Symmetric orbifold CFTs contain twist operators that can join and split copies of the CFT, leading to the creation of pairs from the vacuum. In this paper, we study the pair creation processes involving four twist-2 operators. In addition to the pair creation previously observed purely in the left or right moving sectors, we find a novel mixing between left and right movers during pair creation. This phenomenon arises from nontrivial monodromy conditions that originate from a genus-one covering surface, where left and right movers become coupled through the torus.
Since many thermodynamic properties of black holes are universal, the thermodynamics of their holographic duals should be universal too. We show how this universality is exhibited in the example of symmetric orbifolds of general two dimensional CFTs. We discuss the free energies and phase diagrams of such theories and show that they are indeed universal in the large N limit. We also comment on the implications of our results for the classification of CFTs that can have an interpretation as holographic duals to gravity theories on AdS(3).
We consider general 2D orbifold CFTs of the form M^N/S_N, with M a target space manifold and S_N the symmetric group, and generalize the Lunin-Mathur covering space technique in two ways. First, we consider excitations of twist operators by modes of fields that are not twisted by that operator, and show how to account for these excitations when computing correlation functions in the covering space. Second, we consider non-twist sector operators and show how to include the effects of these insertions in the covering space. We work two examples, one using a simple bosonic CFT, and one using the …
We study the 2D symmetric orbifold CFT of two copies of free bosons. The twist operator can join the two separated copies in the untwisted sector into a joined copy in the twisted sector. Starting with a state with any number of quanta in the untwisted sector, the state in the twisted sector obtained by the action of the twist operator can be computed by using the covering map method. We develop a new method to compute the effect of a twist operator by using the Bogoliubov ansatz and conformal symmetry. This may lead to more efficient tools to compute correlation functions involving twist oper…
We study BPS bound states of little strings in a limit where they realise monopole strings in five dimensional gauge theories. The latter have gauge group $U(M)^N$ and arise from compactification of $(1,0)$ little string theories of type $A_{M-1} \times A_{N-1}$. We find evidence that the partition function of a certain subclass of monopole strings of charge $(k,\ldots,k)$ ($k\geq 1$) is expressible as the partition function of a symmetric orbifold sigma model, whose target space is precisely the symmetric product of the moduli space of monopoles with charge $(1, \ldots, 1)$.
In the framework of AdS$_3$/ CFT$_2$ correspondence, we present a systematic analysis of the late time thermalization of a two dimensional CFT state created by insertion of small number of heavy operators on the vacuum. We show that at late Lorentzian time, the universal features of this thermalization are solely captured by the eigenvalues of the monodromy matrix corresponding to the solutions of the uniformization equation. We discuss two different ways to extract the monodromy eigenvalues while bypassing the need for finding explicitly the full monodromy matrix - first, using a monodromy pr…
We explore the OPE of certain twist operators in symmetric product ($S_N$) orbifold CFTs, extending our previous work arXiv:1804.01562 to the case of $\mathcal{N}=(4,4)$ supersymmetry. We consider a class of twist operators related to the chiral primaries by spectral flow parallel to the twist. We conjecture that at large $N$, the OPE of two such operators contains only fields in this class, along with excitations by fractional modes of the superconformal currents. We provide evidence for this by studying the coincidence limits of two 4-point functions to several non-trivial orders. We show ho…
We consider correlation functions in symmetric product ($S_N$) orbifold CFTs at large $N$ with arbitrary seed CFT. Specifically, we consider correlators of descendant operators constructed using both the full Virasoro generators $L_{m}$ and fractional Virasoro generators $\ell_{m/n_i}$. Using covering space techniques, we show that correlators of descendants may be written entirely in terms of correlators of ancestors, and further that the appropriate set of ancestors are those operators that lift to conformal primaries on the cover. We argue that the covering space data should cancel out in s…
In this paper we study the properties of two-dimensional CFTs defined by cyclic and symmetric orbifolds of free Dirac fermions, especially by focusing on the partition function and entanglement entropy. Via the bosonization, we construct the twist operators which glue two complex planes to calculate the partition function of Z_2 orbifold CFT on a torus. We also find an expression of Z_N cyclic orbifold in terms of Hecke operators, which provides an explicit relation between the partition functions of cyclic orbifolds and those of symmetric ones. We compute the entanglement entropy and Renyi en…
In this work, we explore the twist operator OPEs of a generic bosonic symmetric product ($S_N$) orbifold CFT. We conjecture that at large $N$ the OPE of bare twist operators contains only bare twists and excitations of bare twists with fractional Virasoro modes. These fractionally excited operators are the only ones that depend exclusively on the lengths of the twists and the central charge, agreeing with the general structure of correlators of bare twists found in the literature. To provide evidence for this, we study the coincidence limit of a four point function of bare twist operators to s…
The hypothesis that every theory of quantum gravity in AdS_3 is a dimensional reduction of string/M-theory leads to a natural conjecture for the density of states of two dimensional CFTs with a large central charge limit. We prove this conjecture for 2D CFTs which are orbifolds by permutation groups. In particular, we characterize those permutation groups which give CFTs with well-defined large N limits and can thus serve as holographic duals to bulk gravity theories in AdS_3. We then show that the holographic dual of a permutation orbifold will have a Hagedorn spectrum in the large N limit. T…
We consider correlation functions in symmetric product ($S_N$) orbifold CFTs at large $N$ with arbitrary seed CFT, expanding on our earlier work arXiv:2211.04633 . Using covering space techniques, we calculate descent relations using fractional Virasoro generators in correlators, writing correlators of descendants in terms of correlators of ancestors. We first consider the case three-point functions of the form ($m$-cycle)-($n$-cycle)-($q$-cycle) which lift to arbitrary primaries on the cover, and descendants thereof. In these examples we show that the final descent relations do not depend on …
Symmetric product orbifold theories are valuable due to their universal features at large $N$. Here we will demonstrate that they have features that are not as pervasive: we provide evidence of strange behaviour under deformations within their moduli space. To this end, we consider the symmetric product orbifold of tensor products of $\mathcal{N}=2$ super-Virasoro minimal models, and classify them according to two criteria. The first criterion is the existence of a single-trace twisted exactly marginal operator that triggers the deformation. The second criterion is a sparseness condition on th…
In this work we study the time evolution of Renyi entanglement entropy for locally excited states created by twist operators in cyclic orbifold $(T^2)^n/\mathbb{Z}_n$ and symmetric orbifold $(T^2)^n/S_n$. We find that when the square of its compactification radius is rational, the second Renyi entropy approaches a universal constant equal to the logarithm of the quantum dimension of the twist operator. On the other hand, in the non-rational case, we find a new scaling law for the Renyi entropies given by the double logarithm of time $\log\log t$ for the cyclic orbifold CFT.
We study D-branes wrapping an exceptional four-cycle P(1,a,b) in a blown-up C^3/Z_m non-compact Calabi-Yau threefold with (m;a,b)=(3;1,1), (4;1,2) and (6;2,3). In applying the method of local mirror symmetry we find that the Picard-Fuchs equations for the local mirror periods in the Z_{3,4,6} orbifolds take the same form as the ones in the local E_{6,7,8} del Pezzo models, respectively. It is observed, however, that the orbifold models and the del Pezzo models possess different physical properties because the background NS B-field is turned on in the case of Z_{3,4,6} orbifolds. This is shown …
Relativistic QFTs are in general defined by a collection of effective actions, describing the dynamics of quantum fields at different energy scales. The consequent natural idea of a space of theories is still a rather imprecise notion, since a detailed knowledge or a classification of QFTs is out of reach. In two dimensions the situation is in a better shape: in this context CFTs are under control in many instances, and we know sequences of rational theories emerging as end-points of RG flows. The present thesis explores the behaviour of sequences of rational two-dimensional CFTs when the cent…
We investigate the anomalous creation of fundamental strings using the boundary state formalism of fractional D-branes on ALE spaces in the orbifold limit. The open string Witten index plays a crucial role in this calculation and so the result remains unchanged even if we blow up the orbifold geometrically, matching the anomaly inflow argument. Further we consider the quiver gauge theories on such fractional D3-branes and see that the string creation mechanism determines 1-loop logarithmic monodromy of these gauge theories. Also we comment on the relation of D(-1)-D3 amplitude to the 1-loop be…
We describe a class of theories obtained by fibering a Landau-Ginburg orbifold over a compact Kaehler base. While such theories are often described as phases of some GLSM, our description is independent of such an embedding. We provide a method for computing the massless spectrum. This note is based on the longer paper arXiv:1307.7063.
We explore the structures of light cone and Regge limit singularities of $n$-point Virasoro conformal blocks in $c>1$ two-dimensional conformal field theories with no chiral primaries, using fusion matrix approach. These CFTs include not only holographic CFTs dual to classical gravity, but also their full quantum corrections, since this approach allows us to explore full $1/c$ corrections. As the important applications, we study time dependence of Renyi entropy after a local quench and out-of-time ordered correlator (OTOC) at late time. We first show that, the $n$-th ($n>2$) Renyi entropy afte…
We study interacting features of $S_N$ Orbifold CFTs. Concentrating on characters (associated with $S_N$ Orbifold primaries) we first formulate a novel procedure for evaluating them through $GL(\infty)_+$ tracing. The result is a polynomial formula which we show gives results equivalent to those found by Bantay. From this we deduce a hierarchy of commuting Hamiltonians featuring locality in the induced space, and nonlinear string-type interactions.
We exploit null vectors of the fractional Virasoro algebra of the symmetric product orbifold to compute correlation functions of twist fields in the large $N$ limit. This yields a new method to derive correlation functions in these orbifold CFTs that is purely based on the symmetry algebra. We explore various generalisations, such as subleading (torus) contributions or correlation functions of other fields than the bare twist fields. We comment on the consequences of our computation for the $\text{AdS}_3/\text{CFT}_2$ correspondence.
We explore large-$N$ symmetric orbifolds of the $\mathcal N=2$ minimal models, and find evidence that their moduli spaces each contain a supergravity point. We identify single-trace exactly marginal operators that deform them away from the symmetric orbifold locus. We also show that their elliptic genera exhibit slow growth consistent with supergravity spectra in AdS$_3$. We thus propose an infinite family of new holographic CFTs.
We study two dimensional conformal field theories in the semiclassical limit. In this limit, the four-point function is dominated by intermediate primaries of particular weights along with their descendants, and the crossing equations simplify drastically. For a four-point function receiving sufficiently small contributions from the light primaries, the structure constants involving heavy primaries follow a universal formula. Applying our results to the four-point function of the $\mathbb Z_2$ twist field in the symmetric product orbifold, we produce the Hellerman bound and the logarithmically…
Recent developments in the study of the moonshine phenomenon, including umbral and Conway moonshine, suggest that it may play an important role in encoding the action of finite symmetry groups on the BPS spectrum of K3 string theory. To test and clarify these proposed K3-moonshine connections, we study Landau-Ginzburg orbifolds that flow to conformal field theories in the moduli space of K3 sigma models. We compute K3 elliptic genera twined by discrete symmetries that are manifest in the UV description, though often inaccessible in the IR. We obtain various twining functions coinciding with mo…
We study the spectrum of permutation orbifolds of 2d CFTs. We find examples where the light spectrum grows faster than Hagedorn, which is different from known cases such as symmetric orbifolds. We also describe how to compute their partition functions using a generalization of Hecke operators.
Correlators in symmetric orbifold CFTs are given by a finite sum of admissible branched covers of the 2d spacetime. We consider a Gross-Mende like limit where all operators have large twist, and show that the corresponding branched covers can be described via a Penner-like matrix model. The limiting branched covers are given in terms of the spectral curve for this matrix model, which remarkably turns out to be directly related to the Strebel quadratic differential on the covering space. Interpreting the covering space as the world-sheet of the dual string theory, the spacetime CFT correlator t…
The space of permutation orbifolds is a simple landscape of two dimensional CFTs, generalizing the well-known symmetric orbifolds. We consider constraints which a permutation orbifold with large central charge must obey in order to be holographically dual to a weakly coupled (but possibly stringy) theory of gravity in AdS. We then construct explicit examples of permutation orbifolds which obey these constraints. In our constructions the spectrum remains finite at large N, but differs qualitatively from that of symmetric orbifolds. We also discuss under what conditions the correlation functions…
We study conformal twist field four-point functions on a $\mathbb Z_N$ orbifold. We examine in detail the case $N=3$ and analyze theories obtained by replicated $N$-times a minimal model with central charge $c<1$. A fastly convergent expansion of the twist field correlation function in terms of sphere conformal blocks with central charge $Nc$ is obtained by exploiting covering map techniques. We discuss extensive applications of the formalism to the entanglement of two disjoint intervals in CFT, in particular we propose a conformal block expansion for the partially transposed reduced density m…
We study solutions of type IIB supergravity which are SL(2,R) x SU(2) x U(1)^2 invariant deformations of AdS_3 x S^3 x K3 and take the form of products of self-dual spacelike warped AdS_3 and a deformed three-sphere. One of these backgrounds has been recently argued to be relevant for a derivation of Kerr/CFT from string theory, whereas the remaining ones are holographic duals of two-dimensional dipole theories and their S-duals. We show that each of these backgrounds is holographically dual to a deformation of the DLCQ of the D1-D5 CFT by a specific supersymmetric (1,2) operator, which we wri…
We show how aspects of the R-charge of N=2 CFTs in four dimensions are encoded in the q-deformed Kontsevich-Soibelman monodromy operator, built from their dyon spectra. In particular, the monodromy operator should have finite order if the R-charges are rational. We verify this for a number of examples including those arising from pairs of ADE singularities on a Calabi-Yau threefold (some of which are dual to 6d (2,0) ADE theories suitably fibered over the plane). In these cases we find that our monodromy maps to that of the Y-systems, studied by Zamolodchikov in the context of TBA. Moreover we…
It is of interest to find criteria on a 2d CFT which indicate that it gives rise to emergent gravity in a macroscopic 3d AdS space via holography. Symmetric orbifolds in the large $N$ limit have partition functions which are consistent with an emergent space-time string theory with $L_{\rm string} \sim L_{\rm AdS}$. For supersymmetric CFTs, the elliptic genus can serve as a sensitive probe of whether the SCFT admits a large radius gravity description with $L_{\rm string} \ll L_{\rm AdS}$ after one deforms away from the symmetric orbifold point in moduli space. We discuss several classes of con…
We study a class of Little String Theories (LSTs) of A type, described by $N$ parallel M5-branes spread out on a circle and which in the low energy regime engineer supersymmetric gauge theories with $U(N)$ gauge group. The BPS states in this setting correspond to M2-branes stretched between the M5-branes. Generalising an observation made in arXiv:1706.04425, we provide evidence that the BPS counting functions of special subsectors of the latter exhibit a Hecke structure in the Nekrasov-Shatashvili (NS) limit, i.e. the different orders in an instanton expansion of the supersymmetric gauge theor…
Entwinement was first introduced as the CFT dual to extremal, non-minimal geodesics of quotiented AdS$_3$ spaces. It was heuristically meant to capture the entanglement of internal, gauged degrees of freedom, for instance in the symmetric product orbifold CFT of the D1/D5 brane system. The literature now contains different, and sometimes inequivalent, field theory definitions of entwinement. In this paper, we build a discretized lattice model of symmetric product orbifold CFTs, and explicitly construct a gauge-invariant reduced density matrix whose von Neumann entropy agrees with the holograph…
Prompted by the Mathieu Moonshine observation, we identify a pair of 45-dimensional vector spaces of states that account for the first order term in the massive sector of the elliptic genus of K3 in every Z2-orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a Z2-orbifold CFT yields a well-defined faithful representation on them. Moreover, each such representation is obtained by restriction of the 45-dimensional irreducible representation of the Mathieu group M24 constructed by Margolin. Thus we provide a piece o…
