On the counting of $O(N)$ tensor invariants

Avohou, Remi C.; Geloun, Joseph Ben; Dub, Nicolas

doi:10.4310/atmp.2020.v24.n4.a1

Cited by 12 publications

(16 citation statements)

References 45 publications

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“…The building blocks of the theory are complex and real tensors that we now introduce. This section follows [8,7,5].…”

Section: Notation: Complex and Real Tensorsmentioning

confidence: 99%

“…For tensor models [1]- [5], these observables build from the contractions of multidimensional arrays or tensors that transform covariantly under the action of some classical Lie groups. The most recent studies on tensor models over Lie groups consider U(N), the unitary group of order N, and O(N), the orthogonal group of order N. Note that much less is known about tensor models with Sp(2N)-invariants, Sp(2N) being the (real or complex) symplectic group, see [6] and [7]. Defined by contractions of tensors, the observables or interactions of tensor models simply become polynomial invariants of these classical Lie groups (for short we shall call them tensor invariants).…”

Section: Introductionmentioning

confidence: 99%

“…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%

“…The two-point correlators of complex observables expand in these orthogonal bases. Many of these features of complex tensor models extend to the real case [7]. Once the enumeration of real tensor invariants sorted, their TopFT formulation finds a bijection with the covers of the torus with defects, and their algebra possesses also orthogonal bases and is semi-simple.…”

Section: Introductionmentioning

confidence: 99%

“…This work delivers a summary of three contributions [8], [11], and [7]. We put in parallel the counting of U(N) tensor invariants and its corollaries [8] [11] and that of O(N) tensor invariants [7]. The next section introduces our notation.…”

Section: Introductionmentioning

confidence: 99%

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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)

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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.

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“…The building blocks of the theory are complex and real tensors that we now introduce. This section follows [8,7,5].…”

Section: Notation: Complex and Real Tensorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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)

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RG flows and fixed points of O(N)r models

Jepsen,

2024

J. High Energ. Phys.

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By means of ϵ and large N expansions, we study generalizations of the O(N) model where the fundamental fields are tensors of rank r rather than vectors, and where the global symmetry (up to additional discrete symmetries and quotients) is O(N)r, focusing on the cases r ≤ 5. Owing to the distinct ways of performing index contractions, these theories contain multiple quartic operators, which mix under the RG flow. At all large N fixed points, melonic operators are absent and the leading Feynman diagrams are bubble diagrams, so that all perturbative fixed points can be readily matched to full large N solutions obtained from Hubbard-Stratonovich transformations. The family of fixed points we uncover extend to arbitrary higher values of r, and as their number grows superexponentially with r, these theories offer a vast generalization of the critical O(N) model.We also study sextic O(N)r theories, whose large N limits are obscured by the fact that the dominant Feynman diagrams are not restricted to melonic or bubble diagrams. For these theories the large N dynamics differ qualitatively across different values of r, and we demonstrate that the RG flows possess a numerous and diverse set of perturbative fixed points beginning at rank four.

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Extremal fixed points and Diophantine equations

Herzog,

Jepsen,

Osborn

et al. 2024

J. High Energ. Phys.

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The coupling constants of fixed points in the ϵ expansion at one loop are known to satisfy a quadratic bound due to Rychkov and Stergiou. We refer to fixed points that saturate this bound as extremal fixed points. The theories which contain such fixed points are those which undergo a saddle-node bifurcation, entailing the presence of a marginal operator. Among bifundamental theories, a few examples of infinite families of such theories are known. A necessary condition for extremality is that the sizes of the factors of the symmetry group of a given theory satisfy a specific Diophantine equation, given in terms of what we call the extremality polynomial. In this work we study such Diophantine equations and employ a combination of rigorous and probabilistic estimates to argue that these infinite families constitute rare exceptions. The Pell equation, Falting’s theorem, Siegel’s theorem, and elliptic curves figure prominently in our analysis. In the cases we study here, more generic classes of multi-fundamental theories saturate the Rychkov-Stergiou bound only in sporadic cases or in limits where they degenerate into simpler known examples.

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On the counting of $O(N)$ tensor invariants

Cited by 12 publications

References 45 publications

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

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

RG flows and fixed points of O(N)r models

Extremal fixed points and Diophantine equations

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