Buryak, A, Dubrovin, B, Guéré, J et al. (1 more author) (2018) Tau-Structure for the Double Ramification Hierarchies. Communications in Mathematical Physics, 363 (1). pp. 191-260. ISSN 0010-3616
Abstract
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 semisimple cohomological field theory, the Dubrovin–Zhang and double ramification hierarchies are related by a normal [i.e. preserving the tau-structure (Dubrovin et al. in Adv Math 293:382–435, 2016)] 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 CP¹, 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 (Fan et al. in Ann Math 178(1):1–106, 2013).
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © Springer-Verlag GmbH Germany, part of Springer Nature 2018. This is a post-peer-review, pre-copyedit version of an article published in Communications in Mathematical Physics. The final authenticated version is available online at: https://doi.org/10.1007/s00220-018-3235-4. |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 08 Aug 2018 12:42 |
Last Modified: | 28 Aug 2019 00:43 |
Status: | Published |
Publisher: | Springer Verlag |
Identification Number: | 10.1007/s00220-018-3235-4 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:134317 |