Hopf algebras and combinatorics in QFT

Hopf algebras and combinatorics in QFT

Abstract: A significant result in quantum field theory was the description of the BPHZ renormalization scheme in terms of Hopf algebras. This turned out to work as a more generic realization for other renormalization schemes, and, naturally, the Hopf algebraic formulation invited more combinatorics into the game. In this talk we are going to cover the basics of QFT, we’ll see what is the BPHZ renormalization scheme, and then we will go through some interesting enumeration problems. For the latter, we shall see how the number of connected chord diagrams can be solely used to count one-particle-irreducible (1PI) diagrams in Yukawa theory.
 

On a generalization of root systems.

On a generalization of root systems.

Abstract:

In this talk I will discuss a generalization of the classical notion of a (finite) root system. Namely, we replace the requirement that a root system is invariant under reflections by the weaker requirement that if two roots a and b make a obtuse angle, then a+b is also a root and if and and b are perpendicular, then a+b is a root if and only if a-b is a root. It turns out that this definition encompasses all classical root systems, the root systems of classical Lie superalgebras, and Kostant root systems. In particular, we are able to treat many common properties of these systems can be treated in a uniform way. I will also explain the classification of such systems of rank 2 and will discuss the feasibility of classification in general.

This is a joint work with Rita Fioresi.

 

Augmentations, Fillings, and Clusters

Augmentations, Fillings, and Clusters

Abstract: A Legendrian link is a 1-dimensional closed manifold embedded in R^3 satisfying certain tangent conditions. Rainbow closures of positive braids are natural examples of Legendrian links. In the study of Legendrian links, one important task is to distinguish different exact Lagrangian fillings of a Legendrian link, up to Hamiltonian isotopy, in symplectic R^4. We introduce a cluster K2 structure on the augmentation variety of the Chekanov-Eliashberg dga for the rainbow closure of any positive braid. Using the Eckholm-Honda-Kalman functor from the cobordism category of Legendrian links to the category of dga’s, we prove that a big family of fillings give rise to cluster seeds on the augmentation variety, and these cluster seeds can be used to distinguish non-Hamiltonian isotopic fillings. Moreover, by relating a cyclic rotation concordance on a positive braid closure with the Donaldson-Thomas transformation on the corresponding augmentation variety, we prove that other than a family of positive braids that are associated with finite type quivers, the rainbow closure of all other positive braids admit infinitely many non-Hamiltonian isotopic fillings. This is joint work with H. Gao and L. Shen.
 

Injections from Kronecker Products and the Cohomological Invariants of Half-Spin.

Injections from Kronecker Products and the Cohomological Invariants of Half-Spin.

Abstract: Let G be a linear algebraic group over a field F. As introduced by Serre, degree n cohomological invariants of G with coefficients in a group A, where A is equipped with an action of the absolute Galois group of F, are natural transformations of Galois cohomology functors H^1(-,G) -> H^n(-,A). Commonly studied are the degree three invariants with coefficients in Q/Z tensor Q/Z. These invariants were recently described by Merkurjev for the semisimple adjoint case, and by Bermudez and Ruozzi for semisimple G which are neither simply connected nor adjoint. In particular, they described the structure of the normalized degree three invariants (those which send the trivial object to zero) of the half-spin group HSpin_16. By generalizing a technique of Garibaldi we construct new injections into HSpin induced by the Kronecker tensor product map. In particular we construct an injection PSp_2n X PSp_2m -> HSpin_4nm which we use to describe the normalized invariants of HSpin_4k for any k, generalizing the result of Bermudez and Ruozzi.
 

Towards a Kazhdan-Lusztig theory for Double Affine Hecke Algebras.

Towards a Kazhdan-Lusztig theory for Double Affine Hecke Algebras.

Abstract: Some years ago, it was shown how to attach an Iwahori-Hecke algebra to a p-adic loop group. Although the resulting algebra admits a presentation that identifies it with a variant of the double affine Hecke algebra, its finer algebraic/combinatorial structure is still quite mysterious since it is governed by a certain non-Coxeter (semi-)group. Nonetheless, Dinakar Muthiah has recently introduced a length function on this object, investigated a certain Bruhat-type order on it (in joint work with Dan Orr), and also proposed an approach to defining a “double affine” Kazhdan-Lusztig theory. I will explain how, from a more global perspective (i.e. adeles on the projective line, reduction theory for loop groups), some preliminary steps in Muthiah’s program can be deduced.

Joint work with D. Muthiah.
 

Twisting functors and representations of affine vertex algebras.

Twisting functors and representations of affine vertex algebras.

Abstract: Twisting functors are important tools in the representation theory, in particular in the classification by O.Mathieu of cuspidal representations of a simple finite-dimensional Lie algebra g. Twisting functor is associated with a root of the Lie algebra g. The case of a simple root is well understood, in particular it gives a twisting of the category O. On the other hand, in the case of a non-simple root we obtain a family of Gelfand-Tsetlin g-modules for any commutative subalgebra of the enveloping algebra containing a Cartan subalgebra and the Casimir element of the sl(2)-subalgebra based on the non-simple root. We give explicit D-module realization of such modules using the Beilinson-Bernstein correspondence. Further on, we apply the Wakimoto functor to construct new families of positive energy representations of affine vertex algebras together with their free field realizations. This is a joint talk with Libor Krizka.
 

Transfer of characters for discrete series representations in the equal rank case via the Cauchy-Harish-Chandra integral.

Transfer of characters for discrete series representations in the equal rank case via the Cauchy-Harish-Chandra integral.

Abstract: For every irreducible reductive dual pair (G, G’) in Sp(W), R. Howe proved the existence of an isomorphism between the spaces R(G) and R(G’), where R(G) is the set of infinitesimal equivalence classes of irreducible admissible representations of \tilde{G}; (preimage of G in the metaplectic group) which can be realized as a quotient of the metaplectic representation. All the representations appearing in the correspondence have a distribution character, and characters are analytic objects completely identifying the irreducible representations. In particular, one natural question is to understand the transfer of characters in the theta correspondence (or Howe’s duality). In 2000, T. Przebinda introduced the Cauchy-Harish-Chandra integral and conjectured that the transfer of characters should be obtained via this map. This conjecture is true if G is compact and was proven by Przebinda for unitary representations in the stable range case. In my talk, after recalling carefully the construction of the Cauchy-Harish-Chandra integral and stating Przebinda’s conjecture, I will explain how to prove this conjecture for the pair (G,G’) = (U(p, q), U(r, s)), with p+q = r+s, starting with a discrete series representation of \tilde{G}.