The string geometry behind topological amplitudes
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Springer
Received: October 16, 2019 Accepted: November 29, 2019 Published: January 2, 2020
Carlo Angelantonja,b and Ignatios Antoniadisc,d a
Dipartimento di Fisica, Universit` a di Torino, and INFN Sezione di Torino, Via Pietro Giuria 1, Torino 10125, Italy b Arnold-Regge Center, Via Pietro Giuria 1, Torino 10125, Italy c ´ Laboratoire de Physique Th´eorique et Hautes Energies — LPTHE, Sorbonne Universit´e, CNRS, 4 Place Jussieu, Paris 75005, France d Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, Bern CH-3012, Switzerland
E-mail: [email protected], [email protected] Abstract: It is shown that the generating function of N = 2 topological strings, in the heterotic weak coupling limit, is identified with the partition function of a six-dimensional Melvin background. This background, which corresponds to an exact CFT, realises in string theory the six-dimensional Ω-background of Nekrasov, in the case of opposite deformation parameters ǫ1 = −ǫ2 , thus providing the known perturbative part of the Nekrasov partition function in the field theory limit. The analysis is performed on both heterotic and type I strings and for the cases of ordinary N = 2 and N = 2∗ theories. Keywords: Superstrings and Heterotic Strings, Supersymmetric Gauge Theory, Topological Strings ArXiv ePrint: 1910.03347
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)005
JHEP01(2020)005
The string geometry behind topological amplitudes
Contents 1
2 Glimpses on heterotic topological amplitudes
3
3 The geometry of the string background
5
4 Strings on Melvin space
7
5 Heterotic string on Melvin space
11
6 Heterotic string on Melvin × T 2 × K3 6.1 The N = 2∗ case
12 14
7 Open strings on Melvin space
15
8 Open strings on Melvin × T 2 × K3 8.1 The N = 2∗ case
16 18
9 Comments on Melvin and Ω backgrounds
18
A Dedekind eta function and Jacobi theta functions
20
B Some properties of the two-dimensional Narain lattice
21
1
Introduction
It is known [1] that a series of higher derivative F -terms of N = 2 supersymmetric compactifications of string theory in four dimensions, of the form Fg W 2g with W the Weyl superfield and Fg a function of the vector moduli, is computed by the genus-g partition function of a topological string [2] obtained by an appropriate twist [3, 4] of the corresponding N = 2 superconformal σ-model describing the compactification on a six-dimensional Calabi-Yau (CY) manifold. An important property of Fg ’s is the holomorphic anomaly expressed as a recursive differential equation that can be understood either from boundary contributions in the degeneration limit of Riemann surfaces within the topological theory [2], or from non-local terms in the string effective action due to the propagation of massless states [1]. As a consequence of the heterotic-type II string duality, the Fg ’s can be easily studied on the heterotic (or type I) side at the one loop level, upon identifying the
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