scholarly journals Superintegrability of Kontsevich matrix model

2021 ◽  
Vol 81 (3) ◽  
Author(s):  
Andrei Mironov ◽  
Alexei Morozov
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Schur Functions ◽  
Complex Model ◽  
Tensor Model ◽  
Complex Matrix ◽  
Schur Polynomials ◽  
Hurwitz Numbers ◽  
Complex Tensor ◽  
Coulomb Potentials

AbstractMany eigenvalue matrix models possess a peculiar basis of observables that have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb potentials are distinguished among other central potentials, and we call it superintegrability. As a peculiarity of matrix models, the relevant basis is formed by the Schur polynomials (characters) and their generalizations, and superintegrability looks like a property $$\langle character\rangle \,\sim character$$ ⟨ c h a r a c t e r ⟩ ∼ c h a r a c t e r . This is already known to happen in the most important cases of Hermitian, unitary, and complex matrix models. Here we add two more examples of principal importance, where the model depends on external fields: a special version of complex model and the cubic Kontsevich model. In the former case, straightforward is a generalization to the complex tensor model. In the latter case, the relevant characters are the celebrated Q Schur functions appearing in the description of spin Hurwitz numbers and other related contexts.

scholarly journals Genus expansion of matrix models and ћ expansion of KP hierarchy

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
A. Andreev ◽  
A. Popolitov ◽  
A. Sleptsov ◽  
A. Zhabin
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Special Interest ◽  
Hermitian Matrix ◽  
Geometric Meaning ◽  
Kp Hierarchy ◽  
Hurwitz Numbers ◽  
Kp Equations ◽  
Genus Expansion ◽  
Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

scholarly journals Cut-and-join structure and integrability for spin Hurwitz numbers

2020 ◽  
Vol 80 (2) ◽  
Author(s):  
A. Mironov ◽  
A. Morozov ◽  
S. Natanzon
Keyword(s):  
Matrix Models ◽  
Generating Function ◽  
Schur Functions ◽  
Integrable Hierarchy ◽  
Young Diagrams ◽  
Hurwitz Numbers ◽  
Expansion Coefficients ◽  
Common Eigenfunctions

Abstract Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions $$Q_R$$QR with $$R\in \hbox {SP}$$R∈SP are common eigenfunctions of cut-and-join operators $$W_\Delta $$WΔ with $$\Delta \in \hbox {OP}$$Δ∈OP. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a $$\tau $$τ-function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.

scholarly journals Multiple phases in a generalized Gross-Witten-Wadia matrix model

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Jorge G. Russo ◽  
Miguel Tierz
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Characteristic Polynomial ◽  
Matrix Model ◽  
Phase Structure ◽  
Unitary Matrix ◽  
Two Dimensional ◽  
Toeplitz Determinants ◽  
Dimensional Parameter ◽  
Planar Limit

Abstract We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

scholarly journals W-representation of Rainbow tensor model

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Bei Kang ◽  
Lu-Yao Wang ◽  
Ke Wu ◽  
Jie Yang ◽  
Wei-Zhong Zhao
Keyword(s):  
Partition Function ◽  
Matrix Model ◽  
Crucial Role ◽  
Elementary Function ◽  
Tensor Model ◽  
Witt Algebra ◽  
Virasoro Constraints ◽  
Compact Expression ◽  
Non Gaussian ◽  
Dual Expression

Abstract We analyze the rainbow tensor model and present the Virasoro constraints, where the constraint operators obey the Witt algebra and null 3-algebra. We generalize the method of W-representation in matrix model to the rainbow tensor model, where the operators preserving and increasing the grading play a crucial role. It is shown that the rainbow tensor model can be realized by acting on elementary function with exponent of the operator increasing the grading. We derive the compact expression of correlators and apply it to several models, i.e., the red tensor model, Aristotelian tensor model and r = 4 rainbow tensor model. Furthermore, we discuss the case of the non-Gaussian red tensor model and present a dual expression for partition function through differentiation.

PHASE DIAGRAM AND ORTHOGONAL POLYNOMIALS IN MULTIPLE-WELL MATRIX MODELS

1991 ◽  
Vol 06 (25) ◽  
pp. 4491-4515 ◽  
Author(s):  
OLAF LECHTENFELD ◽  
RASHMI RAY ◽  
ARUP RAY
Keyword(s):  
Phase Diagram ◽  
Orthogonal Polynomials ◽  
Matrix Models ◽  
Matrix Model ◽  
Phase Structure ◽  
Saddle Points ◽  
Tree Level ◽  
Potential Wells ◽  
Large N

We investigate a zero-dimensional Hermitian one-matrix model in a triple-well potential. Its tree-level phase structure is analyzed semiclassically as well as in the framework of orthogonal polynomials. Some multiple-arc eigenvalue distributions in the first method correspond to quasiperiodic large-N behavior of recursion coefficients for the second. We further establish this connection between the two approaches by finding three-arc saddle points from orthogonal polynomials. The latter require a modification for nondegenerate potential minima; we propose weighing the average over potential wells.

scholarly journals Quantum gravity as a multitrace matrix model

2017 ◽  
Vol 32 (31) ◽  
pp. 1750180
Author(s):  
Badis Ydri ◽  
Cherine Soudani ◽  
Ahlam Rouag
Keyword(s):  
Quantum Gravity ◽  
Matrix Models ◽  
Large Class ◽  
Matrix Model ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Topology Change ◽  
New Model ◽  
And Topology ◽  
Emergent Geometry

We present a new model of quantum gravity as a theory of random geometries given explicitly in terms of a multitrace matrix model. This is a generalization of the usual discretized random surfaces of two-dimensional quantum gravity which works away from two dimensions and captures a large class of spaces admitting a finite spectral triple. These multitrace matrix models sustain emergent geometry as well as growing dimensions and topology change.

Correlators in the β-deformed Gaussian Hermitian and complex matrix models

2019 ◽  
Vol 34 (33) ◽  
pp. 1950221 ◽  
Author(s):  
Ying Chen ◽  
Bei Kang ◽  
Min-Li Li ◽  
Li-Fang Wang ◽  
Chun-Hong Zhang
Keyword(s):  
Matrix Models ◽  
Integral Representations ◽  
Witt Algebra ◽  
Complex Matrix ◽  
Virasoro Constraints

We investigate the [Formula: see text]-deformed Gaussian Hermitian and [Formula: see text] complex matrix models which are defined as the eigenvalue integral representations. Their [Formula: see text] constraints are constructed such that the constraint operators yield the same [Formula: see text][Formula: see text]-algebra. When particularized to the Virasoro constraints in the [Formula: see text] constraints, the corresponding constraint operators yield the Witt algebra and null 3-algebra. By solving our Virasoro constraints, we derive the formulas for correlators in these two [Formula: see text]-deformed matrix models, respectively.

β-DEFORMED MATRIX MODELS AND NEKRASOV PARTITION FUNCTION

2013 ◽  
Vol 21 ◽  
pp. 92-100
Author(s):  
TAKESHI OOTA
Keyword(s):  
Gauge Theory ◽  
Partition Function ◽  
Matrix Models ◽  
Matrix Model ◽  
The Matrix

The β-deformed matrix models of Selberg type are introduced. They are exactly calculable by using the Macdonald-Kadell formula. With an appropriate choice of the integration contours and interactions, the partition function of the matrix model can be identified with the Nekrasov partition function for SU(2) gauge theory with Nf = 4. Known properties of good q-expansion basis for the matrix model are summarized.

scholarly journals Complex matrix model and fermion phase space for bubbling AdS geometries

2005 ◽  
Vol 2005 (10) ◽  
pp. 004-004 ◽  
Author(s):  
Yastoshi Takayama ◽  
Asato Tsuchiya
Keyword(s):  
Phase Space ◽  
Matrix Model ◽  
Complex Matrix
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