Abstract
We introduce the \(\mathrm {SL} (2,\mathbb {C})\) group action on a partition function of a cohomological field theory via a certain Givental’s action. Restricted to the small phase space we describe the action via the explicit formulae on a CohFT genus g potential. We prove that applied to the total ancestor potential of a simple-elliptic singularity the action introduced coincides with the transformation of Milanov–Ruan changing the primitive form (cf. Milanov and Ruan in Gromov–Witten theory of elliptic orbifold \(\mathbb {P}^{1}\) and quasi-modular forms, arXiv:1106.2321, 2011).
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
We will comment on the basis and pairing fixing later in the text.
In some articles this function could be also called “prepotential”.
Following the standard convention we use the lower case variables numbering \(t_k\) in the explicit example rather than \(t^k\) used to the general formulae.
Only one particular case of simple-elliptic singularity \(P_8\) was considered in [17]. However the technique used is extended in a straightforward way to all other cases too.
The statement of theorem of Milanov–Ruan has the operator \(\hat{X}_\mathbf s \) rather than the inverse of it, however they also consider a bit different definition of the Givental’s action—the connection to our is by taking the inverse R-action.
References
Basalaev, A.: Orbifold GW theory as the Hurwitz–Frobenius submanifold. J. Geom. Phys. 77, 30–42 (2014)
Basalaev, A.: \(\text{SL}(2, \mathbb{C})\)-action on cohomological field theories and Gromov–Witten theory of elliptic orbifolds. Oberwolfach reports, 22/2015 (2015)
Basalaev, A.: 6-dimensional FJRW theories of the simple-elliptic singularities. arXiv:1610.07428 (2016)
Basalaev, A.: Homepage. http://basalaev.wordpress.com
Basalaev, A., Priddis, N.: Givental-type reconstruction at a non-semisimple point. Mich. Math. J. arXiv:1605.07862 (2016)
Basalaev, A., Takahashi, A.: On rational Frobenius manifolds of rank three with symmetries. J. Geom. Phys. 77, 30–42 (2014)
Chiodo, A., Iritani, H., Ruan, Y.: Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence, pp. 119–127. Publ.math.IHES (2014)
Dubrovin, B: Geometry of 2D topological field theories. In: Lecture Notes in Mathematics, pp. 120–348. Springer (1996)
Dubrovin, B., Zhang, Y.: Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation. arXiv:hep-th/9712232 (2001)
Dunin-Barkowski, P., Shadrin, S., Spitz, L.: Givental graphs and inversion symmetry. Lett. Math. Phys. 103(5), 533–557 (2013)
Faber, C., Shadrin, S., Zvonkine, D.: Tautological relations and the \(r\)-spin Witten conjecture. Ann. Sci. Ec. Norm. Super. 4(43), 621–658 (2006)
Givental, A.: Gromov–Witten invariants and quantization of quadratic hamiltonians. Mosc. Math. J. 1(4), 1–17 (2001)
Hertling, C.: Frobenius Manifolds and Moduli Spaces for Singularities, Cambridge edn. Cambridge University Press, Cambridge (2002)
Kontsevich, M., Manin, Y.: Gromov–Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164(3), 525–562 (1994)
Lee, Y.P. :Notes on axiomatic Gromov–Witten theory and applications. In: Proceedings of the Symposia in Pure Mathematics, p. 80 (2005)
Milanov, T.: Analyticity of the total ancestor potential in singularity theory. Adv. Math. 255, 217–241 (2014)
Milanov, T., Ruan, Y.: Gromov–Witten theory of elliptic orbifold \(\mathbb{P}^{1}\) and quasi-modular forms. arXiv:1106.2321 (2011)
Milanov, T., Shen, Y.: The modular group for the total ancestor potential of Fermat simple elliptic singularities. Commun. Number Theory Phys. 8, 329–368 (2014)
Pandharipande, R., Pixton, A., Zvonkine, D.: Relations on \(\overline{\cal{M}}_{g, n}\) via 3-spin structures. Am. Math. Soc. 28, 279–309 (2015)
Saito, K.: Period mapping associated to a primitive form. Publ. RIMS Kyoto Univ. 19, 1231–1264 (1983)
Satake, I., Takahashi, A.: Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities. Ann. Inst. Fourier 61, 2885–2907 (2011)
Shadrin, S.: BCOV theory via Givental group action on cohomological field theories. Mosc. Math. J. 9, 411–429 (2009)
Shen, Y., Zhou, J.: Ramanujan identities and quasi-modularity in Gromov–Witten theory. arXiv:1411.2078v1 (2014)
Teleman, C.: The structure of 2D semi-simple field theories. Invent. Math. 189, 525–588 (2012)
Acknowledgements
The Alexey Basalaev is grateful to Sergey Shadrin for his help with Givental’s action, to Claus Hertling for many useful comments and to Davide Veniani for the editorial help. The Alexey Basalaev is also very grateful to Maxim Kazarian for sharing his unpublished notes and to the anonymous referees for many valuable comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Basalaev, A. \(\mathrm{SL}(2, \mathbb {C})\) group action on cohomological field theories. Lett Math Phys 108, 161–183 (2018). https://doi.org/10.1007/s11005-017-0995-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-017-0995-2