Minimal Strings Research Articles

A relation between (2,2m − 1) minimal strings and the Virasoro minimal string

We propose a connection between the newly formulated Virasoro minimal string and the well-established (2,2m−1) minimal string via their dual matrix models by deriving the string equation of the Virasoro minimal string using the expansion of its density of states in powers of Em+1/2. The string equation is expressed as a power series involving double-scaled multicritical matrix models, which are dual to (2,2m−1) minimal strings. This reformulation of the Virasoro minimal string enables us to employ matrix theory tools to compute its n-boundary correlators. We analyze the scaling behavior of n-boundary correlators and quantum volumes V0,n(b)(ℓ1,…,ℓn) in the JT gravity limit.

Perturbative unorientable JT gravity and matrix models

We consider an orthogonal polynomial formulation of the double scaling limit of multicritical matrix models in the β = 1 Dyson-Wigner class. They capture the physics of 2D quantum gravity coupled to minimal matter on unorientable surfaces, otherwise called unoriented minimal strings. We derive a formula for the density of states valid to all orders in perturbation theory. We show how to define an interpolation between the multicritical models and that a certain interpolation among an infinite number of them provides an alternative definition of unoriented JT gravity. We discuss the strengths and weaknesses of our formulation.

From minimal strings towards Jackiw–Teitelboim gravity: on their resurgence, resonance, and black holes

Abstract Two remarkable facts about Jackiw–Teitelboim (JT) two-dimensional dilaton-gravity have been recently uncovered: this theory is dual to an ensemble of quantum mechanical theories; and such ensembles are described by a random matrix model which itself may be regarded as a special (large matter-central-charge) limit of minimal string theory. This work addresses this limit, putting it in its broader matrix-model context; comparing results between multicritical models and minimal strings (i.e. changing in-between multicritical and conformal backgrounds); and in both cases making the limit of large matter-central-charge precise (as such limit can also be defined for the multicritical series). These analyses are first done via spectral geometry, at both perturbative and nonperturbative levels, addressing the resurgent large-order growth of perturbation theory, alongside a calculation of nonperturbative instanton-actions and corresponding Stokes data. This calculation requires an algorithm to reach large-order, which is valid for arbitrary two-dimensional topological gravity. String equations—as derived from the Gel’fand–Dikii construction of the resolvent—are analyzed in both multicritical and minimal string theoretic contexts, and studied both perturbatively and nonperturbatively (always matching against the earlier spectral-geometry computations). The resulting solutions, as described by resurgent transseries, are shown to be resonant. The large matter-central-charge limit is addressed—in the string-equation context—and, in particular, the string equation for JT gravity is obtained to next derivative-orders, beyond the known genus-zero case (its possible exact-form is also discussed). Finally, a discussion of gravitational perturbations to Schwarzschild-like black hole solutions in these minimal-string models, regarded as deformations of JT gravity, is included—alongside a brief discussion of quasinormal modes.

Jackiw-Teitelboim supergravity as a double-cut matrix model

We study a Jackiw-Teitelboim (JT) supergravity theory, defined as an Euclidean path integral over orientable supermanifolds with constant negative curvature, that was argued by Stanford and Witten to be captured by a random matrix model in the $\boldsymbol{\beta}{=}2$ Dyson-Wigner class. We show that the theory is a double-cut matrix model tuned to a critical point where the two cuts coalesce. Our formulation is fully non-perturbative and manifestly stable, providing for explicit unambiguous computation of observables beyond the perturbative recursion relations derivable from loop equations. Our construction shows that this JT supergravity theory may be regarded as a particular combination of certain type 0B minimal string theories, and is hence a natural counterpart to another family of JT supergravity theories recently shown to be built from type 0A minimal strings. We conjecture that certain other JT supergravities can be similarly defined in terms of double-cut matrix models.

Jackiw-Teitelboim supergravity, minimal strings, and matrix models

The author studies a family of two-dimensional Jackiw--Teitelboim (JT) supergravites, and, by using string theory techniques, he gives them a complete definition to all orders in the topological expansion. This construction provides a non--perturbative formulation for the JT supergravity that is well--defined and stable. Furthermore, by using a combination of analytical and numerical methods, the author shows explicitly how non--perturbative physics can be extracted for JT gravity within this framework.

Minimal Elements for the Prime Numbers

ABSTRACTWe say a string of symbols s is minimal for a language L if s is a member of L, and it is not possible to obtain another member of L by striking out one or more symbols from s. Although the set M(L) of minimal strings is necessarily finite, determining it explicitly for a given L can be a difficult computational problem. We use some number-theoretic heuristics to compute M(L), where L is the language of base-b representations of the prime numbers, for 2 ≤ b ≤ 30.

The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painlevé II Equation

Resurgent transseries have recently been shown to be a very powerful construction in order to completely describe nonperturbative phenomena in both matrix models and topological or minimal strings. These solutions encode the full nonperturbative content of a given gauge or string theory, where resurgence relates every (generalized) multi-instanton sector to each other via large-order analysis. The Stokes phase is the adequate gauge theory phase where an 't Hooft large N expansion exists and where resurgent transseries are most simply constructed. This paper addresses the nonperturbative study of Stokes phases associated to multi-cut solutions of generic matrix models, constructing nonperturbative solutions for their free energies and exploring the asymptotic large-order behavior around distinct multi-instanton sectors. Explicit formulae are presented for the Z_2 symmetric two-cut set-up, addressing the cases of the quartic matrix model in its two-cut Stokes phase; the "triple" Penner potential which yields four-point correlation functions in the AGT framework; and the Painleve II equation describing minimal superstrings.

Generating fixed density bracelets of arbitrary base

Bracelets are lexicographically minimal k-ary strings symmetric under rotation and reversal. In this paper, we present an algorithm for lexicographic listing of bracelets with fixed density. Our algorithm works for arbitrarily large alphabet size and each successive bracelet is generated in constant amortized time.

Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models

We address the nonperturbative structure of topological strings and c=1 matrix models, focusing on understanding the nature of instanton effects alongside with exploring their relation to the large-order behavior of the 1/N expansion. We consider the Gaussian, Penner and Chern-Simons matrix models, together with their holographic duals, the c=1 minimal string at self-dual radius and topological string theory on the resolved conifold. We employ Borel analysis to obtain the exact all-loop multi-instanton corrections to the free energies of the aforementioned models, and show that the leading poles in the Borel plane control the large-order behavior of perturbation theory. We understand the nonperturbative effects in terms of the Schwinger effect and provide a semiclassical picture in terms of eigenvalue tunneling between critical points of the multi-sheeted matrix model effective potentials. In particular, we relate instantons to Stokes phenomena via a hyperasymptotic analysis, providing a smoothing of the nonperturbative ambiguity. Our predictions for the multi-instanton expansions are confirmed within the trans-series set-up, which in the double-scaling limit describes nonperturbative corrections to the Toda equation. Finally, we provide a spacetime realization of our nonperturbative corrections in terms of toric D-brane instantons which, in the double-scaling limit, precisely match D-instanton contributions to c=1 minimal strings.

Minimal strings and quantum algebraic curves(Fundamental Problems and Applications of Quantum Field Theory)

Minimal strings and quantum algebraic curves(Fundamental Problems and Applications of Quantum Field Theory)

Semi-classical mechanics in phase space: the quantum target of minimal strings

The target space Mp,q of (p,q) minimal strings is embedded into the phase space of an associated integrable classical mechanical model. This map is derived from the matrix model representation of minimal strings. Quantum effects on the target space are obtained from the semiclassical mechanics in phase space as described by the Wigner function. In the classical limit the target space is a fold catastrophe of the Wigner function that is smoothed out by quantum effects. Double scaling limit is obtained by resolving the singularity of the Wigner function. The quantization rules for backgrounds with ZZ branes are also derived.

On Minimal Strings Containing the Elements of S_n by Decimation

The permutations by decimation problem is thought to be applicable to computer graphics, and raises interesting theoretical questions in combinatory theory.We present the results of some theoretical and practical investigation into this problem.We show that sequences of this form are $O(n^2)$ in length, but finding optimal solutions can be difficult.

Min of Mat is not necessarily Mat

We show that the set of lexicographically minimal strings of each length in a matrix language is not necessarily a matrix language. This partially solves a problem formulated by J. Berstel during the Fourth DLT Conference, Aachen, 1999. We also briefly discuss the case of simple matrix languages.

Minimal strings in a regular language with respect to a partial order on the alphabet

Let L be a regular language and “⩽” a partial order on its alphabet Σ; the partial order “⩽” may be extended to Σ ∗ by defining a 1 a 2⋯ a k ⩽ b 1 b 2⋯ b m if k⩽ m and a j ⩽ b j for 1⩽ j⩽ k. We show that L={x: x is a minimal string in L} is a regular language. A similar property does not hold, however, for the context-free languages. We construct a finite-state machine for the language L by showing that a certain type of “marked” finite-state machines are equivalent to the standard finite-state machines. Interestingly, the number of states N in the minimum deterministic finite-state machine for L may be as large as O( N 2), where N is the number of states in the minimum deterministic finite-state machine for L. The motivation for considering the minimal string problem is that it has important applications in “common sense reasoning” of temporal events in artificial intelligence.

Minimal surfaces and strings from spinors a realization of the cartan programme

It is shown ow the old Enneper-Weierstrass integral parametrization ofminimal surfaces inR3 and the Eisenhart ones inR3,1, when expressed through bilinear spinor polynomia, may be considered as deriving from a particularlocal realization of the possibility envisaged by Cartan: to consider ordinary vectors as generated from isotropic planes in complex spaces, in the frame of the bijective Cartan map connecting pure spinor directions to totally null planes in complex spaces. In the case ofR3 the correspondingglobal realization of the Cartan map extends the Enneper-Weierstrass parametrization to the Gauss-conformal map of the minimal surface to S2, which may be identified with the Riemann celestial sphere. Forreal spinors minimal surfaces are substituted bystrings both inR2,1 andR3,1; inR2,1 strings are globally mapped to a torus (inR4). InR3,1 (and its conformal extensions) a prescription is given to obtain strings as integrals of real, bilinear spinor null vectors, from the Enneper-Weierstrass spinor representation of minimal surfaces, through the use of unitary transformations in spinor space which allows its restriction to the real (Majorana spinor-space). It is shown that the Nambu action, or the area of the world surface described by the space-time string, is minimized by the Lagrangian density expressed as a quadrilinear spinor product formally reminding Fermi and Thirring interaction Lagrangians.

Ising systems with random bonds

The thermodynamics at low temperatures of an Ising spin system with random ferromagnetic and antiferromagnetic bonds is considered. It is shown that the replica method leads to expansions for the free energy and correlation functions in terms of the amount of frustration. At low temperatures in two dimensions the Toulouse sum over minimal strings is obtained. An estimate for the entropy at zero temperature and equally probable ferromagnetic and antiferromagnetic bonds is obtained.

The resurgence of instantons in string theory

Nonperturbative eects in string theory are usually associated to D{branes. In many cases it can be explicitly shown that D{brane instantons control the large{order behavior of string perturbation theory, leading to the well{known (2g)! growth of the genus expansion. This paper presents a detailed treatment of nonperturbative solutions in string theory, and their relation to the large{order behavior of perturbation theory, making use of transseries and resurgent analysis. These are powerful techniques addressing general nonperturbative contributions within non{linear systems, which are developed at length herein as they apply to string theory. The cases of topological strings, the Painlev e I equation describing 2d quantum gravity, and the quartic matrix model, are explicitly addressed. These results generalize to minimal strings and general matrix models. It is shown that, in order to completely understand string theory at a fully nonperturbative level, new sectors are required beyond the standard D{brane sector.