Spin Hurwitz theory and Miwa transform for the Schur Q-functions. (English) Zbl 1496.81077
Summary: Schur functions are the common eigenfunctions of generalized cut-and-join operators which form a closed algebra. They can be expressed as differential operators in time-variables and also through the eigenvalues of auxiliary \(N \times N\) matrices \(X\), known as Miwa variables. Relevant for the cubic Kontsevich model and also for spin Hurwitz theory is an alternative set of Schur Q-functions. They appear in representation theory of the Sergeev group, which is a substitute of the symmetric group, related to the queer Lie superalgebras \(\mathfrak{q}(N)\). The corresponding spin \(\hat{\mathcal{W}}\)-operators were recently found in terms of time-derivatives, but a substitute of the Miwa parametrization remained unknown, which is an essential complication for the matrix model technique and further developments. We demonstrate that the Miwa representation, in this case, involves a fermionic matrix \(\Psi\) in addition to \(X\), but its realization using supermatrices is not quite naive.
MSC:
| 81T05 | Axiomatic quantum field theory; operator algebras |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 20G43 | Schur and \(q\)-Schur algebras |
| 35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 20B30 | Symmetric groups |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
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