# New Recursion Formulae and Integrablity for Calabi-Yau Spaces (11w5114)

Arriving in Banff, Alberta Sunday, October 16 and departing Friday October 21, 2011

## Organizers

Vincent Bouchard (University of Alberta)

Tom Coates (Imperial College)

Motohico Mulase (University of California, Davis)

Emma Previato (Boston University)

Jian Zhou (Tsinghua University)

## Objectives

The proposed workshop has a clear set of goals and a focused research plan.

In terms of geometry, the objective is to increase our understanding of the Gromov-Witten invariants of toric Calabi-Yau spaces. Coates and Zhou are leading figures in the recent developments on this topic. Coates and his collaborators Corti, Iritani, and Tseng have studied Gromov-Witten theory of toric varieties from the point of view of the crepant resolution conjecture. The Gromov-Witten theory of toric Calabi-Yau 3-folds has recently seen an explosive development due to Zhou and his collaborators J. Li, C.C.M. Liu, and K. Liu. They have solved the Marino-Vafa conjecture, and established a mathematical foundation for the topological vertex theory of Aganagic, Klemm, Marino, and Vafa.

Many mathematicians are excited about the new mysterious formulas proposed by the random matrix theory community and the topological string theory community on the Gromov-Witten invariants of toric Calabi-Yau 3-folds. The physics predictions suggest the existence of a non-trivial Virasoro-type constraint, an integrable system that governs the Gromov-Witten invariants, and a recursion formula that computes them.

Our first goal is to understand the algebro-geometric structure of the topological recursion of Eynard-Orantin. This recursion takes the form of an integration formula, and has the identical structure of the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces. For the case of mixed intersection numbers of the cotangent psi-classes and the Mumford-Miller-Morita kappa classes, which is a generalization of the Weil-Petersson volume considered by Mirzakhani, we now know the following.

1) There is a Virasoro constraint condition;

2) the generating function satisfies the KdV hierarchy; and

3) there is a topological recursion that computes all these mixed intersections. (K. Liu-Xu and Mulase-Safnuk).

Although these mathematical results were established before the discovery of Eynard-Orantin theory, Eynard has recently shown that these results can be obtained by the Laplace transform of the topological recursion for a particular choice of the spectral curve.

In some cases the integration in the Eynard-Orantin recursion formula can be calculated explicitly. Bouchard and Marino considered a limiting case of one of the predictions of BKMP, and obtained a conjectural recursion formula for Hurwitz numbers as an integration formula. Eynard, Mulase, and Safnuk have shown that after computing the integrals in this formula, the Bouchard-Marino formula becomes equivalent to the Laplace transform of the cut-and-join equation of Goulden, Jackson, and Vakil, and have thus proved the conjecture. We note that the cut-and-join equation is a particular example of a topological recursion. The relation between Hurwitz numbers and the Witten-Kontsevich theory has been discussed by many authors. By now we have achieved a good understanding on the subject (Okounkov-Pandharipande, Kazarian-Lando, Chen-Y. Li-K. Liu, Goulden-Jackson-Vainshtein, Goulden-Jackson-Vakil, Kim-K. Liu, and Mulase-Zhang).

At least for these examples, we have a good grasp of the relation between Virasoro constraints, integrable equations, and the topological recursion. From the point of view of Gromov-Witten theory, however, these are only the simplest examples of the general BKMP conjecture. Our more ambitious goal is to understand this conjecture itself. To understand the BKMP conjecture, the key is the structural analysis of the Gromov-Witten invariants of toric Calabi-Yau 3-folds.

Although the BKMP conjecture does not address any relation to integrable systems, the three questions stated in the ``Overview'' section are tightly related. For example, the relation between higher Hodge integrals and the KP equations has been established by Zhou. The Givental formalism for integrable systems due to Givental and Coates-Givental may be the most promising candidate to answer Question 2.

The psi-kappa mixed intersections and the Hurwitz numbers (linear Hodge integrals) are by now almost completely understood, and we have satisfactory answers to the three questions. From a purely integrable system point of view, it is easy to explain why these particular cases are simpler than the others: It is because the ``spectral curve'' behind the scene has genus 0. For both Witten-Kontsevich theory and Hurwitz theory, the integrable system responsible is the one-component KP hierarchy. By analyzing the way the Virasoro constraint comes in to the theory, we know that one can deal with only the genus 0 Eynard-Orantin theory in one-component KP theory. A higher genus theory inevitably requires a multi-component KP theory. Thus we expect that the generating function of the Gromov-Witten invariants of a toric Calabi-Yau 3-fold satisfies a multi-component KP equations. Previato, as an algebraic geometer, has worked on versions of the multicomponent KP equations (in particular one which by Lapace transform is related to Painleve' VI, in turn related by Yu. Manin to the mirror of the projective plane) and on dispersionless KP (whose tau-function solutions also exhibit a thus far mysterious recursion).

The format of a BIRS workshop is ideal for the proposed research. Although the goals are specific and topics are focused, they require diverse expertise. A mixture of experts from topological string theory, Gromov-Witten theory, and algebraic geometry of integrable systems is crucial. The organizer's group strongly represents these areas.

We also emphasize that the proposed research is timely. The discoveries of physicists were made in 2007, and their conjectures have not been mathematically solved in their most general forms. Much recent mathematical research turns out to be closely related to the predictions of the physicists. One of the conjectures (the Bouchard-Marino conjecture) has been recently solved in terms of a matrix integral (Borot-Eynard-Mulase-Safnuk), and also purely geometrically (Eynard-Mulase-Safnuk).

In terms of geometry, the objective is to increase our understanding of the Gromov-Witten invariants of toric Calabi-Yau spaces. Coates and Zhou are leading figures in the recent developments on this topic. Coates and his collaborators Corti, Iritani, and Tseng have studied Gromov-Witten theory of toric varieties from the point of view of the crepant resolution conjecture. The Gromov-Witten theory of toric Calabi-Yau 3-folds has recently seen an explosive development due to Zhou and his collaborators J. Li, C.C.M. Liu, and K. Liu. They have solved the Marino-Vafa conjecture, and established a mathematical foundation for the topological vertex theory of Aganagic, Klemm, Marino, and Vafa.

Many mathematicians are excited about the new mysterious formulas proposed by the random matrix theory community and the topological string theory community on the Gromov-Witten invariants of toric Calabi-Yau 3-folds. The physics predictions suggest the existence of a non-trivial Virasoro-type constraint, an integrable system that governs the Gromov-Witten invariants, and a recursion formula that computes them.

Our first goal is to understand the algebro-geometric structure of the topological recursion of Eynard-Orantin. This recursion takes the form of an integration formula, and has the identical structure of the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces. For the case of mixed intersection numbers of the cotangent psi-classes and the Mumford-Miller-Morita kappa classes, which is a generalization of the Weil-Petersson volume considered by Mirzakhani, we now know the following.

1) There is a Virasoro constraint condition;

2) the generating function satisfies the KdV hierarchy; and

3) there is a topological recursion that computes all these mixed intersections. (K. Liu-Xu and Mulase-Safnuk).

Although these mathematical results were established before the discovery of Eynard-Orantin theory, Eynard has recently shown that these results can be obtained by the Laplace transform of the topological recursion for a particular choice of the spectral curve.

In some cases the integration in the Eynard-Orantin recursion formula can be calculated explicitly. Bouchard and Marino considered a limiting case of one of the predictions of BKMP, and obtained a conjectural recursion formula for Hurwitz numbers as an integration formula. Eynard, Mulase, and Safnuk have shown that after computing the integrals in this formula, the Bouchard-Marino formula becomes equivalent to the Laplace transform of the cut-and-join equation of Goulden, Jackson, and Vakil, and have thus proved the conjecture. We note that the cut-and-join equation is a particular example of a topological recursion. The relation between Hurwitz numbers and the Witten-Kontsevich theory has been discussed by many authors. By now we have achieved a good understanding on the subject (Okounkov-Pandharipande, Kazarian-Lando, Chen-Y. Li-K. Liu, Goulden-Jackson-Vainshtein, Goulden-Jackson-Vakil, Kim-K. Liu, and Mulase-Zhang).

At least for these examples, we have a good grasp of the relation between Virasoro constraints, integrable equations, and the topological recursion. From the point of view of Gromov-Witten theory, however, these are only the simplest examples of the general BKMP conjecture. Our more ambitious goal is to understand this conjecture itself. To understand the BKMP conjecture, the key is the structural analysis of the Gromov-Witten invariants of toric Calabi-Yau 3-folds.

Although the BKMP conjecture does not address any relation to integrable systems, the three questions stated in the ``Overview'' section are tightly related. For example, the relation between higher Hodge integrals and the KP equations has been established by Zhou. The Givental formalism for integrable systems due to Givental and Coates-Givental may be the most promising candidate to answer Question 2.

The psi-kappa mixed intersections and the Hurwitz numbers (linear Hodge integrals) are by now almost completely understood, and we have satisfactory answers to the three questions. From a purely integrable system point of view, it is easy to explain why these particular cases are simpler than the others: It is because the ``spectral curve'' behind the scene has genus 0. For both Witten-Kontsevich theory and Hurwitz theory, the integrable system responsible is the one-component KP hierarchy. By analyzing the way the Virasoro constraint comes in to the theory, we know that one can deal with only the genus 0 Eynard-Orantin theory in one-component KP theory. A higher genus theory inevitably requires a multi-component KP theory. Thus we expect that the generating function of the Gromov-Witten invariants of a toric Calabi-Yau 3-fold satisfies a multi-component KP equations. Previato, as an algebraic geometer, has worked on versions of the multicomponent KP equations (in particular one which by Lapace transform is related to Painleve' VI, in turn related by Yu. Manin to the mirror of the projective plane) and on dispersionless KP (whose tau-function solutions also exhibit a thus far mysterious recursion).

The format of a BIRS workshop is ideal for the proposed research. Although the goals are specific and topics are focused, they require diverse expertise. A mixture of experts from topological string theory, Gromov-Witten theory, and algebraic geometry of integrable systems is crucial. The organizer's group strongly represents these areas.

We also emphasize that the proposed research is timely. The discoveries of physicists were made in 2007, and their conjectures have not been mathematically solved in their most general forms. Much recent mathematical research turns out to be closely related to the predictions of the physicists. One of the conjectures (the Bouchard-Marino conjecture) has been recently solved in terms of a matrix integral (Borot-Eynard-Mulase-Safnuk), and also purely geometrically (Eynard-Mulase-Safnuk).