The 22nd edition of the ICFT workshop series on integrable and conformal field theories in the UK will be hosted at

**Cardiff University, June 1 and 2, 2018**.

The event location is

School of Mathematics

Cardiff University

Senghennydd Road

CF24 4AG Cardiff (UK)

All talks will take place in the room E/2.20.

To get to the department from Cardiff central station on foot, see this map. Alternatively, take a taxi.

**Local organisers: **Gandalf Lechner, Simon Wood

**Organising committee: **P. Bowcock, P. Dorey, C. Dunning, P. Fendley, A. Fring, C. Korff, N. MacKay, A. Taormina, G. Watts, R. Weston, C. Young

**Contact: ** LechnerG and WoodSi at cardiff.ac.uk

**Registration: **Everybody is welcome to attend. Please inform one of the organisers via email (see above) if you want to participate, and also mention if you want to join for the dinner on Friday evening.

**Programme**

**Friday**

1:30-2:00 arrival, coffee

2:00-3:00 Kasia Rejzner:

Sine Gordon model from the perspective of perturbative algebraic quantum field theory

3:00-3:10 break

3:10-3:50 Julia Cen:

Complex solitons, reality and degenerate structure

3:50-4:20 coffee break

4:20-5:20 Yoh Tanimoto:

Operator-algebraic construction of integrable QFT and CFT

5:20-5:30 break

5:30-6:10 Junpeng Cao:

Off-diagonal Bethe ansatz solution of high spin Heisenberg model with generic integrable boundaries

8:00 dinner at la Cuina

**Saturday**

9:15-10:15 Ana Ros Camacho:

On a first example of tensor equivalence within the Landau-Ginzburg/conformal field theory correspondence

10:15-10:25 break

10:25-11:05 Thomas Gemuenden:

Orbifolds of lattice vertex operator algebras with sparse light spectrum

11:05-11:35 coffee break

11:35-12:15 Sam Fearn:

A supersymmetric index for 2d theories with large N=4 superconformal symmetry

12:15-12:25 break

12:25-1:25 Cornelius Schmidt-Colinet:

Renormalisation group interfaces in conformal field theory

**Abstracts:**

- Junpeng Cao:
**Off-diagonal Bethe ansatz solution of high spin Heisenberg model with generic integrable boundaries**

The off-diagonal Bethe ansatz method is generalized to the high spin integrable systems associated with the su(2) and su(n) algebra by employing the spin-s isotropic Heisenberg chain model with generic integrable boundaries as an example. With the fusion techniques, certain closed operator identities for constructing the functional T-Q relations and the Bethe ansatz equations are derived. It is found that a variety of inhomogeneous T-Q relations obeying the operator product identities can be constructed. Numerical results for two-site s = 1 case indicate that an arbitrary choice of the derived T-Q relations is enough to give the complete spectrum of the transfer matrix. The corresponding eigenstates are also constructed.

- Julia Cen:
**Complex solitons, reality and degenerate structure**

Usually, one only thinks of real soliton solutions to be physically interesting, but we found some particular complex soliton solutions to also be meaningful, due to the fact that they have real energies. We show how to construct complex multi-soliton solutions to the complex Korteweg de-Vries and sine-Gordon equations [1] using Hirota’s method, Bäcklund and Darboux-Crum transformations. Through computing the time-delays from multi-soliton scattering, we can explain how PT-symmetry along with integrability ensures reality of energy for complex multi-solitons [2]. We also discovered how to obtain degenerate multi-solitons with these methods [3,4] and found they have time-dependent displacement limit from multi-soliton scattering rather than the usual constant displacement limit as for the non-degenerate case. We will also briefly introduce some new nonlocal integrable systems we recently found [5].Based on:

[1] J. Cen and A. Fring, Complex solitons with real energies, J. Phys. A 49(36), 365202 (2016)

[2] J. Cen, F. Correa and A. Fring, Time-delay and reality conditions for complex solitons, J. Math. Phys. 58(3), 032901 (2017)

[3] F. Correa and A. Fring, Regularized degenerate multi-solitons, J. High Energy Phys. 2016(9), 8 (2016)

[4] J. Cen, F. Correa and A. Fring, Degenerate multi-solitons in the sine-Gordon equation, J. Phys. A 50(43), 435201 (2017)

[5] J. Cen, F. Correa and A. Fring, Integrable nonlocal Hirota equations, arXiv:1710.11560 (2017) -
Sam Fearn:

**A supersymmetric index for 2d theories with large N=4 superconformal symmetry**

The elliptic genus of K3 is a topological invariant on the moduli space of theories describing superstrings propagating on K3, and is known to exhibit a phenomenon coined 'Mathieu Moonshine'. The elliptic genus may be calculated using data from the underlying 2d N=4 superconformal field theory, which includes the Witten index of irreducible, unitary representations of the corresponding N=4 superconformal algebra. In theories with a specific extension of this N=4 symmetry, called 'large N=4 symmetry', the Witten index of any unitary, irreducible representation is trivial. However, a new supersymmetric index known as I_1 that generalises these previous indices was introduced by Gukov, Martinec, Moore and Strominger. In this talk I will discuss the index I_1 for representations of the large N=4 algebra as well as a method for calculating this index for a specific class of theories which exhibit large N=4 superconformal symmetry. -
Thomas Gemünden:

**Orbifolds of lattice vertex operator algebras with sparse light spectrum**

A holomorphic vertex operator algebra of central charge c is called extremal if its only states of weight less than 1 + [c/24] are Virasoro descendants. Very few examples of such vertex operator algebras are known - most notably the "moonshine module" - and their existence is an open question for c>40. We will give a brief review of lattice vertex operator algebras and the theory of cyclic orbifolds by Scheithauer, et al.. Then we discuss how orbifolds of vertex operator algebras constructed from extremal lattices can be used to find new examples of vertex operator algebras with sparse light spectrum and consider examples at c=48 and c=72. This is based on joint work with Christoph Keller. - Kasia Rejzner:
**Sine Gordon model from the perspective of perturbative algebraic quantum field theory**

In this talk I will explain how the framework of perturbative algebraic quantum field theory (pAQFT) can be used to construct the net of von Neumann algebras for the Sine Gordon model in two dimensions. The idea is to start with the massless scalar field in two dimensions, then write down the perturbation series for the interacting fields (as abstract operators, without fixing the Hilbert space representation) and show its convergence. Next, one considers a convenient representation of the theory that deals with the infrared problem of the massless scalar field in two dimensions (an infravacuum representation) and one studies properties of the net, including the duality between the Sine Gordon model and the Thirring model.

- Ana Ros Camacho:
**On a first example of tensor equivalence within the Landau-Ginzburg/conformal field theory correspondence**

The Landau-Ginzburg/conformal field theory correspondence is a result dating from the late 80s-early 90s which in particular predicts some relation between categories of representations of vertex operator algebras and categories of matrix factorizations. There are only few examples of this correspondence, and in particular, just one tensor equivalence between two such categories. We will review in detail this result, developed joint with A. Davydov and I. Runkel, and present some work in progress joint with D. Ridout (University of Melbourne) on certain features of the VOAs side appearing as well at the matrix factorizations one.

- Cornelius Schmidt-Colinet:
**Renormalisation group interfaces in conformal field theory**

Renormalisation group interfaces are conformal interfaces which correspond to (relevant) renormalisation group flows.

We will consider examples in two dimensional coset CFTs, and an approach by holographic methods to a particular

class of RG interfaces which correspond to double trace deformations of theories containing a generalised free field.

- Yoh Tanimoto:
**Operator-algebraic construction of integrable QFT and CFT**

We review a recent construction of the full family of local observables satisfying the Haag-Kastler axioms, a mathematically satisfying framework of quantum field theory, for a class of factorizing diagonal S-matrix. A subfamily these models are naturally realized on the same Hilbert space as that of some copies of the U(1)-current (the Heisenberg CFT) and can be regarded as deformations of the latter.

Funded by the London Mathematical Society and the IoP (Mathematical and Theoretical Physics Group).