2011/08/31 by Sergei Gukov, Piotr Sułkowski, Piotr Sulkowski · 4 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic curve #Algebraic structures and combinatorial models #Differential geometry #Geometric and Algebraic Topology #Meromorphic function #Polynomial #Quantization (signal processing) #Quantum #Topological quantum field theory #Topological quantum number #Topology (electrical circuits) #hep-th #math.AG #math.QA
paper · pdf · doi:10.1007/jhep02(2012)070
published as JHEP 1202 (2012) 070 · 58 pages, 5 figures, minor modifications, references added
openalex publication_date 2012/02/01 · arxiv created 2012/06/12 · arxiv updated 2012/06/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
A bstract 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 ℏ → 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 \widehatA( \widehatx,\widehaty ) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing \widehatA 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.