Bound on the central charge of CFTs in large dimension

Boundaries not only are fundamental elements in nearly all realistic physical systems, but also greatly enrich the structure of quantum field theories. In this paper, we demonstrate that conformal field theory (CFT) with a boundary, known as surface CFT in three dimensions, can be studied with the setup of fuzzy sphere. We consider the example of surface criticality of the 3d Ising CFT. We propose two schemes by cutting a boundary in the orbital space and the real space respectively to realise the ordinary and the normal surface CFTs on the fuzzy sphere. We obtain the operator spectra through state-operator correspondence. We observe integer spacing of the conformal multiplets, and thus provide direct evidence of conformal symmetry. We identify the ordinary surface primary o o , the displacement operator \mathrm{D} D and their conformal descendants and extract their scaling dimensions. We also study the one-point and two-point correlation functions and extract the bulk-to-surface OPE coefficients, some of which are reported for the first time. In addition, using the overlap of the bulk CFT state and the polarised state, we calculate the boundary central charges of the 3d Ising surface CFTs non-perturbatively. Other conformal data obtained in this way also agrees with prior methods. We also discuss future directions, such as special surface CFT, cone defects, and surfaces of other bulk CFTs.

The dilogarithm identities for the central charges of conformal field theories of simply laced type were conjectured by Bazhanov, Kirillov, and Reshetikhin. Their functional generalizations were conjectured by Gliozzi and Tateo. They have been partly proved by various authors. We prove these identities in full generality for any pair of Dynkin diagrams of simply laced type based on the cluster algebra formulation of the Y-systems.

We expand on the investigation of the universal scaling properties in the\nearly time behaviour of fast but smooth quantum quenches in a general\n$d$-dimensional conformal field theory deformed by a relevant operator of\ndimension $\\Delta$ with a time-dependent coupling. The quench consists of\nchanging the coupling from an initial constant value $\\lambda_1$ by an amount\nof the order of $\\delta \\lambda$ to some other final value $\\lambda_2$, over a\ntime scale $\\delta t$. In the fast quench limit where $\\delta t$ is smaller\nthan all other length scales in the problem, $ \\delta t \\ll\n\\lambda_1^{1/(\\Delta-d)}, \\lambda_2^{1/(\\Delta-d)}, \\delta\n\\lambda^{1/(\\Delta-d)}$, the energy (density) injected into the system scales\nas $\\delta{\\cal E} \\sim (\\delta \\lambda)^2 (\\delta t)^{d-2\\Delta}$. Similarly,\nthe change in the expectation value of the quenched operator at times earlier\nthan the endpoint of the quench scales as $<{\\cal O}_\\Delta> \\sim \\delta\n\\lambda (\\delta t)^{d-2\\Delta}$, with further logarithmic enhancements in\ncertain cases. While these results were first found in holographic studies, we\nrecently demonstrated that precisely the same scaling appears in fast mass\nquenches of free scalar and free fermionic field theories. As we describe in\ndetail, the universal scaling refers to renormalized quantities, in which the\nUV divergent pieces are consistently renormalized away by subtracting\ncounterterms derived with an adiabatic expansion. We argue that this scaling\nlaw is a property of the conformal field theory at the UV fixed point, valid\nfor arbitrary relevant deformations and insensitive to the details of the\nquench protocol. Our results highlight the difference between smooth fast\nquenches and instantaneous quenches where the Hamiltonian abruptly changes at\nsome time.\n

We show that the average null energy condition implies novel lower bounds on the scaling dimensions of highly-chiral primary operators in four-dimensional conformal field theories. Denoting the spin of an operator by a pair of integers left(k,overline{k}right) specifying the transformations under chiral mathfrak{s}mathfrak{u}(2) rotations, we explicitly demonstrate these new bounds for operators transforming in (k, 0) and (k, 1) representations for sufficiently large k. Based on these calculations, along with intuition from free field theory, we conjecture that in any unitary conformal field theory, primary local operators of spin left(k,overline{k}right) and scaling dimension Δ satisfy varDelta ge max left{k,overline{k}right} . If left|k-overline{k}right|>4 , this is stronger than the unitarity bound.

Chirally factorised truncated conformal space approach

It was recently discovered that for a boundary system in the presence of a background magnetic field, the quantum fluctuation of the vacuum would create a non-uniform magnetization density for the vacuum and a magnetization current is induced in the vacuum [1]. It was also shown that this “magnetic Casimir effect” of the vacuum is closely related to another quantum effect of the vacuum, the Weyl anomaly. Furthermore, the phenomena can be understood in terms of the holography of the boundary system [2]. In this paper, we generalize this four dimensional effect to six dimensions. We use the AdS/BCFT holography to show that in the presence of a 3-form magnetic field strength H, a string current is induced in a six dimensional boundary conformal field theory. This allows us to determine the gauge field contribution to the Weyl anomaly in six dimensional conformal field theory in a H-flux background. For the (2,0) superconformal field theory of N M5-branes, the current has a magnitude proportional to N3 for large N. This suggests that the degree of freedoms scales as N3 in the (2,0) superconformal theory of N multiple M5-branes. The prediction we have for the Weyl anomaly is a new criteria that the (2,0) theory should satisfy.

This thesis is dedicated to certain nonlocal aspects of conformal quantum field theories (CFTs). Two main directions are the study of CFTs on a particular globally-nontrivial spacetime corresponding to finite temperature, and the study of particular nonlocal CFT observables localized on light-rays. Specifically, we introduce bootstrap techniques for determining finite-temperature data of CFTs, and make novel predictions for the 2+1-dimensional Ising model. We propose the “stringy equivalence principle”, stating that coincident gravitational shocks commute, as a generalization of the strong equivalence principle of Einstein’s General Relativity that must hold in all consistent theories of gravity. We prove it in Anti-de Sitter (AdS) spacetimes by studying light-ray operators in the holographically dual CFT. We also derive an operator product expansion (OPE) for light-ray operators in CFT, by which two light-ray operators on the same light-sheet can be expanded as a sum of single light-ray operators. Light-ray operators model detectors — such as calorimeters. We use the light-ray OPE to compute energy event shape observables suitable for conformal collider physics. An additional part of this thesis determines the low-energy vacua of two-dimensional maximal super-Yang-Mills theory, which describes the dynamics of stacks of D-strings in Type IIB string theory. By computing an invariant of the renormalization group (RG) flow from high to low energy — a modified thermal partition function named the refined elliptic genus — we prove the existence of multiple vacua, and identify the superconformal field theories capturing their dynamics. The vacua correspond to bound states of (p,q)-strings in Type IIB string theory. Our computation serves as a check of the strong-weak S-duality of the Type IIB string.

We introduce a very early universe model based on the thermodynamics of a gas of closed strings in a background which is non-perturbative in $\alpha'$. Upon considering the fully $\alpha'$-corrected equations extended to include certain anisotropic cosmological backgrounds, we describe the evolution of the system in three different stages parametrized by the gas' equation of state. Using standard string thermodynamical arguments, we start with an isotropic 10-dimensional universe inside the string scale and evolve it towards a universe with four large spacetime dimensions and six stabilized internal dimensions in the Einstein frame.

We study the OPE coefficients cΔ, J for heavy-light scalar four-point functions, which can be obtained holographically from the two-point function of a light scalar of some non-integer conformal dimension ΔL in an AdS black hole. We verify that the OPE coefficient cd,0 = 0 for pure gravity black holes, consistent with the tracelessness of the holographic energy-momentum tensor. We then study the OPE coefficients from black holes involving matter fields. We first consider general charged AdS black holes and we give some explicit low-lying examples of the OPE coefficients. We also obtain the recursion formula for the lowest-twist OPE coefficients with at most two current operators. For integer ΔL, although the OPE coefficients are not fully determined, we set up a framework to read off the coefficients γΔ,J of the log(z overline{z} ) terms that are associated with the anomalous dimensions of the exchange operators and obtain a general formula for γΔ,J. We then consider charged AdS black holes in gauged supergravity STU models in D = 5 and D = 7, and their higher-dimensional generalizations. The scalar fields in the STU models are conformally massless, dual to light operators with ΔL = d − 2. We derive the linear perturbation of such a scalar in the STU charged AdS black holes and obtain the explicit OPE coefficient cd−2,0. Finally, we analyse the asymptotic properties of scalar hairy AdS black holes and show how cd,0 can be nonzero with exchanging scalar operators in these backgrounds.

Duality and the fractional quantum Hall effect

Repeated local measurements of quantum many-body systems can induce a phase transition in their entanglement structure. These measurement-induced phase transitions (MIPTs) have been studied for various types of dynamics, yet most cases yield quantitatively similar critical exponents, making it unclear how many distinct universality classes are present. Here, we probe the properties of the conformal field theories governing these MIPTs using a numerical transfer-matrix method, which allows us to extract the effective central charge, as well as the first few low-lying scaling dimensions of operators at these critical points for (1+1)-dimensional systems. Our results provide convincing evidence that the generic and Clifford MIPTs for qubits lie in different universality classes and that both are distinct from the percolation transition for qudits in the limit of large on-site Hilbert space dimension. For the generic case, we find strong evidence of multifractal scaling of correlation functions at the critical point, reflected in a continuous spectrum of scaling dimensions.

Monopole operators are studied in a large family of quantum critical points between Dirac and topological quantum spin liquids (QSLs): chiral and Z$_{2}$ QSLs. These quantum phase transitions are described by conformal field theories (CFTs): quantum electrodynamics in 2+1 dimensions with 2N flavors of two-component massless Dirac fermions and a four-fermion interaction. For the transition to a chiral spin liquid, it is the Gross-Neveu interaction (QED$_{3}$-GN), while for the transitions to Z$_{2}$ QSLs it is a superconducting pairing term with general spin/valley structure (generalized QED$_{3}$-Z$_{2}$GN). Using the state-operator correspondence, we obtain monopole scaling dimensions to sub-leading order in 1/N. For monopoles with a minimal topological charge q=1/2, the scaling dimension is 2N*0.26510 at leading-order, with the quantum correction being 0.118911(7) for the chiral spin liquid, and 0.102846(9) for the simplest Z$_{2}$ case (the expression is also given for a general pairing term). Although these two anomalous dimensions are nearly equal, the underlying quantum fluctuations possess distinct origins. The analogous result in QED$_{3}$ is also obtained and we find a sub-leading contribution of -0.038138(5), which differs slightly from the value first obtained in the literature. The scaling dimension of a QED$_{3}$-GN monopole with minimal charge is very close to the scaling dimensions of other operators predicted to be equal by a conjectured duality between QED$_{3}$-GN with 2N=2 flavors and the CP$^{1}$ model. Additionally, non-minimally charged monopoles on both sides of the duality have similar scaling dimensions. By studying the large-q asymptotics of the scaling dimensions in QED$_{3}$, QED$_{3}$-GN, and QED$_{3}$-Z$_{2}$GN we verify that the constant O(q$^{0}$) coefficient precisely matches the universal non-perturbative prediction for CFTs with a global U(1) symmetry.

Modular invariance and the spectrum of two dimensional conformal field theories

A very theoretical description is given of research in two- dimensional critical phenomena and conformal field theory. Major progress is reported in the field of fluctuating two-dimensional surfaces. A discretized representation of fluctuating geometry is used where surfaces are represented by triangulations; continuum surfaces are recovered by taking the size of the triangles to zero. One of the central goals of the theory of critical phenomena is to find all possible universality classes of n-dimensional critical phenomena; this goal has been translated into the problem of clasifying all possible scale-invariant euclidean quantum field theories. (RWR)

In various dimensions, we can sometimes compute observables of interacting conformal field theories (CFTs) that are connected to free theories via the renormalization group (RG) flow by computing protected quantities in the free theories. On the other hand, in two dimensions, it is often possible to algebraically construct observables of interacting CFTs using free fields without the need to explicitly construct an underlying RG flow. In this Letter, we begin to extend this idea to higher dimensions by showing that one can compute certain observables of an infinite set of unitary strongly interacting four-dimensional N=2 superconformal field theories (SCFTs) by performing simple calculations involving sets of nonunitary free four-dimensional hypermultiplets. These free fields are distant cousins of the Majorana fermion underlying the two-dimensional Ising model and are not obviously connected to our interacting theories via an RG flow. Rather surprisingly, this construction gives us Lagrangians for particular observables in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity while preserving the full N=2 superconformal algebra. As a by-product, we find relations between characters in unitary and nonunitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction.