Vortex Counting and Lagrangian 3-Manifolds

By Tudor Dimofte, Sergei Gukov, Lotte Hollands. 2010

382 citations recorded by the source corpus.

To every 3-manifold M one can associate a two-dimensional $${\mathcal{N}=(2, 2)}$$ supersymmetric field theory by compactifying five-dimensional $${\mathcal{N}=2}$$ super-Yang–Mills theory on M. This system naturally appears in the study of half-BPS surface operators in four-dimensional $${\mathcal{N}=2}$$ 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 $${\mathcal{N}=(2, 2)}$$ supersymmetric field theories and the refined BPS invariants of the dual geometries. In certain cases, this counting can also be 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.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/01d91f7cdc92aaff8356c3d572b76086753a5aa4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Sergei Gukov
2. C. Vafa
3. Tudor Dimofte
4. E. Witten
5. Pavel Putrov
6. Mina Aganagic
7. F. Benini
8. Marcos Mariño

In Mathematical Physics

1. Sergei Gukov
2. C. Vafa
3. E. Witten
4. Tudor Dimofte
5. Mina Aganagic
6. Marcos Mariño
7. Sh. Shakirov
8. Abhijit Gadde

In Nuclear and High Energy Physics

1. C. Vafa
2. Sergei Gukov
3. Marcos Mariño
4. E. Witten
5. F. Benini
6. Tudor Dimofte
7. Mina Aganagic
8. Pavel Putrov

Citations in the corpus

• Homological algebra of knots and BPS states (2011, methodology): “In the description [12] via equivariant instanton counting on a 4–manifold M4 , the defect (M5–brane) corresponds to a particular ramification along the divisor D M4 , the so-called surface operator.”
• Exact results in D = 2 supersymmetric gauge theories (2012, neutral): “J H E P 0 5 ( 2 0 1 3 ) 0 9 3 contour integral representation [4, 35] (see appendix G for details),”
• Knot Homology from Refined Chern-Simons Theory (2011, background): “7) wrapped by the M5 branes (see [64,25,22]).”
• 4d Index to 3d Index and 2d TQFT (2011, methodology): “Similarly, in the limit S(1) T → 0 (γ → 0 in the notation of section 4), we expect to recover a Chern-Simons theory with a non-compact gauge group [42, 43, 44, 45, 46, 47].”
• Instanton counting with a surface operator and the chain-saw quiver (2011, background): “The latter viewpoint was more suited to the comparison to the topological string approach in the case of the simple surface operator, as was exemplified in many papers [72, 39, 73, 40, 41].”
• Chern-Simons theory and S-duality (2011, methodology): “as in [1], we can simply turn off the Ω-background but put the theory TM on a partially compactified spacetime R2 × S1 R, where the radius of S1 R is R = −1 2 .”
• 2d index and surface operators (2013, background): “The latter can be understood either as Ward identities for line operators or, alternatively, as quantum operators produced via quantization of moduli spaces of SUSY parameters after the theory is compactified on a circle, so that the theory is also effectively two-dimensional [40].”
• Partition Functions of $${\mathcal{N}=(2,2)}$$N=(2,2) Gauge Theories on S2 and Vortices (2012, background): “This expression agrees with [34], up to a redefinition of the masses.”
• Quantum geometry of refined topological strings (2011, background): “The Abranes can be given a purely gauge-theoretic formulation as surface operators, studied recently in [30, 31, 32, 33], for example.”
• A new look at the path integral of quantum mechanics (2010, background): “The fact that the Chern-Simons function arises in this way as a superpotential has been important in recent work [32] (this paper actually involves reduction from five to two dimensions rather than from four to one), and related observations have been made, for example, in [33].”
• Fivebranes and 4-manifolds (2013, methodology): “Specifically, when 3d N = 2 theory T [M3;G] is considered on a circle, its supersymmetric ground states are in one-to-one correspondence with gauge equivalence classes of flat GC connections on M3 [5]: dA+A ∧A = 0 (2.”
• Fivebranes and 3-manifold homology (2016, background): “This is the reason for the effective 2d theory on Σ to localize on complexified flat connections on M3 [27].”
• Fivebranes and Knots (2011, background): “Possibly a hint comes from recent approaches to related problems such as [67].”
• BPS spectra and 3-manifold invariants (2017, background): “One can further reduce it to the 2d N = (2, 2) LG theory on R+ × R with the target space being the space of GC flat connections modulo gauge transformations and the superpotential being the holomorphic Chern-Simons functional CS(A) where A = (Aμ+iφμ)dx is the GC connection 1-form [32].”