Quantum Fields and Strings

In this talk I will review the results of recent work in collaboration with  Cecilia De Fazio, Benjamin Doyon and István M. Szécsényi. We studied the entanglement of excited states consisting of a finite number of particle excitations. More precisely, we studied the difference between the entanglement entropy of such states and that of the ground state in a simple bi-partition of a quantum system, where both the size of the system and of the bi-partition are infinite, but their ratio is finite. We originally studied this problem in massive 1+1 dimensional QFTs where analytic computations were possible. We have found the results to apply more widely, including to higher dimensional free theories. In all cases we find that the increment of entanglement is a simple function of the ratio between region's and system's size only. Such function, turns out to be exactly the entanglement of a qubit state where the coefficients of the state are simply associated with the probabilities of particles being localised in one or the other part of the bi-partition. In this talk I will describe the results in some detail and discuss their domain of applicability. I will also highlight the main QFT techniques that we have used in order to obtain them analytically and  present some numerical data. 

In the context of the AdS(4)/BCFT(3) correspondence, we study the holographic entanglement entropy for spatial regions having arbitrary shape. An analytic expression for the subleading term with respect to the area law is discussed. When the bulk spacetime is a part of AdS(4),
this formula becomes the Willmore functional with a proper boundary term evaluated on the minimal surface viewed as a submanifold of the three dimensional flat Euclidean space with a boundary. 
Numerical checks of this formula are performed through a code which allows to construct minimal area surfaces anchored to generic curves. For some simple regions like infinite strips and disks, analytic results are obtained and they confirm the general expression for the subleading term. In particular, when the spatial region contains corners adjacent to the boundary, a logarithmic divergence occurs whose coefficient is determined by a so-called corner function which depends on the boundary conditions. An analytic expression for the holographic corner function and its checks are also discussed.

I will argue that in the context of string theory, the Big Bang singularity of standard and inflationary cosmology is automatically resolved. To see this at the level of an effective field theory, ideas from "Double Field Theory" are useful.

I will present a brief introduction to non-Lorentzian geometries, an important example of such geometries being Newton-Cartan geometry and its torsionful generalization, which is the natural geometry to which non-relativistic field theories couple to. The talk will subsequently review how such geometries have in recent years appeared in gravity, string theory and holography. In particular, torsional Newton-Cartan geometry has been shown to appear as the boundary geometry for Lifshitz spacetimes. Furthermore, dynamical Newton-Cartan geometry is related to Horava-Lifsthiz gravity theories and appears in novel Chern-Simons theories of gravity in three dimensions. The latter can be obtained from a well-defined limit of the AdS3/CFT2 correspondence. Finally, I will briefly comment on how Newton-Cartan geometry appears in non-relativistic string theory.

Generically, a small amount of matter introduced to anti-de Sitter spacetime leads to formation of a black hole; however, the high degree of symmetry of AdS means that some initial distributions of matter (possibly also technically generic) oscillate indefinitely. Whether a given initial profile leads to a horizon at arbitrarily small amplitudes is of great interest for a number of reasons, not least because horizon formation corresponds holographically to thermalization in CFT.  We will present an overview of approaches the question and show a phase diagram of the stability behavior of AdS in the presence of a scalar field, including an analysis of nonperturbative phases at the border of perturbative stability and instability.

Polyakov’s bootstrap programme aims at solving conformal field theories using 
unitarity and conformal symmetry. Its implementation in two dimensions has been 
highly successful and numerical studies, in particular of the 3-dimensional Ising 
model, have clearly demonstrated the potential for higher dimensional theories. 
Analytical results in higher dimensions, however, require significant insight 
into the conformal group and its representations. Surprisingly little is actually 
known about this important group theory challenge. I  will explain a remarkable 
and unexpected connection with a class of Schroedinger equations that was uncovered 
in recent joint work with M. Isachenkov. The study of the relevant quantum mechanics 
systems has created an entire branch of modern mathematics whose results can now be 
put to use in the conformal bootstrap program. 

I will review recent progress in computing the exact planar spectrum of closed strings in
AdS3 backgrounds with 8+8 supercharges, including the derivation of the protected spectrum.
Such theories have a multi-dimensional moduli space, and I will show what effect varying
the moduli has on the exact closed string spectrum, and how the integrable structure changes
as we do so, paying particular attention to the background supported by NS-NS flux.

String theory provides us with 8d supersymmetric gauge theories with gauge algebras su(N), so(2N), sp(N), e_6, e_7 and e_8, but no construction for so(2N+1), f_4 and g_2 is known. If string theory is universal in 8 dimensions, this pattern requires explanation. I will show that the theories for f_4 and so(2N+1) have a global gauge anomaly in flat space, while g_2 does not have it. Surprisingly, we also find that the sp(N) theories, arising from example from O7^+ planes in string theory, have a subtler gauge anomaly. This subtler anomaly, in contrast to the one in flat space, could in principle be canceled by a topological analogue of the Green-Schwarz mechanism. I will discuss one simple example of such a generalized anomaly cancellation mechanism in three dimensions, and then explain why the generalized Green-Schwarz term required in 8 dimensions to make the O7^+ consistent is necessarily a more subtle generalization of a Chern-Simons coupling

We study the spectrum of local operators living on a defect in a generic conformal field theory, and their coupling to the local bulk operators.  We establish the existence of universal accumulation points in the spectrum at large s, s being the charge of the operators under rotations in the space transverse to the defect. Our tools include a formula that inverts the bulk to defect OPE and is analytic in s, analogous to the Caron-Huot formula for the four-point function. Some important assumptions are made in deriving this result: we comment on them.

In the first part of the seminar I will give a short review of the frame-independent formulation of Newtonian gravity, called Newton-Cartan Gravity, and explain why there is a renewed interest into non-relativistic gravity in general.  In the second part I will discuss, as a particular application,  a recent proposal for an Effective Field Theory describing a massive spin-2 mode (the so-called GMP mode) in the Fractional Quantum Hall Effect.

Topological defect operators are extended operators in a quantum field theory (QFT) whose correlation functions are independent of continuous changes of the ambient space. They satisfy nontrivial fusion relations and put nontrivial constraints on the QFT itself and its deformations (such as renormalization group (RG) flows). Canonical examples of topological defect operators include generators of (higher-form) global symmetries whose constraints on the QFT have been well-studied. However they constitute just a tip of a much richer framework which we explore in detail in two spacetime dimensions. 

More specifically, we study the crossing relations of general topological defect lines (TDL) in two dimensions, discuss their relation to the usual 't Hooft anomalies associated to global symmetries, and use them to constrain RG flows to either conformal fixed points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.