Givental symmetries of Frobenius manifolds and multi-component KP tau-functions

@article{Feigin2010GiventalSO,
  title={Givental symmetries of Frobenius manifolds and multi-component KP tau-functions},
  author={E. Feigin and J. Leur and S. Shadrin},
  journal={Advances in Mathematics},
  year={2010},
  volume={224},
  pages={1031-1056}
}
We establish a link between two different constructions of the action of the twisted loop group on the space of Frobenius structures. The first construction (due to Givental) describes the action of the twisted loop group on the partition functions of formal (axiomatic) Gromov–Witten theories. The explicit formulas for the corresponding tangent action were computed by Y.-P. Lee. The second construction (due to van de Leur) describes the action of the same group on the space of Frobenius… Expand
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References

SHOWING 1-10 OF 41 REFERENCES
Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov - Witten invariants of all genera into theExpand
The structure of 2D semi-simple field theories
I classify the cohomological 2D field theories based on a semi-simple complex Frobenius algebra A. They are controlled by a linear combination of κ-classes and by an extension datum to theExpand
The construction of Frobenius manifolds from KP tau-functions
Frobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux–Egoroff equations. This system of partial differential equations appears as aExpand
Semisimple Frobenius structures at higher genus
In the context of equivariant Gromov-Witten theory of tori actions with isolated fixed points, we compute genus g > 1 Gromov-Witten potentials and their generalizations with gravitationalExpand
N=1 formal genus zero Gromov–Witten theories and Givental’s formalism
Abstract In [A. Givental, Symplectic geometry of Frobenius structures. arxiv: math.AG/0305409 ] Givental introduced and studied a space of formal genus zero Gromov–Witten theories G W 0 , i.e.Expand
GROMOV - WITTEN INVARIANTS AND QUANTIZATION OF QUADRATIC HAMILTONIANS
We describea formalism based on quantizationof quadratichamil- tonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures aboutExpand
Symplectic geometry of Frobenius structures
The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in two-dimensional topological fieldExpand
Invariance of tautological equations I: conjectures and applications
The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, this framework gives an efficient algorithm to calculate all tautologicalExpand
Givental’s Lagrangian Cone and $S^1$-Equivariant Gromov–Witten Theory
In the approach to Gromov--Witten theory developed by Givental, genus-zero Gromov--Witten invariants of a manifold $X$ are encoded by a Lagrangian cone in a certain infinite-dimensional symplecticExpand
Twisted GLn Loop Group Orbit and Solutions of the WDVV Equations
We show that all (n-component) KP tau-functions, which are related to the twisted loop group of GLn, give solutions of the Darboux-Egoroff system of PDE’s. Using the Geometry of the Grassmannian weExpand
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