Counting $$U(N)^{\otimes r}\otimes O(N)^{\otimes q}$$ U ( N ) ⊗ r ⊗ O ( N ) ⊗ q invariants and tensor model observables

Abstract

Abstract $$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ U ( N ) ⊗ r ⊗ O ( N ) ⊗ q invariants are constructed by contractions of complex tensors of order $$r+q$$ 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, $$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$$ 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 tensor fields of arbitrary 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 $$r>1$$ 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 $$r=1$$ r = 1 , the counting maps to the enumeration of branched covers of the 2-sphere with $$q+3$$ 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.