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## Geometric Invariants and Spectral Curves |

Combinatorial
problems related to branched coverings of two-dimensional Riemann surfaces are
being attracting attention of researchers since Hurwitz who considered mappings
from the complex projective plane to spheres branched over a fixed number of
points. The numbers of different combinatorial classes of such mappings are
traditionally called the Hurwitz numbers and they have appeared recently in
relation to many important enumeration and topological problems of
low-dimensional geometry. They turned out to be linked to such, very different
on the first glance, problems as constructing Gromov--Witten
invariants of projective curves, as relation to integrable
models via Kadomtsev--Petviashvili
(KP) hierarchy equations, as relation to matrix models, and relation to quantum
spectral curves and cohomological field theories. It
also happens that partition functions of the corresponding theories admit
explicit or hidden conformal transformation structures. These structures are
often complimentary to structures of integrable
theories and they manifest themselves in the topological recursion procedure
allowing constructing higher genus terms from the spectral curve and a fixed
set of data (commonly a set of one-differentials) on it. We
expect the following topics to be covered on the workshop; each of them will
include a two-hour introductory lecture, a student discussion section and 2-3
research talks. ---
Givental and Frobenius
structures. Givental theory is an advanced tool for studying Gromov--Witten
invariants of target varieties and general cohomological
field theories that allows, in particular, obtaining explicit relations between
partition functions of different theories, reconstructing higher genera
correlation functions from genus zero data, and establishing properties of
semi-simple theories. The general relation between Gromov--Witten
potentials associated to Frobenius structures
associated with a semi-simple quantum cohomology, or
the Gromov--Witten potential, is given by a definite
canonical transformation of the direct product of $r$ Kontsevich--Witten
tau-functions of the KdV hierarchy. ---
Quantum spectral curves and topological recursion. Quantum
spectral curves have appeared naturally in the context of beta ensembles, which
are generalisations of matrix models again admitting
topological recursion procedures. In the community of theoretical physicists a
boom in this field was initiated by the Alday--Gaiotto--Tachikawa conjecture
establishing relations between Nekrasov--Shatashvili generating functions for instanton
counting and beta-ensembles. Chekhov, Eynard, and Ribault explicitly demonstrated that conformal blocks of
quantum Liouville theory can be nonperturbatively
described in terms of quantum analogues of algebraic-geometric quantities,
which we call quantum spectral curves. The topological recursion for these
object lacks the locality property and it is tempting to understand its
relations to structures appearing in other topics of the workshop. ---
Discretizations of moduli spaces. Givental-type formulas had first appeared in the context of discretisation of moduli spaces. The corresponding
generating function counts numbers of integer points inside moduli spaces and
is equivalent to the Hermitian one-matrix model thus
admitting the topological recursion. These correspondences were established in
the case of two branching points ($r=2$). One of the goals of the workshop is
to consider generalisations of these theories to the case of higher $r$ and more general
matrix-model theories and to discuss their relations to quantum spectral
curves. ---
Hurwitz numbers and related generating functions. Since
works of A.Yu.Orlov, Okounkov,
and Pandharipande, generating functions for numbers
of branched coverings of ${\mathbb C}P^1$ (the
Hurwitz numbers) were known to be KP tau-functions. On the other hand, the ELSV
formula relates some class of these generating functions to the Gromov--Witten invariants.It was
understood later that a wide class of these generating functions is
simultaneously partition functions of the generalized Kontsevich
matrix models thus enabling constructing a topological recursion. ---
Quantum Riemann surfaces and knot invariants. In
the knot theory, the spectral curve is determined by A-polynomials. The
corresponding asymptotic expansion gives coloured
Jones polynomials. Here, again, there are two versions of quantisation:
one understood as discretisation (in terms of
possible numbers of polynomials) and the second one as an actual quantisation of A-polynomial spectral curve by Gukov and Su{\l}kowski, which
relates it to algebras of geodesic functions on Riemann surfaces with holes in
the Poincar\'e uniformization
or to algebras of monodromies of certain Fuchsian systems and to generalisations
of these algebras to those of twisted Yangians. [Back] |