Eigenvalue dynamics for multimatrix models

Koch, Robert de Mello; Gossman, David; Nkumane, Lwazi; Tribelhorn, Laila

doi:10.1103/physrevd.96.026011

Cited by 10 publications

(19 citation statements)

References 63 publications

(103 reference statements)

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“…In fact, one can now permute the m tensors M or the m tensorsM as a whole, i.e. multiply all the permutations σ i by two common permutations, from the right and from the left sides, which factorizes S ⊗r m by S m both from the left and from the right sides and provides the double coset [10,12,31,42,43] S r m = S m \S ⊗r m /S m (3.5) An explicit description/parameterization of this coset can begin from violating the "symmetry" S r between different indices (i.e. colorings), for example, by always putting σ 1 = id (this "symmetry" is, in any case, violated by the difference between N i in the gauge groups U (N i )).…”

Section: Models Operators and Gaussian Averagesmentioning

confidence: 99%

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Cut and join operator ring in tensor models

2018

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Recent advancement of rainbow tensor models based on their superintegrability (manifesting itself as the existence of an explicit expression for a generic Gaussian correlator) has allowed us to bypass the longstanding problem seen as the lack of eigenvalue/determinant representation needed to establish the KP/Toda integrability. As the mandatory next step, we discuss in this paper how to provide an adequate designation to each of the connected gauge-invariant operators that form a double coset, which is required to cleverly formulate a tree-algebra generalization of the Virasoro constraints. This problem goes beyond the enumeration problem per se tied to the permutation group, forcing us to introduce a few gauge fixing procedures to the coset. We point out that the permutation-based labeling, which has proven to be relevant for the Gaussian averages is, via interesting complexity, related to the one based on the keystone trees, whose algebra will provide the tensor counterpart of the Virasoro algebra for matrix models. Moreover, our simple analysis reveals the existence of nontrivial kernels and co-kernels for the cut operation and for the join operation respectively that prevent a straightforward construction of the non-perturbative RG-complete partition function and the identification of truly independent time variables. We demonstrate these problems by the simplest non-trivial Aristotelian RGB model with one complex rank-3 tensor, studying its ring of gauge-invariant operators, generated by the keystone triple with the help of four operations: addition, multiplication, cut and join. 1As usual in quantum field theory, the study of any such model consists of several steps: describing the symmetries and the field content of the model, enumeration and classification of operators, introduction of appropriate generating functions and evaluation of correlators/averages. Only at the last of these steps, the action/dynamics of the model is needed, though a clever choice of the generating functions to make can also depend on the action and on a particular phase of the model. Traditional analysis begins from the Gaussian phase, and then the Ward identities are used to express the correlation functions through the Gaussian spectral curve in a functorial way (by the procedure known as topological recursion [35]), and transition to the non-perturbative (Dijkgraaf-Vafa) phases goes through a deformation of the spectral curve. This approach is successfully developed for the one-matrix eigenvalue models (where also integrability properties are revealed and understood), and the present task is to extend it in two directions: to multi-matrix and to (multi-)tensor models. However, such extension is quite sophisticated and can hardly be made by one simple effort. As suggested in [10], we move by small steps, but in a systematic way with the hope that it will be no less straightforward for tensors than it has proven to be for matrices.Accordingly, the very first task is to provide an efficient enumeration of operators. As was alrea...

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Section: Models Operators and Gaussian Averagesmentioning

confidence: 99%

“…[111] = 3 (5.42) and we get 43) which is the case: in the set K 2 , K 2 , K 2 , K 2 1 , the two operators in the middle get identified by symmetrization, reducing the total number from 4 to 3. Similarly 44) which is also true.…”

Section: A Toy Example: S Coloringmentioning

confidence: 99%

Cut and join operator ring in tensor models

2018

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“…while the number of gauge invariant operators in the Aristotelian (rang 3 rainbow) tensor model grows even faster [16,42]: (95) where β n is the number of unlabeled dessins denfants with n edges [43].…”

Section: Resultsmentioning

confidence: 99%

Sum rules for characters from character-preservation property of matrix models

Миронов

Морозов

2018

J. High Energ. Phys.

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One of the main features of eigenvalue matrix models is that the averages of characters are again characters, what can be considered as a far-going generalization of the Fourier transform property of Gaussian exponential. This is true for the standard Hermitian and unitary (trigonometric) matrix models and for their various deformations, classical and quantum ones. Arising explicit formulas for the partition functions are very efficient for practical computer calculations. However, to handle them theoretically, one needs to tame remaining finite sums over representations of a given size, which turns into an interesting conceptual problem. Already the semicircle distribution in the large-N limit implies interesting combinatorial sum rules for characters. We describe also implications to W -representations, including a character decomposition of cut-and-join operators, which unexpectedly involves only single-hook diagrams and also requires non-trivial summation identities. * mironov@lpi.ru; mironov@itep.ru † morozov@itep.ru

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“…To interpret the link between the particle system and the Gauss graph operators, recall the link between giant gravitons and an eigenvalue description of the multi matrix dynamics, which has been pursued in [32,33]. Thus, the fact that the matrix model computations appear to be related to the dynamics of non-interacting particles gives hints as to how matrix model dynamics may simplify, along the line of the proposals of [34,35,36,37]. Any computation of overlaps performed with our wave functions can be mapped into a computation of Gauss graph correlators.…”

Section: Discussionmentioning

confidence: 99%

From Gauss graphs to giants

Koch

Nkumane

2018

J. High Energ. Phys.

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Abstract:We identify the operators in N = 4 super Yang-Mills theory that correspond to 1 8 -BPS giant gravitons in AdS 5 × S 5 . Our evidence for the identification comes from (1) counting these operators and showing agreement with independent counts of the number of giant graviton states, and (2) by demonstrating a correspondence between correlation functions of the super Yang-Mills operators and overlaps of the giant graviton wave functions.

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Eigenvalue dynamics for multimatrix models

Cited by 10 publications

References 63 publications

Cut and join operator ring in tensor models

Cut and join operator ring in tensor models

Sum rules for characters from character-preservation property of matrix models

From Gauss graphs to giants

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