Virasoro constraints for the generating function of Hurwitz numbers. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:43:18Z http://mathoverflow.net/feeds/question/38293 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38293/virasoro-constraints-for-the-generating-function-of-hurwitz-numbers Virasoro constraints for the generating function of Hurwitz numbers. Sasha 2010-09-10T11:27:17Z 2012-01-05T08:51:12Z <p>Generating function of the simple Hurwitz numbers is known to be connected with Gromov-Witten potential of the point (Kontsevich $\tau$-function) (see e.g. <a href="http://arxiv.org/abs/math/9910004" rel="nofollow">Ian Goulden, David Jackson and Ravi Vakil</a>). On the other side Virasoro constraints are known to play an imporatant role in the Gromov-Witten theory, in particular for the point. </p> <p>Is there known the set of Virasoro constraints for the generating function of simple (or double) Hurwitz numbers? Any ideas why it should (or should not) exist? </p> http://mathoverflow.net/questions/38293/virasoro-constraints-for-the-generating-function-of-hurwitz-numbers/39026#39026 Answer by Paul Johnson for Virasoro constraints for the generating function of Hurwitz numbers. Paul Johnson 2010-09-16T21:53:12Z 2010-09-16T21:53:12Z <p>This is something I've long been meaning to think about seriously, so maybe I can get back to you with something better later. In the meantime... </p> <p>I wasn't aware of anything explicitly in the literature about this until googling just a bit ago, when I found the paper "Virasoro constraints for Kontsevich-Hurwitz partition function" by Mironov and Morozov. </p> <p>I haven't fully digested their paper yet, but they seem to be using the viewpoint of the Kazarian paper on Hodge integrals and KP hierarchy I mentioned in my answer to your other question. </p> <p>Briefly: the ELSV formula relates single Hurwitz numbers to Hodge integrals, essentially the GW theory of a point. Kazarian shows how this transformation can be done explicitly by a certain operator M&amp;M call $\hat{U}$ (the quantization of a quadratic function). </p> <p>M and M's point seems to be that since the generating functions are related by this operator, and one of them satisfies Virasoro, we can conjugate the Virasoro operators by $\hat{U}$ and get Virasoro operators for the other generating function. The particular form they take seems to be a bit of a mess, and I worry about some details, but I've only skimmed that paper very quickly.</p> <p>But philosophically, this seems to be going about things backwards: the Hurwitz side is really simpler, and the above setup is often used to show that the GW of a point satisfies Virasoro. I feel we should be able to construct Virasoro operators for single Hurwitz numbers more easily with some kind of direct approach.</p> <p>Hurwitz theory is all about statements about the symmetric group, and there are constructions (I read about it in a Frenkel-Wang paper) that build Virasoro actions out of the symmetric group. </p> <p>I'm not fully motivating this, but single and Hurwitz theory is very conveniently done on a certain Fock space. Basically, you have the operator that multiplies by a transposition, (called $M_0$ by Kazarian, $\mathcal{F}_2$ by Okounkov-Pandharipande, physicists have some other notation for it...), and you have the operators $\alpha_n, n\in\mathbb{Z}$ that add or remove cycles of length $n$ from a conjugacy class, and that form a heisnberg algebra. </p> <p>These operators are exactly what you need to do Hurwitz theory. But they're also what Frenkel and Wang use to construct a Virasoro algebra -- essentially, the commutators $L_n=[\alpha_n, M_0]$. So we might hope that some similar construction would give us easier to understand Virasoro constraints for single Hurwitz numbers. But I haven't spent the necessary time trying to nail it down.</p> <p>As far as double Hurwitz numbers go, I'm a little less hopeful for the above vague ideas. All I know is that Goulden, Jackson and Vakil have a few lines about trying and failing to construct Virasoro operators in their "Towards the Geometry of Double Hurwitz Numbers" paper.</p>