Vortex Counting and Lagrangian 3-manifolds (1006.0977v1)

Abstract: To every 3-manifold M one can associate a two-dimensional N=(2,2) supersymmetric field theory by compactifying five-dimensional N=2 super-Yang-Mills theory on M. This system naturally appears in the study of half-BPS surface operators in four-dimensional N=2 gauge theories on one hand, and in the geometric approach to knot homologies, on the other. We study the relation between vortex counting in such two-dimensional N=(2,2) supersymmetric field theories and the refined BPS invariants of the dual geometries. In certain cases, this counting can be also mapped to the computation of degenerate conformal blocks in two-dimensional CFT's. Degenerate limits of vertex operators in CFT receive a simple interpretation via geometric transitions in BPS counting.

• The paper establishes a dual correspondence by compactifying a 5D supersymmetric gauge theory on a 3-manifold to relate 3-manifold invariants with 2D field theories.
• It details a rigorous methodology for computing vortex partition functions as perturbative series that mirror instanton calculations.
• The study reveals connections to knot theory and geometric transitions, offering new insights into low-dimensional topology and BPS state counting.

Overview of "Vortex Counting and Lagrangian 3-manifolds"

The paper "Vortex Counting and Lagrangian 3-manifolds" by Dimofte, Gukov, and Hollands, offers a detailed investigation into the intricate relationships between three-dimensional topological objects and two-dimensional quantum field theories. This paper is part of a broader effort to explore connections between low-dimensional topology and higher-dimensional quantum field theories, specifically within the context of supersymmetric gauge theories and conformal field theories (CFTs).

Key Concepts and Framework

• 3-Manifold and 2D Supersymmetric Field Theory Correspondence: The paper begins by establishing a theoretical framework that links three-dimensional manifolds with two-dimensional supersymmetric field theories. This connection emerges by compactifying a five-dimensional supersymmetric gauge theory on a 3-manifold $M$. Notably, this framework enables the exploration of relationships between various supersymmetric field theories and topological invariants derived from 3-manifolds.
• Relation to Vortex Counting: The authors explore the computation of vortex partition functions in 2D $\mathcal{N}=(2,2)$ supersymmetric gauge theories, focusing on how these computations relate to refined BPS invariants of associated geometries. The vortex partition function serves as a perturbative series in an equivariant cohomology context, drawing parallels to quantum invariants of 3-manifolds.
• AGT-like Correspondence and Duality: The paper proposes a duality analogous to the AGT correspondence, relating a 3-manifold $M$ to a 2D field theory in which the 3-manifold's geometric data is translated into quantum field theoretical properties. In this duality, partition functions of non-supersymmetric quantum field theories are expressed through counting of BPS states in a supersymmetric theory, enhancing the calculative power in both branches of paper.

Numerical and Theoretical Results

• Vortex Partition Function Formalism: The paper outlines the formalism of calculating vortex partition functions, emphasizing their structure as perturbative series, where each order in the expansion is tied to a specific topological attribute of the space-time under consideration. The explicit dependence of the partition function on coupling parameters is explored numerically to provide insights into the underlying topological invariants.
• Instanton and Vortex Equivalence: Through intricate analytical processes, the paper demonstrates how certain limits of 4D instanton partition functions mirror vortex partition functions of lower-dimensional theories. This correspondence is pivotal in transferring calculative insights gained in higher-dimensional theories to two-dimensional field theories with physical and mathematical significance.

Implications for Knot Theory and Geometric Transitions

• Connections to Knot Theory: A significant aspect of this research is the connection it establishes between field theories and knot homologies. The paper of knots, particularly through the lens of physics, highlights how complex algebraic structures can be translated into field theoretical properties, facilitating new approaches to longstanding problems in topology.
• Geometric Transitions: The dual role of Lagrangian submanifolds in enumerative geometry and partition function computations exemplifies the broader implications of this paper. It suggests a novel type of geometric transition, where physical attributes of certain spaces are literally encoded within mathematical structures such as links and knots, fostering robust mappings between seemingly disparate areas of mathematics and physics.

Future Directions

This paper sets the stage for several impactful lines of research. Immediate directions include exploring more general classes of 3-manifolds and associated field theories, refining numerical techniques for evaluating partition functions, and expanding the current understanding of BPS states in various topological settings. Additionally, the implications for knot theory invite a deeper examination of the quantum field theory-knot homology relationship, potentially leveraging these insights for both quantitative and qualitative advances in the paper of low-dimensional topologies.

Overall, the paper integrates a rich confluence of mathematical concepts with sophisticated physical theories, paving the way for advanced theoretical and computational studies in the field of supersymmetric and conformal field theories as well as topological quantum field theories.