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A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model

Davide Gaiotto, Leonardo Rastelli

TL;DR

The paper presents an exact open/closed duality framework in which topological matrix models, exemplified by the Kontsevich model, arise as cubic open string field theories on N stable Liouville D-branes. A boundary-to-puncture mechanism converts open worldsheet boundaries into closed-string insertions, with Kontsevich parameters z_i encoding closed-string sources t_k via t_k = (g_s/k) Tr Z^{-k}; topological localization further reduces OSFT to a finite-dimensional matrix model. The authors compute Liouville BCFT correlators to match Kontsevich vertices, establish Ward identities linking open and closed sectors, and extend the construction to nonzero bulk cosmological constant, showing background independence through open/closed couplings. They outline generalizations to other (p,1) theories, discuss connections to discretized random surfaces in D = −2, and propose a broader paradigm where open string field theory on branes captures the full closed-string dynamics in solvable low-dimensional settings. The work provides a concrete, solvable realization of open/closed duality with potential implications for holography and the role of OSFT in dual descriptions.

Abstract

We argue that topological matrix models (matrix models of the Kontsevich type) are examples of exact open/closed duality. The duality works at finite N and for generic `t Hooft couplings. We consider in detail the paradigm of the Kontsevich model for two-dimensional topological gravity. We demonstrate that the Kontsevich model arises by topological localization of cubic open string field theory on N stable branes. Our analysis is based on standard worldsheet methods in the context of non-critical bosonic string theory. The stable branes have Neumann (FZZT) boundary conditions in the Liouville direction. Several generalizations are possible.

A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model

TL;DR

The paper presents an exact open/closed duality framework in which topological matrix models, exemplified by the Kontsevich model, arise as cubic open string field theories on N stable Liouville D-branes. A boundary-to-puncture mechanism converts open worldsheet boundaries into closed-string insertions, with Kontsevich parameters z_i encoding closed-string sources t_k via t_k = (g_s/k) Tr Z^{-k}; topological localization further reduces OSFT to a finite-dimensional matrix model. The authors compute Liouville BCFT correlators to match Kontsevich vertices, establish Ward identities linking open and closed sectors, and extend the construction to nonzero bulk cosmological constant, showing background independence through open/closed couplings. They outline generalizations to other (p,1) theories, discuss connections to discretized random surfaces in D = −2, and propose a broader paradigm where open string field theory on branes captures the full closed-string dynamics in solvable low-dimensional settings. The work provides a concrete, solvable realization of open/closed duality with potential implications for holography and the role of OSFT in dual descriptions.

Abstract

We argue that topological matrix models (matrix models of the Kontsevich type) are examples of exact open/closed duality. The duality works at finite N and for generic `t Hooft couplings. We consider in detail the paradigm of the Kontsevich model for two-dimensional topological gravity. We demonstrate that the Kontsevich model arises by topological localization of cubic open string field theory on N stable branes. Our analysis is based on standard worldsheet methods in the context of non-critical bosonic string theory. The stable branes have Neumann (FZZT) boundary conditions in the Liouville direction. Several generalizations are possible.

Paper Structure

This paper contains 22 sections, 94 equations, 3 figures.

Table of Contents

  1. Introduction and Summary
  2. From open to closed worldsheets
  3. Review of ${\bf (2, 2k+1)}$ strings and the Kontsevich model
  4. The Kontsevich model is cubic open string field theory
  5. Extended Liouville D-branes in topological string theory
  6. Closed bosonic strings in ${\bf D=-2}$
  7. Remarks on closed string observables
  8. Open string theory on stable branes
  9. Macroscopic loops
  10. Boundary CFT
  11. Open string field theory and the Kontsevich model
  12. Generalities
  13. Topological localization
  14. Liouville BCFT and the matrix model
  15. Discussion
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Feynman rules for the Kontsevich model.
  • Figure 2: The two fatgraphs with $g=0$ and $h=3$. The indices $i, j, k$ are Chan-Paton labels ranging from 1 to $N$. The sum of the two graphs is ${g_s}/(z_i z_j z_k)$. Upon summing over the Chan-Paton labels, this gives $t_1^3/(6 g_s^2) \longrightarrow \langle {\cal O}_1 {\cal O }_1 {\cal O}_1 \rangle_{g=0} = 1$.
  • Figure 3: Degeneration of the Riemann surface as the closed string operator ${\cal O}_k$ approaches the boundary. The shadowed region represents the hole. As the short neck pinches, the surface factorizes into two surfaces, each with the extra insertion of an open string tachyon, indicated by a cross.