Quantum Fields and Strings

The mathematical notion of moonshine relates the theory of finite groups with that of modular objects. The first example, 'Monstrous Moonshine', was clarified in the context of two dimensional conformal field theory in the 90's. In 2010, interest in moonshine in the physics community was reinvigorated when Eguchi et. al. observed representations of the finite group M24 appearing in the elliptic genus of nonlinear sigma models on K3. In 2013, Cheng, Duncan, and Harvey provided a uniform construction of 23 new examples of moonshine, called 'umbral moonshine', of which M24 moonshine is a special case. 

In this talk, I will describe recent work studying the symmetries of certain Landau-Ginzburg orbifold theories that flow in the IR to c=6 N=(4, 4) superconformal field theories on the moduli space of K3 sigma models. We show that discrete symmetries of the UV theory implicate all 23 instances of umbral moonshine, not just M24 moonshine, in symmetries of K3 CFTs. I will then discuss a particular string theory compactification to three dimensions where we find a precise connection to all 23 umbral groups and type IIA string theory on K3.

Over the last few years it has become increasingly clear that there is a deep connection between quantum gravity and quantum information. The connection goes back to the discovery that black holes carry entropy with an amount given by the horizon area. I will present evidence that this is only the tip of the iceberg, and prove that a similar area law applies to more general Renyi entanglement entropies. To demonstrate the simplicity of this prescription, I will use it to calculate the mutual Renyi information between two disks of arbitrary dimension. Furthermore, I will provide quantum corrections to the area law and use it to solve the following important problem: what region of the dual spacetime is described by a subregion in a holographic theory? The answer to this question lies in a new perspective that I will advocate: holography is a quantum error correcting code.

I will discuss a natural basis of CFT operators for probing dual gravitational physics in a diffeomorphism-invariant manner. On the CFT side, these operators are already well-known: they are 'OPE Blocks' that contribute to the Operator Product Expansion with fixed Casimir. On the gravity side, I will show that these OPE blocks are dual to diff-invariant geodesic or surface operators. Our new entry to the holographic dictionary can be understood as an operator generalization of the Ryu-Takayanagi proposal and I will show it gives a unified description for a host of important results in holography. 

In this talk, I will investigate the structure of certain protected operator algebras that arise in threedimensional N = 4 superconformal field theories. I will show that these algebras can be understood as a quantization of (either of) the half-BPS chiral ring(s). An important feature of this quantization is that it has a preferred basis in which the structure constants of the quantum algebra are equal to the OPE coefficients of the underlying superconformal theory. I will present evidence in examples that for a given choice of quantum algebra (defined up to a certain gauge equivalence), there is at most one choice of canonical basis, and conjecture that this is true in general.

I will discuss recent work on big crunch singularities produced in asymptotic AdS cosmologies using gauge/gravity duality. The dual description consists of a constant mass deformation of ABJM theory on de Sitter space and is well-defined and stable for small deformations.

  There is a critical deformation where the theory becomes unstable at weak and at strong coupling. I will discuss a field theory diagnostic of this instability as well as boundary two-point correlators calculated via the geodesic approximation. Near the critical deformation a second saddle point contribution enters, in which the spacelike geodesics probe the high curvature region near the singularity. Its contribution strongly enhances the long-distance correlations. This has a natural interpretation in the weakly coupled boundary theory where the critical point corresponds to a massless limit.

In the context of class S theories and 4D/2D duality relations there, we discuss the skein

relations of general topological defects on the 2D side which is expected to be counterparts

of composite surface-line operators in 4D class S theory. Such defects are geometrically

interpreted as networks in a three dimensional space. We also propose a conjectural com-

putational procedure for such defects in two dimensional SU(N) topological q-deformed

Yang-Mills theory by interpreting it as a statistical mechanical system associated with

ideal triangulations.

This talk is based partly on arXiv:1504.00121 with Yuji Tachikawa (Phys.Univ.Tokyo) and much on a forthcoming paper in arXiv.

We study two-dimensional (4, 4) superconformal field theories of central charge c = 6, corresponding to nonlinear sigma models on K3 surfaces, using the superconformal bootstrap method. This is made possible through a surprising relation between the BPS N = 4 superconformal blocks of c = 6 and bosonic Virasoro conformal blocks of c = 28, and an exact result on the moduli dependence of a certain integrated BPS 4-point function. Nontrivial bounds on the non-BPS spectrum are obtained in the K3 CFT as functions of the CFT moduli, that interpolate between the free orbifold points and singular CFT points. We observe directly from the CFT perspective the signature of a continuous spectrum above a gap at the singular moduli, and find numerically an upper bound on this gap that is saturated by the A1 N = 4 cigar CFT.