Holomorphic Blocks in Three Dimensions

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.

• The paper demonstrates the factorization of partition functions in 3D N=2 gauge theories into holomorphic blocks, offering a novel non-perturbative computational tool.
• It analyzes the analytical continuation and Stokes phenomena of these blocks, emphasizing the role of symmetries and dualities in supersymmetric theories.
• The study connects holomorphic blocks to Chern-Simons path integrals, opening new avenues for applications in quantum topology and knot invariants.

Holomorphic Blocks in Three Dimensions: An Analytical Perspective

The paper "Holomorphic Blocks in Three Dimensions" explores the intriguing world of three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories, exploring their partition functions through the lens of holomorphic blocks. Authored by Christopher Beem, Tudor Dimofte, and Sara Pasquetti, this paper builds on the resurgence of interest in three-dimensional field theories, particularly in understanding their behavior on curved manifolds and their relation to higher-dimensional theories.

Overview of Holomorphic Blocks

In the context of three-dimensional $\mathcal{N}=2$ gauge theories, holomorphic blocks are defined as fundamental objects that encapsulate the theory's behavior on effectively lower-dimensional geometries. These blocks provide a non-perturbative handle on three-dimensional theories compactified on geometries like the cigar $D^2$, allowing for a refined understanding of their partition functions across more complex topological structures, such as $S^3_b$ (ellipsoids) and $S^2 \times S^1$.

A key insight presented by the authors is the interpretation of these blocks as BPS indices, capturing contributions from states preserved under specific combinations of supercharges. The approach to define these blocks leverages a twisted setup on $D^2$, with mappings to geometric Langlands in four dimensions and even connections to the (2,0) theory in six dimensions.

Main Contributions

1. Factorization of Partition Functions: The authors demonstrate that partition functions on $S^3_b$ and $S^2 \times S^1$ can be factorized into products of these holomorphic blocks. This factorization establishes a profound link between seemingly disparate geometrical configurations and provides a method to compute more intricate path integrals via simpler, lower-dimensional tools.
2. Analytical Continuation and Stokes Phenomena: A novel feature of holomorphic blocks is their behavior across different domains of their parameters. The paper carefully explores the Stokes phenomena associated with transitions across these domains, illuminating the underpinnings of the analytical structure of these blocks. Particularly, it is shown how Stokes matrices, which dictate these transitions, reflect the underlying symmetries and dualities of the parent three-dimensional theory.
3. Connection to Chern-Simons Theory: A significant portion of the discussion is dedicated to relating holomorphic blocks to non-perturbative path integrals in analytically continued Chern-Simons theory. The authors propose that these blocks are, in essence, Chern-Simons partition functions on three-manifolds, which are expanded on certain integration cycles. This proposal aligns with the analytic continuation of Chern-Simons path integrals via Lefschetz thimbles, as formulated in recent works.
4. Example Analyses: The paper meticulously examines several examples to validate the concepts presented, including the $SU(2)$ gauge theory and the $CP^1$ sigma model. These examples serve to elucidate complex phenomena like mirror symmetry and demonstrate the robustness of the block formalism in characterizing the partition functions of more constrained geometries.

Implications and Future Directions

The implications of the authors’ framework are multifold. Holomorphic blocks serve as computational tools that potentially unlock new pathways in the study of higher-dimensional theories by reducing the computational complexity associated with the evaluation of path integrals. Furthermore, their connection to Chern-Simons theory adds an exciting dimension to the study of knot invariants and quantum topology, presenting a bridge between supersymmetric gauge theories and mathematical structures in knot theory.

The paper speculates on future directions, including the possibility of refining the block formalism to incorporate refined BPS indices or Poincaré polynomials, potentially leading to new homological knot invariants. Additionally, understanding the full implications of block factorization for understanding the string-theoretic origins of these field theories could provide deeper insights into the web of dualities connecting various string and field theories.

Overall, "Holomorphic Blocks in Three Dimensions" offers a rich tapestry of theoretical insights, unifying facets of supersymmetry, quantum field theory, and mathematical physics under a coherent framework that highlights the elegance and utility of holomorphic blocks in studying topological and geometrical aspects of three-dimensional gauge theories.