Non-Perturbative Topological Strings And Conformal Blocks

TL;DR

The paper constructs a non-perturbative completion of closed topological strings by decomposing amplitudes into open-string blocks determined by contour data $Z^{ ext{top}}_{\mathcal A}$, with contours identified as Lagrangian submanifolds that fix boundary conditions. Applying this to Penner-type matrix models, it reveals a light-cone diagram interpretation where critical points of $W$ correspond to interaction points and gradient-flow contours define the non-perturbative blocks; these blocks are then connected to Liouville conformal blocks through the AGT correspondence. A precise dictionary is established: $β=-b^2$, $m_\ell=-2b α_\ell$, and $N=(Q- extstyle\sum α)/b$, with internal momenta set by momentum-violation conditions at each vertex, allowing the matrix-model blocks to reproduce DOZZ-type factors and conformal blocks (three- and four-point) up to leg factors. The work also develops a diagonal monodromy basis by combining contours under Dehn twists, enabling a direct match to conformal blocks, and discusses extensions to Toda theories, higher genus, and analytic continuations, along with the physical interpretation of light-cone diagrams in this non-perturbative framework.

Abstract

We give a non-perturbative completion of a class of closed topological string theories in terms of building blocks of dual open strings. In the specific case where the open string is given by a matrix model these blocks correspond to a choice of integration contour. We then apply this definition to the AGT setup where the dual matrix model has logarithmic potential and is conjecturally equivalent to Liouville conformal field theory. By studying the natural contours of these matrix integrals and their monodromy properties, we propose a precise map between topological string blocks and Liouville conformal blocks. Remarkably, this description makes use of the light-cone diagrams of closed string field theory, where the critical points of the matrix potential correspond to string interaction points.

Figures (7)

• Figure 1: A natural set of contours $\tilde{C}_1,\dotsi, \tilde{C}_n$ is given by the pre-images of the straight lines on the $W$-plane. The slope of the straight lines is given by the downward gradient flow of $\text{Re}(W/g_s).$
• Figure 2: An example of a light-cone diagram for a given matrix potential $W$. The locations $z_\ell$ of the poles of $dW$ determine the twist angles $\theta_\ell$ and the interaction times $\tau_\ell$, while the residues $m_\ell$ determine the size of the closed strings. Such a light-cone diagram gives a pants decomposition of the genus zero Riemann surface and thereby determines a tree structure. Together with the data of the distribution $\{N_1,\ldots,N_n\}$ of the matrix eigenvalues among the critical points $\{p_1,\ldots,p_n\}$ of the matrix potential, this tree specifies a corresponding conformal block.
• Figure 3: The momentum violation at each vertex is given by the number of eigenvalues at the corresponding critical point, directed towards increasing light-cone time.
• Figure 4: (a) The contour $\tilde{C}_2$ given by the pre-images of straight line in the $W$ plane. (b) Its image under monodromy transformation $M_{2;1}$ that takes the location $z_2$ of the pole around $z_1$ in a counterclockwise orientation. (c) The combination $C_2$ of the two contours that is invariant under the monodromy transformation.
• Figure 5: An example of the contours for computing a five-point conformal block. The ordering of the contour is given by the time-ordering of the interaction points on the light-cone diagram. Hence one should first perform the integration over the eigenvalue along the contours $C_2$, then $C_1$, and finally $C_3$.