Infinite dimensional Grassmannian structure of two-dimensional quantum gravity

Abstract

We study the infinite dimensional Grassmannian structure of 2D quantum gravity coupled to minimal conformal matters, and show that there exists a large symmetry, the W 1 + O0 symmetry. Using this symmetry structure, we prove that the square root of the partition function, which is a τ function of the p-reduced KP hierarchy, satisfies the vacuum condition of the W 1 + ao algebra. We further show that this condition is reduced to the vacuum condition of the W p algebra when the redundant variables for the p-reduction are eliminated. This mechanism also gives a prescription for extracting the W p algebra from the W 1 + 00 algebra.

Lattice topological field theory in two dimensions

“…17). 6 Then, by the invariance of exp(−δS) under the fusion and bubble transformations, f α (n x ) is determined to have the form 13) which implies that f α (n x ) is proportional to the deficit angle around the vertex x. proof First, the invariance under the fusion transformation ( fig. 18) yields the following identity: (6.14) from which f α (n x ) is known to be a linear function of n x ; f α (n x ) = A α n x +B α .…”

“…Intuitive consideration tells us that each background metric g µν in continuous theory should correspond to a triangulation T in our lattice framework. In fact, in 2D quantum gravity, the summation over all quantum fluctuations of metric can be replaced by the summation over all triangulations [2][3] [4][5] [6]:…”

Wk structure of A from covariant construction of Kac-Moody algebras

“…This implies that a realization of W k in terms of twisted bosons of the kind used by the authors in [18] is needed.…”

“…Therefore, I can give a realization of W 1+∞ algebra (see [17] for more detail) in terms of the well known free boson realization [18] for transverse part tensored with the light-like vertex:…”

Givental formula in terms of Virasoro operators

“…The first example is the point. Its function τ • , which corresponds to the pure topological gravity [4], is the τ -function for the KdV hierarchy [10], [11]. It can be realized as the matrix integral (famous Kontsevich model [12])…”