Counting tensor model observables and branched covers of the 2-sphere

Abstract

Lattice gauge theories of permutation groups with a simple topological action (henceforth permutation-TFTs) have recently found several applications in the combinatorics of quantum field theories (QFTs). They have been used to solve counting problems of Feynman graphs in QFTs and ribbon graphs of large N , often revealing inter-relations between different counting problems. In another recent development, tensor theories generalizing matrix theories have been actively developed as models of random geometry in three or more dimensions. Here, we apply permutation-TFT methods to count gauge invariants for tensor models (colored as well as non-colored), exhibiting a relationship with counting problems of branched covers of the 2-sphere, where the rank d of the tensor gets related to a number of branch points. We give explicit generating functions for the relevant counting and describe algorithms for the enumeration of the invariants. As well as the classic count of Hurwitz equivalence classes of branched covers with fixed branch points, collecting these under an equivalence of permuting the branch points is relevant to the color-symmetrized tensor invariant counting. We also apply the permutation-TFT methods to obtain some formulae for correlators of the tensor model invariants.

Holography for tensor models

“…Our first task is to provide the independent count of the number of U (N 1 ) × U (N 2 ) × U (N 3 ) invariants. This problem has been solved in [59,23]. Invariants constructed using n φ abc s and nφ abc s are in one-to-one correspondence with elements of the double coset…”

“…One approach towards the finite N physics of the tensor model is to use representation theory techniques, as developed in [59,23]. The basic idea is to construct gauge invariant operators as a product of the basic fields multiplied by a projection operator.…”

“…In what follows we will simply state and use the results we need. The reader requiring more details is encouraged to consult [59,23], as well as [83,84,85,86].…”

Generalized cut operation associated with higher order variation in tensor models

“…Some of the insights come from the Virasoro structure of the matrix models [27][28][29][30][31][32][33][34] and its combinatorics and finding its counterpart in tensor models in general is expected to pave the way to bring progress in this field. Recent references include [35][36][37][38][39][40][41][42][43][44][45][46][47][48].…”

Topological string correlators from matrix models

“…With this motivation in mind, a number of connections between the combinatorics of Wick contractions in matrix models and topological theories have been uncovered [4][5][6][7][8][9][10][11]. This particular approach to the interpretation of the matrix models correlators, as well as certain counting problems, is very reminiscent of the string interpretation of 2d-YangMills theory [12][13][14][15].…”