Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2011-07-29
Physics
High Energy Physics
High Energy Physics - Theory
57 pages, 5 figures
Scientific paper
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 $\hbar \to 0$, 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 $\hat{A} (\hat x, \hat y)$ using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing $\what{A}$ 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 geometry," 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.
Gukov Sergei
Sułkowski Piotr
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