# Tau-Structure for the Double Ramification Hierarchies

@article{Buryak2016TauStructureFT, title={Tau-Structure for the Double Ramification Hierarchies}, author={Alexandr Buryak and Boris Dubrovin and J{\'e}r{\'e}my Gu{\'e}r{\'e} and Paolo Rossi}, journal={Communications in Mathematical Physics}, year={2016}, volume={363}, pages={191-260} }

In this paper we continue the study of the double ramification hierarchy of Buryak (Commun Math Phys 336(3):1085–1107, 2015). After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton, and divisor equations plus some important degree constraints). We then formulate a stronger version of the conjecture from Buryak (2015): for any…

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## References

SHOWING 1-10 OF 44 REFERENCES

### Double Ramification Cycles and Quantum Integrable Systems

- Mathematics
- 2015

In this paper, we define a quantization of the Double Ramification Hierarchies of Buryak (Commun Math Phys 336:1085–1107, 2015) and Buryak and Rossi (Commun Math Phys, 2014), using intersection…

### Tautological relations and the r-spin Witten conjecture

- Mathematics
- 2010

A geometric interpretation of Y.P. Lee’s algorithm leads to a much simpler proof of the fact that every tautological relation gives rise to a universal relation, and implies that in any semi-simple Gromov–Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautology relations, the formal and the geometric Gronov– Witten potentials coincide.

### Towards a description of the double ramification hierarchy for Witten's $r$-spin class

- Computer Science, Mathematics
- 2015

### Rational reductions of the 2D-Toda hierarchy and mirror symmetry

- MathematicsJournal of the European Mathematical Society
- 2017

We introduce and study a two-parameter family of symmetry reductions of the two-dimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a…

### The structure of 2D semi-simple field theories

- Mathematics
- 2012

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…

### BCFG Drinfeld–Sokolov hierarchies and FJRW-theory

- Mathematics
- 2013

According to the ADE Witten conjecture, which is proved by Fan, Jarvis and Ruan, the total descendant potential of the FJRW invariants of an ADE singularity is a tau function of the corresponding…

### Witten’s D4 integrable hierarchies conjecture

- Mathematics
- 2010

The authors prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D4 with symmetry group 〈J〉 and for D4T with symmetry group Gmax, respectively, are both…

### A polynomial bracket for the Dubrovin--Zhang hierarchies

- Mathematics
- 2010

We define a hierarchy of Hamiltonian PDEs associated to an arbitrary tau-function in the semi-simple orbit of the Givental group action on genus expansions of Frobenius manifolds. We prove that the…

### Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations

- Mathematics
- 2008

We compute the recently introduced Fan–Jarvis–Ruan–Witten theory of W-curves in genus zero for quintic polynomials in five variables and we show that it matches the Gromov–Witten genus-zero theory of…