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.

56 pages, 14 Figures

References in corpus (15)

• Random tensor models in the large N limit: Uncoloring the colored tensor models
• Diagonal multi-matrix correlators and BPS operators in N=4 SYM
• Melons are branched polymers
• Branes, Anti-Branes and Brauer Algebras in Gauge-Gravity duality
• The gravity duals of N=2 superconformal field theories
• The 1/N Expansion of Tensor Models Beyond Perturbation Theory
• Enhanced symmetries of gauge theory and resolving the spectrum of local operators
• From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group
• From counting to construction of BPS states in N=4 SYM
• Schur-Weyl duality as an instrument of Gauge-String duality
• What is the Simplest Gauge-String Duality?
• New Directions in Bipartite Field Theories
• Correlators in the Simplest Gauge-String Duality
• Renormalizable Models in Rank $d\ge 2$ Tensorial Group Field Theory
• Holographic Hierarchy in the Gaussian Matrix Model via the Fuzzy Sphere

Cited by in corpus (49)

• The Tensor Track, III
• Invitation to Random Tensors
• Correlators in tensor models from character calculus
• The Multi-Orientable Random Tensor Model, a Review
• Ward identities and combinatorics of rainbow tensor models
• Three Dimensional View of Arbitrary $q$ SYK models
• Gauge Invariants, Correlators and Holography in Bosonic and Fermionic Tensor Models
• 2PI effective action for the SYK model and tensor field theories
• On (q,t)-deformation of Gaussian matrix model
• Tensorial Gross-Neveu models
• Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras
• Sum rules for characters from character-preservation property of matrix models
• Random Tensors and Quantum Gravity
• Cut and join operator ring in Aristotelian tensor model
• SYK-like tensor quantum mechanics with $\mathrm{Sp}(N)$ symmetry
• The Tensor Theory Space
• Constructive Tensor Field Theory
• Large $N$ Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension $d\ge 2$
• Multi-critical behaviour of 4-dimensional tensor models up to order 6
• Permutation Centralizer Algebras and Multi-Matrix Invariants
• From Kronecker to tableau pseudo-characters in tensor models
• Complete solution to Gaussian tensor model and its integrable properties
• The full Ward-Takahashi Identity for colored tensor models
• Invariant Operators, Orthogonal Bases and Correlators in General Tensor Models
• Tensor and Matrix models: a one-night stand or a lifetime romance?
• An analogue of Schur functions for the plane partitions
• Permutation Invariant Gaussian Matrix Models
• Combinatoric topological string theories and group theory algorithms
• Topological String Correlators from Matrix Models
• The quantum detection of projectors in finite-dimensional algebras and holography
• Generalized cut operation associated with higher order variation in tensor models
• Permutation invariant Gaussian 2-matrix models
• Permutation combinatorics of worldsheet moduli space
• All-orders asymptotics of tensor model observables from symmetries of restricted partitions
• Correspondence between Feynman diagrams and operators in quantum field theory that emerges from tensor model
• Holography for Tensor models
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• Linguistic Matrix Theory
• A combinatorial matrix approach for the generation of vacuum Feynman graphs multiplicities in ${\varphi }^4$ theory
• Matrix and tensor witnesses of hidden symmetry algebras
• Sextic tensor field theories in rank $3$ and $5$
• Holomorphic primary fields in free CFT4 and Calabi-Yau orbifolds
• Kronecker coefficients from algebras of bi-partite ribbon graphs
• An exact solution method for the enumeration of connected Feynman diagrams
• Backgrounds from Tensor Models: A Proposal
• The Grassmannian VOA
• The Non-melonic Sector of Tensor Models and Gravity
• Chiral symmetry breaking generalizes in tensor theories
• Review on the Operator/Feynman diagram/Dessins d'enfant Correspondence in Tensor Model