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Register a new userConnecting the Kontsevich-Witten and Hodge tau-functions by the $hat{GL(infty)}$ operators
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In this paper, we present an explicit formula that connects the Kontsevich-Witten tau-function and the Hodge tau-function by differential operators belonging to the $hat{GL(infty)}$ group. Indeed, we show that the two tau-functions can be connected using Virasoro operators. This proves a conjecture posted by Alexandrov in [1].
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The Brezin-Gross-Witten (BGW) model is one of the basic examples in the class of non-eigenvalue unitary matrix models. The generalized BGW tau-function $tau_N$ was constructed from a one parametric deformation of the original BGW model using the generalized Kontsevich model representation. It is a tau-function of the KdV hierarchy for any value of $Ninmathbb C$, where the case $N=0$ reduces to the original BGW tau-function. In this paper, we present a bosonic representation of $tau_N$ in terms of the $W_{1+infty}$ operators that preserves the KP integrability. This allows us to establish a connection between the (generalized) BGW and Kontsevich-Witten tau-functions using $GL(infty)$ operators, both considered as the basic building blocks in the theory of matrix models and partition functions.
We give a proof of Alexandrovs conjecture on a formula connecting the Kontsevich-Witten and Hodge tau-functions using only the Virasoro operators. This formula has been confirmed up to an unknown constant factor. In this paper, we show that this factor is indeed equal to one by investigating series expansions for the Lambert W function on different points.
In this paper, we show that the generating function for linear Hodge integrals over moduli spaces of stable maps to a nonsingular projective variety $X$ can be connected to the generating function for Gromov-Witten invariants of $X$ by a series of differential operators ${ L_m mid m geq 1 }$ after a suitable change of variables. These operators satisfy the Virasoro bracket relation and can be seen as a generalization of the Virasoro operators appeared in the Virasoro constraints for Kontsevich-Witten tau-function in the point case. This result is an extension of the work in cite{LW} for the point case which solved a conjecture of Alexandrov.
Using matrix model, Mironov and Morozov recently gave a formula which represents Kontsevich-Witten tau-function as a linear expansion of Schur Q-polynomials. In this paper, we will show directly that the Q-polynomial expansion in this formula satisfies the Virasoro constraints, and consequently obtain a proof of this formula without using matrix model. We also give a proof for Alexandrovs conjecture that Kontsevich-Witten tau-function is a hypergeometric tau-function of the BKP hierarchy after re-scaling.
We prove the equivalence between two explicit expressions for two-point Witten-Kontsevich correlators obtained by M. Bertola, B. Dubrovin, D. Yang and by P. Zograf.
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