Fivebranes and 4-manifolds

TL;DR

The paper establishes a duality between 4-manifolds and 2d ${\mathcal N}=(0,2)$ theories $T[M_4]$, organizing a dictionary that treats Kirby diagrams, plumbing, and cobordisms as building blocks for boundary theories and their domain walls. By pairing a top-down, M-theory–driven view (Vafa–Witten theory and affine algebras) with a bottom-up, boundary-condition construction in 3d ${\mathcal N}=2$ theories, it relates VW partition functions to flavored elliptic genera and interprets gluing as sums over branching functions and cosets. The framework yields concrete predictions for VW invariants on 4-manifolds, provides physical realizations of Kirby moves as dualities, and suggests practical algorithms to construct wider classes of theories $T[M_4]$ by composing simple cobordisms. Altogether, the approach links smooth 4-manifold topology with 2d/3d supersymmetric field theories, with potential implications for smooth structure classification and 4d invariants.

Abstract

We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2) theories, we obtain a number of results, which include new 3d N=2 theories T[M_3] associated with rational homology spheres and new results for Vafa-Witten partition functions on 4-manifolds. In particular, we point out that the gluing measure for the latter is precisely the superconformal index of 2d (0,2) vector multiplet and relate the basic building blocks with coset branching functions. We also offer a new look at the fusion of defect lines / walls, and a physical interpretation of the 4d and 3d Kirby calculus as dualities of 2d N=(0,2) theories and 3d N=2 theories, respectively

Figures (19)

• Figure 1: A Kirby diagram and the corresponding plumbing graph for the plumbing 4-manifold associated to the string $(a_1, a_2, \ldots, a_n)$.
• Figure 2: A Kirby diagram and the corresponding plumbing graph for the $E_8$ manifold with $b_2 = - \sigma = 8$ and $\partial E_8 \approx \Sigma (2,3,5)$.
• Figure 3: Kirby diagram of a 4-manifold bounded by a 3-torus $T^3$.
• Figure 4: For a plumbing tree, the eigenvalues (and, therefore, the determinant) of the intersection form $Q$ can be computed by orienting the edges toward a single vertex and then successively eliminating them using the two rules shown here.
• Figure 5: Plumbing tree of a 4-manifold bounded by a Seifert fibration. We assume $b \le -1$ and $a_{ij} \le -2$.