In this paper we continue the study of the double ramification hierarchy of [Bur15].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 [Bur15]: for any semisimple cohomological field theory, the Dubrovin-Zhang and double ramification hierarchies are related by a normal (i.e. preserving the tau-structure [DLYZ16]) Miura transformation which we completely identify in terms of the partition function of the CohFT. In fact, using only the partition functions, the conjecture can be formulated even in the non-semisimple case (where the Dubrovin-Zhang hierarchy is not defined). We then prove this conjecture for various CohFTs (trivial CohFT, Hodge class, Gromov-Witten theory of CP 1 , 3-, 4-and 5-spin classes) and in genus 1 for any semisimple CohFT. Finally we prove that the higher genus part of the DR hierarchy is basically trivial for the Gromov-Witten theory of smooth varieties with non-positive first Chern class and their analogue in Fan-Jarvis-Ruan-Witten quantum singularity theory [FJR13].
Abstract. We provide a mirror symmetry theorem in a range of cases where the state-of-the-art techniques relying on concavity or convexity do not apply. More specifically, we work on a family of FJRW potentials named after Fan, Jarvis, Ruan, and Witten's quantum singularity theory and viewed as the counterpart of a non-convex Gromov-Witten potential via the physical LG/CY correspondence. The main result provides an explicit formula for Polishchuk and Vaintrob's virtual cycle in genus zero. In the non-concave case of the so-called chain invertible polynomials, it yields a compatibility theorem with the FJRW virtual cycle and a proof of mirror symmetry for FJRW theory. In the last two decades, mirror symmetry has been a central statement in theoretical physics and a fundamental driving force for several developments in mathematics. For instance it can be phrased mathematically as a prediction on GromovWitten invariants, namely the intersection numbers attached to curves traced on a Calabi-Yau variety. In this form, it has been proven in a vast range of concrete cases: the most famous example provides a full computation of the genus-zero invariants enumerating rational curves on the quintic threefold [21,29]. Even in this case, it is often pointed out how we still lack a complete computation in higher genus (only the genus-one case was completely proven by Zinger [35]).But even in genus zero the problem of computing Gromov-Witten invariants of projective varieties is far from being completely solved; indeed, most known techniques focus on computing Gromov-Witten invariants attached to cohomology classes which lie in the so-called ambient part of cohomology: the restriction to classes from the ambient projective space. For the quintic threefold, working with ambient cohomology classes turns out to determine the entire theory; however, in general, this scheme covers only a tiny portion of quantum cohomology. Remarkably, even the ambient cohomology classes may pose problem as soon as we work with orbifolds.It is interesting to notice that these gaps in Gromov-Witten computation all arise from the same phenomena: as we argue below, certain positivity or negativity conditions named convexity and concavity are not always satisfied, making the virtual cycle 1 challenging to compute. In genus one, this difficulty was overcome by Zinger after a great deal of hard work, but we still lack a comprehensive approach for higher genus. Guided by the frame of ideas of mirror symmetry and the Landau-Ginzburg/Calabi-Yau correspondence, we switch to the quantum theory of singularities introduced by Fan, Jarvis, and Ruan [17,18] based on ideas of Witten [34] (FJRW theory). Single non-concave quantum invariants were inferred from concave ones using tautological relation (e.g. using WDVV equation as in [16,17,26]), but so far no systematic approach tackling directly the virtual cycle has been taken. Polishchuk and Vaintrob recently opened the way to an algebraic computation: their construction [32] of a virtual cycle is given by applyin...
In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the first author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus 1 quantum correction and, as an application, compute completely the quantization of the 3-and 4-KdV hierarchies (the DR hierarchies for Witten's 3-and 4-spin theories). We then focus on the recursion relation satisfied by the DR Hamiltonian densities and, abstracting from its geometric origin, we use it to characterize and construct a new family of quantum and classical integrable systems which we call of double ramification type, as they satisfy all of the main properties of the DR hierarchy. In the second part, we obtain new insight towards the Miura equivalence conjecture between the DR and Dubrovin-Zhang hierarchies, via a geometric interpretation of the correlators forming the double ramification tau-function. We then show that the candidate Miura transformation between the DR and DZ hierarchies (which we uniquely identified in our previous paper) indeed turns the Dubrovin-Zhang Poisson structure into the standard form. Eventually, we focus on integrable hierarchies associated with rank-1 cohomological field theories and their deformations, and we prove the DR/DZ equivalence conjecture up to genus 5 in this context.
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