Multicritical Models Research Articles

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.

Unitary matrix models and random partitions: Universality and multi-criticality

The generating functions for the gauge theory observables are often represented in terms of the unitary matrix integrals. In this work, the perturbative and non-perturbative aspects of the generic multi-critical unitary matrix models are studied by adopting the integrable operator formalism, and the multi-critical generalization of the Tracy-Widom distribution in the context of random partitions. We obtain the universal results for the multi-critical model in the weak and strong coupling phases. The free energy of the instanton sector in the weak coupling regime, and the genus expansion of the free energy in the strong coupling regime are explicitly computed and the universal multi-critical phase structure of the model is explored. Finally, we apply our results in concrete examples of supersymmetric indices of gauge theories in the large N limit.

Functional renormalization and the MS¯ scheme

Working with scalar field theories, we discuss choices of regulator that, inserted in the functional renormalization group equation, reproduce the results of dimensional regularization at one and two loops. The resulting flow equations can be seen as nonperturbative extensions of the $\overline{\text{MS}}\,$ scheme. We support this claim by recovering all the multicritical models in two dimensions. We discuss a possible generalization to any dimension. Finally, we show that the method also preserves nonlinearly realized symmetries, which is a definite advantage with respect to other regulators.

Multicritical Landau-Potts field theory

We investigate a perturbatively renormalizable $S_{q}$ invariant model with $N=q-1$ scalar field components below the upper critical dimension $d_c=\frac{10}{3}$. Our results hint at the existence of multicritical generalizations of the critical models of spanning random clusters and percolations in three dimensions. We also discuss the role of our multicritical model in a conjecture that involves the separation of first and second order phases in the $(d,q)$ diagram of the Potts model.

Critical Wess-Zumino models with four supercharges in the functional renormalization group approach

We analyze the $\mathcal{N}=1$ supersymmetric Wess-Zumino model dimensionally reduced to the $\mathcal{N}=2$ supersymmetric model in three Euclidean dimensions. As in the original model in four dimensions and the $\mathcal{N}=(2,2)$ model in two dimensions the superpotential is not renormalized. This property puts severe constraints on the non-trivial fixed-point solutions, which are studied in detail. We admit a field-dependent wave function renormalization that in a geometric language relates to a K\"ahler metric. The K\"ahler metric is not protected by supersymmetry and we calculate its explicit form at the fixed point. In addition we determine the exact quantum dimension of the chiral superfield and several critical exponents of interest, including the correction-to-scaling exponent $\omega$, within the functional renormalization group approach. We compare the results obtained at different levels of truncation, exploring also a momentum-dependent wave function renormalization. Finally we briefly describe a tower of multicritical models in continuous dimensions.

Functional perturbative RG and CFT data in the epsilon -expansion

We show how the use of standard perturbative RG in dimensional regularization allows for a renormalization group-based computation of both the spectrum and a family of coefficients of the operator product expansion (OPE) for a given universality class. The task is greatly simplified by a straightforward generalization of perturbation theory to a functional perturbative RG approach. We illustrate our procedure in the epsilon -expansion by obtaining the next-to-leading corrections for the spectrum and the leading corrections for the OPE coefficients of Ising and Lee-Yang universality classes and then give several results for the whole family of renormalizable multi-critical models phi ^{2n}. Whenever comparison is possible our RG results explicitly match the ones recently derived in CFT frameworks.

New universality class in three dimensions: The critical Blume-Capel model

We study the Blume-Capel universality class in $d=\frac{10}{3}-\epsilon$ dimensions. The RG flow is extracted by looking at poles in fractional dimension of three loop diagrams using $\overline{\rm MS}$. The theory is the only nontrivial universality class which admits an expansion to three dimensions with $\epsilon=\frac{1}{3}<1$. We compute the relevant scaling exponents and estimate some of the OPE coefficients to the leading order. Our findings agree with and complement CFT results. Finally we discuss a family of nonunitary multicritical models which includes the Lee-Yang and Blume-Capel classes as special cases.

Leading CFT constraints on multi-critical models in d > 2

We consider the family of renormalizable scalar QFTs with self-interacting potentials of highest monomial ϕm below their upper critical dimensions {d}_c=frac{2m}{m-2} , and study them using a combination of CFT constraints, Schwinger-Dyson equation and the free theory behavior at the upper critical dimension. For even integers m ≥ 4 these theories coincide with the Landau-Ginzburg description of multi-critical phenomena and interpolate with the unitary minimal models in d = 2, while for odd m the theories are non-unitary and start at m = 3 with the Lee-Yang universality class. For all the even potentials and for the Lee-Yang universality class, we show how the assumption of conformal invariance is enough to compute the scaling dimensions of the local operators ϕk and of some families of structure constants in either the coupling’s or the ϵ-expansion. For all other odd potentials we express some scaling dimensions and structure constants in the coupling’s expansion.

Multimeson Yukawa interactions at criticality

The critical behavior of a relativistic $\mathbb{Z}_2$-symmetric Yukawa model at zero temperature and density is discussed for a continuous number of fermion degrees of freedom and of spacetime dimensions, with emphasis on the role played by multi-meson exchange in the Yukawa sector. We argue that this should be generically taken into account in studies based on the functional renormalization group, either in four-dimensional high-energy models or in lower-dimensional condensed-matter systems. By means of the latter method, we describe the generation of multi-critical models in less then three dimensions, both at infinite and finite number of flavors. We also provide different estimates of the critical exponents of the chiral Ising universality class in three dimensions for various field contents, from a couple of massless Dirac fermions down to the supersymmetric theory with a single Majorana spinor.

Multicritical continuous random trees

We introduce generalizations of Aldous’s Brownian continuous random treeas scaling limits for multicritical models of discrete trees. These discretemodels involve trees with fine-tuned vertex-dependent weights ensuring akth root singularity in their generating function. The scaling limitinvolves continuous trees with branching points of order up tok+1. We derive explicit integral representations for the average profile of thiskth order multicritical continuous random tree, as well as for its history distributions measuringmulti-point correlations. The latter distributions involve non-positive universal weights atthe branching points together with fractional derivative couplings. We prove universality byrederiving the same results within a purely continuous axiomatic approach based on theresolution of a set of consistency relations for the multi-point correlations. The averageprofile is shown to obey a fractional differential equation whose solution involveshypergeometric functions and matches the integral formula of the discrete approach.

Nonperturbative 2D gravity, punctured spheres and Θ-vacua in string theories

We consider a model of 2D gravity with the coefficient of the Euler characteristic having an imaginary part π 2 . This is equivalent to introduce a Θ-vacuum structure in the genus expansion whose effect is to convert the expansion into a series of alternating signs, presumably Borel summable. We show that the specific heat of the model has a physical behaviour. It can be represented nonperturbatively as a series in terms of integrals over moduli spaces of punctured spheres and the sum of the series can be rewritten as a unique integral over a suitable moduli spaces of punctured spheres and the an explicit realization à la Friedan-Shenker of 2D quantum gravity. We conjecture that the expansion in terms of punctures and the genus expansion can be derived using the Duistermaat-Heckman theorem. We briefly analyze expansions in terms of punctured spheres also for multicritical models.

Two-dimensional semi-infinite multicritical Ising models

The order-parameter correlation functions of the Z2-invariant multicritical points in the unitary minimal series of conformal field theory are derived for a semi-infinite plane constrained to fixed or free boundary conditions. These yield the corresponding universal surface exponents which distinguish the behaviour between even- and odd-critical models. We also make explicit the interplay between duality and boundary conditions.

Multicritical phases of the O( n) model on a random lattice

We exhibit the multicritical phase structure of the loop gas model on a random surface. The dense phase is reconsidered, with special attention paid to the topological points g = 1/ p. This phase is complementary to the dilute and higher multicritical phases in the sense that dense models contain the same spectrum of bulk operators (found in the continuum by Lian and Zuckerman) but a different set of boundary operators. This difference illuminates the well-known ( p, q) asymmetry of the matrix chain models. Higher multicritical phases are constructed, generalizing both Kazakov's multicritical models as well as the known dilute phase models. They are quite likely related to multicritical polymer theories recently considered independently by Saleur and Zamolodchikov. Our results may be of help in defining such models on flat honeycomb lattices; an unsolved problem in polymer theory. The phase boundaries correspond again to “topological” points with g = p/1 integer, which we study in some detail. Two qualitatively different types of critical points are discovered for each such g. For the special point g = 2 we demonstrate that the dilute phase O(−2) model does not correspond to the Parisi-Sourlas model, a result likely to hold as well for the flat case. Instead it is proven that the first multicritical O(−2) point possesses the Parisi-Sourlas supersymmetry.

Integrable systems in topological field theory

Integrability of the system of PDE for dependence on coupling parameters of the (tree-level) primary partition function in massive topological field theories, being imposed by the associativity of the perturbed primary chiral algebra, is proved. In the conformal case it is shown that all the topological field theories are classified as solutions of a universal high-order Painlevé-type equation. Another integrable hierarchy (of systems of hydrodynamic type) is shown to describe coupling to gravity of the matter sector of any topological field theory. Different multicritical models with the given structure of primary correlators are identified with particular self-similar solutions of the hierarchy. The partition function of any of the models is calculated as the corresponding τ-function of the hierarchy.

A well-defined multicritical series in hermitian one-matrix models

A new multicritical series is described for hermitian one-matrix models with even potential U( φ) = Σ d i = 1 g 2 i φ 2 i . It is characterized by a negative g 2 and a positive g 2 d . The latter condition ensures that the matrix model is well-defined. The corresponding value of γ 0 is −1/( d − 1). The specific heat is F″ = 1 4 u 2 , where u satisfies a string equation of the mKdV type. For the case d=3, a numerical solution is presented which is pole free and completely determined by asymptotic boundary conditions. This new sequence, as well as the other two sequences previously found for hermitian one-matrix models with even potentials, are shown to be special cases of a two-parameter family of multicritical models characterized by an engenvalue distribution with n ⩽ 2 cuts.

MATRIX MODELS WITH NONEVEN POTENTIALS

We study examples of Hermitian one-matrix models with even and odd terms present in the potential. A definition of criticality is presented which in these cases leads to multicritical models falling into the same universality classes as those of the purely even potentials. We also show that (in our examples) for polynomial potentials ending in odd powers (unbounded), the coupling constants, in addition to their expected real critical values, also admit critical values which alternate between imaginary/real values in the odd/even terms. We find that, remarkably, the ensuing statistical models are insensitive to the real/imaginary nature of these critical values. This feature may be of relevance in the recently studied connection between matrix models and the moduli space of Riemann surfaces.

CRITICAL MODELS COUPLED TO TWO-DIMENSIONAL GRAVITY IN THE CONTINUUM

We discuss minimal conformal field theories M(p,q) coupled to 2-dimensional gravity in the continuum. We find a transformation that enables us to relate all models with p=2 to each other. We identify the Fock space in each model and calculate correlation functions. We thus show that the k-th multi-critical matrix model corresponds to the non-unitary minimal model with q=2k−1.

Loop equations and Virasoro constraints in non-perturbative two-dimensional quantum gravity

We give a derivation of the loop equation for two-dimensional gravity from the KdV equations and the string equation of the one-matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their extension to multi-matrix models. For the multi-critical models the loop equation naturally singles out the operators corresponding to the primary fields of the minimal models.

Bigeneric nonperturbative strings

The zero dimensional hermitian matrix models with asymmetric potentials include a family of bigeneric ( p, q) multicritical models which have nontrivial double-scaling continuum limits. These exhibit more general critical behaviors than the k multi-critical models that occur for even potentials. They are related to a simpler construction which we call the k bigeneric models. The specific heat of the kth bigeneric model coincides with that of the kth multicritical model for odd genus, but vanishes for all even genus. We provide a number of explicit examples of bigeneric string equations.