• Corpus ID: 119173130

Frobenius manifolds and quantum groups

@article{Xu2017FrobeniusMA,
  title={Frobenius manifolds and quantum groups},
  author={Xiaomeng Xu},
  journal={arXiv: Mathematical Physics},
  year={2017}
}
  • Xiaomeng Xu
  • Published 30 December 2017
  • Mathematics
  • arXiv: Mathematical Physics
We introduce an isomonodromic Knizhnik-Zamolodchikov connection with respect to the quantum Stokes matrices, and prove that the semiclassical limit of the KZ type connection gives rise to the Dubrovin connections of semisimple Frobenius manifolds. This quantization procedure of Dubrovin connections is parallel to the quantization from Poisson Lie groups to quantum groups, and is conjecturally formulated as a deformation of symplectic transformations on loop spaces in the sense of Givental. 
2 Citations

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  • Xiaomeng Xu
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We study the Stokes phenomenon of the generalized Knizhnik–Zamolodchikov (gKZ) equations, and prove that their Stokes matrices satisfy the Yang–Baxter equations. In particular, the monodromy of the

References

SHOWING 1-10 OF 51 REFERENCES

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 field

A Kohno-Drinfeld theorem for quantum Weyl groups

Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W and let V be a finite–dimensional representation of g. In [MTL], we introduced a new, W –equivariant flat connection

G-bundles, isomonodromy, and quantum Weyl groups

It is now twenty years since Jimbo, Miwa, and Ueno [23] generalized Schlesinger’s equations (governing isomonodromic deformations of logarithmic connections on vector bundles over the Riemann sphere)

Frobenius manifolds, quantum cohomology, and moduli spaces

Introduction: What is quantum cohomology? Introduction to Frobenius manifolds Frobenius manifolds and isomonodromic deformations Frobenius manifolds and moduli spaces of curves Operads, graphs, and

The Structure of 2 D Semisimple 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 the

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 the

Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations

The Hamiltonian theory of isomonodromy equations for meromorphic connections with irregular singularities on algebraic curves is constructed. An explicit formula for the symplectic structure on the

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 the

Frobenius manifolds and formality of Lie algebras of polyvector fields

We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0 Gromov-Witten invariants

Quantum Riemann–Roch, Lefschetz and Serre

Given a holomorphic vector bundle E over a compact Kahler manifold X, one defines twisted Gromov-Witten invariants of X to be intersection numbers in moduli spaces of stable maps f :Σ → X with the
...