no. 7
A group action on Losev-Manin cohomological field theories
[Une action de groupe sur les théories de champs cohomologiques de Losev-Manin]
Shadrin, Sergey1 ; Zvonkine, Dimitri2
1 Korteweg-de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, Postbus 94248, 1090 GE Amsterdam, Nederland
2 Institut mathématique de Jussieu, Université Paris VI, 175, rue du Chevaleret, 75013 Paris, France and Stanford University Department of Mathematics Building 380, Sloan Hall Stanford, California 94305, USA
Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2719-2743.

Nous introduisons un analogue de l’action du groupe de Givental sur l’espace des solutions de l’équation de commutativité. Nous proposons une construction de cette action en cohomologie de la compactification de Losev-Manin des espaces des modules en genre 0 ; une autre utilisant juste de l’algèbre linéaire sur l’espace des séries de Laurent ; une troisième en termes d’opérateurs différentiels agissant sur des potentiels de Gromov-Witten ; et une quatrième en termes des fonctions tau de la hiérarchie multi-KP. La dernière approche est équivalente à  la classification de Losev-Polyubin obtenue par la technique des transformations d’habillage (dressing transformations).

We discuss an analog of the Givental group action for the space of solutions of the commutativity equation. There are equivalent formulations in terms of cohomology classes on the Losev-Manin compactifications of genus 0 moduli spaces; in terms of linear algebra in the space of Laurent series; in terms of differential operators acting on Gromov-Witten potentials; and in terms of multi-component KP tau-functions. The last approach is equivalent to the Losev-Polyubin classification that was obtained via dressing transformations technique.

DOI : 10.5802/aif.2791
Classification : 53D45, 14H10
Keywords: cohomological field theory, commutativity equation, Losev-Manin space, Givental’s group, Gromov-Witten theory, Kadomtsev-Petviashvili hierarchy.
Mot clés : théorie de champs cohomologique, équation de commutativité, espace de Losev-Manin, groupe de Givental, théorie de Gromov-Witten, hiérarchie de Kadomtsev-Petviashvili

Shadrin, Sergey 1 ; Zvonkine, Dimitri 2

1 Korteweg-de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, Postbus 94248, 1090 GE Amsterdam, Nederland
2 Institut mathématique de Jussieu, Université Paris VI, 175, rue du Chevaleret, 75013 Paris, France and Stanford University Department of Mathematics Building 380, Sloan Hall Stanford, California 94305, USA 
@article{AIF_2011__61_7_2719_0,
     author = {Shadrin, Sergey and Zvonkine, Dimitri},
     title = {A group action on {Losev-Manin} cohomological field theories},
     journal = {Annales de l'Institut Fourier},
     pages = {2719--2743},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {7},
     year = {2011},
     doi = {10.5802/aif.2791},
     mrnumber = {3112505},
     zbl = {1275.53085},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2791/}
}
TY  - JOUR
AU  - Shadrin, Sergey
AU  - Zvonkine, Dimitri
TI  - A group action on Losev-Manin cohomological field theories
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 2719
EP  - 2743
VL  - 61
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2791/
DO  - 10.5802/aif.2791
LA  - en
ID  - AIF_2011__61_7_2719_0
ER  - 
%0 Journal Article
%A Shadrin, Sergey
%A Zvonkine, Dimitri
%T A group action on Losev-Manin cohomological field theories
%J Annales de l'Institut Fourier
%D 2011
%P 2719-2743
%V 61
%N 7
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2791/
%R 10.5802/aif.2791
%G en
%F AIF_2011__61_7_2719_0
Shadrin, Sergey; Zvonkine, Dimitri. A group action on Losev-Manin cohomological field theories. Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2719-2743. doi : 10.5802/aif.2791. https://aif.centre-mersenne.org/articles/10.5802/aif.2791/

[1] Bakalov, B.; Milanov, T. W N+1 -constraints for singularities of type A N (arXiv: 0811.1965)

[2] Barannikov, S. Non-commutative periods and mirror symmetry in higher dimensions, Commun. Math. Phys., Volume 228 (2002) no. 2, pp. 281-325 | MR | Zbl

[3] Bayer, Yu. A. ands Manin Stability Conditions, Wall-Crossing and Weighted Gromov-Witten Invariants, Mosc. Math. J., Volume 9 (2009) no. 1, pp. 3-32 | MR | Zbl

[4] Chiodo, A.; Zvonkine, D. Twisted Gromov-Witten r-spin potential and Givental’s quantization (arXiv:0711.0339)

[5] Coates, T.; Ruan, Y. Quantum Cohomology and Crepant Resolutions: A Conjecture (arXiv:0710.5901)

[6] Dijkgraaf, R.; Verlinde, E.; Verlinde, H. Topological strings in d<1, Nuclear Phys. B, Volume 352 (1991) no. 1, pp. 59-86 | MR

[7] Dubrovin, B. Geometry of 2D topological field theories, Integrable systems and quantum groups (Montecatini Terme, 1993) (Lecture Notes in Math.), Volume 1620, Springer, Berlin, 1996, pp. 120-348 | MR | Zbl

[8] Dubrovin, B.; Zhang, Y. Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants (arXiv:math/0108160)

[9] Faber, C.; Shadrin, S.; Zvonkine, D. Tautological relations and the r-spin Witten conjecture (arXiv:math/0612510)

[10] Feigin, E.; van de Leur, J.; Shadrin, S. Givental symmetries of Frobenius manifolds and multi-component KP tau-functions (arXiv:0905.0795) | Zbl

[11] Givental, A. Symplectic geometry of Frobenius structures, Mosc. Math. J., Volume 1 (2001) no. 4, pp. 551-568 | Zbl

[12] Givental, A. Gromov–Witten invariants and quantization of quadratic hamiltonians, Frobenius manifolds (Aspects Math., E36), Vieweg, Wiesbaden, 2004, pp. 91-112

[13] Hertling, C. Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2002 | MR | Zbl

[14] Kac, V. G.; van de Leur, J. W. The n-component KP hierarchy and representation theory, Important developments in soliton theory (Springer Ser. Nonlinear Dynam.), Springer, Berlin, 1993, pp. 302-343 | MR | Zbl

[15] Kac, V. G.; van de Leur, J. W. The n-component KP hierarchy and representation theory, J. Math. Phys., Volume 44 (2003) no. 8, pp. 3245-3293 | MR | Zbl

[16] Kontsevich, M.; Manin, Y. Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., Volume 164 (1994) no. 3, pp. 525-562 | MR | Zbl

[17] Lee, Y.-P. Invariance of tautological equations I: conjectures and applications, J. Eur. Math. Soc. (JEMS), Volume 10 (2008) no. 2, pp. 399-413 | MR | Zbl

[18] Lee, Y.-P. Invariance of tautological equations II: Gromov–Witten theory (with Appendix A by Y. Iwao and Y.-P. Lee), J. Amer. Math. Soc., Volume 22 (2009) no. 2, pp. 331-352 | MR | Zbl

[19] van de Leur, J. Twisted GL n loop group orbit and solutions of the WDVV equations, J. Amer. Math. Soc., Volume 2001 no. 11, pp. 551-573 | MR | Zbl

[20] Losev, A. On “Hodge” topological strings at genus zero, JETP Lett., Volume 65 (1997) no. 5, pp. 386-392

[21] Losev, A. Hodge strings and elements of K. Saito’s theory of primitive form, Topological field theory, primitive forms and related topics (Prog. Math.), Volume 160, Birkhäuser, Boston, 1999, pp. 305-335 | MR | Zbl

[22] Losev, A.; Manin, Y. New moduli spaces of pointed curves and pencils of flat connections, Mich. Math. J., Volume 48 (2000), Spec. Vol., pp. 443-472 | MR | Zbl

[23] Losev, A.; Polyubin, I. On compatibility of tensor products on solutions to commutativity and WDVV equations, JETP Lett., Volume 73 (2001) no. 2, pp. 53-58

[24] Losev, A.; Polyubin, I. Commutativity equations and dressing transformations, JETP Lett., Volume 77 (2003) no. 2, pp. 53-57

[25] Manin, Yu. I. Frobenius manifolds, quantum cohomology, and moduli spaces, Mathematical Society Colloquium Publications, 47, American Mathematical Society, Providence, RI, 1999 | MR | Zbl

[26] Shadrin, S. BCOV theory via Givental group action on cohomological field theories, Mosc. Math. J., Volume 9 (2009) no. 2, pp. 411-429 | MR | Zbl

[27] Witten, E. On the structure of the topological phase of two-dimensional gravity, Nuclear Phys. B, Volume 340 (1990) no. 2-3, pp. 281-332 | MR

Cité par Sources :