Non-unitary Minimal Models Research Articles

Three-Dimensional Topological Field Theories and Nonunitary Minimal Models.

We find an intriguing relation between a class of three-dimensional nonunitary topological field theories (TFTs) and Virasoro minimal models M(2,2r+3) with r≥1. The TFTs are constructed by topologically twisting 3D N=4 superconformal field theories (SCFTs) of rank-0, i.e., having zero-dimensional Coulomb and Higgs branches. We present ultraviolet (UV) field theory descriptions of the SCFTs with manifest N=2 supersymmetry, which we argue is enhanced to N=4 in the infrared. From the UV description, we compute various partition functions of the TFTs and reproduce some basic properties of the minimal models, such as their characters and modular matrices. We expect more general correspondence between topologically twisted 3d N=4 rank-0 SCFTs and 2D nonunitary rational conformal field theories.

Resurgence, conformal blocks, and the sum over geometries in quantum gravity

In two dimensional conformal field theories the limit of large central charge plays the role of a semi-classical limit. Certain universal observables, such as conformal blocks involving the exchange of the identity operator, can be expanded around this classical limit in powers of the central charge c. This expansion is an asymptotic series, so — via the same resurgence analysis familiar from quantum mechanics — necessitates the existence of non-perturbative effects. In the case of identity conformal blocks, these new effects have a simple interpretation: the CFT must possess new primary operators with dimension of order the central charge. This constrains the data of CFTs with large central charge in a way that is similar to (but distinct from) the conformal bootstrap. We study this phenomenon in three ways: numerically, analytically using Zamolodchikov’s recursion relations, and by considering non-unitary minimal models with large (negative) central charge. In the holographic dual to a CFT2, the expansion in powers of c is the perturbative loop expansion in powers of ћ. So our results imply that the graviton loop expansion is an asymptotic series, whose cure requires the inclusion of new saddle points in the gravitational path integral. In certain cases these saddle points have a simple interpretation: they are conical excesses, particle-like states with negative mass which are not in the physical spectrum but nevertheless appear as non-manifold saddle points that control the asymptotic behaviour of the loop expansion. This phenomenon also has an interpretation in SL(2, ℝ) Chern-Simons theory, where the non-perturbative effects are associated with the non-Teichmüller component of the moduli space of flat connections.

Ginzburg-Landau description and emergent supersymmetry of the (3, 8) minimal model

A pair of the 2D non-unitary minimal models M(2, 5) is known to be equivalent to a variant of the M(3, 10) minimal model. We discuss the RG flow from this model to another non-unitary minimal model, M(3, 8). This provides new evidence for its previously proposed Ginzburg-Landau description, which is a ℤ2 symmetric theory of two scalar fields with cubic interactions. We also point out that M(3, 8) is equivalent to the (2, 8) superconformal minimal model with the diagonal modular invariant. Using the 5-loop results for theories of scalar fields with cubic interactions, we exhibit the 6 − ϵ expansions of the dimensions of various operators. Their extrapolations are in quite good agreement with the exact results in 2D. We also use them to approximate the scaling dimensions in d = 3, 4, 5 for the theories in the M(3, 8) universality class.

Multicriticality in Yang-Lee edge singularity

In this paper we study the non-unitary deformations of the two-dimensional Tricritical Ising Model obtained by coupling its two spin ℤ2 odd operators to imaginary magnetic fields. Varying the strengths of these imaginary magnetic fields and adjusting correspondingly the coupling constants of the two spin ℤ2 even fields, we establish the presence of two universality classes of infrared fixed points on the critical surface. The first class corresponds to the familiar Yang-Lee edge singularity, while the second class to its tricritical version. We argue that these two universality classes are controlled by the conformal non-unitary minimal models mathcal{M} (2, 5) and mathcal{M} (2, 7) respectively, which is supported by considerations based on PT symmetry and the corresponding extension of Zamolodchikov’s c-theorem, and also verified numerically using the truncated conformal space approach. Our results are in agreement with a previous numerical study of the lattice version of the Tricritical Ising Model [1]. We also conjecture the classes of universality corresponding to higher non-unitary multicritical points obtained by perturbing the conformal unitary models with imaginary coupling magnetic fields.

Thermodynamic Bethe Ansatz past turning points: the (elliptic) sinh-Gordon model

We analyze the Thermodynamic Bethe Ansatz (TBA) for various integrable S-matrices in the context of generalized T overline{mathrm{T}} deformations. We focus on the sinh-Gordon model and its elliptic deformation in both its fermionic and bosonic realizations. We confirm that the determining factor for a turning point in the TBA, interpreted as a finite Hagedorn temperature, is the difference between the number of bound states and resonances in the theory. Implementing the numerical pseudo-arclength continuation method, we are able to follow the solutions to the TBA equations past the turning point all the way to the ultraviolet regime. We find that for any number k of resonances the pair of complex conjugate solutions below the turning point is such that the effective central charge is minimized. As k → ∞ the UV effective central charge goes to zero as in the elliptic sinh-Gordon model. Finally we uncover a new family of UV complete integrable theories defined by the bosonic counterparts of the S-matrices describing the Φ1,3 integrable deformation of non-unitary minimal models mathcal{M} 2,2n+3.

The semiclassical gravitational path integral and random matrices (toward a microscopic picture of a dS2 universe)

We study the genus expansion on compact Riemann surfaces of the gravitational path integral {mathcal{Z}}_{mathrm{grav}}^{(m)} in two spacetime dimensions with cosmological constant Λ > 0 coupled to one of the non-unitary minimal models ℳ2m − 1, 2. In the semiclassical limit, corresponding to large m, {mathcal{Z}}_{mathrm{grav}}^{(m)} admits a Euclidean saddle for genus h ≥ 2. Upon fixing the area of the metric, the path integral admits a round two-sphere saddle for h = 0. We show that the OPE coefficients for the minimal weight operators of ℳ2m − 1, 2 grow exponentially in m at large m. Employing the sewing formula, we use these OPE coefficients to obtain the large m limit of the partition function of ℳ2m − 1, 2 for genus h ≥ 2. Combining these results we arrive at a semiclassical expression for {mathcal{Z}}_{mathrm{grav}}^{(m)} . Conjecturally, {mathcal{Z}}_{mathrm{grav}}^{(m)} admits a completion in terms of an integral over large random Hermitian matrices, known as a multicritical matrix integral. This matrix integral is built from an even polynomial potential of order 2m. We obtain explicit expressions for the large m genus expansion of multicritical matrix integrals in the double scaling limit. We compute invariant quantities involving contributions at different genera, both from a matrix as well as a gravity perspective, and establish a link between the two pictures. Inspired by the proposal of Gibbons and Hawking relating the de Sitter entropy to a gravitational path integral, our setup paves a possible path toward a microscopic picture of a two-dimensional de Sitter universe.

Matrix integrals & finite holography

We explore the conjectured duality between a class of large N matrix integrals, known as multicritical matrix integrals (MMI), and the series (2m − 1, 2) of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an S2 topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an S2 and a T2 topology using BRST cohomology considerations. Matrix integrals support this finiteness.

Thermal correlation functions of KdV charges in 2D CFT

Two dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function. We determine their modular transformation properties, give explicit expressions in a number of cases, and give an expression for an arbitrary correlation function which is determined up to a finite number of functions of the central charge. We show that these modular differential operators annihilate the characters of the (2m + 1, 2) family of non-unitary minimal models. We also show that the distribution of KdV charges becomes sharply peaked at large level.

Massive ODE/IM correspondence and nonlinear integral equations for -type modified affine Toda field equations

The massive ordinary differential equation/integrable model (ODE/IM) correspondence is a relation between the linear problem associated with modified affine Toda field equations and 2D massive integrable models. We study the massive ODE/IM correspondence for the -type modified affine Toda field equations. Based on the ψ-system satisfied by the solutions of the linear problem, we derive the Bethe ansatz equations and determine the asymptotic behavior of the Q-functions for large values of the spectral parameter. We derive the non-linear integral equations for the Q-functions from the Bethe ansatz equations. We compute the effective central charge in the UV limit, which is identified by one of the non-unitary WAr minimal models when the solution has trivial monodromy around the origin of the complex plane.

Two-dimensional RCFT’s without Kac-Moody symmetry

Using the method of modular-invariant differential equations, we classify a family of Rational Conformal Field Theories with two and three characters having no Kac-Moody algebra. In addition to unitary and non-unitary minimal models, we find "dual" theories whose characters obey bilinear relations with those of the minimal models to give the Moonshine Module. In some ways this relation is analogous to cosets of meromorphic CFT's. The theory dual in this sense to the Ising model has central charge 47/2 and is related to the Baby Monster Module.

Entanglement entropy from corner transfer matrix in Forrester–Baxter non-unitary RSOS models

Using a corner transfer matrix approach, we compute the bipartite entanglement Rényi entropy in the off-critical perturbations of non-unitary conformal minimal models realised by lattice spin chains Hamiltonians related to the Forrester–Baxter RSOS models (Bianchini et al 2015 J. Stat. Mech. P03010) in regime III. This allows to show on a set of explicit examples that the Rényi entropies for non-unitary theories rescale near criticality as the logarithm of the correlation length with a coefficient proportional to the effective central charge. This complements a similar result, recently established for the size rescaling at the critical point (Bianchini et al 2015 J. Phys. A: Math. Theor. 48 04FT01), showing the expected agreement of the two behaviours. We also compute the first subleading unusual correction to the scaling behaviour, showing that it is expressible in terms of expansions of various fractional powers of the correlation length, related to the differences between the conformal dimensions of fields in the theory and the minimal conformal dimension. Finally, a few observations on the limit leading to the off-critical logarithmic minimal models of Pearce and Seaton (2012 J. Stat. Mech. P09014) are put forward.

RSOS quantum chains associated with off-critical minimal models and parafermions

We consider the φ1,3 off-critical perturbation of the general non-unitary minimal models where 2 ⩽ m ⩽ m′ and m, m′ are coprime and t measures the departure from criticality corresponding to the φ1,3 integrable perturbation. We view these models as the continuum scaling limit in the ferromagnetic Regime III of the Forrester–Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. We also consider the RSOS models in the antiferromagnetic Regime II related in the continuum scaling limit to parfermions with n = m′ − 2. Using an elliptic Yang–Baxter algebra of planar tiles encoding the allowed face configurations, we obtain the Hamiltonians of the associated quantum chains defined as the logarithmic derivative of the transfer matrices with periodic boundary conditions. The transfer matrices and Hamiltonians act on a vector space of paths on the Am′−1 Dynkin diagram whose dimension is counted by generalized Fibonacci numbers.

Coset construction of logarithmic minimal models: branching rules and branching functions

Working in the Virasoro picture, it is argued that the logarithmic minimal models can be extended to an infinite hierarchy of logarithmic conformal field theories at higher fusion levels . From the lattice, these theories are constructed by fusing together n × n elementary faces of the appropriate models. It is further argued that all of these logarithmic theories are realized as diagonal cosets where n is the integer fusion level and is a fractional level. These cosets mirror the cosets of the higher fusion level minimal models of the form , but are associated with certain reducible representations. We present explicit branching rules for characters in the form of multiplication formulas arising in the logarithmic limit of the usual Goddard–Kent–Olive coset construction of the non-unitary minimal models . The limiting branching functions play the role of Kac characters for the theories.

Finite-Size scaling at Yang-Lee singularities of 2D discrete spin models

We numerically study the Hamiltonian limits of the two dimensional (2D) Ising and 3-state Potts models in complex magneticfields. From the Phenomenological Renormalization Group, we find the criticalfield values associated with the Yang-Lee singularity (YLS) in these models. We also determine the low-lying part of the excitation spectrum at the YLS. Finally, we compare the resulting patterns of energy levels to predictions for conformal non-unitary minimal models.

Non-unitary minimal models, Bailey's Lemma and N=1,2 Superconformal algebras

Using the Bailey flow construction, we derive character identities for the N=1 superconformal models SM(p′,2p+p′) and SM(p′,3p′−2p), and the N=2 superconformal model with central charge c=3 from the nonunitary minimal models M(p,p′). A new Ramond sector character formula for representations of N=2 superconformal algebras with central element c=3 is given.

New families of flows between two-dimensional conformal field theories

We present evidence for the existence of infinitely-many new families of renormalisation group flows between the nonunitary minimal models of conformal field theory. These are associated with perturbations by the $\phi_{21}$ and $\phi_{15}$ operators, and generalise a family of flows discovered by Martins. In all of the new flows, the finite-volume effective central charge is a non-monotonic function of the system size. The evolution of this effective central charge is studied by means of a nonlinear integral equation, a massless variant of an equation recently found to describe certain massive perturbations of these same models. We also observe that a similar non-monotonicity arises in the more familiar $\phi_{13}$ perturbations, when the flows induced are between nonunitary minimal models.

Excited states in some simple perturbed conformal field theories

The method of analytic continuation is used to find exact integral equations for a selection of finite-volume energy levels for the non-unitary minimal models M 2,2 N+3 perturbed by their ϕ 13 operators. The N = 2 case is studied in particular detail. Along the way, we find a number of general results which should be relevant to the study of excited states in other models.

A new RSOS restriction of the Zhiber-Mikhailov-Shabat model and Φ(1,5) perturbations of non-unitary minimal models

The RSOS restriction of the Zhiber-Mikhailov-Shabat (ZMS) model is investigated. It is shown that in addition to the usual RSOS restriction, corresponding to Φ (1,5) and Φ (2,1) perturbations of minimal CFT, there is another one which yields Φ (1,5) perturbations of non-unitary minimal models. The new RSOS restriction is carried out and the particular case of the minimal models M (3,10), M (3,14) and M (3,16) is discussed in detail. In the first two cases, while the mass spectra of the two RSOS restrictions are the same, the bootstrap systems and the detailed amplitudes are different. In the third case, even the spectra of the two RSOS restrictions are different. In addition, for M (3,10) an interpretation in terms of the tensor product of two copies of M (2,5) is given.

Turbulent Two-Dimensional Magnetohydrodynamics and Conformal Field Theory

We show that an infinite number of non-unitary minimal models may describe two dimensional turbulent magnetohydrodynamics (MHD), both in the presence and absence of the Alf 'ven effect. We argue that the existence of a critical dynamical index results in the Alf 'ven effect or equivalently the equipartition of energy. We show that there are an infinite number of conserved quantities in 2D−MHDturbulent systems both in the limit of vanishing the viscocities and in force free case. In the force free case, using the non-unitary minimal modelM2, 7we derive the correlation functions for the velocity stream function and magnetic flux function. Generalising this simple model we find the exponents of the energy spectrum in the inertial range for a class of conformal field theories.

INTEGRABLE PERTURBATIONS OF CFT WITH COMPLEX PARAMETER: THE M3/5 MODEL AND ITS GENERALIZATIONS

By using the thermodynamic Bethe ansatz approach, we give evidence of the existence of both massive and massless behaviors for the ϕ2,1 perturbation of the M3,5 nonunitary minimal model, thus resolving apparent contradictions in the previous literature. The two behaviors correspond to changing the perturbing bare coupling constant from real values to imaginary ones. Generalizations of this picture to the whole class of nonunitary minimal models Mp,2p±1, perturbed by their least relevant operator, lead to a cascade of flows similar to that of unitary minimal models perturbed by ϕ1,3. Various aspects and generalizations of this phenomenon and the links with the Izergin–Korepin model are discussed.