Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs

Gravitational shockwaves are insensitive to higher-curvature corrections in the action. Recent work found that the OPE coefficients of lowest-twist multi-stress-tensor operators, computed holographically in a planar black hole background, are insensitive as well. In this paper, we analyze the relation between these two limits. We explicitly evaluate the two-point function on a shockwave background to all orders in a large central charge expansion. In the geodesic limit, we find that the ANEC exponentiates in the multi-stress-tensor sector. To compare with the black hole limit, we obtain a recursion relation for the lowest-twist products of two stress tensors in a spherical black hole background, letting us efficiently compute their OPE coefficients and prove their insensitivity to higher curvature terms. After resumming the lowest-twist stress-tensors and analytically continuing their contributions to the Regge limit, we find a perfect agreement with the shockwave computation. We also discuss the role of double-trace operators, global degenerate states, and multi-stress-tensor conformal blocks. These holographic results suggest the existence of a larger universal structure in higher-dimensional CFTs.

A bstract We develop a resurgence analysis for large central charge (large C ) expansions in two-dimensional CFTs using the Coulomb gas formalism. Through the exact Borel-Laplace representations of the conformal blocks I 1 ( C, z ) and I 2 ( C, z ) associated with the four-point correlation function $$\langle {\phi }_{2,1}\left(0\right){\phi }_{2,1}\left(z, \overline{z }\right){\phi }_{2,1}\left(1\right){\phi }_{2,1}(\infty )\rangle $$ , we demonstrate that I 1 ( C, z ) participates in the Stokes phenomenon of I 2 ( C, z ) (and vice versa), and establish that monodromy in z arises from alien calculus in the Borel plane variable ζ (Borel dual to C ). From a given conformal block, resurgence theory thus enables us to discover other internal operators (conformal blocks). This approach establishes a non-perturbative connection between conformal blocks, shedding light on the resurgence phenomena in more general quantum field theories.

Thermal states of quantum systems with many degrees of freedom are subject to a bound on the rate of onset of chaos, including a bound on the Lyapunov exponent, $\lambda_L\leq 2\pi /\beta$. We harness this bound to constrain the space of putative holographic CFTs and their would-be dual theories of AdS gravity. First, by studying out-of-time-order four-point functions, we discuss how $\lambda_L=2\pi/\beta$ in ordinary two-dimensional holographic CFTs is related to properties of the OPE at strong coupling. We then rule out the existence of unitary, sparse two-dimensional CFTs with large central charge and a set of higher spin currents of bounded spin; this implies the inconsistency of weakly coupled AdS$_3$ higher spin gravities without infinite towers of gauge fields, such as the $SL(N)$ theories. This fits naturally with the structure of higher-dimensional gravity, where finite towers of higher spin fields lead to acausality. On the other hand, unitary CFTs with classical $W_{\infty}[\lambda]$ symmetry, dual to 3D Vasiliev or hs[$\lambda$] higher spin gravities, do not violate the chaos bound, instead exhibiting no chaos: $\lambda_L=0$. Independently, we show that such theories violate unitarity for $|\lambda|>2$. These results encourage a tensionless string theory interpretation of the 3D Vasiliev theory. We also perform some CFT calculations of chaos in Rindler space in various dimensions.

The four-point conformal block plays an important part in the analysis of the conformally invariant operator algebra in two-dimensional space. The behavior of the conformal block is calculated in the present paper in the limit in which the dimension ..delta.. of the intermediate operator tends to infinity. This makes it possible to construct a recursion relation for this function that connects the conformal block at arbitrary ..delta.. to the blocks corresponding to the dimensions of the zero vectors in the degenerate representations of the Virasoro algebra. The relation is convenient for calculating the expansion of the conformal block in powers of the uniformizing parameters q = i ..pi.. tau.

In this paper we study the constraints imposed by conformal invariance on extended objects a.k.a. defects in a conformal field theory. We identify a particularly nice class of defects that is closed under conformal transformations. Correlation function of the defect with a bulk local operator is fixed by conformal invariance up to an overall constant. This gives rise to the notion of defect expansion, where the defect itself is expanded in terms of local operators. This expansion generalizes the idea of the boundary state. We will show how one can fix the correlation function of two defects from the knowledge of the defect expansion. The defect correlator admits a number of conformal cross-ratios depending on their dimensionality. We find the differential equation obeyed by the conformal block and solve them in certain special cases.

We introduce the software blocks_3d for computing four-point conformal blocks of operators with arbitrary Lorentz representations in 3d CFTs. It uses Zamolodchikov-like recursion relations to numerically compute derivatives of blocks around a crossing-symmetric configuration. It is implemented as a heavily optimized, multi-threaded, C++ application. We give performance benchmarks for correlators containing scalars, fermions, and stress tensors. As an example application, we recompute bootstrap bounds on four-point functions of fermions and study whether a previously observed sharp jump can be explained using the “fake primary” effect. We conclude that the fake primary effect cannot fully explain the jump and the possible existence of a “dead-end” CFT near the jump merits further study.

Let M(r) be the moduli space of rank r vector bundles with trivial determinant on a Riemann surface X . This space carries a natural line bundle, the determinant line bundle L . We describe a canonical isomorphism of the space of global sections of L^k with a space known in conformal field theory as the ``space of conformal blocks, which is defined in terms of representations of the Lie algebra sl(r, C((z))).

Infinite conformal symmetry in two-dimensional quantum field theory

Gravity and general relativity are considered as an Effective Field Theory (EFT) at low energies and macroscopic distances. The effective action of the conformal anomaly of light or massless quantum fields has significant effects on macroscopic scales, due to associated light cone singularities that are not captured by an expansion in local curvature invariants. A compact local form for the Wess-Zumino effective action of the conformal anomaly and stress tensor is given, requiring the introduction of a new light scalar field, which it is argued should be included in the low energy effective action for gravity. This scalar conformalon couples to the conformal part of the spacetime metric and allows the effective value of the vacuum energy, described as a condensate of an exact 4-form abelian gauge field strength F = dA, to change in space and time. This is achieved by the identification of the torsion dependent part of the Chern-Simons 3-form of the Euler class with the gauge potential A, which enters the effective action of the conformal anomaly as a J · A interaction analogous to electromagnetism. The conserved 3-current J describes the worldtube of 2-surfaces that separate regions of differing vacuum energy. The resulting EFT thus replaces the fixed constant Λ of classical gravity, and its apparently unnaturally large sensitivity to UV physics, with a dynamical condensate whose ground state value in empty flat space is Λeff = 0 identically. By allowing Λeff to vary rapidly near the 2-surface of a black hole horizon, the proposed EFT of dynamical vacuum energy provides an effective Lagrangian framework for gravitational condensate stars, as the final state of complete gravitational collapse consistent with quantum theory. The possible consequences of dynamical vacuum dark energy for cosmology, the cosmic coincidence problem, and the role of conformal invariance for other fine tuning issues in the Standard Model are discussed.

The simultaneous breaking of conformal and chiral symmetry is investigated within the framework of nonlinear realizations and effective Lagrangians. The explicit introduction of a massless dilaton field, $\ensuremath{\chi}$ enables conformal invariance to be preserved in Lagrangians for massive matter fields. It is shown that the equation of Callan, Coleman, and Jackiw, ${\ensuremath{\partial}}_{\ensuremath{\mu}}{D}_{\ensuremath{\mu}}={\ensuremath{\theta}}_{\ensuremath{\mu}\ensuremath{\mu}}$, remains valid notwithstanding the introduction of this particle, and also that it is possible to construct Lagrangians which are simultaneously invariant under the chiral and conformal groups. If we introduce a term which explicitly violates both symmetries, then the dilaton acquires a definite (bare) mass which can be expressed in terms of the masses of the chiral bosons ---the pion in the case of chiral $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$, the pion and kaon in the case of chiral $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$. The precise form of this mass relation depends upon the type of symmetry-breaking term adopted.

The equivalence principle is a perennial subject of controversy, especially in connection with radiation by a uniformly accelerated classical charge, or a freely falling charge observed by a supported detector. Recently, related issues have been raised in connection with the Unruh radiation associated with accelerated detectors (including two-level atoms and resonant cavities). A third type of system, very easy to analyze because of conformal invariance, is a two-dimensional scalar field interacting with perfectly reflecting boundaries (mirrors). After reviewing the issues for atoms and cavities, we investigate a stationary mirror from the point of view of an accelerated detector in ‘Rindler space’. In keeping with the conclusions of earlier authors about the electromagnetic problem, we find that a radiative effect is indeed observed; from an inertial point of view, the process arises from a collision of the negative vacuum energy of Rindler space with the mirror. There is a qualitative symmetry under interchange of accelerated and inertial subsystems (a vindication of the equivalence principle), but it hinges on the accelerated detector’s being initially in its own ‘Rindler vacuum’. This observation is consistent with the recent work on the Unruh problem.

In this thesis we introduce the S-theory. We apply the S-theory to variational problems of immersed Riemann surfaces, Elasticity theory and Spin theory. 
\nFrom classical mechanics it is known that geometric problems can be substantially simplified by symmetry considerations. The Noether theorem describes in this case how to find conservation laws from the invariance property of the observed energy. At first, the motivation was to give the space of immersions M = {f : M → R3}, for an oriented Riemannian manifold M of dimension less than or equal to three, an Euclidean invariant description. For geometric energies, i.e. energies which are invariant under Euclidean transformations, one would be able to derive conservation laws for the respective variational problems. It is a well known fact that immersed space curves are uniquely determined by their curvature functions (κ1,κ2,τ) up to Euclidean transformations. In the case of immersed surfaces and immersed 3-manifolds the situation becomes more complicated and leads to a non conformal deformation theory (S-Theorie). 
\nThe S-Theory builds the foundations of this thesis. Starting from a Riemannian manifold M and a reference metric ⟨ , ⟩ one can model any other Riemmanian metric g through a positive definite and self adjoint operator S via g = ⟨S,S⟩. 
\nThe operator S is an isometry, i.e. g = S∗⟨,⟩. From this point of view we compute the Levi- Civita connection with respect to g out of Levi-Civita connection of the reference metric and the operator S. 
\nFurther we introduce Spin bundles over two and three dimensional oriented Riemannian manifolds. Thereby Spin bundles, in contrast to most of the literature, have additionally a quaternionic structure. Many formulas become clearer and more accessible. We show that any Spin bundle Σ over (M, ⟨ , ⟩) has a unique Spin connection and compute via the S-Theory the deformed Spin connection with respect to the metric g. Eventually we consider the induced Spin bundle of an immersion f : M → R3 and compute the corresponding Spin connection and find a new interpretation of the Gauss-Codazzi equation. We introduce the Dirac operator and compute its deformation with respect to the metric g. 
\nWe are now able to give the space of immersed surfaces M = {f : M → R3} the desired geometric description. We compute the normal space of M and formulate the Noether theorem. As an application we compute conservation laws for the Willmore functional. 
\nFinally we introduce the intrinsic version of the famous distance-squared energy on a n-dimensional Riemannian manifold. We introduce the corresponding stress tensor and show that it’s closeness is a characterization for critical points of the energy. Thereby the S-theory provides a fundamental concept for the understanding of the stress tensor. In the case of a Riemann surface, the S-theory provides a canonical harmonic rotation 1-form for critical points of the distance-squared energy.

The paper analyses a possible occurrence of soft and semi-soft viscous modes in slow (low Reynolds number) flows of uniaxially anisotropic nematic liquids as described by the five parametric Leslie-Ericksen-Parodi (LEP) constitutive equations (CEs). As in the similar elastic case, the soft viscous modes theoretically cause no resistance to flow, nullifying the corresponding components of the viscous part of the total stress tensor, and do not contribute to the dissipation. That is why these modes can also be called dissipative soft modes. In some flows, these dissipative soft modes may cause the effect of “nematic superfluidity”. As in the theories of nematic elastic solids, this effect is caused by a marginal thermodynamic stability. The analysis is simplified in a specific local, rotating orthogonal coordinate system whose one axis is directed along the director. We demonstrate that depending on closeness of material parameters to the marginal stability conditions, LEP CEs describe the entire variety of soft, semi-soft and harder behaviors of nematic viscous liquids. When the only shearing dissipative modes are soft, the viscous part of stress tensor is symmetric, and LEP CE for stress, scaled with isotropic viscosity is reduced to a one-parametric, stress-strain rate anisotropic relation. When additionally the elongation dissipative mode is also soft, this scaled relation has no additional parameters and shows that the dissipation is always less than that in isotropic phase. Simple shearing and simple elongation flows illustrate these possible effects.

A free massless field coupled to a background charge Q is considered. The conformal blocks of the theory are adequately described by the N-point vertex VN;0 which is, up to a multiplicative constant, completely determined by the Knizhnik–Zamolodchikov equations and a set of Ward identities on the sphere. The N-point g-loop vertex VN;g for compact Riemann surfaces Σg of genus g≠0 is constructed by a sewing procedure. The main new result is a rigorous proof of the fact that VN;0 is trace class for nonoverlapping disks. This allows one to show that the set of all N-point vertices (for all N) is a modular functor.

We develop a recursive approach to computing Neveu-Schwarz conformal blocks associated with n-punctured Riemann surfaces. This work generalizes the results of [1] obtained recently for the Virasoro algebra. The method is based on the analysis of the analytic properties of the superconformal blocks considered as functions of the central charge c. It consists of two main ingredients: the study of the singular behavior of the conformal blocks and the analysis of their asymptotic properties when c tends to infinity. The proposed construction is applicable for computing multi-point blocks in different topologies. We consider some examples for genus zero and one with different numbers of punctures. As a by-product, we propose a new way to solve the recursion relations, which gives more efficient computational procedure and can be applied to SCFT case as well as to pure Virasoro blocks.