Virasoro constraints for Kontsevich-Hurwitz partition function

TL;DR

The work shows that the Kontsevich-Hurwitz partition function ${\cal Z}$ satisfies a deformed Virasoro system obtained by conjugating the ordinary Virasoro generators with a twist operator ${\hat U}$ that depends on the interpolation parameter $u$. This deformation shifts the lowest Virasoro mode by $u^2/24$ and preserves the Virasoro subalgebra, yielding ${\cal Z} = {\hat U} Z_K$, while its genus-zero sector is annihilated by a new operator ${\hat N}_1$. By connecting time variables through a $T(p)$ mapping and leveraging ELSV, the authors relate the Kontsevich-Hurwitz free energy to Hurwitz numbers, show that $e^H$ is a KP tau-function, and demonstrate consistency with AMM-Eynard-type formulations on the Lambert curve. The results illuminate how KH sits as a distinguished phase in the matrix-model landscape, with explicit constructions of the deformed Virasoro constraints, an enhanced algebra of annihilators, and a pathway to additional integral representations. Overall, the paper provides concrete, testable structures linking Hodge integrals, Hurwitz numbers, and KP tau-functions through a controlled Virasoro deformation.

Abstract

M.Kazarian and S.Lando found a 1-parametric interpolation between Kontsevich and Hurwitz partition functions, which entirely lies within the space of KP tau-functions. V.Bouchard and M.Marino suggested that this interpolation satisfies some deformed Virasoro constraints. However, they described the constraints in a somewhat sophisticated form of AMM-Eynard equations for the rather involved Lambert spectral curve. Here we present the relevant family of Virasoro constraints explicitly. They differ from the conventional continuous Virasoro constraints in the simplest possible way: by a constant shift u^2/24 of the L_{-1} operator, where u is an interpolation parameter between Kontsevich and Hurwitz models. This trivial modification of the string equation gives rise to the entire deformation which is a conjugation of the Virasoro constraints L_m -> U L_m U^{-1} and "twists" the partition function, Z_{KH}= U Z_K. The conjugation U is expressed through the previously unnoticed operators which annihilate the quasiclassical (planar) free energy of the Kontsevich model, but do not belong to the symmetry group GL(\infty) of the universal Grassmannian.

Figures (7)

• Figure 1: The covering $y\rightarrow x$ of the Riemann sphere in the simplest case of the curve $Q_N(y) = x$. Left picture: the real section. Right picture: symbolical complex view. All critical points (zeroes of $Q'(y)$ are assumed different. The $N$ sheets merge together at infinity.
• Figure 2: The covering $y\rightarrow x$ of the Riemann sphere in the case of generic $P_N(x,y)=0$. Left picture: a fully reducible symbol, no branching at infinity. Actually in the picture $N=3$ and $M=6$, so that $p=1$ (this is the cubic representation of a torus, like $x^3+y^3+\alpha xy = 0$). Right picture: generic branching at infinity, with $n$ groups of merging $m_1$, $m_2$, $\ldots$, $m_n$ sheets. Actually in the picture $n=2$, $m_1=2$, $m_2=3$, $M=5$, $p = 0$.
• Figure 3: The two vertices in diagram technique which describes the action of $\hat{W}_0$ on $e^{p_1}$. Arrows denote derivatives with respect to $p$, ends without arrows carry $p$ themselves. Vertices contain factors of $ij$ and $i+j$. In what follows we often write $i$ instead of $p_i$.
• Figure 4: The lowest-order diagrams for Hurwitz function $H(p)$. All free arrows at the right hand are supposed to act on $e^{p_1}$, i.e. they carry index $1$ (from $p_1$) and come with weight $1$. Each diagram is a monomial in $p_k$'s, where relevant values of $k$ are indices of the incoming lines at the left. The sum of $k$'s is equal to the number of free arrows in the diagram. Expression for diagram is made out of $ij/2$ and $(i+j)/2$ factors at the vertices and $\frac{u^{3m}}{m!}$ where $m$ is the total number of vertices. Diagrams with $p$ loops contribute only to $F^{(p)}$. Selection rule for $q$ is more complicated, because $u$ enters not only through $u^{3m}$ but also through the $T(p)$ dependence. For $m=3$ we do not draw identical diagrams, instead their multiplicity are shown.
• Figure 5: The simplest diagrams contributing to $r^{(n)}$: connected rooted trees with exactly one external arrow carrying index $n$. The coupling constant is $u^3=t$. The sum of all diagrams is $R^{(n)} = p_n + np_{n+1}t + \frac{n(n+2)}{2}p_{n+2}t^2 + \frac{n(n+3)^2}{6}p_{n+3}t^3 + \ldots$ The zeroth-order contribution $p_n$ corresponds to the diagram with no vertices, not shown on the picture. The combinatorial coefficient $8$ in the last diagram is made from a "naive" factor $4$, counting the places to attach the outgoing $n$ arrow (or, what is the same, the "up-down" orientations of two vertices with different incoming lines) and an extra, perhaps less familiar $2$, counting the "time"-ordering of the two most right vertices: a phenomenon illustrated also by appearance of $B$ and $C$ diagrams in Fig.\ref{['p1p3']} below.