S-duality and supersymmetry on curved manifolds
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Springer
Received: August 11, 2020 Accepted: August 24, 2020 Published: September 21, 2020
Guido Festuccia and Maxim Zabzine Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden
E-mail: [email protected], [email protected] Abstract: We perform a systematic study of S-duality for N = 2 supersymmetric nonlinear abelian theories on a curved manifold. Localization can be used to compute certain supersymmetric observables in these theories. We point out that localization and S-duality acting as a Legendre transform are not compatible. For these theories S-duality should be interpreted as Fourier transform and we provide some evidence for this. We also suggest the notion of a coholomological prepotential for an abelian theory that gives the same partition function as a given non-abelian supersymmetric theory. Keywords: Supersymmetric Gauge Theory, Differential and Algebraic Geometry, Duality in Gauge Field Theories, Extended Supersymmetry ArXiv ePrint: 2007.12001
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)128
JHEP09(2020)128
S-duality and supersymmetry on curved manifolds
Contents 1 Introduction
1
2 N = 2 theory on curved manifolds 2.1 N = 2 supersymmetry 2.2 Cohomological description 2.3 Ward identities and localization
3 3 4 5 8 9 10
4 Non-linear N = 2 theory 4.1 S-duality in the non-linear theory 4.2 Gravitational corrections
13 13 15
5 S-duality in cohomological variables 5.1 Naive derivation 5.2 S-duality as Fourier transform 5.3 Examples 5.3.1 S 4 5.3.2 CP2
15 16 18 22 22 23
6 Effective N = 2 abelian theory
25
7 Summary
28
A Notations for spinors
30
B N = 2 rigid supergravity
30
C N = 2 chiral and vector multiplets C.1 Chiral multiplet C.2 Anti-chiral multiplet C.3 Vector multiplet
31 31 33 34
D Cohomohological description of chiral multiplet
35
E Legendre transform
36
F Fourier transform
38
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JHEP09(2020)128
3 S-duality for abelian N = 2 theory 3.1 N = 2 supersymmetric theory 3.2 S-duality in cohomological variables
1
Introduction
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JHEP09(2020)128
Equivariant localization of quantum field theories on compact manifolds has gained considerable attention since [1] (for a review of the field see [2]). The localization technique was widely applied to 2D-7D supersymmetric theories on different manifolds that additionally admit some torus action. In this respect two issues need to be addressed: the first problem is to construct supersymmetric field theories on various spaces and to determine which geometrical properties are necessary for supersymmetry. This problem is mainly within classical field theory. The second issue is the implementation of localization for a given supersymmetric problem and involves determining the localization locus and calculating certain superdeterminants. This appears to be hard in four and higher dimensions where the path integral is often dominated by highly singular configurations that are hard to control on compact manifolds. For exam
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