Workshop on Representation Theory and Mathematical Physics

Let g be a finite-dimensional, simple Lie algebra over the field of complex numbers, and U be the quantum, untwisted affine algebra, associated to g. Via Lusztig's q-exponential formulae, it is well known that the affine braid group of g acts on any integrable representation of U. In particular, one obtains an action of the coroot lattice of g on such a representation. 

In this talk, I will present an explicit formula for these lattice operators on finite-dimensional representations of U, in terms of the generators of its maximal commutative subalgebra in Drinfeld's loop presentation. This formula was obtained in a joint work in progress with V. Toledano Laredo.

I discuss the oscillator construction for Q-operators of rational spin chains in the closed and open setting. Some focus will be given on the closed case of type BCD and the orthosymplectic case osp(N|2m).

A distinguishing feature of six-dimensional superconformal field theories (SCFTs) is the existence of solitonic, string-like excitations among their degrees of freedom. In this talk I will discuss how these solitonic strings can be employed to determine an interesting physical observable of 6d SCFTs: their ALE partition function, for which one takes the 6d spacetime to be the product of a two-dimensional torus and an ALE space (that is, an orbifold of C^2 by a discrete subgroup of SU(2)). This quantity is ultimately expected to have an interpretation as the generating function of higher-rank Donaldson-Thomas invariants of elliptic CY threefolds.

We will focus on 6d SCFTs with unitary gauge group which are known to admit a dual description in terms of 5d unitary quiver gauge theories. As a consequence of duality, the ALE partition function of the 6d theory is expected to match with the 5d theory’s partition function on S^1xALE. The latter quantity can be determined by employing a mathematically rigorous approach by Bruzzo, Pedrini, Sala, and Szabo. Upon making a subtle modification to the results of Bruzzo et al, we will be able to verify in a concrete example the matching between dual partition functions.

We consider the monodromy-free opers corresponding to solutions of the Affine Gaudin Bethe Ansatz equations. We define and study the spectral determinants (called Q functions) for these opers. We conjecture that the Q-functions obtained from the Affine Gaudin Bethe Ansatz coincide with the Q-functions of the Bazhanov-Lukyanov-Zamolodchikov opers with the monster potential, which are related to the quantum KdV flows according to the ODE/IM correspondence. We give supporting evidence for this conjecture.

There is a well-known relationship between finite W-algebras and Yangians. The work of Rogoucy and Sorba on the "rectangular case" in type A eventually led Brundan and Kleshchev to introduce shifted Yangians, which surject onto the finite W-algebras for general linear Lie algebras. Thus, these W-algebras can be realised as trucated shifted Yangians. In parallel, the work of Ragoucy and then Brown showed that truncated twisted Yangians is isomorphic to the finite W-algebra associated to a rectangular nilpotent element in a Lie algebra of type B, C or D. For many years there has been a hope that this relationship can be extended to other nilpotent elements.

I will report on a joint work with Lukas Tappeiner in which we introduced the shifted twisted Yangians, following the work of Lu-Wang-Zhang, and described Poisson isomorphisms between their truncated semiclassical degenerations and the Slodowy slices associated with even nilpotent elements in classical simple Lie algebras. I will also mention a work in progress with Lu-Peng-Tappeiner-Wang which deals with the quantum analogue of our theorem.

In this talk, I will introduce a conjectural “new” half of the affine Yangian of type ADE and explain how it arises naturally from the theory of cohomological Hall algebras.

Integrability in quantum systems is usually attributed to a quantum algebra structure. One of the prominent examples of the former is the integrability in AdS_5 /CFT_4 correspondence. Although the integrability in the model was established already two decades ago, the underlying quantum algebra is still incomplete. In the current work, motivated by the classical analysis of the Lie bialgebras, we explore a possible approach to derive the quantum algebra starting from quantum affine algebras of semi-simple Lie (super) algebras by applying Hopf algebra maps that we call the contraction and reduction.

Finite-dimensional representations of Lie algebras (and quantum groups) have characters that are invariant under the Weyl group action. However, this invariance does not hold for infinite-dimensional representations in the category $\mathcal{O}$. Recent studies by E. Frenkel and D. Hernandez have revealed that the Weyl group's action on the q-characters of representations of quantum affine Borel algebras in the category $\mathcal{O}$ exhibits intricate structures and has significant applications for previously unsolved questions.

In this talk, we explore various categories $\mathcal{O}^w$ of representations of Borel subalgebras of quantum affine algebras, labeled by elements of the Weyl group. We propose a conjectural method for calculating q-characters of representations in these new categories, and we explain their connection to the extended $TQ$ and $Q\widetilde{Q}$ systems.

Prefactorisation algebras (PFAs) axiomatise the algebraic structure of observables in QFTs. In the case of locally constant PFAs, which correspond to topological QFTs, I will describe how the PFA induces an algebraic structure on the gauge invariant observables of the TQFT. This involves an ordinary associative, unital product but typically also "higher" operations known as Massey products. In the case of holomorphic PFAs in one complex dimension, which correspond to 2d CFTs, I will describe how the PFA encodes the full vertex algebra structure of the CFT.

We consider conformal four-point integrals to investigate how much of their mathematical structure in two spacetime dimensions carries over to higher dimensions. In particular, we discuss recursions in the loop order and spacetime dimension. This results e.g. in new expressions for conformal ladder integrals with generic propagator powers in all even dimensions and allows us to lift results on 2d fishnet integrals with underlying Calabi-Yau geometry to higher dimensions. Moreover, we demonstrate that the Basso-Dixon generalizations of these integrals obey different variants of the Toda equations of motion, thus establishing a connection to classical integrability and the family of so-called tau-functions. We then show that all of these integrals can be written in a double copy form that combines holomorphic and anti-holomorphic building blocks. Here integrals in higher dimensions are constructed from an intersection pairing of two-dimensional "periods'' together with their derivatives. 

Quantum toroidal algebras Uq(g_tor) occur as the quantum affinizations of affine quantum groups. In particular, they contain (and are generated by) horizontal and vertical quantum affine subalgebras Uh and Uv.

After recalling the definition and basic properties of these algebras, our first main result will be constructing an action of the extended double affine braid group. (Here, a surprising finite presentation for Uq(g_tor) is a crucial ingredient in our proof.)

We shall then exploit this action to exhibit horizontal-vertical symmetries of Uq(g_tor) in untwisted types, in the form of anti-involutions and 'Miki automorphisms' which exchange Uh and Uv. We'll finish by discussing representation theoretic applications of these results (work in progress), and deducing congruence group actions of the central extension of SL(2,Z) on Uq(g_tor).

On Wednesday afternoon, there will be a poster session with short presentations given by:

• Alessandro Carotenuto (Università di Bologna): "Convex Orderings on Positive Roots and Quantum Differential Calculi"
• Henrik Jürgens (Università di Modena e Reggio Emilia): "On the properties of the correlation functions of the sl(n+1)-invariant model"
• Matteo Misurati (Università di Ferrara): "Free Biproduct quasi-Hopf algebras of rank 2"
• Andrea Rivezzi (Charles University): "On the structure of the universal Drinfeld-Yetter algebra"
• Lucas Tappeiner (University of Bath): "Mixed Twisted Yangians"