FERMIONS IN THE TOPOLOGICAL EXPANSION OF MATRIX MODELS

1991 ◽  
Vol 06 (09) ◽  
pp. 781-787
Author(s):  
G. FERRETTI
Keyword(s):  
Free Energy ◽  
Matrix Models ◽  
Matrix Model ◽  
Hermitian Matrix ◽  
Coupling Constant ◽  
Counting Problem ◽  
Planar Limit ◽  
Quartic Interaction ◽  
Hermitian Matrix Model ◽  
Topological Expansion

The hermitian matrix model with quartic interaction is studied in presence of fermionic variables. We obtain the contribution to the free energy due to the presence of fermions. The first two terms beyond the planar limit are explicitly found for all values of the coupling constant g. These terms represent the solution of the counting problem for vacuum diagrams with one or two fermionic loops.

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 The planar limit of $$ \mathcal{N} $$ = 2 chiral correlators

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Bartomeu Fiol ◽  
Alan Rios Fukelman
Keyword(s):  
Free Energy ◽  
Infinite Number ◽  
Correlation Functions ◽  
Hermitian Matrix ◽  
Hooft Coupling ◽  
Planar Limit ◽  
Single Trace ◽  
Hermitian Matrix Model ◽  
Combinatorial Expression ◽  
Do So

Abstract We derive the planar limit of 2- and 3-point functions of single-trace chiral primary operators of $$ \mathcal{N} $$ N = 2 SQCD on S4, to all orders in the ’t Hooft coupling. In order to do so, we first obtain a combinatorial expression for the planar free energy of a hermitian matrix model with an infinite number of arbitrary single and double trace terms in the potential; this solution might have applications in many other contexts. We then use these results to evaluate the analogous planar correlation functions on ℝ4. Specifically, we compute all the terms with a single value of the ζ function for a few planar 2- and 3-point functions, and conjecture general formulas for these terms for all 2- and 3-point functions on ℝ4.

PROPERTIES OF LOOP EQUATIONS FOR THE HERMITIAN MATRIX MODEL AND FOR TWO-DIMENSIONAL QUANTUM GRAVITY

1990 ◽  
Vol 05 (22) ◽  
pp. 1753-1763 ◽  
Author(s):  
J. AMBJØRN ◽  
YU. M. MAKEENKO
Keyword(s):  
Quantum Gravity ◽  
Matrix Model ◽  
Hermitian Matrix ◽  
General Procedure ◽  
Iterative Solution ◽  
Two Dimensional ◽  
Coupling Constant ◽  
Continuum Equation ◽  
Hermitian Matrix Model ◽  
The Continuum

We study the properties of the loop equations for the N × N Hermitian matrix model with arbitrary (even) interaction as well as of their continuum limit, associated with the two-dimensional quantum gravity. We apply the general procedure of iterative solution proposed recently by David. We relate the specific heat to the singular behavior of the connected correlator of two loops. We solve the continuum equation to a few lower orders in the string coupling constant, obtaining results for macroscopic loops including the case of a multicritical fixed point.

GENERALIZED c=1 MATRIX MODELS AND SYMMETRIC SPACES

1992 ◽  
Vol 07 (18) ◽  
pp. 4259-4274 ◽  
Author(s):  
M.A. OLSHANETSKY
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Symmetric Spaces ◽  
Hermitian Matrix ◽  
Scaling Limit ◽  
Natural Generalization ◽  
Target Space ◽  
Approximation Procedure ◽  
Hartree Fock ◽  
Hermitian Matrix Model

Random matrix models with a line as a target space are considered. The models are a natural generalization of the Hermitian matrix model and connected with the classical symmetric spaces of the Euclidean type which were classified by Cartan. Ten different types of these spaces exist. Three models on a line related to these models are reduced to one-dimensional free N-fermion problems which have special symmetric configurations. The solutions in the double scaling limit to all orders of perturbation are the same as for the Hermitian matrix model. In the general case the fermions interact with the Calogero-Mozer integrable potential. Due to this fact only the planar limit can be calculated by applying the Hartree-Fock approximation procedure.

scholarly journals FROM ONE-MATRIX MODEL TO KONTSEVICH MODEL

1993 ◽  
Vol 08 (30) ◽  
pp. 2875-2890 ◽  
Author(s):  
J. AMBJØRN ◽  
C. F. KRISTJANSEN
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Matrix Model ◽  
Correlation Functions ◽  
Hermitian Matrix ◽  
Scaling Limit ◽  
Genus Zero ◽  
Kdv Hierarchy ◽  
Hermitian Matrix Model ◽  
Double Scaling Limit

Loop equations of matrix models express the invariance of the models under field redefinitions. We use loop equations to prove that it is possible to define continuum times for the generic Hermitian one-matrix model such that all correlation functions in the double scaling limit agree with the corresponding correlation functions of the Kontsevich model expressed in terms of KdV times. In addition the double scaling limit of the partition function of the Hermitian matrix model agrees with the τ-function of the KdV hierarchy corresponding to the Kontsevich model (and not the square of the τ-function) except for some complications at genus zero.

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.

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