2D CFT Research Articles

Brane profiles of non-supersymmetric strings

We connect the indications of 2D CFT for branes in non-supersymmetric strings to actual spacetime profiles, taking the bulk tadpole potentials into account. We find exact solutions for the uncharged branes that spread in the internal intervals of Dudas-Mourad vacua for non-tachyonic orientifolds and heterotic strings. These solutions involve suitable dressings of the uncharged branes of the nine-dimensional tadpole-free theory. Similar exact results for uncharged branes that are transverse to the internal space, or for charged branes, appear more challenging. Nevertheless, we identify their large-distance behavior, which is determined by linearized equations, and show that it is compatible with the CFT analysis in all expected cases. We also provide some hints on the leading back-reaction of the form-fields.

Hawking-Page and entanglement phase transition in 2d CFT on curved backgrounds

The thermodynamics and the entanglement properties of two-dimensional conformal field theories (2d CFTs) on curved backgrounds are studied. By means of conformal mapping we study the equivalent system on flat space governed by the deformed Hamiltonian, which is a spatial integral of the Hamiltonian density modulated by an enveloping function. Focusing on holographic CFTs, we observe Hawking-Page like phase transition for the thermal and the entanglement entropy as we vary the background metric. We also compute the mutual information to study the information theoretic correlation between parts of the curved spacetime. The gravity dual of 2d CFTs on curved background is also discussed.

A hidden 2d CFT for self-dual Yang-Mills on the celestial sphere

Self-dual Yang-Mills theory admits an underlying infinite dimensional symmetry algebra, which has been obtained from mode expansion of Mellin transformed 4d scattering amplitudes and separately, Koszul duality on twistor space. In this paper, we propose to derive an explicit 2d realization of the algebra by performing a particular gauge transformation on the twistor action for self-dual Yang-Mills. The gauge parameter used in the transformation generates pure gauge connections corresponding to large gauge transformations on 4d Minkowski space, which localises part of the twistor action to a ℂℙ1 on ℂ* reduction of twistor space. Under a projection, it can be mapped to the celestial sphere at the light-cone cut of the origin on I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}. Geometrically, this is the common boundary celestial sphere shared by Euclidean AdS3 or Lorentzian dS3 slices of Minkowski space. We comment on the geometric meaning of the derivation from the perspective of minitwistor spaces of the 3d slices embedded in 4d Minkowski space. Using the action functional of this 2d CFT, we compute its stress-energy tensor and central charge. By a further marginal deformation, we calculate correlation functions of current algebra generators purely from the 2d side which reproduce full 4d tree-level form factors.

Heavy states in 3d gravity and 2d CFT

We discuss correlators of light fields in heavy states in AdS3 gravity and holographic 2d CFTs. In the bulk, the propagator of free fields in AdS backgrounds containing a conical defect or a BTZ black hole can be obtained by solving a wave equation, as well as by the method of images. On the boundary, these geometries are sourced by heavy operator insertions, and the propagator is dual to a heavy-light (HHLL) correlator. By matching its expansion in Virasoro blocks to our bulk results, we determine the OPE coefficients of all contributing states in both the s and t channels. In the s channel, these states are excitations of the light field on top of the heavy state, and their OPE coefficients are the amplitudes to create them. The t-channel OPE is dominated by the Virasoro vacuum block, but there is also an infinite family of light two-particle states that contribute to the correlator. The OPE coefficients that couple these states to heavy operators represent their expectation values in heavy backgrounds. We determine them exactly, derive their asymptotic form at large twist, and discuss their behavior near and above the BTZ threshold, where they become thermal one-point functions.

Behind-the-horizon excitations from a single 2d CFT

In this work, we consider the atypical non-equilibrium state found in [1708.06328] which holographically represents a behind-the-horizon excitation in a black hole spacetime. The special feature of this state is that it looks like an equilibrium state when probed by a class of low-energy operators. First, we retrieve this property using the uniformization mapping in the limit of a large central charge, in the process we are able to derive rather than presume approximate thermal physics. Furthermore, in the large-c and high-energy limit, we realize these excitations as elements of the commutant algebra of a GNS-representation of the light operator algebra. Instead of analytically continuing a mixed heavy-light Euclidean correlator to a Lorentzian correlator, we identify the Euclidean correlator as a GNS-linear form and interpret the Lorentzian correlator as a vacuum expectation value of representatives of the light operator algebra on the GNS-vacuum.

Lightcone Modular Bootstrap and Tauberian Theory: A Cardy-Like Formula for Near-Extremal Black Holes

AbstractWe show that for a unitary modular invariant 2D CFT with central charge $$c>1$$ c > 1 and having a nonzero twist gap in the spectrum of Virasoro primaries, for sufficiently large spin J, there always exist spin-J operators with twist falling in the interval $$(\frac{c-1}{12}-\varepsilon ,\frac{c-1}{12}+\varepsilon )$$ ( c - 1 12 - ε , c - 1 12 + ε ) with $$\varepsilon =O(J^{-1/2}\log J)$$ ε = O ( J - 1 / 2 log J ) . We establish that the number of Virasoro primary operators in such a window has a Cardy-like, i.e., $$\exp \left( 2\pi \sqrt{\frac{(c-1)J}{6}}\right) $$ exp 2 π ( c - 1 ) J 6 growth. A similar result is proven for a family of holographic CFTs with the twist gap growing linearly in c and a uniform boundedness condition, in the regime $$J\gg c^3\gg 1$$ J ≫ c 3 ≫ 1 . From the perspective of near-extremal rotating BTZ black holes (without electric charge), our result is valid when the Hawking temperature is much lower than the “gap temperature.”

Stress tensors of 3d Carroll CFTs

We discuss kinematical features of conformal Carroll field theories in three dimensions (3d). Conformal extension of Carroll algebra is infinite dimensional even in 3d unlike its relativistic counterpart, and hence 3d Carroll CFTs share similarities with 2d CFT. We provide a construction for the conserved charges for Carrollian CFTs and an expression for stress tensor OPEs consistent with the algebra of charges. We discuss a free field model where these symmetries are realised and explicitly compute the OPEs verifying our construction. In addition we comment on the possibility of extending the conformal symmetries to diffeomorphisms of spatial slice for these Carrollian theories.

The Singularities of Selberg- and Dotsenko–Fateev-Like Integrals

We discuss the meromorphic continuation of certain hypergeometric integrals modeled on the Selberg integral, including the 3-point and 4-point functions of BPZ’s minimal models of 2D CFT as described by Felder & Silvotti and Dotsenko & Fateev (the “Coulomb gas formalism”). This is accomplished via a geometric analysis of the singularities of the integrands. In the case that the integrand is symmetric (as in the Selberg integral itself) or, more generally, what we call “DF-symmetric,” we show that a number of apparent singularities are removable, as required for the construction of the minimal models via these methods.

AdS3/RMT2 duality

We introduce a framework for quantifying random matrix behavior of 2d CFTs and AdS3 quantum gravity. We present a 2d CFT trace formula, precisely analogous to the Gutzwiller trace formula for chaotic quantum systems, which originates from the SL(2, ℤ) spectral decomposition of the Virasoro primary density of states. An analogy to Berry’s diagonal approximation allows us to extract spectral statistics of individual 2d CFTs by coarse-graining, and to identify signatures of chaos and random matrix universality. This leads to a necessary and sufficient condition for a 2d CFT to display a linear ramp in its coarse-grained spectral form factor.Turning to gravity, AdS3 torus wormholes are cleanly interpreted as diagonal projections of squared partition functions of microscopic 2d CFTs. The projection makes use of Hecke operators. The Cotler-Jensen wormhole of AdS3 pure gravity is shown to be extremal among wormhole amplitudes: it is the minimal completion of the random matrix theory correlator compatible with Virasoro symmetry and SL(2, ℤ)-invariance. We call this MaxRMT: the maximal realization of random matrix universality consistent with the necessary symmetries. Completeness of the SL(2, ℤ) spectral decomposition as a trace formula allows us to factorize the Cotler-Jensen wormhole, extracting the microscopic object ZRMT(τ) from the coarse-grained product. This captures details of the spectrum of BTZ black hole microstates. ZRMT(τ) may be interpreted as an AdS3 half-wormhole. We discuss its implications for the dual CFT and modular bootstrap at large central charge.

Celestial holography and AdS3/CFT2 from a scaling reduction of twistor space

Celestial amplitudes obtained from Mellin transforming 4d momentum space scattering amplitudes contain distributional delta functions, hindering the application of conventional CFT techniques. In this paper, we propose to bypass this problem by recognizing Mellin transforms as integral transforms projectivizing certain components of the angular momentum. It turns out that the Mellin transformed wavefunctions in the conformal primary basis can be regarded as representatives of certain cohomology classes on the minitwistor space of the hyperbolic slices of 4d Minkowski space. Geometrically, this amounts to treating 4d Minkowski space as the embedding space of AdS3. By considering scattering of such on-shell wavefunctions on the projective spinor bundle ℙ\U0001d54a of Euclidean AdS3, we bypass the difficulty of the distributional properties of celestial correlators using the traditional AdS3/CFT2 dictionary and find conventional 2d CFT correlators for the scaling reduced Yang-Mills theory living on the hyperbolic slices. In the meantime, however, one is required to consider action functionals on the auxiliary space ℙ\U0001d54a, which introduces additional difficulties. Here we provide a framework to work on the projective spinor bundle of hyperbolic slices, obtained from a careful scaling reduction of the twistor space of 4d Minkowski spacetime.

Three-point functions of conserved currents in 4D CFT: General formalism for arbitrary spins

Three-point functions of conserved currents in 4D CFT: General formalism for arbitrary spins

The regularization of Dotsenko–Fateev integrals

We discuss the regularization of certain hypergeometric integrals appearing in 2D CFT, a step needed in the construction of the BPZ minimal models via the Coulomb gas formalism. The method is a generalization of Pochhammer’s regularization of the Euler Beta-function. The constructions of the relevant homology classes are inspired by a recent singular-geometric analysis of the Dotsenko–Fateev integrand.

State dependence of Krylov complexity in 2d CFTs

We compute the Krylov Complexity of a light operator O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}L in an eigenstate of a 2d CFT at large central charge c. The eigenstate corresponds to a primary operator O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}H under the state-operator correspondence. We observe that the behaviour of K-complexity is different (either bounded or exponential) depending on whether the scaling dimension of O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}H is below or above the critical dimension hH = c/24, marked by the 1st order Hawking-Page phase transition point in the dual AdS3 geometry. Based on this feature, we hypothesize that the notions of operator growth and K-complexity for primary operators in 2d CFTs are closely related to the underlying entanglement structure of the state in which they are computed, thereby demonstrating explicitly their state-dependent nature. To provide further evidence for our hypothesis, we perform an analogous computation of K-complexity in a model of free massless scalar field theory in 2d, and in the integrable 2d Ising CFT, where there is no such transition in the spectrum of states.

Integrable crosscap states: from spin chains to 1D Bose gas

The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.

Fake Z

Recently introduced connections between quantum codes and Narain CFTs provide a simple ansatz to express a modular-invariant function Zleft(tau, overline{tau}right) in terms of a multivariate polynomial satisfying certain additional properties. These properties include algebraic identities, which ensure modular invariance of Zleft(tau, overline{tau}right) , and positivity and integrality of coefficients, which imply positivity and integrality of the \U0001d532(1)n × \U0001d532(1)n character expansion of Zleft(tau, overline{tau}right) . Such polynomials come naturally from codes, in the sense that each code of a certain type gives rise to the so-called enumerator polynomial, which automatically satisfies all necessary properties, while the resulting Zleft(tau, overline{tau}right) is the partition function of the code CFT — the Narain theory unambiguously constructed from the code. Yet there are also “fake” polynomials satisfying all necessary properties, that are not associated with any code. They lead to Zleft(tau, overline{tau}right) satisfying all modular bootstrap constraints (modular invariance and positivity and integrality of character expansion), but whether they are partition functions of any actual CFT is unclear. We consider the group of the six simplest fake polynomials and denounce the corresponding Z’s as fake: we show that none of them is the torus partition function of any Narain theory. Moreover, four of them are not partition functions of any unitary 2d CFT; our analysis for other two is inconclusive. Our findings point to an obvious limitation of the modular bootstrap approach: not every solution of the full set of torus modular bootstrap constraints is due to an actual CFT. In the paper we consider six simple examples, keeping in mind that thousands more can be constructed.

Toward random tensor networks and holographic codes in CFT

In holographic CFTs satisfying eigenstate thermalization, there is a regime where the operator product expansion can be approximated by a random tensor network. The geometry of the tensor network corresponds to a spatial slice in the holographic dual, with the tensors discretizing the radial direction. In spherically symmetric states in any dimension and more general states in 2d CFT, this leads to a holographic error-correcting code, defined in terms of OPE data, that can be systematically corrected beyond the random tensor approximation. The code is shown to be isometric for light operators outside the horizon, and non-isometric inside, as expected from general arguments about bulk reconstruction. The transition at the horizon occurs due to a subtle breakdown of the Virasoro identity block approximation in states with a complex interior.

Chaos and operator growth in 2d CFT

We study the out-of-time-ordered correlator (OTOC) in a zero temperature 2d large-c CFT under evolution by a Liouvillian composed of the Virasoro generators. A bound was conjectured in [1] on the growth of the OTOC set by the Krylov complexity which is a measure of operator growth. The latter grows as an exponential of time with exponent 2α, which sets an upper bound on the Lyapunov exponent, ΛL≤ 2α. We find that for a two dimensional zero temperature CFT, the OTOC decays exponentially with a Lyapunov exponent which saturates this bound. We show that these Virasoro generators form the modular Hamiltonian of the CFT with half space traced out. Therefore, evolution by this modular Hamiltonian gives rise to thermal dynamics in a zero temperature CFT. Leveraging the thermal dynamics of the system, we derive this bound in a zero temperature CFT using the analyticity and boundedness properties of the OTOC.

Asymptotic density of states in 2d CFTs with non-invertible symmetries

It is known that the asymptotic density of states of a 2d CFT in an irreducible representation ρ of a finite symmetry group G is proportional to (dim ρ)2. We show how this statement can be generalized when the symmetry can be non-invertible and is described by a fusion category mathcal{C} . Along the way, we explain what plays the role of a representation of a group in the case of a fusion category symmetry; the answer to this question is already available in the broader mathematical physics literature but not yet widely known in hep-th. This understanding immediately implies a selection rule on the correlation functions, and also allows us to derive the asymptotic density.

Celestial conformal colliders

We start by observing that the light-ray operators featured in the conformal collider literature are celestial primaries. This allows us to rephrase the corresponding 4D CFT correlators as probing a conformally soft matter sector of the 2D celestial CFT (CCFT). To demonstrate the utility of this perspective we show how the recent w1+∞ symmetry observed in CCFT suggests a natural extension of the conformal collider operators.

On triality defects in 2d CFT

We consider the triality fusion category discovered in the c = 1 Kosterlitz-Thouless theory [1]. We analyze this fusion category using the tools from the group theoretical fusion category and compute the simple lines, fusion rules and F-symbols. We then studied the physical implication of this fusion category including deriving the spin selection rule, computing the asymptotic density of states of irreducible representations of the fusion category symmetries, and analyzing its anomaly and constraints under the renormalization group flow. There is another set of F-symbols for the fusion categories with the same fusion rule known in the literature [2]. We find these two solutions are different as they lead to different spin selection rules. This gives a complete list of the fusion categories with the same fusion rule by the classification result in [3].