We show that two natural approaches to quantum gravity coincide. This identity is nontrivial and relies on the equivalence of each approach to KdV equations. We also investigate related mathematical problems.

/pdf/intersection-theory-on-the-moduli-space-of-curves-and-the-2zk9zv56ec.pdf

Intersection theory on the moduli space of curves and the matrix Airy function

2D gravity and random matrices

We consider the topological B-model on local Calabi-Yau geometries. We show how one can solve for the amplitudes by using W-algebra symmetries which encodes the symmetries of holomorphic diffeomorphisms of the Calabi-Yau. In the highly effective fermionic/brane formulation this leads to a free fermion description of the amplitudes. Furthermore we argue that topological strings on Calabi-Yau geometries provide a unifying picture connecting non-critical (super)strings, integrable hierarchies, and various matrix models. In particular we show how the ordinary matrix model, the double scaling limit of matrix models, and Kontsevich-like matrix model are all related and arise from studying branes in specific local Calabi-Yau three-folds. We also show how A-model topological string on P 1 and local toric threefolds (and in particular the topological vertex) can be realized and solved as B-model topological string amplitudes on a Calabi-Yau manifold.

/pdf/topological-strings-and-integrable-hierarchies-356uc0krr8.pdf

Topological Strings and Integrable Hierarchies

On geometry and matrix models

Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinary KP and Toda hierarchy An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quantization of the similar construction of solutions of dispersionless hierarchies These results as well as those obtained in previous papers are presented with proofs and necessary technical details

http://cds.cern.ch/record/263140/files/9405096.pdf

Integrable hierarchies and dispersionless limit

Conformal matrix models as an alternative to conventional multi-matrix models

Towards unified theory of 2d gravity

The Kazakov-Migdal model, if considered as a functional of external fields, can always be represented as an expansion over characters of the GL group The integration over “matter fields” can be interpreted as going over the model (the space of all highest weight representations) of GL In the case of compact unitary groups the integrals should be substituted by discrete sums over the weight lattice The D=0 version of the model is the generalized Kontsevich integral, which in the above-mentioned unitary (discrete) situation coincides with the partition function of 2D Yang-Mills theory with the target space of genus g=0 and m=0, 1, 2 holes This particular quantity is always a bilinear combination of characters and appears to be a Toda lattice τ function (This is a generalization of the classical statement that individual GL characters are always singular KP τ functions) The corresponding element of the universal Grassmannian is very simple and somewhat similar to the one arising in investigations of the c=1 string models However, in certain circumstances the formal sum over representations should be evaluated by the steepest descent method, and this procedure leads to some more-complicated elements of the Grassmannian This “Kontsevich phase,” as opposed to the simple “character phase,” deserves further investigation

Generalized kazakov-migdal-kontsevich model: group theory aspects

Generalized Kontsevich model versus Toda hierarchy and discrete matrix models

We introduce a new 1-matrix model with arbitrary potential and the matrix-valued background field. Its partition function is a $\tau$-function of KP-hierarchy, subjected to a kind of ${\cal L}_{-1}$-constraint. Moreover, partition function behaves smoothly in the limit of infinitely large matrices. If the potential is equal to $X^{K+1}$, this partition function becomes a $\tau$-function of $K$-reduced KP-hierarchy, obeying a set of ${\cal W} _K$-algebra constraints identical to those conjectured in \cite{FKN91} for double-scaling continuum limit of $(K-1)$-matrix model. In the case of $K=2$ the statement reduces to the early established \cite{MMM91b} relation between Kontsevich model and the ordinary $2d$ quantum gravity . Kontsevich model with generic potential may be considered as interpolation between all the models of $2d$ quantum gravity with $c<1$ preserving the property of integrability and the analogue of string equation.

S. Kharchev

Papers

Conformal matrix models as an alternative to conventional multi-matrix models

Towards unified theory of 2d gravity

Generalized kazakov-migdal-kontsevich model: group theory aspects

Generalized Kontsevich model versus Toda hierarchy and discrete matrix models

Towards unified theory of $2d$ gravity