Quantum Fields and Strings

The analogy between Multi-scale Entanglement Renormalization
Ansatz (MERA) and the spatial slice of three-dimensional anti-de
Sitter space (AdS3) has motivated a great interest in tensor networks
among holographers. I discuss a way to promote this analogy to a
rigorous, quantitative, and constructive relation. A key quantitative
ingredient is the way the strong subadditivity of entanglement entropy
is encoded in MERA and in a holographic spacetime. The upshot is that
the map between MERA and the spatial slice of AdS3 is mediated through
an additional integral transform. Interpreted directly, MERA is a
discretization not of the spatial slice of AdS3, but of the space of
geodesics on the spatial slice of AdS3.

As an alternative to the Holographic Renormalization procedure, we

introduce a regularization scheme for AdS gravity based on the addition

boundary terms which are a given polynomial of the extrinsic and 

intrinsic curvatures (Kounterterms).

Since these terms are closely related to either topological invariants 

or Chern-Simons densities in the corresponding dimension, they can be 

easily generalized to other gravity theories (Einstein-Gauss-Bonnet, 

Lovelock, etc.).

Finally, a general prescription on how to obtained the standard 

counterterm series in AdS gravity is given.​ 

Via the AdS/CFT correspondence, fundamental constraints on the entanglement structure of quantum systems translate to constraints on spacetime geometries that must be satisfied in any consistent theory of quantum gravity. In this talk, we describe some of the constraints arising from strong subadditivity and from the positivity and monotonicity of relative entropy. Our results may be interpreted as a set of energy conditions restricting the possible form of the stress-energy tensor in consistent theories of Einstein gravity coupled to matter."

The moduli space of k G instantons on C^2, where G is a classical gauge group, has a well known HyperKahler quotient formulation known as the ADHM construction. The extension to exceptional groups is an open problem.
In string theory this is realized using a system of branes, and the moduli space of instantons is identified with the Higgs branch of a particular supersymmetric gauge theory with 8 supercharges.
A less known, and less studied aspect of moduli spaces of instantons is that they can be realized as the Coulomb branch of a supersymmetric gauge theory in 2+1 dimensions.
Recent developments on the understanding of the Coulomb branch gives us a nice solution to the problem where G is an exceptional group, thus allowing a systematic study of these moduli spaces which was unavailable so far.
I will discuss these developments, and present the corresponding quivers, and the Coulomb branch Hilbert Series - the main tool which lead to the recent progress.

Recent research has suggested deep connections between geometry and entropy. This connection was first seen in black hole thermodynamics, but has been more fully realized in the Ryu-Takayanagi proposal for calculating entanglement entropies in AdS/CFT. We suggest that this connection is even broader: entropy, and in particular compression, are the fundamental building blocks of emergent geometry. We demonstrate how spatial geometry can be derived from the properties of a recursive compression algorithm for the boundary CFT. We propose a general algorithm for constructing MERA-like tensor networks and elucidate connections to the mathematical field of integral geometry.

The tt* equations define a flat connection on the moduli spaces of 2d, N=2 quantum field theories. For conformal theories with c=3d, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. I will show that the non-holomorphic content of the tt* equations in the cases d=1,2,3 is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with generators of a Lie algebra. This Lie algebra replaces the anti-holomorphic derivatives of tt* and provides these with a finer and algebraic meaning. The generators of the differential rings of special functions are given by quasi-modular forms for d=1 and their generalizations in d=2,3. For d=3, this can be used to provide a purely Lie algebraic formulation of the higher genus topological string theory amplitudes and of the BCOV holomorphic anomaly equations which govern them.

In this talk, I will prove the Landau-Ginzburg mirror symmetry conjecture for general quasi-homogenous singularities, i.e., the FJRW theory (LG A-model) of such polynomials is equivalent to the Saito-Givental theory (LG B-model) of the mirror polynomial. This is joint work with Weiqiang He, Rachel Webb and Yefeng Shen. 

The Ryu-Takayanagi formula relates the entanglement entropy in a conformal field theory to the area of a minimal surface in its holographic dual. I will show that this relation can be inverted to reconstruct the bulk stress-energy tensor near the boundary of the bulk spacetime, from the entanglement on the boundary. I will also show that the positivity and monotonicity of the relative entropy for small spherical domains between the reduced density matrices of an excited state and of the ground state of the CFT, translate to energy conditions in the bulk.

The Truncated Conformal Space Approach (TCSA) is a method that allows for controlled calculations in strongly coupled quantum field theories. It can be used whenever a theory is conformally invariant at short distances, and allows infrared quantities to be calculated in terms of the CFT data of this UV theory.

As an example, I will discuss how the TCSA applies to phi^4 theory in D dimensions and I will show that the method reliably reproduces several nonperturbative phenomena.

> I talk about a method to determine the anomaly polynomials of genera 6d N=(2,0) and N=(1,0) SCFTs, in terms of the anomaly matching on their tensor branches. This method is almost purely field theoretical, and can be applied to all known 6d SCFTs. Green-Schwarz mechanism plays the crucial role.

In this talk I will present some results of an upcoming paper where we study four-dimensional N=2  superconformal field theories using the conformal bootstrap.
We focus on two different four-point functions, involving either the superconformal primary of the flavor current multiplet or the one of the chiral multiplet.
Numerical analysis of the crossing equations yields lower bounds on the allowed central charges, and upper bounds on the dimensions of unprotected operators (for unitary theories).