# Defects in topological and conformal field theory

King's College London
27 - 28 June 2019

### Venue

The meeting took place at the Strand Campus of King's College London. The meeting started on Thursday 27th of June at 1pm and ended on Friday 28th at around 4pm.

Here is a map of the campus. The Strand building is marked B [in green] and the King's building is marked A [in red]. The main entrance is on the strand [the red arrow with the pedestrian and wheelchair symbols].

Local travel information may also be found here

London transport information is available here. The nearest tube station is Temple; other stations within walking distance are Embankment, Charing Cross, Covent Garden and Holborn.

Train information may be obtained from here. The nearest national rail stations are Charing Cross, Waterloo & Waterloo East, Blackfriars and City Thameslink.

### Programme

The programme was as follows:

 Thursday, 27.6 All talks were in room S-1.06 12:30 - 13:00 Registration [Room K0.16] 13:00 - 14:00 Carqueville Orbifolds via defects 14:00 - 14:30 Belletête Fusion quotients and topological defects in Temperley-Lieb lattice models 14:30 - 15:00 coffee [Room K0.16] 15:00 - 16:00 Konechny Boundary renormalisation group interfaces 16:00 - 16:30 Distler Conformal Manifolds in Class-S 16:30 - 17:00 coffee [Room K0.16] 17:00 - 17:30 Shen A Defect Verlinde Formula 17:30 - 18:00 Meineri Colliders and conformal interfaces 18:00 - 18:30 Rodgers The Weyl anomaly of Wilson surface defects in N=(2,0) theory Friday, 28.6: All talks were in room K-1.56 9:00 - 10:00 Henriques Notions of defects between chiral CFTs 10:00 - 10:30 Woike Boundary Conditions and the Swiss-Cheese Operad 10:30 - 11:00 coffee [Room K0.16] 11:00 - 12:00 Beem Holomorphic twists, vertex algebras, and conformal defects in four dimensions 12:00 - 12:30 Northe Interface flows in the D1/D5 system 12:30 - 14:00 lunch 14:00 - 15:00 Meusburger Mapping class group actions in Kitaev lattice model 15:00 - 15:30 Müller Defects and Orbifolds of 2-dimensional Yang-Mills theory 15:30 - 16:00 Schulz Boundaries and supercurrent multiplets in 3D Landau-Ginzburg models

### Abstracts and slides

The abstracts (and slides) for the talks are as follows:

 Beem Holomorphic twists, vertex algebras, and conformal defects in four dimensions Belletête Fusion quotients and topological defects in Temperley-Lieb lattice models The Temperley-Lieb lattice models are an infinite family of integrable lattice models which are often thought to approximate certain 2D conformal field theories on finite lattices. A construction of topological defects in these models has been suggested in terms of a relatively simple recipe: one takes the transfer matrix of one of those models and after following a series of algebraic manipulations one is left with a transfer matrix describing the same model now carrying a topological defect. However, it turns out that this recipe is a concrete realization of a more general construction which we dubbed the fusion quotients. I will explain how these operators are constructed and how to interpret them in terms of fusion rules. Carqueville Orbifolds via defects I will review the orbifold construction for n-dimensional defect TQFTs and discuss examples for n=2 and n=3, based on joint work with Meusburger, Runkel and Schaumann. Distler Conformal Manifolds in Class-S Conformal Field Theories sometimes come in continuous families, parametrized by a "conformal manifold". The most basic example is a supersymmetric gauge theory with exactly-vanishing beta-function(s). Naively, the conformal manifold is the quotient of the space of gauge coupling by some group of identifications ("S-dualities"). This picture is too naive. I will review some lessons, gleaned from theories of Class-S, about the global structure of their conformal manifolds. Henriques Notions of defects between chiral CFTs The question what is a defect between chiral CFTs does not admit a unique answer. I will investigate a couple different notions, and explain how they are related. Konechny Boundary renormalisation group interfaces Renormalisation group (RG) interfaces were introduced by I. Brunner and D. Roggenkamp in 2007. To construct such an interface consider perturbing a UV fixed point, described by a conformal field theory (CFT), by a relevant operator on a half space. Renormalising and letting the resulting QFT flow along the RG flow we obtain a conformal interface between the UV and IR fixed point CFTs. Although enjoying a full conformal symmetry this interface carries information about the RG flow it originated from. In this talk I will consider a rather special case of the RG interface between two boundary conditions of a 2D CFT which is obtained from a boundary RG flow interpolating between two conformal boundary conditions. This interface is zero-dimensional and is thus described by a local boundary-condition changing operator. I investigate its properties in concrete models and formulate some general conjectures that can help charting phase diagrams of boundary RG flows. Meineri Colliders and conformal interfaces We set up a scattering experiment of matter against an impurity which separates two generic one-dimensional critical quantum systems. We compute the flux of reflected and transmitted energy, thus defining a precise measure of the transparency of the interface between the related two-dimensional conformal field theories. If the largest symmetry algebra is Virasoro, we find that the reflection and transmission coefficients are independent of the details of the initial state, and are fixed in terms of the central charges and of the two-point function of the displacement operator. The situation is more elaborate when extended symmetries are present. Positivity of the total energy flux at infinity imposes bounds on the coefficient of the two-point function of the displacement operator, which controls the free-energy cost of a small deformation of the interface. Meusburger Mapping class group actions in Kitaev lattice model Müller Defects and Orbifolds of 2-dimensional Yang-Mills theory In this talk we discuss symmetries of 2-dimensional Yang-Mills theory corresponding to outer automorphisms of the structure group G and the corresponding defects. We argue that in the topological limit the partition function with defects computes the symplectic volume of the moduli space of twisted $$G$$-bundles. Using the defect approach to orbifolds we construct the corresponding orbifold theory. We present our results using lattice renormalization and in the functorial approach to area-dependent QFTs via regularised Frobenius algebras introduced by Runkel and Szegedy. The talk is based on joint work in progress with R. Szabo and L. Szegedy. Northe Interface flows in the D1/D5 system Not only has the Kondo model played a major role in condensed matter physics, but it has also sharpened our techniques for dealing with boundaries and interfaces in CFT and string theory. Within CFT, the Kondo effect is described via branes which acquire additional dimensions. Starting from the D1/D5 system, we have found the BPS solutions to the DBI system describing RG flows between D1- and D3-brane solutions. Using a class of half BPS solutions provided we find corresponding backreacted supergravity interface solutions for both types of branes and confirm the g-theorem. Our approach provides an explicit example of a Kondo-like CFT defect, with an explicit gravitational dual. Rodgers The Weyl anomaly of Wilson surface defects in N=(2,0) theory N=(2,0) theory is an interacting six-dimensional superconformal field theory, of importance in M-theory. It is believed to admit two-dimensional defects called Wilson surfaces, similar to Wilson lines in Yang-Mills theory. An important quantity characterising these defects is the Weyl anomaly, which may count the massless degrees of freedom on Wilson surface. I will describe the calculation of two coefficients in the contribution of these defects to the Weyl anomaly, using the AdS/CFT correspondence. Schulz Boundaries and supercurrent multiplets in 3D Landau-Ginzburg models Theories with 3D N=2 bulk supersymmetry may preserve a 2D N=(0,2) subalgebra when a boundary is introduced, possibly with localized degrees of freedom. We propose generalized supercurrent multiplets with bulk and boundary parts adapted to such setups. Using their structure, we comment on implications for the $$\bar Q_+$$-cohomology. As an example, we apply the developed framework to Landau-Ginzburg models. In these models, we study the role of boundary degrees of freedom and matrix factorizations. We verify our results using quantization. Shen A Defect Verlinde Formula We revisit the problem of boundary excitations at a topological boundary or junction defects between topological boundaries in non-chiral bosonic topological orders in 2+1 dimensions. Based on physical considerations, we derive a formula that relates the fusion rules of the boundary excitations, and the "half-linking" number between condensed anyons and confined boundary excitations. This formula is a direct analogue of the Verlinde formula. We also demonstrate how these half-linking numbers can be computed in explicit Abelian and non-Abelian examples. As a fundamental property of topological orders and their allowed boundaries, this should also find applications in finding suitable platforms realizing quantum computing devices. Woike Boundary Conditions and the Swiss-Cheese Operad Through the machinery of factorization homology, little disk algebras offer a way to construct fully extended topological field theories. In dimension two, this naturally leads to (ribbon) braided monoidal categories. The little disk operad can be modified to account for various types of stratifications, the easiest case being the one of a boundary. In that case the little disk operad is replaced the Swiss-Cheese operad. When computing its categorical algebras, one recovers an ansatz made by Fuchs, Schweigert and Valentino for the description of boundary conditions and surface defects in three-dimensional topological field theory based on physical assumptions. Based on joint work with Lukas Müller on the little bundles operad one can generalize this to equivariant field theories.

### Registration

There was a registration fee of £20 for the meeting to cover coffee/tea/biscuits, payable on registration; this was reduced to £10 for members of the Institute of Physics.

### Support

There were limited funds available to support UK-based Research students and those with caring responsibilities. The Institute of Physics provides support to PhD students through its own internal funding mechanism.

### Accommodation

Accommodation in central London can be expensive and cheaper options can fill quickly.

Student halls of residence can provide cheap accommodation, but King's itself is not available during this period. You could try the London School of Economics residences or the University of London residences; others may also be available.

Some hotels that are close to King's and which are often used by the Mathematics department the Crescent Hotel and the Bedford Hotel. Another option is the Goodenough. The closest hotel is the Strand Palace Hotel but this is also more expensive.