[Paper Review] Topological correlators of $SU(2)$, $\mathcal{N}=2^*$ SYM on four-manifolds

Jan Manschot, Gregory W. Moore|arXiv (Cornell University)|Apr 13, 2021

Black Holes and Theoretical Physics137 references4 citations

TL;DR

This paper derives explicit formulae for topological correlators in $\mathcal{N}=2^*$, $SU(2)$ supersymmetric Yang-Mills theory on smooth, compact four-manifolds with $b_2^+ \geq 1$, showing they transform correctly under $S$-duality and are mock modular forms when $b_2^+ = 1$. It establishes a consistent formulation via a $\mathrm{Spin}^c$ structure, recovers known results in the $m \to 0$ (Vafa-Witten) and $m \to \infty$ (Donaldson-Witten) limits, and identifies the holomorphic part of the partition function as a generating function for Euler numbers of the matter bundle over the instanton moduli space.

ABSTRACT

We consider topologically twisted $\mathcal{N}=2$, $SU(2)$ gauge theory with a massive adjoint hypermultiplet on a smooth, compact four-manifold $X$. A consistent formulation requires coupling the theory to a ${ m Spin}^c$ structure, which is necessarily non-trivial if $X$ is non-spin. We derive explicit formulae for the topological correlation functions when $b_2^+\geq 1$. We demonstrate that, when the ${ m Spin}^c$ structure is canonically determined by an almost complex structure and the mass is taken to zero, the path integral reproduces known results for the path integral of the $\mathcal{N}=4$ gauge theory with Vafa-Witten twist. On the other hand, we reproduce results from Donaldson-Witten theory after taking a suitable infinite mass limit. The topological correlators are functions of the UV coupling constant $τ_{ m uv}$ and we confirm that they obey the expected $S$-duality transformation laws. The holomorphic part of the partition function is a generating function for the Euler numbers of the matter (or obstruction) bundle over the instanton moduli space. For $b_2^+=1$, we derive a non-holomorphic contribution to the path integral, such that the partition function and correlation functions are mock modular forms rather than modular forms. We comment on the generalization of this work to the large class of $\mathcal{N}=2$ theories of class $S$.

Motivation & Objective

• To derive explicit formulae for topological correlators in $\mathcal{N}=2^*$, $SU(2)$ SYM on four-manifolds with $b_2^+ \geq 1$.
• To establish a consistent formulation of the topologically twisted theory by coupling to a $\mathrm{Spin}^c$ structure, which is essential for non-spin manifolds.
• To demonstrate that the partition function and correlators transform correctly under $S$-duality, confirming modular properties.
• To show that in the $m \to 0$ limit, the path integral reproduces the Vafa-Witten result for $\mathcal{N}=4$ theory.
• To recover Donaldson-Witten invariants in the $m \to \infty$ limit, linking $\mathcal{N}=2^*$ theory to Donaldson theory.

Proposed method

• The path integral is formulated using a $\mathrm{Spin}^c$ structure with characteristic class $c_{\rm uv}$, which is required for consistent coupling in the presence of the massive adjoint hypermultiplet.
• Topological correlators are computed as formal power series in homology classes, with the generating function expressed via Coulomb branch integrals involving the modified prepotential $\tilde{\mathcal{F}}$.
• For $b_2^+ = 1$, a non-holomorphic correction is derived to ensure the partition function transforms as a mock modular form rather than a modular form.
• The holomorphic part of the partition function is identified as a generating function for the Euler numbers of the matter bundle over the instanton moduli space.
• The $S$-duality transformation laws are verified by analyzing the modular properties of the partition function under $\tau_{\rm uv} \to -1/\tau_{\rm uv}$.
• The $m \to 0$ and $m \to \infty$ limits are taken using asymptotic analysis of the Coulomb branch integral, recovering known results from $\mathcal{N}=4$ and Donaldson-Witten theories.

Experimental results

Research questions

• RQ1 How do topological correlators in $\mathcal{N}=2^*$, $SU(2)$ SYM on four-manifolds depend on the UV coupling $\tau_{\rm uv}$ and mass parameter $m$?
• RQ2 What is the role of the $\mathrm{Spin}^c$ structure in formulating the topological theory, especially on non-spin manifolds?
• RQ3 How do the partition function and correlators behave under $S$-duality, and do they transform as modular or mock modular forms?
• RQ4 What happens in the $m \to 0$ and $m \to \infty$ limits, and do they reproduce known results from $\mathcal{N}=4$ and Donaldson-Witten theories?
• RQ5 What is the precise relation between the holomorphic part of the partition function and the Euler numbers of the matter bundle over the instanton moduli space?

Key findings

• The topological correlators are shown to be invariant under metric deformations up to a total derivative, confirming their status as smooth manifold invariants.
• The partition function is a mock modular form when $b_2^+ = 1$, requiring a non-holomorphic correction to achieve modular properties.
• The holomorphic part of the partition function generates the Euler numbers of the matter bundle over the instanton moduli space.
• In the $m \to 0$ limit, the path integral reproduces the known result for the Vafa-Witten twisted $\mathcal{N}=4$ theory.
• In the $m \to \infty$ limit, the theory reduces to Donaldson-Witten theory, recovering the Donaldson invariants via the Seiberg-Witten map.
• The $S$-duality transformation laws are confirmed: the partition function transforms correctly under $\tau_{\rm uv} \to -1/\tau_{\rm uv}$, with the coupling $\tau_{\rm uv}$ transforming as a modular parameter.