3-Manifolds and 3d Indices (1112.5179v1)

Abstract: We identify a large class R of three-dimensional N=2 superconformal field theories. This class includes the effective theories T_M of M5-branes wrapped on 3-manifolds M, discussed in previous work by the authors, and more generally comprises theories that admit a UV description as abelian Chern-Simons-matter theories with (possibly non-perturbative) superpotential. Mathematically, class R might be viewed as an extreme quantum generalization of the Bloch group; in particular, the equivalence relation among theories in class R is a quantum-field-theoretic "2-3 move." We proceed to study the supersymmetric index of theories in class R, uncovering its physical and mathematical properties, including relations to algebras of line operators and to 4d indices. For 3-manifold theories T_M, the index is a new topological invariant, which turns out to be equivalent to non-holomorphic SL(2,C) Chern-Simons theory on M with a previously unexplored "integration cycle."

• The paper introduces class R, a new framework connecting 3-manifolds with 3d N=2 supersymmetric field theories via their indices.
• It details methods to compute the supersymmetric index using tetrahedral decompositions and 2--3 moves to explore quantum invariants.
• The study unveils connections between abelian Chern-Simons-matter theories and topological quantum invariants, paving the way for future research.

An Expert Analysis of "3-Manifolds and 3d Indices"

The paper "3-Manifolds and 3d Indices" by Tudor Dimofte, Davide Gaiotto, and Sergei Gukov explores the intricate relationship between three-dimensional superconformal field theories (SCFTs), as characterized by their indices, and the mathematical description of three-manifolds. The authors present a comprehensive framework for constructing and analyzing SCFTs derived from M5-branes wrapped on 3-manifolds, consequently linking topological properties of these manifolds to physical properties of the corresponding field theories. This connection is mathematically realized through the concept of class $R$ of SCFTs, which admits UV descriptions as abelian Chern-Simons matter theories with potential non-perturbative contributions. Here, we analyze the paper's central claims, methods, and implications for future research.

Central Thesis

At the heart of this work is the introduction of class $R$, a class of theories that includes an extensive range of 3-dimensional $N=2$ SCFTs, potentially serving as effective descriptions of wrapped M5-branes. The theories in class $R$ are typically described in terms of abelian Chern-Simons-matter theories that allow for the addition of superpotential terms, which can significantly augment their rich mathematical structures. A pivotal concept introduced in the paper is the "2--3 move," a transformation in quantum-field-theoretic terms, which maintains the equivalence among theories in class $R$.

The Role of the Supersymmetric Index

The authors explore the supersymmetric index as a key tool for understanding the properties of 3d theories. This index acts similarly to a topological invariant, representing both physical characteristics, such as the space of vacua, and mathematical insights, such as spectral properties of associated manifolds. A salient result is that for the theories $T_M$, the index offers a new invariant that is equivalent to the non-holomorphic $SL(2,)$ Chern-Simons theory on $M$, a relationship previously unexplored in this context.

Techniques and Results

The research primarily revolves around the following components:

1. Dualities and Mirror Symmetry: The paper extensively builds on known mirror symmetries for $N=2$ theories, using them to interrelate different descriptions of SCFTs that result from different M5-brane compactifications.
2. Supersymmetric Indices Calculation: Detailed prescriptions for calculating the supersymmetric index of a 3-manifold theory from those of its constituent tetrahedral theories are given, underpinning the ability to derive precise SCFT descriptions from manifold triangulations.
3. Quantum Lagrangian: The paper outlines a method for associating a quantum Lagrangian submanifold with every SCFT in class $R$, using it to establish connections between geometric structures and field-theoretic phenomena.

Implications and Future Directions

The established connection to Chern-Simons theory deeply impacts both physical and mathematical understanding of SCFTs and 3-manifolds. The insight that the index serves as a richer invariant than previously realized opens new avenues for studying knot invariants and three-manifold topology. Additionally, the perspective that class $R$ could generalize to higher-rank theories promises future developments in the intersection of gauge theory and algebraic geometry, potentially enriching both fields.

In summary, this paper presents a transformative perspective on understanding SCFTs associated with three-manifolds through indices and quantum topology. It establishes a fertile ground for future exploration of multidimensional field theories and invites further investigations into the powerful connections between quantum field theory and topological quantum field theory. Such explorations will likely continue to yield significant mathematical insights and illuminate further aspects of fundamental physics.