Rémi Avohou: Counting UO-tensorial invariants

mercredi 26 juin 2024, 14:00 → 15:00 Europe/Paris

Invariants under the group  $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ are generated through the contraction of complex tensors, which have an order of $r+q$, also denoted  $(r,q)$.  These tensors undergo transformations according to $r$ fundamental representations of the unitary group $U(N)$ and $q$ fundamental representations of the orthogonal group $O(N)$. Consequently, $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants serve as observables in tensor models, possessing a tensor field with an order of $(r,q)$. In this presentation, I will show the enumeration of these observables using group theoretic formulae, for tensor fields with any given order $(r,q)$. For a general order $(r,q)$, such enumeration can be viewed as the partition function for a topological quantum field theory (TQFT), where the symmetric group acts as the gauge group. The discussion will include the identification of the $2$-complex pertaining to the enumeration of the invariants, which consequently defines the TQFT, and establish a correspondence with counting associated with covers of various topologies.