Parisi-Sourlas Supersymmetry in Random Field Models.

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.

We present a new type of generalization of the Rényi entropy that follows naturally from its representation as a thermodynamic quantity. We apply it to the case of [Formula: see text]-dimensional conformal field theories (CFTs) reduced on a region bounded by a sphere. It is known how to compute their Rényi entropy as an integral of the thermal entropy of hyperbolic black holes in [Formula: see text]-dimensional anti-de Sitter spacetime. We show how this integral fits into the framework of extended gravitational thermodynamics, and then point out the natural generalization of the Rényi entropy that suggests itself in that light. In the field theory terms, the new generalization employs aspects of the physics of Renormalization Group (RG) flow to define a refined version of the reduced vacuum density matrix. For [Formula: see text], it can be derived directly in terms of twist operators in field theory. The framework presented here may have applications beyond this context, perhaps in studies of both quantum and classical information theoretic properties of a variety of systems.

We study the process of relaxation back to thermal equilibrium in $(1+1)$-dimensional conformal field theory at finite temperature. When the size of the system is much larger than the inverse temperature, perturbations decay exponentially with time. On the other hand, when the inverse temperature is large, the relaxation is oscillatory with characteristic period set by the size of the system. We then analyse the intermediate regime in two specific models, namely free fermions, and a strongly coupled large $\tt k$ conformal field theory which is dual to string theory on $(2+1)$-dimensional anti-de Sitter spacetime. In the latter case, there is a sharp transition between the two regimes in the ${\tt k}=\infty$ limit, which is a manifestation of the gravitational Hawking-Page phase transition. In particular, we establish a direct connection between quasinormal and normal modes of the gravity system, and the decaying and oscillating behaviour of the conformal field theory.

Modular invariance and the spectrum of two dimensional conformal field theories

We have investigated the hidden conformal symmetry of generic non-extremal rotating black holes in the five-dimensional Godel universe. In a range of parameters, the low-frequency massless scalar wave equation in the “near region” can be described by an SL(2, R)L × SL(2, R)R conformal symmetry. We further found that the microscopic entropy via Cardy formula matches the macroscopic Bekenstein-Hawking entropy and the absorption cross section for the massless scalar also agrees with the one for the two dimensional finite temperature conformal field theory (CFT). All these evidences support the conjecture that the generic non-extremal rotating black hole immersed in the Godel universe can be dual to a two dimensional finite temperature CFT. In addition, we have reformulated the first laws of thermodynamics associated with the inner and outer horizons of the rotating Godel-type black holes into the forms of conformal thermodynamics.

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.

This paper investigates the dynamical phases of Floquet Conformal Field Theories (CFTs) in space-time dimensions greater than two. Building upon our previous work which introduced quaternionic representations for studying Floquet dynamics in higher-dimensional CFTs, we now explore more general square pulse drive protocols that go beyond a single SU(1,1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> subgroup. We demonstrate that, for multi-step drive protocols, the system exhibits distinct dynamical phases characterized by the nature of the eigenvalues of the quaternionic matrix representing time evolution in a single cycle, leading to different stroboscopic responses. Our analysis establishes a fundamental geometric interpretation where these dynamical phases directly correspond to the presence or absence of Killing horizons in the base space of the CFT and in a higher dimensional AdS space on which a putative dual lives. The heating phase is associated with a non-extremal horizon, the critical phase with an extremal horizon which disappears in the non-heating phase. We develop perturbative approaches to compute the Floquet Hamiltonians in different regimes and show, how tuning drive parameters can lead to horizons, providing a geometric framework for understanding heating phenomena in driven conformal systems.

We study quantum chaos in two dimensional conformal field theories, building on the work analyzing the out-of-time order thermal correlation functions using large-c Virasoro blocks. Our work investigates the contribution of light intermediate channels and smearing length scales to the four-point function and scrambling. Precise relations for how light intermediate channels increase the scrambling time and how smearing length scales smaller than the thermal length scale decrease the scrambling time are derived.

In this work, we study the time evolution of the entanglement Hamiltonian during the process of thermalization in a (1+1)-dimensional conformal field theory (CFT) after a quantum quench from a special class of initial states. In particular, we focus on a subsystem which is a finite interval at the end of a semi-infinite line. Based on conformal mappings, the exact forms of both entanglement Hamiltonian and entanglement spectrum of the subsystem can be obtained. Aside from various interesting features, it is found that in the infinite time limit the entanglement Hamiltonian and entanglement spectrum are exactly the same as those in the thermal ensemble. The entanglement spectrum approaches the steady state spectrum exponentially in time. We also study the modular flows generated by the entanglement Hamiltonian in Minkowski spacetime, which provides us with an intuitive picture of how the entanglement propagates and how the subsystem is thermalized. Furthermore, the effect of a generic initial state is also discussed.

We review the imaginary time path integral approach to the quench dynamics of conformal field theories. We show how this technique can be applied to the determination of the time dependence of correlation functions and entanglement entropy for both global and local quenches. We also briefly review other quench protocols. We carefully discuss the limits of applicability of these results to realistic models of condensed matter and cold atoms.

Topological order in Quantum hall effect and two-dimensional conformal field theory

The first law for the holographic entanglement entropy of spheres in a boundary CFT (Conformal Field Theory) with a bulk Lovelock dual is extended to include variations of the bulk Lovelock coupling constants. Such variations in the bulk correspond to perturbations within a family of boundary CFTs. The new contribution to the first law is found to be the product of the variation δ a of the “A”-type trace anomaly coefficient for even dimensional CFTs, or more generally its extension δ a * to include odd dimensional boundaries, times the ratio S / a * . Since a * is a measure of the number of degrees of freedom N per unit volume of the boundary CFT, this new term has the form μ δ N , where the chemical potential μ is given by the entanglement entropy per degree of freedom.

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)

We study the critical behavior and the ground-state entanglement of a large class of su(1|1) supersymmetric spin chains with a general (not necessarily monotonic) dispersion relation. We show that this class includes several relevant models, with both short- and long-range interactions of a simple form. We determine the low temperature behavior of the free energy per spin, and deduce that the models considered have a critical phase in the same universality class as a (1+1)-dimensional conformal field theory (CFT) with central charge equal to the number of connected components of the Fermi sea. We also study the Rényi entanglement entropy of the ground state, deriving its asymptotic behavior as the block size tends to infinity. In particular, we show that this entropy exhibits the logarithmic growth characteristic of (1+1)-dimensional CFTs and one-dimensional (fermionic) critical lattice models, with a central charge consistent with the low-temperature behavior of the free energy. Our results confirm the widely believed conjecture that the critical behavior of fermionic lattice models is completely determined by the topology of their Fermi surface.

We introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain.