# String-matrix duality There is a remarkable duality between certain 2D string theories and [[0197 Matrix model|matrix models]]. A well-known example is the relation between [[0562 Liouville theory|Liouville gravity]] coupled to a $(2,p)$ minimal model (with $p$ odd) and a matrix integral. Taking the $p\to\infty$ limit, the string side becomes [[0050 JT gravity|JT gravity]], and the matrix side has a simpler spectral density. The duality is a form of open-closed duality, in the sense that it is related to two ways of viewing the theory, one in the closed-string picture and one in the open-string picture. ## Topological recursion Topological recursion allows one to obtain all matrix model partition function correlators$\left\langle Z\left(\beta_1\right) \cdots Z\left(\beta_n\right)\right\rangle_{\text {Matrix }}=\int d H e^{-L \operatorname{Tr} V(H)}\left(\operatorname{Tr} e^{-\beta_1 H}\right) \cdots\left(\operatorname{Tr} e^{-\beta_n H}\right)$from just the leading order result. If we write$\left\langle Z\left(\beta_1\right) \cdots Z\left(\beta_n\right)\right\rangle_{\text {conn.,Matrix }}=\sum_{g=0}^{\infty} e^{-(2 g+n-2) S_0} Z_{g, n}^{\text {Matrix }}\left(\beta_1, \cdots, \beta_n\right),$then$Z_{0,1}^{\text{Matrix}}(\beta)$is all that is needed. ## JT recursion The leading order matrix quantity $Z_{0,1}^{\text{Matrix}}(\beta)$ corresponds to the disk partition function in JT gravity. The topological recursion of the matrix model corresponds to the Mirzahkani recursion relation. ## FZZT vs ZZ branes FZZT branes contribute if we insert a determinant quantity to the matrix integral; ZZ branes give rise to instantons that correspond to non-perturbative corrections to all quantities we compute. ## Refs - reviews - [[1993#Ginsparg, Moore (Lectures)]] - [[1993#Di Francesco, Ginsparg, Zinn-Justin (Review)]] - $c=1$ strings - [[2003#McGreevy, Verlinde]]: reinterpretation of the (double-scaled) matrix model as open string field theory for an infinite number of D0-branes - the relation between AdS$_2$ the IR sector of [[0201 Sachdev-Ye-Kitaev model|SYK]] (before SSS) - [[2016#Maldacena, Stanford]] - [[2016#Maldacena, Stanford, Yang]] - original paper on the JT-matrix duality: - [[2019#Saad, Shenker, Stanford]] (aka SSS) - [[0197 Matrix model|matrix model]] recursion - [[1983#Migdal]]: the idea that loop equations determine all orders - [[2004#Eynard]]: implements the loop equations as recursion relations - [[2007#Eynard, Orantin]]: general theory of topological recursion - non-perturbative extensions: - [[2019#Johnson]] - [[2022#Post, van der Heijden, Verlinde]] - [[2022#Altland, Post, Sonnor, van der Heijden, Verlinde]] - JT with matter - [[2022#Jafferis, Kolchmeyer, Mukhametzhanov, Sonner (Long)]] and [[2022#Jafferis, Kolchmeyer, Mukhametzhanov, Sonner (Short)]] - other extensions - [[2019#Stanford, Witten]]: time-reversal symmetry, fermions, supersymmetry - [[2020#Witten (June, b)]]: deformation by a general potential - [[2020#Turiaci, Usatyuk, Weng]]: conical defects - [[2020#Maxfield, Turiaci]]: conical defects from dimensionally reducing Seifert manifolds - large-genus expansion - [[2024#Griguolo, Papalini, Russo, Seminara]]: refined expression for the large genus asymptotics of the Weil-Petersson volumes of the moduli space of super-Riemann surfaces with an arbitrary number of boundaries - non-perturbative effects - [[1994#Polchinski]]: suggests that non-perturbative effects of the matrix side appear as holes on the string side - periodic potential - [[2024#Blommaert, Levine, Mertens, Papalini, Parmentier]] - topological gravity - [[2011#Gopakumar]] - [[2022#Gopakumar, Mazenc]] - [[0657 Virasoro minimal string|Virasoro minimal string]]