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Universal Functions for Topological Correlators

Elias Furrer, Jan Manschot

Abstract

We consider correlation functions of topologically twisted, $\mathcal{N}=2$ supersymmetric Yang-Mills theory with gauge group ${\rm SU}(2)$ and $N_f\leq 3$ massive hypermultiplets in the fundamental representation. For a smooth, compact, oriented four-manifold $X$ with $b_2^+>1$, the correlation functions are expressed in terms of a finite set of universal functions. The mass dependence of these functions encodes intersection numbers of the moduli space of instantons. We determine closed expressions for the universal functions by combining techniques of the Seiberg-Witten geometry, $u$-plane integral and the blowup formula. If $X$ is specialised to a complex algebraic surface $S$, the correlation functions can be identified with generating functions of Segre invariants for moduli spaces of sheaves on $S$. We verify that our results agree with the results by Göttsche and Kool for these generating functions.

Universal Functions for Topological Correlators

Abstract

We consider correlation functions of topologically twisted, supersymmetric Yang-Mills theory with gauge group and massive hypermultiplets in the fundamental representation. For a smooth, compact, oriented four-manifold with , the correlation functions are expressed in terms of a finite set of universal functions. The mass dependence of these functions encodes intersection numbers of the moduli space of instantons. We determine closed expressions for the universal functions by combining techniques of the Seiberg-Witten geometry, -plane integral and the blowup formula. If is specialised to a complex algebraic surface , the correlation functions can be identified with generating functions of Segre invariants for moduli spaces of sheaves on . We verify that our results agree with the results by Göttsche and Kool for these generating functions.
Paper Structure (24 sections, 2 theorems, 217 equations, 1 figure, 1 table)

This paper contains 24 sections, 2 theorems, 217 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Topological SQCD
  3. Brief review of Coulomb branch geometry
  4. Topological twist
  5. The effective theory
  6. Blowup formula
  7. Universal functions from topological partition functions
  8. Segre numbers and universal functions
  9. Comparison for all Nf
  10. Contribution from monopole singularity
  11. Contribution from instanton branch
  12. Universal functions from SW geometry
  13. Nf=1
  14. Nf=2
  15. Nf=3
  16. ...and 9 more sections

Key Result

Proposition 1

Let $X$ be a smooth, oriented, compact and simply-connected four-manifold of Seiberg--Witten simple type, with $b_2^+(X)>1$. Let $Z_{{\rm SW},-}$SW all Nf be the contribution of the monopole singularity $\mathtt u_{N_f}^-$ to the topological partition function of $\mathcal{N}=2$$\rm{SU}(2)$ SQCD wit The functions agree in the decoupling limit $m^{s}\Lambda_{s}^{4-s}=\Lambda_0^4$ as series in with

Figures (1)

  • Figure 1: The large mass singular structure on the Coulomb branch of ${\rm SU}(2)$ SQCD with $N_f=1,2,3$ hypermultiplets is as follows. For $m\gg \Lambda_0$ and positive, the two singularities $\mathtt u_{N_f}^\pm$ approach the two $N_f=0$ singularities $\pm\Lambda_0^2$ from below, while the third singularity $\mathtt u_{N_f}^*$ moves to $+\infty$. In the limit $m\to\infty$, this recovers the pure $N_f=0$$u$-plane.

Theorems & Definitions (3)

  • Conjecture 1: Göttsche--Kool
  • Proposition 1
  • Theorem 1