On gravitational dressing of 2D field theories in chiral gauge

After giving a pedagogical review of the chiral gauge approach to 2D gravity, with particular emphasis on the derivation of the gravitational Ward identities, we discuss in some detail the interpretation of matter correlation functions coupled to gravity in chiral gauge. We argue that in chiral gauge no explicit gravitational dressing factor, analogue to the Lionville exponential in conformal gauge, is necessary for left-right symmetric matter operators. In particular, we examine the gravitationally dressed four-point correlation function of products of left and right fermions. We solve the corresponding gravitational Ward identity exactly: in the presence of gravity this four-point function exhibits a logarithmic short-distance singularity, instead of the power-law singularity in the absence of gravity. This rather surprising effect is non-perturbative in the gravitational coupling and is a sign for logarithms in the gravitationally dressed operator product expansions. We also discuss some perturbative evidence that the chiral Gross-Neveu model may remain integrable when coupled to gravity.     

Towers of gravitational theories

In this essay we introduce a theoretical framework designed to describe black hole dynamics. The difficulties in understanding such dynamics stems from the proliferation of scales involved when one attempts to simultaneously describe all of the relevant dynamical degrees of freedom. These range from the modes that describe the black hole horizon, which are responsible for dissipative effects, to the long wavelength gravitational radiation that drains mechanical energy from macroscopic black hole bound states. We approach the problem from a Wilsonian point of view, by building a tower of theories of gravity each of which is valid at different scales. The methodology leads to multiple new results in diverse topics including phase transitions of Kaluza-Klein black holes and the interactions of spinning black hole in non-relativistic orbits. Moreover, our methods tie together speculative ideas regarding dualities for black hole horizons to real physical measurements in gravitational wave detectors.     

Toward classification of conformal theories

By studying the representations of the mapping class groups which arise in 2D conformal theories we derive some restrictions on the value of the conformal dimension h_i of operators and the central charge c of the Virasoro algebra. As a simple application we show that when there are a finite number of operators in the conformal algebra, the h_i and c are all rational.     

Stochastic quantization of linearized gravitational theories

A generalization of stochastic quantization is proposed in order to deal with higher order derivative gravitational theories. In the linear approximation the correct graviton propagator is recovered, while for the higher order derivative case, the resulting propagator has pole structure, which exhibits the impossibility of Wick rotation. The unitarity of the models is briefly discussed.     

Chiral deformations of conformal field theories

We study general perturbations of two-dimensional conformal field theories by holomorphic fields. It is shown that the genus one partition function is controlled by a contact term (pre-Lie) algebra given in terms of the operator product expansion. These models have applications to vertex operator algebras, two-dimensional QCD, topological strings, holomorphic anomaly equations and modular properties of generalized characters of chiral algebras such as the W_1+∞ algebra, that is treated in detail.     

Integrability of coupled conformal field theories

The massive phase of two-layer integrable systems is studied by means of RSOS restrictions of affine Toda theories. A general classification of all possible integrable perturbations of coupled minimal models is pursued by an analysis of the (extended) Dynkin diagrams. The models considered in most detail are coupled minimal models which interpolate between magnetically coupled Ising models and Heisenberg spin ladders along the c < 1 discrete series.     

Nearly conformal gauge theories in finite volume

We report new results on nearly conformal gauge theories with fermions in the fundamental representation of the SU(3)$\mathit{SU}(3)$ color gauge group as the number of fermion flavors is varied in the Nf=4–16$N_f=4\text{–}16$ range. To unambiguously identify the chirally broken phase below the conformal window we apply a comprehensive lattice tool set in finite volumes which includes the test of Goldstone pion dynamics, the spectrum of the fermion Dirac operator, and eigenvalue distributions of random matrix theory. We also discuss the theory inside the conformal window and present our first results on the running of the renormalized gauge coupling and the renormalization group beta function. The importance of understanding finite volume zero momentum gauge field dynamics inside the conformal window is illustrated. Staggered lattice fermions are used throughout the calculations.     

Asymptotic mass degeneracies in conformal field theories

By applying a method of Hardy and Ramanujan to characters of rational conformal field theories, we find an asymptotic expansion for degeneracy of states in the limit of large mass which isexact for strings propagating in more than two uncompactified space-time dimensions. Moreover we explore how the rationality of the conformal theory is reflected in the degeneracy of states. We also consider the one loop partition function for strings, restricted to physical states, for arbitrary (irrational) conformal theories, and obtain an asymptotic expansion for it in the limit that the torus degenerates. This expansion depends only on the spectrum of (physical and unphysical) relevant operators in the theory. We see how rationality is consistent with the smoothness of mass degeneracies as a function of moduli.     

Singular vectors in logarithmic conformal field theories

Null vectors are generalized to the case of indecomposable representations which are one of the main features of logarithmic conformal field theories. This is done by developing a compact formalism with the particular advantage that the stress energy tensor acting on Jordan cells of primary fields and their logarithmic partners can still be represented in form of linear differential operators. Since the existence of singular vectors is subject to much stronger constraints than in regular conformal field theory, they also provide a powerful tool for the classification of logarithmic conformal field theories.     

Boundary conditions in rational conformal field theories

We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformal boundary conditions are labelled by the nodes of G. This approach is carried to completion for sl(2) theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the A-D-E classification. We also review the current status for WZW sl(3) theories. Finally, a systematic generalisation of the formalism of Cardy–Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk–boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.     

Canonical dimensions in asymptotically conformal invariant theories

It is shown that canonical dimensions in the product of currents near the light-cone imply a free field structure for the Wightman functions of currents in the underlying skeleton theory.     

Origin of logarithmic operators in conformal field theories

We study logarithmic operators in Coulomb gas models, and show that they occur when the “puncture” operator of the Liouville theory is included in the model. We also consider WZNW models for SL(2,R), and for SU(2) at level 0, in which we find logarithmic operators which form Jordan blocks for the current as well as the Virasoro algebra.     

Lower-dimensional theories of gravity and gravitational waves

We demonstrate that there are no gravitational waves in vacuum in 1+1 gravity, with or without a cosmological constant, and a wave-like field in regions of non-vanishing T_v is quite unlike a conventional gravitational wave in 3+1 gravity. This, and other dissimilarities, lend a word of caution to the extraction of information from lower-dimensional theories for the understanding of 3+1 gravity.     

Tests of gravitational theories using space probes

Several high-precision tests of gravitational theories applied to the field of a spherical mass particle (the Sun) are formulated and analyzed. These tests mainly involve the accurate timing by (on-board) atomic clocks of two-way radio signals between two spacecraft (transponders) in various solar orbits. Measures of the gravitational field to at least 1 part in 10⁸, or orderm/R, wherem is the Schwarzschild radius of the Sun andR is the astronomical unit, are aimed at, with an assumed round trip time precision of 10^–2 s or 1.5 m distance. Since it is important to distinguish between the radial coordinate parameters occurring in the theory of the tests and their common astronomical measurements based on classical assumptions, two of the tests include the means in principle of determining these coordinate parameters.