# Programme

# 15th UK meeting on Integrable Models, Conformal Field Theory and related Topics

## Scientific Programme

The programme will start with the registration and coffee at 14.30 on the Friday and finish on Saturday at 13:00.

**Timetable**

Friday 15/04/2011 |
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14:30 - 15:00 | registration/coffee | |

15:00 - 15:50 | Jean Sebastien Caux |
Dynamics in one dimension: Integrabilty in and out of equilibrium |

15:50 - 16:40 | Paulo Assis |
PT-symmetry and integrable models |

16:40 - 17:10 | coffee/tea | |

17:10 - 17:40 | Vidas Regelskis |
Integrable boundaries in AdS/CFT |

17:40 - 18:10 | Peng Zhao |
Integrability of High Spin Operators in N=4 supersymmetric Yang-Mills Theory |

20:00 | conference dinner | at Rodizio Rico |

Saturday 16/04/2011 |
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9:30 - 10:20 | Ingo Runkel |
Defects and integrable deformations |

10:20 - 11:10 | Roman Zwicky |
Identifying conformal gauge theories on the lattice |

11:10 - 11:40 | coffee/tea | |

11:40 - 12:30 | Charles Young |
Extended T systems |

12:30 - 13:00 | James Silk |
Twist Fields in Massive Dirac Theory |

**Abstracts**

**Jean-Sebastien Caux **

*Dynamics in one dimension: integrability in and out of equilibrium* **(pdf of slides)**

Over the last few years, integrability has become a method of choice for the calculation of equilibrium dynamical correlation functions of systems such as spin chains and interacting atomic gases, its main strength being its ability to go beyond commonly-used low-energy effective theories. After a brief review, this talk will present some new results on simple and more elaborate observables, with a discussion of their relevance to new types of experiments such as resonant inelastic x-ray scattering, and of their relation to new developments in Luttinger liquid theory.

For out-of-equilibrium dynamics, most importantly following a sudden change in one of the global parameters of a system (quantum quench), integrability will be shown to be able to offer reliable results on (re)equilibration phenomena with potential applications in quantum dots, quantum magnets and atomic gases, this approach having the unique advantage of being applicable to arbitrary time scales.

**Paulo Goncalves de Assis**

*PT-symmetry and integrable models ***(pdf of slides)**

In this talk I will revise the basic ideas underlying a powerful correspondence between integrable models and ordinary differential equations (or IM/ODE correspondence, for short). This spectral equivalence played an important role in the study of PT-symmetry, providing rigorous proofs about the spectra of an important class of non-Hermitian Hamiltonians based on known integrable results. My intention is to make progress in the opposite direction, namely to extract information about integrable systems by using differential equations. The comparison between the eigenvalues of the ODEs and the spectral parameters in certain Quantum Integrable Models is considerably well understood for the vacuum state of the latter. For excited states, however, results are only known for the problem with su(2) symmetry.

Here I will show this can be extended to the su(2|1) super-symmetric problem in order to establish the exact correspondence also for excited states.

**Vidas Regelskis **

*Integrable boundaries in AdS/CFT ***(pdf of slides)**

The recent progress in solving the planar limit of AdS/CFT was

accelerated a lot by underlying integrable structures. The striking

success in finding the scattering S-matrices and reflection K-matrices from various boundaries was achieved due to underlying psu(2, 2|4) symmetry which factorizes into two copies, left and right, of the centrally-extended su(2|2) algebra and it's Yangian extension.

In this talk I shall discuss the reflection from D3, D7 and D5 branes.

Boundary conditions depends crucially on the orientation of the brane inside the AdS_5 x S^5 space and some cases allow a `twisted boundary Yangian' structure which is a co-ideal subalgebra of the usual Yangian of the bulk S-matrix. In particular, I will consider the twisted Yangian of the Y=0 D3 brane and the achiral twisted Yangian of D5 brane.

**Ingo Runkel **

*Defects and integrable deformations ***(pdf of slides)**

An integrable field theory possesses an infinite number of mutually commuting conserved charges. I would like to explain how in certain two-dimensional conformal field theories (in particular in minimal models), these charges can be expressed as one-parameter families of defect line operators. This approach offers a new way to understand the T-system functional relations satisfied by the conserved charges and to give a condition under which they survive a perturbation of the conformal field theory away from the critical point.

**James Silk **

*Twist Fields in the Massive Dirac Theory ***(pdf of slides)**

The massless limit of the Dirac theory is a CFT and by examining this CFT we can infer relations in the massive theory. I will outline how twist fields appear in the CFT and how we can verify that the relations we find can be carried across to the massive case. With this consistent construction a simple equation for the vacuum expectation values of twist fields presents itself and we can also find differential equations for certain correlation functions of the twist fields. This talk is based on the work presented in arXiv:1103.2328.

**Charles Young**

*Extended T-systems ***(pdf of slides)**

The T-system is a set of recurrence relations with importance in many areas of integrable systems. Its original setting is the category of finite-dimensional representations of quantum affine algebras, or Yangians, where it can be seen as a system of short exact sequences among tensor products of Kirillov-Reshetikhin modules*. *

Using the theory of q-characters, I will show that, from this viewpoint, the T-system is contained within several larger systems of short exact sequences. I discuss in detail examples in types A and B.

In particular, in type A there is system which closes on the class of all minimal affinizations, while in type B the analogous system closes not on minimal affinizations but on a larger class of modules that "wrap" the Dynkin diagram. This is joint work with E. Mukhin*.*

**Peng Zhao **

*Integrability of High Spin Operators in N=4 supersymmetric Yang-Mills Theory ***(pdf of slides)**

The discovery of integrability in gauge theory and string theory allows us to exactly compute observables on both theories and thus directly test the AdS/CFT correspondence. I will describe how to calculate the anomalous dimension of high spin operators in the SL(2) subsector of the N=4 supersymmetric Yang-Mills theory via Bethe ansatz techniques. This is dual to the spectrum of energy of a long rotating string in AdS3. I will discuss how to compare the weak coupling result with the strong coupling result. The spectrum of excitations outside the SL(2) subsector is of particular interest and investigated by embedding the configuration in the full PSU(2,2|4) spin chain. Finally, I will comment on its relation to null polygonal Wilson loops.

**Roman Zwicky**

*Identifying conformal gauge theories on the lattice (pdf of slides)*

I will introduce conformal gauge theories, argue for their existence and discuss SUSY CGT briefly. I will present a variety of scaling laws that can help to identify the phase diagram in the number of flavours and colours of gauge theories relying on the renormalization group. The trajectory of the Spectrum and the formation of the quark condensate will be discussed. I will briefly summarize some of the current lattice results.

**Posters**

**Emanuele Levi**

*Higher particle form factors of branch point twist fields in integrable quantum field theories*

We compute higher particle form factors of branch point twist fields. These fields were first described in the context of massive 1+1-dimensional integrable quantum field theories and their correlation functions are related to the bi-partite entanglement entropy. We find analytic expressions for some form factors and check those expressions for consistency, mainly by evaluating the conformal dimension of the corresponding twist field in the underlying conformal field theory. We find that solutions to the form factor equations are not unique so that various techniques need to be used to identify those corresponding to the branch point twist field we are interested in. The models for which we carry out our study are characterized by staircase patterns of various physical quantities as functions of the energy scale. As the latter is varied, the beta-function associated to these theories comes close to vanishing at several points between the deep infrared and deep ultraviolet regimes. In other words, renormalisation group flows approach the vicinity of various critical points before ultimately reaching the ultraviolet fixed point. This feature provides an optimal way of checking the consistency of higher particle form factor solutions, as the changes on the conformal dimension of the twist field at various energy scales can only be accounted for by considering higher particle form factor contributions to the expansion of certain correlation functions.

**Jacopo Viti**

*Universal properties of two dimensional percolation*

I will discuss how universal (lattice independent) properties of two dimensional percolation can be computed within the formalism of factorized S matrix and Conformal Field Theory. In particular I will address the problem of determining the clusters size ratio above and below the percolative threshold and of factorization of the three-point connectivity at criticality. Universal ratios are considered also for correlated percolation in the two dimensional Ising model.