Holomorphic Blocks in Three Dimensions

TL;DR

The paper introduces holomorphic blocks as universal, factorizing components for 3d N=2 theories, mapping sphere partition functions and indices to sums of products of blocks labeled by massive vacua. It develops a QM-inspired framework to compute blocks via block integrals, line-operator identities, and gradient-flow contours, revealing Stokes phenomena, mirror symmetry, and fusion as core structural features. Through detailed examples (including the CP^1 sigma-model and knot-complement theories), it demonstrates how blocks reproduce ellipsoid and index observables and connect to analytically continued Chern-Simons theory and open topological strings. The work provides a versatile, non-perturbative toolkit for understanding 3d-3d correspondences, dualities, and knot invariants, with broad implications for non-perturbative QFT and topological string theory.

Abstract

We decompose sphere partition functions and indices of three-dimensional N=2 gauge theories into a sum of products involving a universal set of "holomorphic blocks". The blocks count BPS states and are in one-to-one correspondence with the theory's massive vacua. We also propose a new, effective technique for calculating the holomorphic blocks, inspired by a reduction to supersymmetric quantum mechanics. The blocks turn out to possess a wealth of surprising properties, such as a Stokes phenomenon that integrates nicely with actions of three-dimensional mirror symmetry. The blocks also have interesting dual interpretations. For theories arising from the compactification of the six-dimensional (2,0) theory on a three-manifold M, the blocks belong to a basis of wavefunctions in analytically continued Chern-Simons theory on M. For theories engineered on branes in Calabi-Yau geometries, the blocks offer a non-perturbative perspective on open topological string partition functions.

Figures (21)

• Figure 1: Stretching $S^2\!\times_q\!S^1$ and $S^3_b$ into a union of semi-infinite $D^2\!\times_q\!S^1$ geometries.
• Figure 2: A holomorphic block
• Figure 3: Holonomies $i\oint A$ for any connection of the form $A=A_0+\kappa\omega$ around various cycles in the $D^2\!\times_q\!S^1$ geometry.
• Figure 4: The asymptotic torus of $D^2\!\times_q\!S^1$.
• Figure 5: Matching holonomies $i\oint A$ of a background gauge field during S-fusion.