Quasi-characters are vector-valued modular functions having an integral, but not necessarily positive, q-expansion. Using modular differential equations, a complete classification has been provided in arXiv:1810.09472 for the case of two characters. These in turn generate all possible admissible characters, of arbitrary Wronskian index, in order two. Here we initiate a study of the three-character case. We conjecture several infinite families of quasi-characters and show in examples that their linear combinations an generate admissible characters with arbitrarily large Wronskian index. The structure is completely different from the order two case, and the novel coset construction of arXiv:1602.01022 plays a key role in discovering the appropriate families. Using even unimodular lattices, we construct some explicit three-character CFT corresponding to the new admissible characters.
We investigate the Poincaré approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)$_k$ WZW models provide unitary examples for which the Poincare series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT's sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all …
We investigate the limit of minimal model conformal field theories where the central charge approaches one. We conjecture that this limit is described by a non-rational CFT of central charge one. The limiting theory is different from the free boson but bears some resemblance to Liouville theory. Explicit expressions for the three point functions of bulk fields are presented, as well as a set of conformal boundary states. We provide analytic and numerical arguments in support of the claim that this data forms a consistent CFT.
Two dimensional conformal field theories with central charge one are discussed. After a short review of theories based on one free boson, a different CFT is described, which is obtained as a limit of minimal models.
Given two two-dimensional conformal field theories, a domain wall -- or defect line -- between them is called invertible if there is another defect with which it fuses to the identity defect. A defect is called topological if it is transparent to the stress tensor. A conformal isomorphism between the two CFTs is a linear isomorphism between their state spaces which preserves the stress tensor and is compatible with the operator product expansion. We show that for rational CFTs there is a one-to-one correspondence between invertible topological defects and conformal isomorphisms if both preserv…
This is an introduction to the microscopic techniques of non-rational bulk and boundary conformal field theory which are needed to describe strings moving in non-compact curved backgrounds. The latter arise e.g. in the context of AdS/CFT-like dualities and for studies of time-dependent processes. After a general outline of the central concepts, we focus on two specific but rather prototypical models: Liouville field theory and the 2D cigar. Rather than following the historical path, the presentation attempts to be systematic and self-contained.
We construct a map between a class of codes over $F_4$ and a family of non-rational Narain CFTs. This construction is complementary to a recently introduced relation between quantum stabilizer codes and a class of rational Narain theories. From the modular bootstrap point of view we formulate a polynomial ansatz for the partition function which reduces modular invariance to a handful of algebraic easy-to-solve constraints. For certain small values of central charge our construction yields optimal theories, i.e. those with the largest value of the spectral gap.
In this article, we work out the microscopic statistical foundation of the supergravity description of the simplest 1/2 BPS sector in the AdS(5)/CFT(4). Then, all the corresponding supergravity observables are related to thermodynamical observables, and General Relativity is understood as a mean-field theory. In particular, and as an example, the Superstar is studied and its thermodynamical properties clarified.
Recently, the modular linear differential equation (MLDE) for level-two congruence subgroups $Γ_θ, Γ^{0}(2)$ and $Γ_0(2)$ of $\text{SL}_2(\mathbb{Z})$ was developed and used to classify the fermionic rational conformal field theories (RCFT). Two character solutions of the second-order fermionic MLDE without poles were found and their corresponding CFTs are identified. Here we extend this analysis to explore the landscape of three character fermionic RCFTs obtained from the third-order fermionic MLDE without poles. Especially, we focus on a class of the fermionic RCFTs whose Neveu-Schwarz secto…
Among (conformal) quantum field theories, the rational conformal field theories are singled out by the fact that their correlators can be constructed from a modular tensor category C with a distinguished object, a symmetric special Frobenius algebra A in C, via the so-called TFT-construction. These correlators satisfy in particular all factorization constraints, which involve gluing homomorphisms relating correlators of world sheets of different topology. We review the action of the gluing homomorphisms and discuss the implications of the factorization constraints for boundary conditions. The …
We explore the formulation of non-rational 2D quantum gravity in terms of a chiral CFT on a Riemann surface associated with the target space. The CFT in question is constructed as the collective theory for a matrix chain, which is dual to a statistical height model on dynamical triangulations. The heights are associated with the sheets of the Riemann surface, which represents an infinite branched cover of the spectral plane. We consider two examples of height models: the SOS model and the semi-restricted SOS (SRSOS) model, in which the heights are restricted from below. Both models are describ…
Motivated by the search for rational points in moduli spaces of two-dimensional conformal field theories, we investigate how points with enhanced symmetry algebras are distributed there. We first study the bosonic sigma-model with $S^1$ target space in detail and uncover hitherto unknown features. We find for instance that the vanishing of the twist gap, though true for the $S^1$ example, does not automatically follow from enhanced symmetry points being dense in the moduli space. We then explore the supersymmetric sigma-model on K3 by perturbing away from the torus orbifold locus. Though we do…
We address the problem of computing the tachyon correlation functions in Liouville gravity with generic (non-rational) matter central charge c<1. We consider two variants of the theory. The first is the conventional one in which the effective matter interaction is given by the two matter screening charges. In the second variant the interaction is defined by the Liouville dressings of the non-trivial vertex operator of zero dimension. This particular deformation, referred to as "diagonal'', is motivated by the comparison with the discrete approach, which is the subject of a subsequent paper. In…
We study the sewing constraints for rational two-dimensional conformal field theory on oriented surfaces with possibly non-empty boundary. The boundary condition is taken to be the same on all segments of the boundary. The following uniqueness result is established: For a solution to the sewing constraints with nondegenerate closed state vacuum and nondegenerate two-point correlators of boundary fields on the disk and of bulk fields on the sphere, up to equivalence all correlators are uniquely determined by the one-, two,- and three-point correlators on the disk. Thus for any such theory every…
Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in 4-dimensional Minkowski space-time. We consider in detail a model of a neutral scalar field $ϕ$ of dimension 2. It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of $ϕ$ is shown to coincide with a simpler one, generated by a bilocal scalar field $V(x_1,x…
We extend the scope of the Klein bottle entropy, originally introduced by [Tu, Phys. Rev. Lett. 119, 261603 (2017)] in the rational conformal field theory (CFT), to the compactified boson CFT, which are relevant to the studies of Luttinger liquids. We first review the Klein bottle entropy in rational CFT and discuss details of how to extract the Klein bottle entropy from lattice models using the example of the transverse field Ising model. We then go beyond the scope of rational CFT and study the Klein bottle entropy $\ln g$ in the compactified boson CFT, which turns out to have a straightforw…
We use simple current techniques and their relation to orbifolds with discrete torsion for studying the (0,2) CFT/geometry duality with non-rational internal N=2 SCFTs. Explicit formulas for the charged spectra of heterotic SO(10) GUT models are computed in terms of their extended Poincaré polynomials and the complementary Poincaré polynomial which can be computed in terms of the elliptic genera. While non-BPS states contribute to the charged spectrum, their contributions can be determined also for non-rational cases. For model building, with generalizations to SU(5) and SM gauge groups, one c…
It is well-known that the quantum double $D(N\subset M)$ of a finite depth subfactor $N\subset M$, or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. Thus should arise in conformal field theory. We show that for every subfactor $N\subset M$ with index $[M:N]<4$ the quantum double $D(N\subset M)$ is realized as the representation category of a completely rational conformal net. In particular, the quantum double of $E_6$ can be realized as a $\mathbb Z_2$-simple current extension of $\mathrm{SU}(2)_{10}\times \mathrm{Spin}(11)_1$ and thus …
We study simple examples of ensemble-averaged holography in free compact boson CFTs with rational values of the radius squared. These well-known rational CFTs have an extended chiral algebra generated by three currents. We consider the modular average of the vacuum character in these theories, which results in a weighted average over all modular invariants. In the simplest case, when the chiral algebra is primitive (in a sense we explain), the weights in this ensemble average are all equal. In the non-primitive case the ensemble weights are governed by a semigroup structure on the space of mod…
G. M. T. Watts derived that in two dimensional critical percolation the crossing probability Pi_hv satisfies a fifth order differential equation which includes another one of third order whose independent solutions describe the physically relevant quantities 1, Pi_h, Pi_hv. We will show that this differential equation can be derived from a level three null vector condition of a rational c=-24 CFT and motivate how this solution may be fitted into known properties of percolation.
We revisit certain aspects of thermalization in 2D CFT. In particular, we consider similarities and differences between the time dependence of correlation functions in various states in rational and non-rational CFTs. We also consider the distinction between global and local thermalization and explain how states obtained by acting with a diffeomorphism on the ground state can appear locally thermal, and we review why the time-dependent expectation value of the energy-momentum tensor is generally a poor diagnostic of global thermalization. Since all 2D CFTs have an infinite set of commuting con…
We study topological defect lines in two-dimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting one-dimensional phase boundaries can be used to extract symmetries and order-disorder dualities of the CFT. The case of central charge c=4/5, for which there are two inequivalent world sheet phases corresponding to the tetra-critical Ising model and the critical three-states Pot…
We propose a two-parameter family of modular invariant partition functions of two-dimensional conformal field theories (CFTs) holographically dual to pure three-dimensional gravity in anti de Sitter space. Our two parameters control the central charge, and the representation of $SL(2,\mathbb{Z})$. At large central charge, the partition function has a gap to the first nontrivial primary state of $\frac{c}{24}$. As the $SL(2,\mathbb{Z})$ representation dimension gets large, the partition function exhibits some of the qualitative features of an irrational CFT. This, for instance, is captured in t…
Two-dimensional conformal field theory (CFT) has several sources: the search for simple examples of quantum field theory, the description of surface critical phenomena, the study of (super)string vacua. In the present overview of the subject we emphasize the role of CFT in bridging the gap between mathematics and quantum field theory and discuss some new physical concepts that emerged in the study of CFT models: anomalous dimensions, rational CFT, braid group statistics. In an aside, at the end of the paper, we share the misgivings, recently expressed by Penrose, about some dominant trends in …
These pedagogical lectures present some material, classical or more recent, on (Rational) Conformal Field Theories and their general setting ``in the bulk'' or in the presence of a boundary. Two well posed problems are the classification of modular invariant partition functions and the determination of boundary conditions consistent with conformal invariance. It is shown why the two problems are intimately connected and how graphs -ADE Dynkin diagrams and their generalizations- appear in a natural way.
We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a - not necessarily semisimple - modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: …
We review how modular categories, and commutative and non-commutative Frobenius algebras arise in rational conformal field theory. For Euclidean CFT we use an approach based on sewing of surfaces, and in the Minkowskian case we describe CFT by a net of operator algebras.
In rational conformal field theory, the Verlinde formula computes the fusion coefficients from the modular S-transformations of the characters of the chiral algebra's representations. Generalising this formula to logarithmic models has proven rather difficult for a variety of reasons. Here, a recently proposed formalism (arXiv:1303.0847 [hep-th]) for the modular properties of certain classes of logarithmic theories is reviewed, and refined, using simple examples. A formalism addressing fusion rules in simple current extensions is also reviewed as a means to tackle logarithmic theories to which…
Some general properties of perturbed (rational) CFT in the background metric of symmetric 2D sphere of radius $R$ are discussed, including conformal perturbation theory for the partition function and the large $R$ asymptotic. The truncated conformal space scheme is adopted to treat numerically perturbed rational CFT's in the spherical background. Numerical results obtained for the scaling Lee-Yang model lead to the conclusion that the partition function is an entire function of the coupling constant. Exploiting this analytic structure we are able to describe rather precisely the ``experimental…
Two-dimensional rational CFT are characterised by an integer $\ell$, related to the number of zeroes of the Wronskian of the characters. For two-character RCFT's with $\ell<6$ there is a finite number of theories and most of these are classified. Recently it has been shown that for $\ell \ge 6$ there are infinitely many admissible characters that could potentially describe CFT's. In this note we examine the $\ell=6$ case, whose central charges lie between 24 and 32, and propose a classification method based on cosets of meromorphic CFT's. We illustrate the method using theories on Kervaire lat…
We construct a one-parameter family of lattice models starting from a two-dimensional rational conformal field theory on a torus with a regular lattice of holes, each of which is equipped with a conformal boundary condition. The lattice model is obtained by cutting the surface into triangles with clipped-off edges using open channel factorisation. The parameter is given by the hole radius. At finite radius, high energy states are suppressed and the model is effectively finite. In the zero-radius limit, it recovers the CFT amplitude exactly. In the touching hole limit, one obtains a topological…
The Wen-Wu c=-2 model describes the d-wave paired singlet factor of the Haldane-Rezayi (quantum Hall) wave function in terms of an SU(2) doublet of dimension 1 fermions. The resulting CFT involves an infinite sequence of Virasoro primary fields. The superalgebra generated by these fields, on the other hand, admits a set of ground state representations giving rise to a rational c=-2 CFT. We compute the corresponding characters as well as the characters of the bosonic subalgebra in the Z_2 twisted sector and write down a modular invariant partition function for the ensuing 2-dimensional conforma…
In this paper we study a connection between Jackiw-Teitelboim (JT) gravity on two-dimensional anti de-Sitter spaces and a semiclassical limit of $c<1$ two-dimensional string theory. The world-sheet theory of the latter consists of a space-like Liouville CFT coupled to a non-rational CFT defined by a time-like Liouville CFT. We show that their actions, disk partition functions and annulus amplitudes perfectly agree with each other, where the presence of boundary terms plays a crucial role. We also reproduce the boundary Schwarzian theory from the Liouville theory description. Then, we identify …
Under the assumption that degenerate fields exist, diagonal CFTs such as Liouville theory can be solved analytically using the conformal bootstrap method. Here we generalize this approach to non-diagonal CFTs, i.e. CFTs whose primary fields have nonzero conformal spins. Assuming generic values of the central charge, we find that the non-diagonal sector of the spectrum must be parametrized by two integer numbers. We then derive and solve the equations that determine how three- and four-point structure constants depend on these numbers. In order to test these results, we numerically check crossi…
Globally conformal invariant quantum field theories in a D-dimensional space-time (D even) have rational correlation functions and admit an infinite number of conserved (symmetric traceless) tensor currents. In a theory of a scalar field of dimension D-2 they were demonstrated to be generated by bilocal normal products of free massless scalar fields with an O(N), U(N), or Sp(2N) (global) gauge symmetry [BNRT]. Recently, conformal field theories "with higher spin symmetry" were considered for D=3 in [MZ] where a similar result was obtained (exploiting earlier study of CFT correlators). We sugge…
We solve the recently proposed T- and Y-systems (Hirota equation) for the exact spectrum of AdS/CFT in the strong coupling scaling limit for an arbitrary quasiclassical string state. The corresponding T-functions appear to be super-characters of the SU(2,2|4) group in unitary representations with a highest weight, with the classical AdS5xS5 superstring monodromy matrix as the group element. We propose a concise first Weyl-type formula for these characters and show that they correctly reproduce the results of quasiclassical one-loop quantization in all sectors of the superstring, under some nat…
We study asymptotics of three point coefficients (light-light-heavy) and two point correlators in heavy states in unitary, compact $2$D CFTs. We prove an upper and lower bound on such quantities using numerically assisted Tauberian techniques. We obtain an optimal upper bound on the spectrum of operators appearing with fixed spin from the OPE of two identical scalars. While all the CFTs obey this bound, rational CFTs come close to saturating it. This mimics the scenario of bounds on asymptotic density of states and thereby pronounces an universal feature in asymptotics of 2D CFTs. Next, we cla…
This is part one of a two-part work that relates two different approaches to two-dimensional open-closed rational conformal field theory. In part one we review the definition of a Cardy algebra, which captures the necessary consistency conditions of the theory at genus 0 and 1. We investigate the properties of these algebras and prove uniqueness and existence theorems. One implication is that under certain natural assumptions, every rational closed CFT is extendable to an open-closed CFT. The relation of Cardy algebras to the solutions of the sewing constraints is the topic of part two.
Two different conformal field theories can be joined together along a defect line. We study such defects for the case where the conformal field theories on either side are single free bosons compactified on a circle. We concentrate on topological defects for which the left- and right-moving Virasoro algebras are separately preserved, but not necessarily any additional symmetries. For the case where both radii are rational multiples of the self-dual radius we classify these topological defects. We also show that the isomorphism between two T-dual free boson conformal field theories can be descr…
The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy `identity brane'). We apply the general method to the c_1,p triplet models and reproduce the previously known bulk theory for p=2 at c=-2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-prod…
Rational CFT's are classified by an integer $\ell$, the number of zeroes of the Wronskian of their characters in moduli space. For $\ell=0$ they satisfy non-singular modular-invariant differential equations, while for $\ell>0$ the corresponding equations have singularities. We survey CFT's with two characters and $\ell=0,2,3,4$ and verify the consistency, at the level of characters, of some candidate theories with $\ell\ne 0$. For $\ell=2$ there are seven consistents sets of characters. We identify specific combinations of level-1 current algebras that are potential symmetries of the correspon…
Complex rational Ruijsenaars model. The two-particle case
Beijing Institute of Mathematical Sciences and Applications（北京数学与应用科学研究所）；Laboratory of Theoretical Physics, JINR（约旦联合核子研究所理论物理实验室）；Yerevan Physics Institute（埃里扬物理研究所）；National Research University Higher School of Economics（国立研究型高等经济大学）
Some relations between families of two-character CFTs are explained using a slightly generalised coset construction, and the underlying theories (whose existence was only conjectured based on the modular differential equation) are constructed. The same method also gives rise to interesting new examples of CFTs with three and four characters.
We study symmetries of quantum field theories involving topologically distinct sectors of the field space. To exhibit these symmetries we define special gauge invariant observables, which we call the $qq$-characters. In the context of the BPS/CFT correspondence, using these observables, we derive an infinite set of Dyson-Schwinger-type relations. These relations imply that the supersymmetric partition functions in the presence of $Ω$-deformation and defects obey the Ward identities of two dimensional conformal field theory and its $q$-deformations. The details will be discussed in the companio…
In the modular linear differential equation (MLDE) approach to classifying rational conformal field theories (RCFTs) both the MLDE and the RCFT are identified by a pair of non-negative integers $\textbf{[n,l]}$. $\mathbf{n}$ is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and $\mathbf{l}$, the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study $\textbf{[3,0]}$ and $\textbf{[3,2]}$ MLDEs in order to classify the corresponding CFTs. We reduce the problem to a "finite" problem…
Gauge origami is a generalized supersymmetric gauge theory defined on several intersecting space-time components. It provides a systematic way to consider generalizations of instantons. In this thesis, we explore the gauge origami system in $\mathbb{R}^{1,1}\times \mathbb{C}^{4}$ and its BPS/CFT correspondence. String theoretically, instantons of the gauge origami system arise from D0-branes bound to D$(2p)$-branes wrapping cycles in $\mathbb{C}^{4}$. The low energy theory of the D0-branes is understood as an $\mathcal{N}=2$ supersymmetric quiver quantum mechanical system and the Witten index …
We study topological defect lines in two character rational conformal field theories. Among them one set of two character theories are commutant pairs in $E_{8,1}$ conformal field theory. Using these defect lines we construct defect partition function in the $E_8$ theory. We find that the defects preserve only a part of the $E_8$ current algebra symmetry. We also determine the defect partition function in $c=24$ CFTs using these defects lines of 2 character theories and we find that, with appropriate choice of commutant pairs, these defects preserve all current algebra symmetries of c = 24 CFT…
We consider the late time behavior of the analytically continued partition function $Z(β+ it) Z(β- it)$ in holographic $2d$ CFTs. This is a probe of information loss in such theories and in their holographic duals. We show that each Virasoro character decays in time, and so information is not restored at the level of individual characters. We identify a universal decaying contribution at late times, and conjecture that it describes the behavior of generic chaotic $2d$ CFTs out to times that are exponentially large in the central charge. It was recently suggested that at sufficiently late times…
We show the vertex operator formalism for the quiver gauge theory partition function and the $qq$-character of highest-weight module on quiver, both associated with the integral over the quiver variety.
