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Double ramification cycles and integrable hierarchies.(English)Zbl 1329.14103

Around 1990,E. Witten [in: Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology, held at Harvard University, Cambridge, MA, USA, April 27-29, 1990. Providence, RI: American Mathematical Society; Bethlehem, PA: Lehigh University. 243–310 (1991;Zbl 0757.53049)] conjectured remarkable recurrence relations in the cohomology of the moduli space of curves, that could be captured by combining the intersection numbers into a generating function that one could show satisfied the KdV equations.M. Kontsevich [Commun. Math. Phys. 147, No. 1, 1–23 (1992;Zbl 0756.35081)] proved Witten’s conjecture not long after it was made using a matrix model. Since then it has become apparent that many Gromov-Witten invariants can be combined together into integrable hierarchies. This paper introduces a new hierarchy for cohomological field theories that is distinct from the Dubrovin-Zhang hierarchy, but apparently related to it via a certain transformation.
This new hierarchy is named after the double ramification cycle. To define this cycle, start with a finite list of integers \(a_1, a_2, \dots, a_n\) that sum to zero. Once ordered, the positive elements of \(a_1, a_2, \dots, a_n\) form a partition and the absolute value of the negative elements form a second partition. One may consider the moduli space of genus \(g\) surfaces mapping to \(\mathbb{C}P^1\) with branching data over \(0\) and \(\infty\) given by the two partitions and as many marked points as there are zeros in \(a_1, a_2, \dots, a_n\). A further quotient of this moduli space is taken under the \(\mathbb{C}^\times\) action on the target. There is a natural forgetful map from this space to the moduli space of genus \(g\) curves with marked points, and this map is the double ramification cycle.
The double ramification cycle hierarchy is a generating function encoding evaluations of natural cohomology classes on this cycle. This paper begins with a description of a formal algebraic approach to partial differential equations and Hamiltonian systems in particular. The author then proceeds to use this theory to show that the double ramification cycle hierarchy forms an integrable system. The paper concludes with the computation of this hierarchy in two different cases. He conjectured that the Dubrovin-Zhang hierarchy is related to the double ramification cycle hierarchy via a Miura transformation. This is verified for the two sample computations.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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