A-polynomial, B-model, and quantization

By Sergei Gukov, P. Sułkowski. 2011

181 citations recorded by the source corpus.

A bstractExact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as $ \hbar \to 0 $, and becomes non-commutative or “quantum” away from this limit. For a classical curve defined by the zero locus of a polynomial A(x, y), we provide a construction of its non-commutative counterpart $ \widehat{A}\left( {\widehat{x},\widehat{y}} \right) $ using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing $ \widehat{A} $ that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that come from knots or topological strings, their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a K-theory criterion for a curve to be “quantizable,” and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e89465f52059b09b898486fd53354aff2d2a2eb0

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In Discrete Mathematics and Combinatorics

1. Pavel Putrov
2. Sergey Shadrin
3. L. Spitz
4. Dimitri Zvonkine
5. Marcos Mariño
6. B. Eynard

In Geometry and Topology

1. Pavel Putrov
2. Dimitri Zvonkine
3. L. Spitz
4. Sergey Shadrin
5. Marcos Mariño
6. C. Vafa
7. Mina Aganagic
8. A. Klemm

Citations in the corpus

• Equivalence of ELSV and Bouchard–Mariño conjectures for $$r$$r-spin Hurwitz numbers (2013, result): “Furthermore, it was also shown in that paper that the quantization of that curve leads to an operator annihilating the principal specialization of the generating function for Hurwitz numbers, in accordance with another conjecture, formulated by Gukov and Su lkowski in [21], providing further evidenc”
• ABJM theory as a Fermi gas (2011, background): “However, the corrections found in this way are not in the string coupling constant gs, as in conventional topological string theory [56, 59] but rather in the inverse string coupling 1/gs.”