Coset Conformal Field Theories Research Articles

Z3 and (×Z3)3 symmetry protected topological paramagnets

We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by (×Z3)3 ≡ Z3 × Z3 × Z3 symmetry and smaller Z3 symmetry. We derive microscopic models for the gapless edge, uncover their symmetries and analyze the conformal properties. We study the properties of the gapless edge by employing the numerical density-matrix renormalization group (DMRG) simulation and exact diagonalization. We discuss the corresponding conformal field theory, its central charge, and the scaling dimension of the corresponding primary field. We argue, that the low energy limit of our edge modes defined by the SUk(3)/SUk(2) coset conformal field theory with the level k = 2. The discussed two-dimensional models realize a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.

Holography in de Sitter Space via Chern-Simons Gauge Theory.

In this Letter, we propose a holographic duality for classical gravity on a three-dimensional de Sitter space. We first show that a pair of SU(2) Chern-Simons gauge theories reproduces the classical partition function of Einstein gravity on a Euclidean de Sitter space, namely S^{3}, when we take the limit where the level k approaches -2. This implies that the conformal field theory (CFT) dual of gravity on a de Sitter space at the leading semiclassical order is given by an SU(2) Wess-Zumino-Witten model in the large central charge limit k→-2. We give another evidence for this in the light of known holography for coset CFTs. We also present a higher spin gravity extension of our duality.

Novel integrable interpolations

A novel class of integrable σ-models interpolating between exact coset conformal field theories in the IR and hyperbolic spaces in the UV is constructed. We demonstrate the relation to the asymptotic limit of λ-deformed models for cosets of non-compact groups. An integrable model interpolating between two spacetimes with cosmological and black hole interpretations and exact conformal field theory descriptions is also provided. In the process of our work, a new zoom-in limit, distinct from the well known non-Abelian T-duality limit, is found.

Parton construction of particle-hole-conjugate Read-Rezayi parafermion fractional quantum Hall states and beyond

The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling $\nu = k/(k+2)$. Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted "$\bar{2}^{k}1^{k+1}$," and show that they lie in the same universality classes as the particle-hole-conjugate RR ("anti-RR") states. Our analytical results imply that a $(U(1)_{k+1} \times U(2k)_{-1})/(SU(k)_{-2} \times U(1)_{-1})$ coset conformal field theory describes the edge excitations of the $\bar{2}^{k}1^{k+1}$ state, suggesting non-trivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence "$\bar{n}\bar{2}^{2}1^{4}$," with $n=1,2,3$, to describe the FQH states observed at $\nu=2+1/2,~2+2/5$ and $2+3/8$.

Chiral ring of strange metals: The multicolor limit

The low energy limit of a dense 2D adjoint QCD is described by a family of N=(2,2) supersymmetric coset conformal field theories. In previous work we constructed chiral primaries for a small number N<6 of colors. Our aim in the present note is to determine the chiral ring in the multicolor limit where N is sent to infinity. We shall find that chiral primaries are labeled by partitions and identify the ring they generate as the ring of Schur polynomials. Our findings impose strong constraints on the possible dual description through string theory in an AdS3 compactification.

Arc spaces and the vertex algebra commutant problem

Given a vertex algebra V and a subalgebra A⊂V, the commutant Com(A,V) is the subalgebra of V which commutes with all elements of A. This construction is analogous to the ordinary commutant in the theory of associative algebras, and is important in physics in the construction of coset conformal field theories. When A is an affine vertex algebra, Com(A,V) is closely related to rings of invariant functions on arc spaces. We find strong finite generating sets for a family of examples where A is affine and V is a βγ-system, bc-system, or bcβγ-system.

On spectrum of ILW hierarchy in conformal field theory II: coset CFT’s

We study integrable structure of the coset conformal field theory and define the system of Integrals of Motion which depends on external parameters. This system can be viewed as a quantization of the ILW type hierarchy. We propose a set of Bethe anzatz equations for its spectrum.

$${\mathcal{N} = 2}$$ N = 2 Quiver Gauge Theories on A-type ALE Spaces

We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in $\mathcal{N}=2$ gauge theories on ALE spaces based on quiver varieties and the minimal resolution $X_k$ of the $A_{k-1}$ toric singularity $\mathbb{C}^2/\mathbb{Z}_k$, in light of their recently conjectured duality with two-dimensional coset conformal field theories. We review and elucidate the rigorous constructions of gauge theories for a particular family of ALE spaces, using their relation to the cohomology of moduli spaces of framed torsion free sheaves on a suitable orbifold compactification of $X_k$. We extend these computations to generic $\mathcal{N}=2$ superconformal quiver gauge theories, obtaining in these instances new constraints on fractional instanton charges, a rigorous proof of the Nekrasov master formula, and new quantizations of Hitchin systems based on the underlying Seiberg-Witten geometry.

Coset conformal field theory and instanton counting on ℂ2/ℤ p

Abstract We study conformal field theory with the symmetry algebra $ \mathcal{A}\left( {2,\ p} \right)={{{\widehat{\mathfrak{gl}}{(n)_2}}} \left/ {{\widehat{\mathfrak{gl}}{{{\left( {n-p} \right)}}_2}}} \right.} $ . In order to support the conjecture that this algebra acts on the moduli space of instantons on ℂ2/ℤ p , we calculate the characters of its representations and check their coincidence with the generating functions of the fixed points of the moduli space of instantons. We show that the algebra $ \mathcal{A}\left( {2,\ p} \right) $ can be realized in two ways. The first realization is connected with the cross-product of p Virasoro and p Heisenberg algebras: $ {{\mathcal{H}}^p} $ × Vir p . The second realization is connected with: $ {{\mathcal{H}}^p} \times \widehat{\mathfrak{sl}}{(p)_2}\times \left( {\widehat{\mathfrak{sl}}{(2)_p}\times {{{\widehat{\mathfrak{sl}}{(2)_{n-p }}}} \left/ {{\widehat{\mathfrak{sl}}{(2)_n}}} \right.}} \right) $ . The equivalence of these two realizations provides the non-trivial identity for the characters of $ \mathcal{A}\left( {2,\ p} \right) $ . The moduli space of instantons on ℂ2/ℤ p admits two different compactifications. This leads to two different bases for the representations of $ \mathcal{A}\left( {2,\ p} \right) $ . We use this fact to explain the existence of two forms of the instanton pure partition functions.

Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity

We continue our study of the AGT correspondence between instanton counting on C^2/Z_p and Conformal field theories with the symmetry algebra A(r,p). In the cases r=1, p=2 and r=2, p=2 this algebra specialized to: A(1,2)=H+sl(2)_1 and A(2,2)=H+sl(2)_2+NSR. As the main tool we use a new construction of the algebra A(r,2) as the limit of the toroidal gl(1) algebra for q,t tend to -1. We claim that the basis of the representation of the algebra A(r,2) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q,t go to -1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the $\mathcal{N}=1$ Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q,t tend to -1.

Instanton Moduli Spaces and Bases in Coset Conformal Field Theory

The recently proposed relation between conformal field theories in two dimensions and supersymmetric gauge theories in four dimensions predicts the existence of the distinguished basis in the space of local fields in CFT. This basis has a number of remarkable properties: one of them is the complete factorization of the coefficients of the operator product expansion. We consider a particular case of the U(r) gauge theory on \({\mathbb{C}^{2}/\mathbb{Z}_{p}}\) which corresponds to a certain coset conformal field theory and describe the properties of this basis. We argue that in the case p = 2, r = 2 there exist different bases. We give an explicit construction of one of them. For another basis we propose the formula for matrix elements.

Strange metals in one spatial dimension

We consider $1+1$ dimensional $SU(N)$ gauge theory coupled to a multiplet of massive Dirac fermions transforming in the adjoint representation of the gauge group. The only global symmetry of this theory is a $U(1)$ associated with the conserved Dirac fermion number, and we study the theory at variable, nonzero densities. The high density limit is characterized by a deconfined Fermi surface state with Fermi wave vector equal to that of free gauge-charged fermions. Its low energy fluctuations are described by a coset conformal field theory with central charge $c=({N}^{2}\ensuremath{-}1)/3$ and an emergent $\mathcal{N}=(2,2)$ supersymmetry: the $U(1)$ fermion number symmetry becomes an $R$-symmetry. We determine the exact scaling dimensions of the operators associated with Friedel oscillations and pairing correlations. For $N&gt;2$, we find that the symmetries allow relevant perturbations to this state. We discuss aspects of the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, and its possible dual description in ${\mathrm{AdS}}_{3}$ involving string theory or higher-spin gauge theory. We also discuss the low density limit of the theory by computing the low lying bound state spectrum of the large $N$ gauge theory numerically at zero density, using discretized light cone quantization.

Schramm-Loewner evolution martingales in coset conformal field theory

Schramm-Loewner evolution (SLE) and conformal field theory (CFT) are popular and widely used instruments to study critical behavior of two-dimensional models, but they use different objects. While SLE has natural connection with lattice models and is suitable for strict proofs, it lacks computational and predictive power of conformal field theory. To provide a way for the concurrent use of SLE and CFT, CFT correlation functions, which are martingales with respect to SLE, are considered. A relation between parameters of Schramm-Loewner evolution on coset space and algebraic data of coset conformal field theory is revealed. The consistency of this approach with the behavior of parafermionic and minimal models is tested. Coset models are connected with off-critical massive field theories and implications of SLE are discussed.

Integrability in anyonic quantum spin chains via a composite height model

Recently, properties of collective states of interacting non-Abelian anyons have attracted considerable attention. We study an extension of the ``golden chain model,'' where two- and three-body interactions are competing. Upon fine tuning the interaction, the model is integrable. This provides an additional integrable point of the model, on top of the integrable point, when the three-body interaction is absent. To solve the model, we construct a new, integrable height model, in the spirit of the restricted solid-on-solid model solved by Andrews et al. [J. Stat. Phys. 35, 193 (1984)]. The heights in our model live on both the sites and links of the square lattice. The model is solved by means of the corner transfer matrix method. We find a connection between local height probabilities and characters of a conformal field theory governing the critical properties at the integrable point. In the antiferromagnetic regime, the criticality is described by the ${Z}_{k}$ parafermion conformal field theory, while the $\frac{\text{su}{(2)}_{1}\ifmmode\times\else\texttimes\fi{}\text{su}{(2)}_{1}\ifmmode\times\else\texttimes\fi{}\text{su}{(2)}_{k\ensuremath{-}2}}{\text{su}{(2)}_{k}}$ coset conformal field theory describes the ferromagnetic regime.

Parafermionic Liouville field theory and instantons on ALE spaces

In this paper we study the correspondence between the $\hat{\textrm{su}}(n)_{k}\oplus \hat{\textrm{su}}(n)_{p}/\hat{\textrm{su}}(n)_{k+p}$ coset conformal field theories and $\mathcal{N}=2$ SU(n) gauge theories on $\mathbb{R}^{4}/\mathbb{Z}_{p}$. Namely we check the correspondence between the SU(2) Nekrasov partition function on $\mathbb{R}^{4}/\mathbb{Z}_{4}$ and the conformal blocks of the $S_{3}$ parafermion algebra (in $S$ and $D$ modules). We find that they are equal up to the U(1)-factor as it was in all cases of AGT-like relations. Studying the structure of the instanton partition function on $\mathbb{R}^4/\mathbb{Z}_p$ we also find some evidence that this correspondence with arbitrary $p$ takes place up to the U(1)-factor.

Exactly marginal parafermions

We describe several families of non-unitary coset conformal field theories that possess truly marginal couplings. These generalize the known examples of Wess-Zumino-Witten models on supergroups such as PSU(n|n) or OSP(2n+2|2n). Our extension includes coset space sigma models, affine Toda theories or Gross-Neveu models which are believed to arise in certain limits.

A twisted non-compact elliptic genus

We give a detailed path integral derivation of the elliptic genus of a supersymmetric coset conformal field theory, further twisted by a global U(1) symmetry. It gives rise to a Jacobi form in three variables, which is the modular completion of a mock modular form. The derivation provides a physical interpretation to the non-holomorphic part as arising from a difference in spectral densities for the continuous part of the right-moving bosonic and fermionic spectrum. The spectral asymmetry can also be read off directly from the reflection amplitudes of the theory. By performing an orbifold, we show how our twisted elliptic genus generalizes an existing example.

The non-compact elliptic genus: mock or modular

We analyze various perspectives on the elliptic genus of non-compact supersymmetric coset conformal field theories with central charge larger than three. We calculate the holomorphic part of the elliptic genus via a free field description of the model, and show that it agrees with algebraic expectations. The holomorphic part of the elliptic genus is directly related to an Appell-Lerch sum and behaves anomalously under modular transformation properties. We analyze the origin of the anomaly by calculating the elliptic genus through a path integral in a coset conformal field theory. The path integral codes both the holomorphic part of the elliptic genus, and a non-holomorphic remainder that finds its origin in the continuous spectrum of the non-compact model. The remainder term can be shown to agree with a function that mathematicians introduced to parameterize the difference between mock theta functions and Jacobi forms. The holomorphic part of the elliptic genus thus has a path integral completion which renders it non-holomorphic and modular.

Critical points in coupled Potts models and critical phases in coupled loop models

We show how to couple two critical Q-state Potts models to yield a new self-dual critical point. We also present strong evidence of a dense critical phase near this critical point when the Potts models are defined in their completely packed loop representations. In the continuum limit, the new critical point is described by an SU(2) coset conformal field theory, while in this limit of the critical phase, the two loop models decouple. Using a combination of exact results and numerics, we also obtain the phase diagram in the presence of vacancies. We generalize these results to coupling two Potts models at different Q.

D-branes in unoriented non-critical strings and duality in SO(N) and Sp(N) gauge theories

We exhibit exact conformal field theory descriptions of SO(N) and Sp(N) pairs of Seiberg-dual gauge theories within string theory. The N=1 gauge theories with flavour are realized as low energy limits of the worldvolume theories on D-branes in unoriented non-critical superstring backgrounds. These unoriented backgrounds are obtained by constructing exact crosscap states in the SL(2,R)/U(1) coset conformal field theory using the modular bootstrap method. Seiberg duality is understood by studying the behaviour of the boundary and crosscap states under monodromy in the closed string parameter space.