Double Ramification Cycles and Quantum Integrable Systems

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• 作者：Alexandr Buryak ; Paolo Rossi
• 关键词：moduli space of curves ; cohomological field theories ; quantum integrable systems ; double ramification cycle
• 刊名：Letters in Mathematical Physics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：106
• 期：3
• 页码：289-317
• 全文大小：643 KB
• 参考文献：1.Buryak A.: Dubrovin–Zhang hierarchy for the Hodge integrals. Commun. Number Theory Phys. 9(2), 239–271 (2015)CrossRef MathSciNet
  2.Buryak A.: Double ramification cycles and integrable hierarchies. Commun. Math. Phys. 336(3), 1085–1107 (2015)CrossRef ADS MathSciNet MATH
  3.Buryak, A., Rossi, P.: Recursion relations for double ramification hierarchies. Commun. Math. Phys. (2014). arXiv:​1411.​6797
  4.Buryak A., Shadrin S., Spitz L., Zvonkine D.: Integrals of psi-classes over double ramification cycles. Am. J. Math. 137(3), 699–737 (2015)CrossRef MathSciNet
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  6.Cavalieri R., Marcus S., Wise J.: Polynomial families of tautological classes on \({\mathcal{M}_{g,n}^{rt}}\) . J. Pure Appl. Algebra 216(4), 950–981 (2012)CrossRef MathSciNet MATH
  7.Dubrovin, B.A., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, p. 295. (a new 2005 version of). arXiv:​math/​0108160v1
  8.Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory, GAFA 2000 Visions in Mathematics special volume, part II, pp. 560–673 (2000)
  9.Fabert, O., Rossi, P.: String, dilaton and divisor equation in symplectic field theory. Int. Math. Res. Not. 19, 4384–4404 (2011)MathSciNet
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  11.Pandharipande R., Pixton A., Zvonkine D.: Relations on \({\overline{\mathcal{M}}_{g,n}}\) via 3-spin structures. J. Am. Math. Soc. 28(1), 279–309 (2015)CrossRef MathSciNet MATH
  12.Rossi P.: Gromov-Witten invariants of target curves via symplectic field theory. J. Geom. Phys. 58(8), 931–941 (2008)CrossRef ADS MathSciNet MATH
  13.Rossi, P.: Integrable systems and holomorphic curves. In: Proceedings of the Gökova Geometry-topology conference 2009, pp. 34–57. Int. Press, Somerville (2010)
  14.Zvonkine, D.: Intersection of double loci with boundary strata (2015, unpublished note)
  
• 作者单位：Alexandr Buryak (1)
  Paolo Rossi (2)
  
  1. Department of Mathematics, ETH Zurich, HG G 27.1, Ramistrasse 101, 8092, Zurich, Switzerland
  2. IMB, UMR 5584 CNRS, Université de Bourgogne 9, avenue Alain Savary, 21078, Dijon Cedex, France
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Statistical Physics
  Geometry
  Group Theory and Generalizations
  
• 出版者：Springer Netherlands
• ISSN：1573-0530

文摘

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 numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. We provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with the first descendant of the unit of the cohomological field theory only. We study various examples which provide, in very explicit form, new (1+1)-dimensional integrable quantum field theories whose classical limits are well-known integrable hierarchies such as KdV, Intermediate Long Wave, extended Toda, etc. Finally, we prove polynomiality in the ramification multiplicities of the integral of any tautological class over the double ramification cycle.