Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

VOA[M4]

Boris Feigin, Sergei Gukov

TL;DR

This work proposes a dictionary between smooth 4-manifolds and vertex operator algebras, where gluing of manifolds corresponds to VOA extensions and BRST reductions. By identifying VOA$[M_4]$ as the left-moving chiral algebra of the 2d theory $T[M_4]$ arising from compactification of the 6d $(0,2)$ theory, the authors connect 4-manifold invariants like Vafa–Witten partition functions to VOA characters and lattice VOAs, while modular tensor categories $MTC[M_3]$ encode boundary data for gluing along 3-manifolds. They develop gluing via both extensions (e.g., toric patches) and BRST reduction (e.g., product ruled surfaces, trisections), and they illustrate this with explicit constructions for blow-ups, orientation reversals, Kirby moves, and trialities, as well as with trisection-based building blocks. The framework yields new pathways to 4-manifold invariants and dualities, enriched by the interplay of 2d/3d/6d theories, and motivates further study of associated varieties, wall-crossing phenomena, and higher-rank generalizations. Overall, the paper advances a cohesive algebraic-topological program to extract and compare invariants across manifold decompositions via VOA and MTC data, with concrete examples and several promising conjectures for broader applicability.

Abstract

We take a peek at a general program that associates vertex (or, chiral) algebras to smooth 4-manifolds in such a way that operations on algebras mirror gluing operations on 4-manifolds and, furthermore, equivalent constructions of 4-manifolds give rise to equivalences (dualities) of the corresponding algebras.

VOA[M4]

TL;DR

This work proposes a dictionary between smooth 4-manifolds and vertex operator algebras, where gluing of manifolds corresponds to VOA extensions and BRST reductions. By identifying VOA as the left-moving chiral algebra of the 2d theory arising from compactification of the 6d theory, the authors connect 4-manifold invariants like Vafa–Witten partition functions to VOA characters and lattice VOAs, while modular tensor categories encode boundary data for gluing along 3-manifolds. They develop gluing via both extensions (e.g., toric patches) and BRST reduction (e.g., product ruled surfaces, trisections), and they illustrate this with explicit constructions for blow-ups, orientation reversals, Kirby moves, and trialities, as well as with trisection-based building blocks. The framework yields new pathways to 4-manifold invariants and dualities, enriched by the interplay of 2d/3d/6d theories, and motivates further study of associated varieties, wall-crossing phenomena, and higher-rank generalizations. Overall, the paper advances a cohesive algebraic-topological program to extract and compare invariants across manifold decompositions via VOA and MTC data, with concrete examples and several promising conjectures for broader applicability.

Abstract

We take a peek at a general program that associates vertex (or, chiral) algebras to smooth 4-manifolds in such a way that operations on algebras mirror gluing operations on 4-manifolds and, furthermore, equivalent constructions of 4-manifolds give rise to equivalences (dualities) of the corresponding algebras.

Paper Structure

This paper contains 23 sections, 105 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Transgression and QFT-valued topological invariants
  3. Useful tools
  4. Chiral correlators and 4-manifold invariants
  5. Gluing via extensions: Toric $M_4$
  6. Plumbing graphs
  7. Toric manifolds and toroidal algebras
  8. Operations and relations
  9. Generalized blow-ups
  10. Orientation reversal
  11. Chiral algebras and ${\mathcal{N}}=(0,2)$ boundary conditions
  12. Simple Kirby moves
  13. Triality
  14. Vertex algebras and trisections
  15. ${\mathcal{N}}=(0,2)$ theories: 4d lift of 2d disk amplitudes
  16. ...and 8 more sections

Figures (8)

  • Figure 1: An illustration of a Hopf link on the $S^3$ boundary of a 4-ball.
  • Figure 2: A Kirby diagram and the corresponding plumbing graph.
  • Figure 3: Toric diagram for a linear plumbing.
  • Figure 4: $(a)$ Two 4-manifolds glued along a common boundary $M_3 = \pm \partial M_4^{\pm}$ correspond to $(b)$ three-dimensional ${\mathcal{N}}=2$ theory $T[M_3]$ sandwiched by the 2d ${\mathcal{N}}=(0,2)$ boundary conditions $T[M_4^-]$ and $T[M_4^+]$.
  • Figure 5: $(a)$ A cobordism between 3-manifolds $M_3^-$ and $M_3^+$ corresponds to $(b)$ a 2d domain wall (interface) between 3d ${\mathcal{N}}=2$ theories $T[M_3^-]$ and $T[M_3^+]$.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Example 2.1