Quantum cohomology and Virasoro algebra

Abstract

We propose that the Virasoro algebra controls quantum cohomologies of general Fano manifolds M (c 1 (M ) > 0) and determines their partition functions at all genera. We construct Virasoro operators in the case of complex projective spaces and show that they reproduce the results of Kontsevich-Manin, Getzler etc. on the genus-0,1 instanton numbers. We also construct Virasoro operators for a wider class of Fano varieties. The central charge of the algebra is equal to χ(M ), the Euler characteristic of the manifold M .

Gromov–Witten theory, Hurwitz theory, and completed cycles

“…The Gromov-Witten theory of P 1 was conjectured to be governed by the Toda equation by Eguchi and Yang [8], and also by Dubrovin [5]. The Toda conjecture was further studied in in [6], [7], [16], [30], [34].…”

Virasoro and KdV

“…Regarding the Virasoro conjecture our main references are the works of Dubrovin-Zhang, Eguchi-Hori-Xiong, Getzler, Givental and Liu-Tian ( [6,9,13,14,26]). …”

“…From this point of view, the Witten-Kontsevich case corresponds to the situation when X is a point and it was shown that the exponential of the generating function of intersection numbers on the moduli space of curves was a common solution of the Virasoro constraints and of the KdV hierarchy. Therefore, following the generalization for the case of the projective space proposed in [9], on the one hand one wonders if the generating function fulfills a generalization of the Virasoro constraints. On the other hand, one also wants to know if the generating function is given by (the logarithm of) a particular tau-function of an integrable hierarchy.…”

Tautological algebras of moduli spaces of curves

“…The proportionalities were derived by Getzler and Pandharipande [25] from the Virasoro conjecture of Eguchi, Hori, and Xiong [9]. They were first proved in [15]; see also [40,27].…”

“…1 The third part is completely proved, and there are in fact three proofs: by Givental [26], following earlier work of Eguchi-Hori-Xiong [9] and Getzler-Pandharipande [25]; by Liu and Xu [41]; and by Buryak and Shadrin [6].…”