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I will discuss entanglement negativity, an entanglement measure for mixed quantum states, in many-body systems, including lattice quantum systems and quantum field theories. I will also discuss the possible holographic dual description of entanglement negativity in field theories and tensor networks.

The shape deformation of conformal defects is implemented by the displacement operator. In this talk we consider superconformal defects and we provide evidence of a general relation between the two-point function of the displacement and the one-point function of the stress tensor operator. We then discuss the available techniques for the computation of this one-point function. First, we show how it can be related to a deformation of the background geometry. Then we conclude with a discussion of surface defects in four-dimensional N=2 theories, where the chiral algebra provides an additional powerful tool.

The g-theorem is a prominent example of C-theorems in two-dimensional boundary CFT and the extensions are conjectured to hold in higher-dimensional BCFTs. On the other hand, much less is known for C-theorems in a CFT with conformal defects of higher codimensions. I will investigate the entanglement entropy across a sphere and sphere free energy as a candidate for a C-function in DCFT, and show they differ by a universal term proportional to the vev of the stress tensor. Based on this relation, I will propose to use the sphere free energy as a C-function in DCFT. This proposal unifies the previously known theorems and conjectures, and passes several checks, including a few examples in field theories and a holographic proof in simple gravity dual models of DCFTs.

I will review recent results concerning a general class of parametric BPS Wilson loops in ABJM theory. In particular, I will present a proposal for their exact quantum expression in terms of a parametric Matrix Model and discuss their role in the exact calculation of physical quantities like the Bremsstrahlung function and in testing the AdS4/CFT3 correspondence.

We observe that boundary correlators of the elementary scalar field of the Liouville theory defined on rigid AdS2 background are the same as the correlators of the chiral stress tensor of the Liouville CFT on the complex plane restricted to the real line. The same relation generalizes to the conformal abelian Toda theory: correlators of Toda scalars on AdS2 are directly related to those of the chiral W-symmetry generators in the Toda CFT and thus are essentially controlled by the underlying infinite-dimensional symmetry. These may be viewed as examples of AdS2/CFT1 duality where the CFT1 is the chiral half of a 2d CFT. Similar relation applies also to a non-abelian Toda theory containing a Liouville scalar coupled to a 2d sigma-model originating from the SL(2,R)/U(1) gauged WZW model. Here the Liouville scalar is again dual to the chiral stress tensor T while the other two scalars are dual to the parafermionic operators V± of the non-abelian Toda CFT. The duality is checked at the next-to-leading order in the large central charge expansion by matching the chiral CFT correlators of (T,V+,V−) with the boundary correlators of the three Toda scalars given by the tree-level and one-loop Witten diagrams in AdS2. Based on arXiv:1904.12753,1907.01357.

I will overview recent results on the defect CFT corresponding to Wilson loop operators in N=4 SYM theory. In particular, I will review the calculation of defect correlators at strong coupling using the AdS2 string worldsheet, and I will present exact results for correlation functions in a subsector of the defect CFT using localization. I will also discuss a defect RG flow from the BPS to the ordinary Wilson loop, which can be used to provide a test of the "defect F-theorem" for one-dimensional defects.

In this talk, I will briefly discuss the construction of semiclassical 1/2-BPS boundary conditions and duality interfaces in 3d N=2 theories, following work with T. Dimofte and D. Gaiotto. Then, I will sketch some mathematical applications of these codimension-1 defects to the geometry of triangulated 4-manifolds and chiral algebras, based on work with T. Dimofte and building off related advances by Gadde, Gukov, and Putrov. I will also mention recent and ongoing work on generating 2d dualities from dual "sandwich" configurations of 3d theories sandwiched by both left and right boundary conditions.

A four-dimensional abelian gauge theory can be coupled to a 3d CFT with a U(1) symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFTs) parametrized by the gauge coupling \tau and by the choice of the CFT in the decoupling limit. Upon performing an Electric-Magnetic duality in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten's SL(2, Z) action. In particular the cusps on the real \tau axis correspond to the 3d gauging of the original CFT. We study general properties of this family of BCFTs. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. Finally, upon assuming particle-vortex duality (and its fermionic version), we show how to turn this machinery into a powerful computational tool to study 3d gauge theories.

The partnership between 3D gravity and Chern-Simons theory is well-known and powerful, but many aspects of this relation are unclear. In this talk I'll discuss Wilson lines in Chern-Simons theory and their role in AdS3 gravity. Wilson lines are interesting objects to explore locality in Chern-Simons theory, and how they affect the partnership at the quantum level.