Trisecting non-Lagrangian theories

TL;DR

This work develops a framework to define and compute smooth 4-manifold invariants using topological twists of non-Lagrangian 4d $\mathcal N=2$ and $\mathcal N=3$ theories, reducing the computation to disk amplitudes in a 2d A-model with targets that are highly nontrivial (e.g., Gepner-model–like or asymmetric orbifold spaces). Central to the construction is a trisection of the 4-manifold into three handlebodies, yielding a 2d theory $\mathcal M(F_g)$ on a disk together with three Heegaard boundary conditions whose gluing is governed by $\mathrm{MCG}(F_g)$. The paper explicates the map between 4d chiral rings and 2d chiral rings, observables inherited from the $Q$-cohomology, and the role of anomalies in organizing the invariants, including Seiberg-Witten–like structures manifested as sums over vacua and vortex contributions. It then develops concrete AD-theory examples, particularly rank-1 Argyres-Douglas theories, showing how their elliptic genera and boundary branes encode the 4-manifold data for higher-genus trisections and discuss potential connections to non-compact Gepner models and VOAs, with a view toward broadening the catalog of 4-manifold invariants derived from non-Lagrangian QFTs.

Abstract

We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d N=2 and N=3 theories in which the problem is reduced to a fairly standard computation in topological A-model, albeit with rather unusual targets, such as compact and non-compact Gepner models, asymmetric orbifolds, N=(2,2) linear dilaton theories, "self-mirror" geometries, varieties with complex multiplication, etc.

Figures (8)

• Figure 1: Cutting a 4-manifold is like cutting a cake: with only three skillful cuts, any 4-manifold can be trisected into three basic pieces.
• Figure 2: Disk amplitude in the A-model of ${\mathcal{M}} (F_g)$, with three Heegaard boundary conditions, dual to the trisection in Figure \ref{['fig:trisectMMM']}.
• Figure 3: The standard genus-$g$ Heegaard diagram for $\#^k (S^1 \times S^2)$, with $g=3$ and $k=1$.
• Figure 4: The toric diagram for $M_4 = {\mathbb{C}}{\mathbf{P}}^2$ is a simple example of a genus-1 trisection. Its trisection diagram comprises a torus $F_g = T^2$ with three curves $\alpha$, $\beta$, and $\gamma$ in homology classes $(1,0)$, $(0,1)$, and $(1,1)$, respectively.
• Figure 5: Example of a disk instanton in 2d sigma-model associated with a genus-1 trisection of $M_4 = {\mathbb{C}}{\mathbf{P}}^2$ in Figure \ref{['fig:toricCP2']}. Shown here is a covering space of ${\mathcal{M}} (F_g)$ on which each of the Heegaard branes ${\mathcal{B}}_{\alpha}$, ${\mathcal{B}}_{\beta}$, ${\mathcal{B}}_{\gamma}$ lifts to an infinite set of parallel straight lines.