Quantum Fields and Strings

Carefully studying the singularity structure of the moduli space of vacua of super conformal field theories, provides an incredibly powerful tool to extract information about such theories. In particular I will outline a concrete, yet technically challenging, strategy to go about carrying out a classification for SCFTs with extended supersymmetry in 4d.

I will explain in details how this classification has been worked out successfully for N=2 SCFTs in rank-1 and it is very promising in rank-2. And how with even higher Supersymmetry (N≥3) we can obtain a classification valid for all ranks making a few reasonable, yet key, assumptions.

We establish a direct connection between scattering amplitudes for bi-adjoint scalar theories and a classic polytope--the "associahedron"--known to mathematicians since the 1960s. We find an associahedron naturally living in kinematic space. The tree level scattering amplitude is simply a geometric invariant of the associahedron called its "canonical form" [2], which is a differential form on kinematic space with logarithmic singularities on the boundaries of the associahedron. We show that basic physical principles like locality and unitarity are "rediscovered" as properties of the geometry, and certain "soft" limits can be converted to geometric statements as well.

The associahedron in kinematic space plays an important role in the context of scattering equations. We discover that the scattering equations act as a diffeomorphism between the interior of the open string moduli space (yet another associahedron) and the associahedron in kinematic space. This observation provides the key to a novel derivation of the bi-adjoint CHY formula. Finally, we emphasize the importance of the scattering amplitude as a differential form, as suggested by the "canonical form" construction. We argue on general grounds that every color-dressed amplitude is dual to a form on kinematic space. This is motivated by the surprising observation that "projective" forms on kinematic space satisfy Jacobi identities and therefore have novel implications for color-kinematics duality.

References:

[1] Nima Arkani-Hamed, Yuntao Bai, Song He & Gongwang Yan. In preparation.

[2] Nima Arkani-Hamed, Yuntao Bai & Thomas Lam. Positive Geometries and Canonical Forms. arXiv 1703.04541

I will describe a holographic solution corresponding to a traversable wormhole in AdS/CFT.  The construction involves directly coupling the boundary CFT's, and corresponds to chaos-facilitated quantum teleportation between the two sides.  It does not violate boundary causality or other fundamental physics principles.

Known N=4 theories in four dimensions are characterized by a choice of gauge group, and in some cases some "discrete theta angles", as classified by Aharony, Seiberg and Tachikawa. I will review how this data, for the theories with algebra su(N), is encoded in various familiar realizations of the theory, in particular in the holographic AdS_5 \times S^5 dual and in the compactification of the (2,0) A_N theory on T^2. I will then show how the resulting structure, given by a choice of polarization of an appropriate cohomology group, admits additional choices that, unlike known theories, generically preserve SL(2,Z) invariance in four dimensions.

I discuss some aspects of boundary conformal field theories (bCFTs). I will demonstrate that free bCFTs have a universal way of satisfying crossing symmetry constraints. I will introduce a simple class of interacting bCFTs where the interaction is restricted to the boundary. Finally, I will discuss relationships between boundary trace anomalies and boundary limits of stress-tensor correlation functions.

We review recent developments concerning the dynamics of QCD in 2+1 dimensions. We will discuss the phases of the theory depending on the matter representations and the Chern-Simons level. We present several new dualities and conjectures about the behaviour of these theories in their strongly coupled phases.

We work out constraints imposed by channel duality and analyticity on tree-level amplitudes of four identical real scalars, with the assumptions of a linear spectrum of exchanged particles and Regge asymptotic behaviour. We reduce the requirement of channel duality to a countably infinite set of equations in the general case. We show that channel duality uniquely fixes the soft Regge behaviour of the amplitudes to that found in String theory, (-s)^(2t). Specialising to the case of tachyonic external particles, we use channel duality to show that the amplitude can be any one in an infinite-dimensional parameter space, and present evidence that unitarity doesn't significantly reduce the dimension of the space of amplitudes.
This talk is based on 1707.08135 by Pranjal Nayak, Rohan R. Poojary and RMS.

Branch point twist fields play an important role in the study of measures of entanglement such as the Rényi entropies and the Negativity. In 1+1 dimensions such measures can be written in terms of multi-point functions of branch point twist fields. For 1+1-dimensional integrable quantum field theories and also in conformal field theory much is known about how to compute correlation functions and, with the help of the twist field, this knowledge can be exploited in order to gain new insights into the properties of various entanglement measures. In this talk I will review some of our main results in this context.

I will then go on to introduce a new (related) class of fields we have recently named conical twist fields. These are fields whose two-point functions have (surprisingly) been found to describe gluon amplitudes in the strong coupling limit of super Yang-Mills theories and therefore have featured in a completely different context from that of entanglement measures. Interestingly, at critical points, branch point and conical twist fields have the same conformal dimension and beyond criticality they also have very similar form factors, however they are different in many other respects.  In my talk I will discuss and justify some of their similarities and differences.

A state is called a Markov state if it fulfil the important condition of saturating the Strong Subadditivity inequality. I will show how the vacuum state of any relativistic QFT is a Markov state when reduced to certain geometric regions of spacetime. A characterisation of this regions will be presented as well as two independent proofs of the Markov condition in QFT. 

For the CFT vacuum, the Markov property is the key ingredient to prove the a-theorem (irreversibility of the RG flow in QFT in d=4 spacetime dimensions) using vacuum entanglement entropy. This extends the entropic proofs of the c and F theorems in dimensions d=2 and d=3 and gives a unified picture of all the known irreversibility theorems in QFT. 

I will also comment on the relation of this Markov property with the unitarity bound and other information theory inequalities.

After a small review on divergent series and Borel resummation I will discuss a geometric approach based on Picard-Lefschetz theory to study the interplay between perturbative and non-perturbative effects in the QM path integral.

Such approach can be used to characterize when the perturbative series gives the full answer and when the inclusion of non-trivial saddles--instantons--is mandatory. I will then show how a simple deformation of the original perturbation theory allows to recover the full non perturbative answer from the perturbative coefficients alone, without the need of including instanton corrections. I will illustrate this technique in examples which are known to contain non-perturbative effects, such as the (supersymmetric) double-well potential, the pure anharmonic oscillator, and the perturbative expansion around a false vacuum.

After reviewing the three things in the title I will argue that they represent the same physical phenomenon. In details, Jackiw–Teitelboim (JT) gravity coupled to an arbitrary quantum field theory results in a gravitational dressing of field theoretical scattering amplitudes. The exact expression for the dressed S-matrix was previously known as a solvable example of a novel UV asymptotic behavior, dubbed asymptotic fragility. This dressing is equivalent to the T\barT deformation of the initial quantum field theory compactified on a circle. The same dressed S-matrix is also obtained as a flat space limit of the near AdS2 holography based on (JT) gravity. In order to preserve the flat space unitarity, however, the conventional Schwarzian dressing of boundary correlators needs to be slightly extended. As an intermediate result I will present a new simple expression for flat space amplitudes of massive particles in terms of correlators of holographic CFT’s. 

In quantum theory, there can exist correlations between subsystems of a new kind that are absent in classical systems. These correlations are nowadays called "entanglement".

An entanglement measure is a functional on states quantifying the amount of entanglement across two subsystems (i.e. causally disjoint regions in the context of quantum field theory). A reasonable measure should satisfy certain general properties: for example, it should assign zero entanglement to separable states, and be monotonic under separable, completely positive maps ("LOCC-operations"). The v. Neumann entropy of the "reduced state" (to one of the subsystems) is one such measure if the state for the total system is pure. But for mixed states, it is not, and one has to consider other measures. In particular, one has to consider other measures if the subsystems have a finite non-zero distance.

In this talk I will present several good measures, and in particular analyze the « relative entanglement entropy", $E_R$, defined as the "distance" of the given state to the set of separable states, where "distance" is defined using Araki's relative entropy. I will show several features of this measure for instance: (i) charged states, where the relative entanglement entropy is related to the quantum dimension of the charge, (ii) vacuum states in 1+1 dimensional integrable models, (iii) general upper bounds for certain special regions in general CFTs in d dimensions, (iv) area law type bounds. I will also explain the relationship between $E_R$ and other entanglement measures, such as distillable entropy.

[Based on joint work with Jacobus Sanders.]