2024/04/25 by Rémi Cocou Avohou, Joseph Ben Geloun, Avohou, Remi Cocou +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Computational Physics and Python Applications #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2404.16404
openalex publication_date 2024/04/25 · openalex created_date 2024/04/27 · openalex updated_date 2026/07/28
U(N)⊗ r ⊗ O(N)⊗ q invariants are constructed by contractions of complex tensors of order r+q, also denoted (r,q). These tensors transform under r fundamental representations of the unitary group U(N) and q fundamental representations of the orthogonal group O(N). Therefore, U(N)⊗ r ⊗ O(N)⊗ q invariants are tensor model observables endowed with a tensor field of order (r,q). We enumerate these observables using group theoretic formulae, for arbitrary tensor fields of order (r,q). Inspecting lower-order cases reveals that, at order (1,1), the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order (r,q), the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For r>1, the number of invariants matches the number of (q-dependent) weighted equivalence classes of branched covers of the 2-sphere with r branched points. At r=1, the counting maps to the enumeration of branched covers of the 2-sphere with q+3 branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments.