S. Groot Nibbelink and F. Ruehle, Green-Schwarz Mechanism in Heterotic (2, 0) Gauged Linear σ-models: Torsion and NS5 Branes, JHEP 08 (2011) 083 [arXiv:1107.0320] [INSPIRE].
C. Quigley and S. Sethi, Linear σ-models with Torsion, JHEP 11 (2011) 034 [arXiv:1107.0714] [INSPIRE].
C. Quigley, S. Sethi and M. Stern, Novel Branches of (0, 2) Theories, JHEP 09 (2012) 064 [arXiv:1206.3228] [INSPIRE].
A. Adams, E. Dyer and J. Lee, GLSMs for non-Kähler Geometries, JHEP 01 (2013) 044 [arXiv:1206.5815] [INSPIRE].
D. Israël, T-duality in Gauged Linear σ-models with Torsion, JHEP 11 (2013) 093 [arXiv:1306.6609] [INSPIRE].
D. Israel and M. Sarkis, New supersymmetric index of heterotic compactifications with torsion, JHEP 12 (2015) 069 [arXiv:1509.05704] [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 \( \mathcal{N}=2 \) Gauge Theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
K. Becker, M. Becker, J.-X. Fu, L.-S. Tseng and S.-T. Yau, Anomaly cancellation and smooth non-Kähler solutions in heterotic string theory, Nucl. Phys. B 751 (2006) 108 [hep-th/0604137] [INSPIRE].
T. Eguchi, H. Ooguri and Y. Tachikawa, Notes on the K3 Surface and the Mathieu group M 24, Exper. Math. 20 (2011) 91 [arXiv:1004.0956] [INSPIRE].
E. Kiritsis, C. Kounnas, P.M. Petropoulos and J. Rizos, Universality properties of N = 2 and N = 1 heterotic threshold corrections, Nucl. Phys. B 483 (1997) 141 [hep-th/9608034] [INSPIRE].
M.C.N. Cheng, X. Dong, J. Duncan, J. Harvey, S. Kachru and T. Wrase, Mathieu Moonshine and N = 2 String Compactifications, JHEP 09 (2013) 030 [arXiv:1306.4981] [INSPIRE].
S.M. Harrison, D. Israel, N.M. Paquette and M. Sarkis, Mathieu moonshine and flux compactifications, work in progress.
C. Angelantonj, D. Israel and M. Sarkis, One-loop corrections to heterotic compactifications with torsion, to appear.
J. Distler, Notes on (0, 2) superconformal field theories, hep-th/9502012 [INSPIRE].
S. Elitzur, E. Gross, E. Rabinovici and N. Seiberg, Aspects of Bosonization in String Theory, Nucl. Phys. B 283 (1987) 413 [INSPIRE].
I.V. Melnikov, R. Minasian and S. Theisen, Heterotic flux backgrounds and their IIA duals, JHEP 07 (2014) 023 [arXiv:1206.1417] [INSPIRE].
E. Witten, Elliptic Genera and Quantum Field Theory, Commun. Math. Phys. 109 (1987) 525.
O. Alvarez, T.P. Killingback, M.L. Mangano and P. Windey, The Dirac-Ramond operator in string theory and loop space index theorems, Nucl. Phys. Proc. Suppl. B 1 (1987) 189 [INSPIRE].
O. Alvarez, T.P. Killingback, M.L. Mangano and P. Windey, String Theory and Loop Space Index Theorems, Commun. Math. Phys. 111 (1987) 1 [INSPIRE].
T. Kawai, Y. Yamada and S.-K. Yang, Elliptic genera and N = 2 superconformal field theory, Nucl. Phys. B 414 (1994) 191 [hep-th/9306096] [INSPIRE].
T. Kawai and K. Mohri, Geometry of (0, 2) Landau-Ginzburg orbifolds, Nucl. Phys. B 425 (1994) 191 [hep-th/9402148] [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].
J.A. Harvey and G.W. Moore, Algebras, BPS states and strings, Nucl. Phys. B 463 (1996) 315 [hep-th/9510182] [INSPIRE].
N.P. Skoruppa, Developments in the theory of Jacobi forms, In N. Kuznetsov and V. Bykovsky eds., International Conference on Automorphic Functions and their Applications, Khabarovsk, 27 June – 4 July 1988, pp. 167–185, The USSR Academy of Science, Khabarovsk’ (1990).
V. Gritsenko, Complex vector bundles and Jacobi forms, math/9906191 [INSPIRE].
A. Dabholkar, S. Murthy and D. Zagier, Quantum Black Holes, Wall Crossing and Mock Modular Forms, arXiv:1208.4074 [INSPIRE].
M.R. Gaberdiel, D. Persson, H. Ronellenfitsch and R. Volpato, Generalized Mathieu Moonshine, Commun. Num. Theor Phys. 07 (2013) 145 [arXiv:1211.7074] [INSPIRE].
T. Gannon, Much ado about Mathieu, Adv. Math. 301 (2016) 322 [arXiv:1211.5531] [INSPIRE].
C. Angelantonj, I. Florakis and B. Pioline, A new look at one-loop integrals in string theory, Commun. Num. Theor. Phys. 6 (2012) 159 [arXiv:1110.5318] [INSPIRE].
C. Angelantonj, I. Florakis and B. Pioline, One-Loop BPS amplitudes as BPS-state sums, JHEP 06 (2012) 070 [arXiv:1203.0566] [INSPIRE].
S. Hosono, B.H. Lian, K. Oguiso and S.-T. Yau, Classification of c = 2 rational conformal field theories via the Gauss product, Commun. Math. Phys. 241 (2003) 245 [hep-th/0211230] [INSPIRE].
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Israël, D., Sarkis, M. Dressed elliptic genus of heterotic compactifications with torsion and general bundles. J. High Energ. Phys. 2016, 176 (2016). https://doi.org/10.1007/JHEP08(2016)176
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DOI: https://doi.org/10.1007/JHEP08(2016)176