Topological strings on singular elliptic Calabi-Yau 3-folds and minimal 6d SCFTs
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Springer
Received: February 13, 2018 Accepted: March 12, 2018 Published: March 26, 2018
Topological strings on singular elliptic Calabi-Yau 3-folds and minimal 6d SCFTs
a
Simons Center for Geometry and Physics, SUNY, Stony Brook, NY, 11794-3636 U.S.A. b LPTENS, CNRS, PSL Research University, Sorbonne Universit´es, UPMC, 75005 Paris, France c ICTS, University of Science and Technology of China, Hefei, Anhui 230026, China d Bethe Center for Theoretical Physics, Physikalisches Institut, Universit¨ at Bonn, 53115 Bonn, Germany e Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands
E-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: We apply the modular approach to computing the topological string partition function on non-compact elliptically fibered Calabi-Yau 3-folds with higher Kodaira singularities in the fiber. The approach consists in making an ansatz for the partition function at given base degree, exact in all fiber classes to arbitrary order and to all genus, in terms of a rational function of weak Jacobi forms. Our results yield, at given base degree, the elliptic genus of the corresponding non-critical 6d string, and thus the associated BPS invariants of the 6d theory. The required elliptic indices are determined from the chiral anomaly 4-form of the 2d worldsheet theories, or the 8-form of the corresponding 6d theories, and completely fix the holomorphic anomaly equation constraining the partition function. We introduce subrings of the known rings of Weyl invariant Jacobi forms which are adapted to the additional symmetries of the partition function, making its computation feasible to low base wrapping number. In contradistinction to the case of simpler singularities, generic vanishing conditions on BPS numbers are no longer sufficient to fix the modular ansatz at arbitrary base wrapping degree. We show that to low degree, imposing exact vanishing conditions does suffice, and conjecture this to be the case generally. Keywords: F-Theory, Topological Strings, Anomalies in Field and String Theories, Global Symmetries ArXiv ePrint: 1712.07017
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP03(2018)156
JHEP03(2018)156
Michele Del Zotto,a Jie Gu,b Min-xin Huang,c Amir-Kian Kashani-Poor,b Albrecht Klemmd and Guglielmo Lockharte
Contents 1 Introduction
2 5 5 6 9 12 13
3 The 3.1 3.2 3.3
15 15 18 23
toric geometry of elliptic Calabi-Yau threefolds with base Fn Elliptic fibrations over Hirzebruch surfaces The A2 geometry: a resolution of the elliptic fibration over F3 The D4 geometry: a resolution of the elliptic fibration over F4
4 Expressing the partition function in terms of D(b g) invariant Jacobi forms 4.1 Ansatz for the elliptic genus 4.2 Properties of the denominator of the ansatz 4.3 D(b g) invariant Jacobi forms
24 24 27 30
5 Computing the partition function 34 5.1 The Q expansion 34 5.2 Non-sufficiency of generic vanishing
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