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Virasoro constraints for quantum cohomology. (English) Zbl 0943.58008

The authors give a proof of the genus 0 Virasoro conjecture. The Virasoro conjecture (due to T. Eguchi, K. Hori and C.-S. Xiong [Phys. Lett. B 402, No. 1-2, 71-80 (1997; Zbl 0933.81050)] and Katz [unpublished]) is an infinite series of conjectural relations among the Gromov-Witten invariants (and their gravitational descendants) of a projective manifold \(V\). The relations are given in an appealing form. One first packages all the Gromov-Witten invariants and their descendants into a generating function \(Z\) (the “partition function”) and then defines a sequence of differential operators \(L_{-1}\), \(L_{0}\), \(L_{1}\), \(L_{2}\),…which conjecturally annihilate \(Z\). The coefficients of \(L_{i}Z=0\) are relations among the invariants, for example \(L_{-1}Z=0\) is equivalent to the string equation while \(L_{0}Z=0\) is a combination of the selection rule, the divisor equation, and the dilaton equation. Moreover, the \(L_{i}\)’s define a representation of the Virasoro algebra with central charge \(\chi (V)\), the Euler characteristic of the manifold \(V\). In other words, they satisfy the relation \[ [L_{m},L_{n}]= (m-n)L_{m+n} + \delta _{m,-n}\frac{m (m^{2}-1)}{12}\chi (V). \] If one writes \((L_{n}Z)/Z\) as a Laurent series in \(\lambda \) (the formal variable indexing genus), then one finds that the \(\lambda ^{-2}\) term only involves the genus 0 invariants. Thus this term gives a series of conjectural relations among the genus 0 invariants (and their descendants) that is collectively referred to as the “genus 0 Virasoro conjecture” and is proved in this paper. The author’s proof of this conjecture does not require \(V\) to be Fano (as Eguchi-Hori-Xiong [loc. cit.] assume) but only that \(V\) has only even cohomology (and they point out that even this assumption is not essential).
For a general exposition of the Virasoro conjectures and an independent proof of the genus 0 Virasoro conjecture, see the paper by E. Getzler [Contemp. Math. 241, 147-176 (1999)].

MSC:

58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
14N99 Projective and enumerative algebraic geometry
32G99 Deformations of analytic structures

Citations:

Zbl 0933.81050
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