Minimal Conformal Matter Research Articles

D-type minimal conformal matter: quantum curves, elliptic Garnier systems, and the 5d descendants

We study the quantization of the 6d Seiberg-Witten curve for D-type minimal conformal matter theories compactified on a two-torus. The quantized 6d curve turns out to be a difference equation established via introducing codimension two and four surface defects. We show that, in the Nekrasov-Shatashvili limit, the 6d partition function with insertions of codimension two and four defects serve as the eigenfunction and eigenvalues of the difference equation, respectively. We further identify the quantum curve of D-type minimal conformal matters with an elliptic Garnier system recently studied in the integrability community. At last, as a concrete consequence of our elliptic quantum curve, we study its RG flows to obtain various quantum curves of 5d Sp(N) + Nf F, Nf ≤ 2N + 5 theories.

Star Shaped Quivers in Four Dimensions.

We discuss 4D Lagrangian descriptions, across dimensions IR duals, of compactifications of the 6D (D, D) minimal conformal matter theory on a sphere with arbitrary number of punctures and a particular value of flux as a gauge theory with a simple gauge group. The Lagrangian has the form of a "star shaped quiver" with the rank of the central node depending on the 6D theory and the number and type of punctures. Using this Lagrangian one can construct across dimensions duals for arbitrary compactifications (any, genus, any number and type of USp punctures, and any flux) of the (D, D) minimal conformal matter gauging only symmetries which are manifest in the ultraviolet.

Two 6D origins of 4D SCFTs: Class S and 6D (1, 0) on a torus

We consider all 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ arising from the compactification of exceptional 6d $(2,0)$ SCFTs on a three-punctured sphere with a simple puncture. We find that each of these 4d theories has another origin as a 6d $(1,0)$ SCFT compactified on a torus, which we check by identifying and comparing the central charges and the flavor symmetry. Each 6d theory is identified with a complex structure deformation of $(\mathfrak{e}_n,\mathfrak{e}_n)$ minimal conformal matter, which corresponds to a Higgs branch renormalization group flow. We find that this structure is precisely replicated by the partial closure of the punctures in the class $\mathcal{S}$ construction. We explain how the plurality of origins makes manifest some aspects of 4d SCFTs, including flavor symmetry enhancements and determining if it is a product SCFT. We further highlight the string theoretic basis for this identification of 4d theories from different origins via mirror symmetry.

Flavored surface defects in 4d $$\mathcal{N}=1$$ N = 1 SCFTs

We discuss supersymmetric surface defects in compactifications of six dimensional minimal conformal matter of type SU(3) and SO(8) to four dimensions. The relevant field theories in four dimensions are N=1 quiver gauge theories with SU(3) and SU(4) gauge groups respectively. The defects are engineered by giving space-time dependent vacuum expectation values to baryonic operators. We find evidence that in the case of SU(3) minimal conformal matter the defects carry SU(2) flavor symmetry which is not a symmetry of the four dimensional model. The simplest case of a model in this class is SU(3) SQCD with nine flavors and thus the results suggest that this admits natural surface defects with SU(2) flavor symmetry. We analyze the defects using the superconformal index and derive analytic difference operators introducing the defects into the index computation. The duality properties of the four dimensional theories imply that the index of the models is a kernel function for such difference operators. In turn, checking the kernel property constitutes an independent check of the dualities and the dictionary between six dimensional compactifications and four dimensional models.

A new 5d description of 6d D-type minimal conformal matter

We propose a new 5d description of the circle-compactified 6d $(D_{N+4}, D_{N+4})$ minimal conformal matter theory which can be approached by the 6d $\mathcal{N}=(1,0)$ $Sp(N)$ gauge theory with $N_f=2N+8$ flavors and one tensor multiplet. Compactifying the brane set-up for the 6d theory, we arrive at a 5-brane Tao diagram for 5d $\mathcal{N}=1$ $SU(N+2)$ theory of the vanishing Chern-Simons level with $2N+8$ flavors. We conjecture that the 6d theory is recovered as the UV fixed point of this 5d theory. We show that the global symmetry of this 5d theory is $SO(4N+16)$ identical to that of the 6d theory by analyzing the 7-brane monodromy. By using the Tao diagram, we also find the instanton fugacity is exactly given by the circle radius. By decoupling flavors in this 5d theory, one can obtain all the 5d $SU(N+2)$ gauge theories of various Chern-Simons levels and corresponding enhanced global symmetries at the 5d UV fixed point.

SO(2,C) invariant discrete gauge states in liouville gravity coupled to minimal conformal matter

We contruct the general formula for a set of discrete gauge states (DGS) in c<1 Liouville theory. This formula reproduces the previously found c=1 DGS in the appropriate limiting case. We also demonstrate the SO(2,C) invariant structure of these DGS in the old covariant quantization of the theory. This is in analogy to the SO(2,C) invariant ring structure of BRST cohomology of the theory.

SCALING DIMENSIONS OF MANIFESTLY GENERALLY COVARIANT OPERATORS IN TWO-DIMENSIONAL QUANTUM GRAVITY

Using (2+∊)-dimensional quantum gravity recently formulated by Kawai, Kitazawa and Ninomiya, we calculate the scaling dimensions of manifestly generally covariant operators in two-dimensional quantum gravity coupled to (p, q) minimal conformal matter. Although the spectrum includes all the scaling dimensions of the scaling operators in the matrix model except the boundary operators, there are also many others which do not appear in the matrix model. We argue that the partial agreement of the scaling dimensions should be considered as accidental and that the operators considered give a new series of operators in two-dimensional quantum gravity.

GENERAL CONSTRUCTION OF PHYSICAL STATES IN 2D QUANTUM GRAVITY

The BRST cohomology of two-dimensional gravity coupled to 0≤c&lt;1 minimal conformal matter is investigated in the conformal gauge. We formulate a general procedure to write down the physical states via the technique of spectral sequences. Paying attention to the ghost sectors, we show that the physical states correspond to the microscopic loops, and the macroscopic loops are described by b-ghosts and c-ghosts respectively. For pure gravity, several physical states are first calculated explicitly and, in addition, the general formula for macroscopic loops is obtained by relating it to a simpler Lie algebra cohomology. We also discuss the local operator algebras of microscopic loops in Liouville theory, the so-called ground ring, in the relation to those from the Schwinger-Dyson equation in the matrix model.

Loops in the curvature matrix model

Macroscopic loop correlators are investigated in the hermitian one matrix model with the potential perturbed by the higher order curvature term. In the phase of smooth surfaces the model is equivalent to the minimal conformal matter coupled to gravity. The properties of the model in the intermediate phase are similar to that of the discretized bosonic string with the central charge C > 1. Loop correlators describe the effect of splitting the random surfaces. It is shown that the properties of the surfaces are changed in the intermediate phase because the perturbation modifies the spectrum of the scaling operators.

TWO-DIMENSIONAL GRAVITY COUPLED TO (p, q) MINIMAL MATTER AND VIRASORO CONSTRAINTS

We treat 2D gravity coupled to (p, q) minimal conformal matter as 2D string theory. Then its massless tachyon field can be identified with the Zp-twisted boson. We derive the Virasoro constraints on the tachyon S-matrix from the conformal invariance of its general interaction.

Ward Identities of Liouville Gravity Coupled to Minimal Conformal Matter

The Ward identities of the Liouville gravity coupled to the minimal conformal matter are investigated. We introduce the pseudo-null fields and the generalized equations of motion, which are classified into series of the Liouville charges. These series have something to do with the W and Virasoro constraints. The pseudo-null fields have non-trivial contributions at the boundaries of the moduli space. We explicitly evaluate the several boundary contributions. Then the structures similar to the W and the Virasoro constraints appearing in the topological and the matrix methods are realized. Although our Ward identities have some different features from the other methods, the solutions of the identities are consistent to the matrix model results.

Matrix models for (p, q) matter on unoriented surfaces

Abstract It is shown that a chain of hermitian matrices with a symmetric matrix attached to one of its ends is exactly soluble and generalizes relations previously found in single symmetric matrix systems. The mathematical structure of the solution suggests that these models have critical points representing the coupling to unoriented random surfaces of minimal conformal matter of a more general type than attainable through single symmetric matrix ensembles.

( p, q) string actions

We investigate a recently proposed nonperturbative formulation of two-dimensional quantum gravity coupled to ( p, q) minimal conformal matter. The coupled differential equations for the partition function summed over topologies are shown to follow from an action principle. The basic action for a ( p, q) model takes the general form S ( p, q) = ∫Res[ Q p/ q+ 1 + Σ k = 0 Σ α = 0 q − 2 t ( k), α Q k + ( α + 1)/ q ], where Q is a qth-order differential operator and the t ( k), α are sources for operator insertions. We illustrate our results with the explicit examples of the Ising (4,3) and tricritical Ising (5,4) models. The action S ( p, q) embodies the essential features of the problem (including the relation to generalized KdV hierarchies) in a most compact form.