Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 052, 34 pages      arXiv:1110.0729

Algebro-Geometric Solutions of the Generalized Virasoro Constraints

Francisco José Plaza Martín
Departamento de Matemáticas and IUFFYM, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain

Received December 20, 2014, in final form July 02, 2015; Published online July 07, 2015

We will describe algebro-geometric solutions of the KdV hierarchy whose $\tau$-functions in addition satisfy a generalization of the Virasoro constraints (and, in particular, a generalization of the string equation). We show that these solutions are closely related to embeddings of the positive half of the Virasoro algebra into the Lie algebra of differential operators on the circle. Our results are tested against the case of Witten-Kontsevich $\tau$-function. As by-products, we exhibit certain links of our methods with double covers of the projective line equipped with a line bundle and with ${\rm Gl}(n)$-opers on the punctured disk.

Key words: string equation; Virasoro constraints; KP hierarchy; ${\rm Gl}(n)$-opers; Sato Grassmannian; topological recursion.

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