On deformations of quasi-Miura transformations and the Dubrovin–Zhang bracket

Buryak, Alexandr; Posthuma, Hessel; Shadrin, Sergey

doi:10.1016/j.geomphys.2012.03.006

Cited by 43 publications

(77 citation statements)

References 12 publications

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“…They have recently confirmed that such an integrable hierarchy exists, provided that a certain conjecture about polynomiality of the Poisson brackets holds [31]. This conjecture was partially proved by Buryak-Posthuma-Shadrin [15,16] (the polynomiality of the second bracket is still an open problem). Another approach is to derive Hirota's bilinear equations for the taufunction; see [91,87,88,89,90,58,60,45].…”

Section: N I=1mentioning

confidence: 87%

-constraints for the total descendant potential of a simple singularity

2013

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Abstract. Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finitedimensional Lie algebra, and W its Weyl group. The set of ginvariants in the basic representation of the affine Kac-Moody algebraĝ is known as a W-algebra and is a subalgebra of the Heisenberg vertex algebra F . Using period integrals, we construct an analytic continuation of the twisted representation of F . Our construction yields a global object, which may be called a W -twisted representation of F . Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest weight vector for the W-algebra.1. Introduction 1.1. Motivation from Gromov-Witten theory. Recall that the Gromov -Witten (GW) invariants of a projective manifold X consist of correlators (1.1) τ k 1 (v 1 ), . . . , τ kn (v n ) g,n,dwhere v 1 , . . . , v n ∈ H * (X; C) are cohomology classes and the enumerative meaning of the correlator is the following. Let C 1 , . . . , C n be n cycles in X in a sufficiently generic position that are Poincaré dual to v 1 , . . . , v n , respectively. Then the GW invariant (1.1) counts the number of genus-g, degree-d holomorphic curves in X that are tangent (in an appropriate sense) to the cycles C i with multiplicities k i . For the precise definition we refer to [103,75,8,83]. After A. Givental [57], we organize the GW invariants in a generating series D X called the total descendant potential of X and defined as follows. Choose a basis {v i } N i=1of the vector (super)space H = H * (X; C) and let t k = τ k 1 (t k 1 ), . . . , τ kn (t kn ) g,n,d , where t = (t 0 , t 1 , . . . ) = (t i k ) and the definition of the correlator is extended multi-linearly in its arguments. The function D X is interpreted as a formal power series in the variables t i k with coefficients formal Laurent series in whose coefficients are elements of the Novikov ring C [Q].When X is a point and hence d = 0, the potential D pt (also known as the partition function of pure gravity) is a generating function for certain intersection numbers on the Deligne-Mumford moduli space of Riemann surfaces M g,n . It was conjectured by Witten [103] and proved by Kontsevich [74] that D pt is a tau-function for the KdV hierarchy of soliton equations. (We refer to [19,102] for excellent introductions to soliton equations.) In addition, D pt satisfies one more constraint called the string equation, which together with the KdV hierarchy determines uniquely D pt (see [103]). It was observed in [22,54,69] that the taufunction of KdV satisfying the string equation is characterized as the unique solution of L n D pt = 0 for n ≥ −1, where L n are certain differential operators representing the Virasoro algebra. This means that D pt is a highest-weight vector for the Virasoro algebra and in addition satisfies the string equation L −1 D pt = 0.One of the fundamental open questions in Gromov-Witten theory is the Virasoro conjecture suggested by S. Katz and the physicists Eguchi, Hori, Xiong, and...

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Section: N I=1mentioning

confidence: 87%

-constraints for the total descendant potential of a simple singularity

2013

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“…The theory was later generalized in [BPS12b]. In this section we follow the approach from [BPS12b] (see also [BPS12a]). …”

Section: Dubrovin-zhang Hierarchy For Cpmentioning

confidence: 99%

Recursion Relations for Double Ramification Hierarchies

Buryak

Rossi

2015

Commun. Math. Phys.

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Abstract. In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced in [Bur15a] using intersection theory of the double ramification cycle in the moduli space of stable curves. In particular, we prove a recursion formula that recovers the full hierarchy starting from just one of the Hamiltonians, the one associated to the first descendant of the unit of a cohomological field theory. Moreover, we introduce analogues of the topological recursion relations and the divisor equation both for the hamiltonian densities and for the string solution of the double ramification hierarchy. This machinery is very efficient and we apply it to various computations for the trivial and Hodge cohomological field theories, and for the r-spin Witten's classes. Moreover we prove the Miura equivalence between the double ramification hierarchy and the Dubrovin-Zhang hierarchy for the Gromov-Witten theory of the complex projective line (extended Toda hierarchy).

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“…In [14] the above mentioned Hamiltonian structure of the deformed hierarchy is obtained from the first Hamiltonian structure of the principal hierarchy via the quasi-Miura transformation (6.5). A proof of the polynomiality of the topological deformation of the principal hierarchy and of the Hamiltonian operator P 1 are given in [26] and [27]. For the second Hamiltonian structure of the principal hierarchy, the following conjecture is given in [14].…”

Section: The Topological Deformationsmentioning

confidence: 99%

The inversion symmetry of the WDVV equations and tau functions

Liu

Zhang

2012

Physica D: Nonlinear Phenomena

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For two solutions of the WDVV equations that are related by the inversion symmetry, we show that the associated principal hierarchies of integrable systems are related by a reciprocal transformation, and the tau functions of the hierarchies are related by a Legendre type transformation. We also consider relationships between the Virasoro constraints and the topological deformations of the principal hierarchies.

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On deformations of quasi-Miura transformations and the Dubrovin–Zhang bracket

Cited by 43 publications

References 12 publications

-constraints for the total descendant potential of a simple singularity

-constraints for the total descendant potential of a simple singularity

Recursion Relations for Double Ramification Hierarchies

The inversion symmetry of the WDVV equations and tau functions

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