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4-manifolds and topological modular forms

Sergei Gukov, Du Pei, Pavel Putrov, Cumrun Vafa

TL;DR

The work establishes a physical bridge from 6d (1,0) superconformal theories to invariants of smooth 4-manifolds via twisted compactifications on M4, producing 2d (0,1) theories whose topological Witten genus lands in (equivariant) TMF. It introduces an equivariant TMF framework, clarifying how background flavor fields regularize partition functions and yield modular information controlled by defect groups and modularity levels. The paper provides explicit maps T and σ from 4-manifolds to TMF data, analyzes anomalies across dimensions, and presents concrete examples (M5-branes on C2/Zk and E-strings) where these invariants can be computed. It also develops BPS-equation analysis and vanishing theorems for 5d/4d theories, connecting higher-dimensional physics to 4-manifold invariants and suggesting links to Bauer–Furuta-type torsion invariants. Overall, the work proposes a rich, higher-dimensional origin for TMF-valued invariants of 4- and 3-manifolds, with a robust framework for equivariant refinements and potential future connections to categorified invariants.

Abstract

We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1,0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0,1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a few simple examples. As a byproduct, we obtain some new results on 't Hooft anomalies of 6d (1,0) theories and a better understanding of the relation between 2d (0,1) theories and TMF spectra.

4-manifolds and topological modular forms

TL;DR

The work establishes a physical bridge from 6d (1,0) superconformal theories to invariants of smooth 4-manifolds via twisted compactifications on M4, producing 2d (0,1) theories whose topological Witten genus lands in (equivariant) TMF. It introduces an equivariant TMF framework, clarifying how background flavor fields regularize partition functions and yield modular information controlled by defect groups and modularity levels. The paper provides explicit maps T and σ from 4-manifolds to TMF data, analyzes anomalies across dimensions, and presents concrete examples (M5-branes on C2/Zk and E-strings) where these invariants can be computed. It also develops BPS-equation analysis and vanishing theorems for 5d/4d theories, connecting higher-dimensional physics to 4-manifold invariants and suggesting links to Bauer–Furuta-type torsion invariants. Overall, the work proposes a rich, higher-dimensional origin for TMF-valued invariants of 4- and 3-manifolds, with a robust framework for equivariant refinements and potential future connections to categorified invariants.

Abstract

We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1,0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0,1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a few simple examples. As a byproduct, we obtain some new results on 't Hooft anomalies of 6d (1,0) theories and a better understanding of the relation between 2d (0,1) theories and TMF spectra.

Paper Structure

This paper contains 48 sections, 237 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. From six to two dimensions, via 4-manifolds
  3. Topological twist
  4. Flux compactifications of 6d $(1,0)$ theories
  5. 4-manifolds with special holonomy
  6. Anomaly polynomial reduction
  7. Central charges and Kähler 4-manifolds
  8. Turning on flavor symmetry background
  9. A toy model
  10. $T[M_4]$ for 6d $(1,0)$ hyper, vector and tensor multiplets
  11. A cure for non-compactness: equivariant partition functions
  12. Defect group, relativeness and modularity level
  13. Relativeness and quantization of coefficients of the anomaly polynomials
  14. Relation between anomalies in various dimensions
  15. 3-manifolds
  16. ...and 33 more sections

Figures (1)

  • Figure 1: The figure displays two regions in the 3d projective space of 6d anomaly polynomials in the affine chart where $\alpha\neq 1$, parametrized by $\beta/\alpha,\gamma/\alpha,\delta/\alpha$. The red one is the region carved out by the inequalities (\ref{['6d-bounds']}). The blue one is the region given by condition that $c_\text{L},c_\text{R}>0$ for almost all minimal surfaces of general type (meaning that we take $\chi$ and $\sigma$ to be large that the inequalities (\ref{['min-surface-bounds']}) can be replaced by homogeneous ones).