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Register a new userSchur Q-Polynomials and Kontsevich-Witten Tau Function
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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.
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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.
In this paper, we prove a conjecture of Alexandrov that the generalized Brezin-Gross-Witten tau-functions are hypergeometric tau functions of BKP hierarchy after re-scaling. In particular, this shows that the original BGW tau-function, which has enumerative geometric interpretations, can be represented as a linear combination of Schur Q-polynomials with simple coefficients.
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.
This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group $L$ as positive and negative subgroups L_+, L_- and a commuting sequence in the Lie algebra of L_+. Given f in L_-, there is a formal inverse scattering solution u_f of the hierarchy. When there is a 2 co-cycle that vanishes on both subalgebras of L_+ and L_-, Wilson constructed for each f in L_- a tau function tau_f for the hierarchy. In this third paper, we prove the following results for the nxn KdV hierarchy: (1) The second partials of ln(tau_f) are differential polynomials of the formal inverse scattering solution u_f. Moreover, u_f can be recovered from the second partials of ln(tau_f). (2) The natural Virasoro action on ln(tau_f) constructed in the second paper is given by partial differential operators in ln(tau_f). (3) There is a bijection between phase spaces of the nxn KdV hierarchy and the Gelfand-Dickey (GD_n) hierarchy on the space of order n linear differential operators on the line so that the flows in these two hierarchies correspond under the bijection. (4) Our Virasoro action on the nxn KdV hierarchy is constructed from a simple Virasoro action on the negative group. We show that it corresponds to the known Virasoro action on the GD_n hierarchy under the bijection.
This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theories.
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