Sheer El-Showk

Solving the 3D Ising Model with the Conformal Bootstrap

We study the constraints of crossing symmetry and unitarity in general 3D conformal field theories. In doing so we derive new results for conformal blocks appearing in four-point functions of scalars and present an efficient method for their computation in arbitrary space-time dimension. Comparing the resulting bounds on operator dimensions and product-expansion coefficients in 3D to known results, we find that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space. We also derive general upper bounds on the dimensions of higher spin operators, relevant in the context of theories with weakly broken higher spin symmetries.

Bootstrapping Conformal Field Theories with the Extremal Functional Method

The existence of a positive linear functional acting on the space of (differences between) conformal blocks has been shown to rule out regions in the parameter space of conformal field theories (CFTs). We argue that at the boundary of the allowed region the extremal functional contains, in principle, enough information to determine the dimensions and operator product expansion (OPE) coefficients of an infinite number of operators appearing in the correlator under analysis. Based on this idea we develop the extremal functional method (EFM), a numerical procedure for deriving the spectrum and OPE coefficients of CFTs lying on the boundary (of solution space). We test the EFM by using it to rederive the low lying spectrum and OPE coefficients of the two-dimensional Ising model based solely on the dimension of a single scalar quasiprimary--no Virasoro algebra required. Our work serves as a benchmark for applications to more interesting, less known CFTs in the near future.

Black holes as effective geometries

Gravitational entropy arises in string theory via coarse graining over an underlying space of microstates. In cases with enough supersymmetry, it has been possible to explicitly construct such microstates in spacetime and understand how coarse graining of non-singular, horizon-free objects can lead to an effective description as an extremal black hole. We discuss how these results arise in type II string theory on AdS5 × S5 and on AdS3 × S3 × T4 that preserve 16 and eight supercharges, respectively. For such a picture of black holes as effective geometries to extend to cases with a finite horizon area, the scale of quantum effects in gravity would have to extend well beyond the vicinity of the singularities in the effective theory. By studying examples in M-theory on AdS3 × S2 × CY that preserve four supersymmetries, we show how this can happen.

Quantizing N = 2 multicenter solutions

N = 2 supergravity in four dimensions, or equivalently N = 1 supergravity in five dimensions, has an interesting set of BPS solutions that each correspond to a number of charged centers. This set contains black holes, black rings and their bound states, as well as many smooth solutions. Moduli spaces of such solutions carry a natural symplectic form which we determine, and which allows us to study their quantization. By counting the resulting wavefunctions we come to an independent derivation of some of the wall-crossing formulae. Knowledge of the explicit form of these wavefunctions allows us to find quantum resolutions to some apparent classical paradoxes such as solutions with barely bound centers and those with an infinitely deep throat. We show that quantum effects seem to cap off the throat at a finite depth and we give an estimate for the corresponding mass gap in the dual CFT. This is an interesting example of a system where quantum effects cannot be neglected at macroscopic scales even though the curvature is everywhere small.

Conformal Field Theories in Fractional Dimensions

We study the conformal bootstrap in fractional space-time dimensions, obtaining rigorous bounds on operator dimensions. Our results show strong evidence that there is a family of unitary conformal field theories connecting the 2D Ising model, the 3D Ising model, and the free scalar theory in 4D. We give numerical predictions for the leading operator dimensions and central charge in this family at different values of D and compare these to calculations of ϕ4 theory in the ϵ expansion.

What Maxwell Theory in D 4 teaches us about scale and conformal invariance

The free Maxwell theory in d ≠ 4 dimensions provides a physical example of a unitary, scale invariant theory which is NOT conformally invariant. The easiest way to see this is that the field strength operator F_(μν) is neither a primary nor a descendant. We show how conformal multiplets can be completed, and conformality restored, by adding new local operators to the theory. In d ≥ 5, this can only be done by sacrificing unitarity of the extended Hilbert space. We analyze the full symmetry structure of the extended theory, which turns out to be related to the OSp(d,2|2) superalgebra.

Bootstrapping the Three Dimensional Supersymmetric Ising Model

We implement the conformal bootstrap program for three dimensional conformal field theories with N=2 supersymmetry and find universal constraints on the spectrum of operator dimensions in these theories. By studying the bounds on the dimension of the first scalar appearing in the operator product expansion of a chiral and an antichiral primary, we find a kink at the expected location of the critical three dimensional N=2 Wess-Zumino model, which can be thought of as a supersymmetric analog of the critical Ising model. Focusing on this kink, we determine, to high accuracy, the low-lying spectrum of operator dimensions of the theory, as well as the stress-tensor two-point function. We find that the latter is in an excellent agreement with an exact computation.

Scaling BPS solutions and pure-Higgs states

A bstractDepending on the value of the coupling, BPS states of type II string theory compactified on a Calabi-Yau manifold can be described as multicenter supergravity solutions or as BPS states in a quiver gauge theory. While states that spread into the Coulomb-branch states can be mapped one-to-one to supergravity states, this is not automatically so for the majority of Higgs-branch states. In this paper we explicitly compute the BPS spectrum of the Higgs branch of a three-center quiver with a closed loop, and identify the subset of states that are in one-to-one correspondence with Coulomb/supergravity multicenter states. We also show that there exist additional “pure-Higgs” states, that exist if and only if the charges of the centers can form a scaling solution. Using generating function techniques we compute the large charge degeneracy of the “pure-Higgs” sector and show that it is always exponential. We also construct the map between Higgs- and Coulomb-branch states, discuss its relation to the Higgs-Coulomb map of one of the authors and Verlinde, and argue that the pure Higgs states live in the kernel of this map. Given that these states have no obvious description on the Coulomb branch or in supergravity, we discuss whether they can correspond to a single-center black hole or can be related to more complicated horizonless configurations.

A bound on the entropy of supergravity

We determine, in two independent ways, the number of BPS quantum states arising from supergravity degrees of freedom in a system with fixed total D4D0 charge. First, we count states generated by quantizing the spacetime degrees of freedom of “entropyless” multicentered solutions consisting of