A group action on Losev-Manin cohomological field theories

Shadrin, Sergey; Zvonkine, Dimitri

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.

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

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