Equivariant K-theory and refined Vafa-Witten invariants
File(s)Thomas2020_Article_EquivariantK-TheoryAndRefinedV.pdf (702.05 KB)
Published version
Author(s)
Thomas, RP
Type
Journal Article
Abstract
In [MT2] the Vafa-Witten theory of complex projective surfaces is lifted to oriented $\mathbb C^*$-equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in $t^{1/2}$ invariant under $t^{1/2}\leftrightarrow t^{-1/2}$ which specialise to numerical Vafa-Witten invariants at $t=1$. On the "instanton branch" the invariants give the virtual $\chi_{-t}^{}$-genus refinement of G\"ottsche-Kool. Applying modularity to their calculations gives predictions for the contribution of the "monopole branch". We calculate some cases and find perfect agreement. We also do calculations on K3 surfaces, finding Jacobi forms refining the usual modular forms, proving a conjecture of G\"ottsche-Kool. We determine the K-theoretic virtual classes of degeneracy loci using Eagon-Northcott complexes, and show they calculate refined Vafa-Witten invariants. Using this Laarakker [Laa] proves universality results for the invariants.
Date Issued
2020-09-01
Date Acceptance
2020-05-07
ISSN
0010-3616
Publisher
Springer (part of Springer Nature)
Start Page
1451
End Page
1500
Journal / Book Title
Communications in Mathematical Physics
Volume
378
Copyright Statement
© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URI
Identifier
http://arxiv.org/abs/1810.00078v3
Subjects
Science & Technology
Physical Sciences
Physics, Mathematical
Physics
ABELIAN CATEGORIES
CONFIGURATIONS
math.AG
math.AG
hep-th
math.AG
math.AG
hep-th
0101 Pure Mathematics
0105 Mathematical Physics
0206 Quantum Physics
Mathematical Physics
Notes
Various updates and edits. 51 pages
Publication Status
Published
Date Publish Online
2020-07-20