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Lectures on Vafa-Witten invariants
 
16:00-18:20, October 6 - 27, 2016
R440, Astronomy-Mathematics Building, NTU

Speaker:
Yuji Tanaka (Nagoya University)


Organizers:
River Chiang (National Cheng Kung University)
Mao-Pei Tsui (National Taiwan University)


Date

Vafa-Witten invariants and their partition functions  (I) : Oct. 6 (Th) 16:00-18:20 
Vafa-Witten invariants and their partition functions  (II) : Oct. 20 (Th) 16:00-18:20 
Vafa-Witten invariants and their partition functions  (III) : Oct. 27 (Th) 16:00-18:20 
 
Abstract
 
The Hitchin equations on compact Riemann surfaces have been great sources of many studies in mathematics, such as integrable systems, conformal field theory, etc. In this series of talks, we consider a set of gauge-theoretic equations introduced by Vafa and Witten on closed four-manifolds, which can be seen as an analogue of the Hitchin equations in four dimensions. After mentioning some motivation and backgrounds for this, we describe both analytic and algebraic aspects of the moduli space of solutions to the equations. The latter is joint work with Richard P. Thomas at Imperial College London. This series of talks mainly focuses on describing this algebraic exploration of the theory, which includes modular properties of the partition functions. On the other hand, the analytic studies that we partially discuss involve another kind of analogues of the Hitchin equations, the Kapustin-Witten equations on closed four-manifolds; and a huge breakthrough for the analysis of solutions to them by Cliff Taubes.
 
Outlines
 
1.Backgrounds, and some analytic aspects.
2.Hitchin-Kobayashi correspondence and the algebraic formulation. 
3.Obstruction theory and the invariants.
4.Modularity of the partition functions.
 
References
 
[AG] L. Alvarez-Consul and O. Garcia-Prada, "Hitchin-Kobayashi correspondence, quivers and vortices", Commun. Math. Phys. 23 (2003), 1-33.
[Be] K. Behrend, "Donaldson-Thomas invariants via microlocal geometry", Ann. of Math. 170 (2009), 1307-1338.
[BF] K. Behrend and B. Fantechi, "The intrinsic normal cone", Invent. Math. 128 (1997), 45-88.
[GP] T. Graber and R. Pandharipande, "Localization of virtual classes", Invent. Math. 135 (1999), 487-518.
[Hi] N. J. Hitchin, "The self-duality equations on a Riemann surface", Proc. London Math. Soc. 55 (1987), 59-126.
[HT] D. Huybrechts and R. P. Thomas, "Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes", Math. Ann. 346 (2010), 545-569.
[Ji] Y. Jiang, "Note on MacPherson's local Euler obstruction", preprint, arXiv:1412.3720.
[JT] Y. Jiang and R. P. Thomas, "Virtual signed Euler characteristics", preprint, arXiv:1408.2541.
[JS] D. Joyce and Y. Song, "A theory of generalized Donaldson-Thomas invariants", Memoirs of the AMS 217, no. 1020 (2012).
[KL] Y.-H. Kiem and J. Li, "Localizing virtual cycles by cosections", Jour. A.M.S. 26 (2013), 1025-1050.
[KW] A. Kapustin and E. Witten, "Electric-magnetic duality and the geometric Langlands program", Commun. Number Theory Phys. 1 (2007), 1-236.
[Tan1] Y. Tanaka, "Some boundedness properties of the Vafa-Witten equations on closed four-manifolds", preprint, arXiv:1308.0862v5.
[Tan2] Y. Tanaka, "Stable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces", Manuscripta Math. 146 (2015), 351-363.
[Tan3] Y. Tanaka, "A perturbation and generic smoothness of the Vafa-Witten moduli spaces on closed symplectic four-manifolds, preprint, arXiv:1410.1691,
[Tan4] Y. Tanaka, "On the singular sets of solutions to the
Kapustin-Witten equations on compact Kahler surfaces", preprint, arXiv:1510.07739.
[Tau1] C. H. Taubes, "PSL(2;C)-connections on 3-manifolds with L^2 bounds on curvature", Camb. J. Math. 1 (2013), 239-397.
[Tau2] C. H. Taubes, "Compactness theorems for SL(2;C) generalizations of the 4-dimensional anti-self-dual equations", arXiv:1307.6447v4.
[Tau3] C. H. Taubes, "The zero loci of Z=2 harmonic spinors in dimension 2, 3 and 4", arXiv:1407.6206.
[Th] R. P. Thomas, "A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations", Jour. Diff. Geom. 54  (2000), 367-438.
[VW] C. Vafa and E. Witten, "A strong coupling test of S-duality", Nucl. Phys. B, 432, (1994), 484-550.
 





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