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A-polynomial, B-model, and quantization
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DOI：10.1007/JHEP02(2012)070.png)
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Abstract
En 中文
Exact 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 (h) over bar -> 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 (A) over cap((x) over cap, (y) over cap) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing (A) over cap 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.
Keywords:
Matrix Models
Non-Commutative Geometry
Chern-Simons Theories
Topological Strings
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