Generalized cut operation associated with higher order variation in tensor models

Itoyama, H.; Yoshioka, Reiji

doi:10.1016/j.nuclphysb.2019.114681

Cited by 9 publications

(6 citation statements)

References 80 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…q 3 = 1: q ′ ′ ⊢ k − 1, with k − 1 even so the induction hypothesis (2.24) applies to q ′′ , and so we write: Sym(q) = 3 Sym(q ′ ′ ) ⩽ 3 Sym[2 (k−1)/2 ] = 3 Sym[2 (k+2−3)/2 ] = Sym[3, 2 (k+2−3)/2 ] (A. 19) q 3 = 3: q ′ ′ ⊢ k + 2 − 3 * 3 = k − 7, with k − 7 even. Thus, the induction (2.24) applies to q ′′ : Sym(q) = 3 3 3!…”

Section: Appendix a Proof Of Lemmamentioning

confidence: 99%

See 1 more Smart Citation

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

Geloun

Ramgoolam

2022

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

The counting of the dimension of the space of $U(N) \times U(N) \times U(N)$ polynomial invariants of a complex $3$-index tensor as a function of degree $n$ is known in terms of a sum of squares of Kronecker coefficients. For $n \le N$, the formula can be expressed in terms of a sum of symmetry factors of partitions of $n$ denoted $Z_3(n)$, which also counts the number of bipartite ribbon graphs with $n$ edges. We derive the large $n$ all-orders asymptotic formula for $ Z_3(n)$ making contact with high order results previously obtained numerically. We explain how the different terms in the asymptotics are associated with probability distributions over ribbon graphs. The derivation of the asymptotics relies on the dominance in the sum, of partitions with many parts of length $1$. This large $n$ dominance of small parts also leads to conjectured formulae for the asymptotics of invariants for general $d$-index tensors. The coefficients of powers of $ 1/n$ in these expansions involve Stirling numbers of the second kind along with restricted partition sums.

show abstract

Section: Appendix a Proof Of Lemmamentioning

confidence: 99%

“…The algebra has implications for the structure of tensor model correlators [6,16]. The algebraic perspectives on tensor model correlators have been developed in [17][18][19]. Similar techniques have been applied to orthogonal invariants [20].…”

Section: Introductionmentioning

confidence: 99%

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

Geloun

Ramgoolam

2022

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…, where R 1 , R 2 , R 3 are Young diagrams or partition of n and τ i , i = 1, 2, range over Clebsch-Gordan multiplicities, also known as Kronecker coefficients (the explicit formula is given in [73] and developed in detail in [9]). Further investigations of tensor models from this algebraic perspective are in [25,4,7,34,33,31,53,55,54,2,24]. A known connection between bipartite ribbon graphs and Belyi maps [66,87] gives a topological version of gauge-string duality between tensor models and string theory [8], generalizing analogous correspondences between two-dimensional Yang Mills theory and topological string theory [44,21,50].…”

Section: Introductionmentioning

confidence: 99%

Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients

Geloun¹,

Ramgoolam²

2023

Algebraic Combinatorics

View full text Add to dashboard Cite

We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group products. The existence and structure of this Hilbert space algebra has a number of consequences. The algebra product, which can be expressed in terms of integer ribbon graph reconnection coefficients, is used to define solvable Hamiltonians with eigenvalues expressed in terms of normalized characters of symmetric group elements and degeneracies given in terms of Kronecker coefficients, which are tensor product multiplicities of symmetric group representations. The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs. This leads to an answer to the long-standing question of a combinatorial interpretation of the Kronecker coefficients. As avenues for future research, we discuss applications of the ribbon graph quantum mechanics in algorithms for quantum computation. We also describe a quantum membrane interpretation of these quantum mechanical systems.

show abstract

“…We can easily foresee that the quantum field theory calculations heavily rely on the diagrammatics and combinatorics of these objects. Hence, a systematic combinatorial study of U(N) and O(N) classical invariants has been launched in the recent years bringing already a wealth of core results [8,9,10,11,12,13,14,15,16,7,17,18] A preferred way of enumerating these invariants mainly rests on algebraic techniques of the symmetric groups. There are a lot of reasons why the use symmetric groups has become a natural reflex and a dominating tool in the combinatorial study of tensor invariants.…”

Section: Introductionmentioning

confidence: 99%

On the counting tensor model observables as U(N) and O(N) classical invariants

Geloun¹

2020

Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2019)

View full text Add to dashboard Cite

Real or complex tensor model observables, the backbone of the tensor theory space, are classical (unitary, orthogonal, symplectic) Lie group invariants. These observables represent as colored graphs, and that representation gives an handle to study their combinatorial, topological and algebraic properties. We give here an overview of the symmetric group-theoretic formulation of the enumeration of unitary and orthogonal invariant observables which turns out to bear a rich structure. From their counting formulae, one finds a correspondence with topological field theory on 2-cellular complexes that brings other interpretations of the same countings. Furthermore, tensor model observables span an algebra that turns out to be semi-simple. Dealing with complex tensors, we discuss the representation theoretic base of the algebra making explicit its Wedderburn-Artin decomposition. The real case is more subtle as a base of its Wedderburn-Artin decomposition is yet unknown.

show abstract

Generalized cut operation associated with higher order variation in tensor models

Cited by 9 publications

References 80 publications

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients

On the counting tensor model observables as U(N) and O(N) classical invariants

Contact Info

Product

Resources

About