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. Ian Goulden, David Jackson and Ravi Vakil). On the other side Virasoro constraints are known to play an imporatant role in the Gromov-Witten theory, in particular for the point.
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?
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...
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.
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.
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&M call $\hat{U}$ (the quantization of a quadratic function).
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.
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.
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.
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.
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.
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.
