Questions tagged [matrix-model]
A matrix model is a non-peturbative formulation of a theory, such as string theory based on Matrix quantum mechanics
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Basis and inner product of the Dirac (Clifford) algebra
The Dirac algebra in 4D spacetime is composed of four $4\times 4$ gamma matrices $\{\gamma^\mu\}=\{\gamma^0,\gamma^1,\gamma^2,\gamma^3\}$ satisfying the following anticommutation relation:
$$\{\gamma^\...
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Phase transition of shift of energy level under small perturbation in random matrix theory
I'm a mathematician and I'm thinking about a question in random matrix theory.
Suppose $H_0$ is a $N\times N$ GUE random matrix (variance of each element is $\frac{1}{N}$). We consider a small ...
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Fuzzy Spheres in the Large-$N$ Limit
My question concerns fuzzy sphere solutions and how they fit into the large-$N$ limit of gauge theory.
The Berenstein-Maldacena-Natasse (BMN) model is a matrix model that arises in string theory. The ...
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Reference request: scalar $O(N)$ gauge theory
I am interested in scalar $O(N)$ gauge theory and what you can do with it. Is there a standard reference section in a textbook/monograph/paper/whatever that has a decent overview?
Wikipedia has a ...
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Different definitions of resolvent in matrix model
When I study the matrix models, I get confused of different definations of resolvent. After we define the partition function as
$$Z=\int[dM]e^{-NTrV(M)},$$
where $V(M)$ is a matrix valued function of $...
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What kind of combinations of field components are equal under $SO(9)$ symmetry?
My question is a bit long and chaotic since I haven't learnt group theory systematically.
I am looking at the Banks-Fischler-Shenker-Susskind (BFSS) matrix model. It consists of 9 bosonic matrices $...
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Implementation of Hamiltonian coupling to a bath
I want to study a system coupled to a bath, however I do not fully understand how to implement/think of the Hamiltonian. For simplicity say the bath is given by a spin chain (PBC), e.g. Ising-like
$$...
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How does the matrix model simplify path integral?
While I'm reading the introduction of matrix models in Chapter 8 in Mariño's book(https://doi.org/10.1017/CBO9781107705968), I notice this description of matrix model:
We will begin by a drastic ...
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Does any matrix integral with arbitrary potential has a 2D gravity dual?
Regarding the duality between matrix ensembles and gravity, the relationship has indeed been discussed in various papers, including arXiv:1903.11115, arXiv:1907.03363, and arXiv:2006.13414, among ...
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How to calculate the component term in BFSS matrix model?
I'm reading articles about BFSS, and confused by the calculation.
The Hamiltonian is
$$
H=\frac{g^2}{2}TrP_{I}^{2}-\frac{1}{4g^2}Tr[X_{I},X_{J}]^2
-\frac{1}{2}Tr\psi_{\alpha}\gamma_{\alpha \beta}^{I}[...
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What does the matrix mean in matrix models?
I'm learning what a matrix model means, for example, in Yang–Mills matrix models, IKKT matrix model and BFSS matrix model. I have consulted a large amount of information but still not sure what the ...
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Do matrix models capture the string landscape?
Essentially what the title asks-- are matrix models, such as BFSS, believed to capture in any way the large possible space of false string vacua, for instance as saddles in the action with nonminimal ...
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Why does the eigenvalues of an angular frequency matrix are the natural frequency? (INTUITION)
lest say we have a system of differential equations of some coupled oscillator such that:
$$\overrightarrow a = [w^2]\overrightarrow x$$
if we find the eigenvalues of $[w^2] = \lambda$ why those ...
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What are the 9 matrices used in the BFSS model of quantum mechanics?
The BFSS matrix model (Wikipedia) "describes the behavior of nine large matrices", using:
$$ H = Tr\left(\frac{1}{2}\{ \ \dot X^i \dot X^i - \frac{1}{2}[X^i,X^j] + \theta^T \gamma_i[X^i,\...
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Dirac delta of matrix argument - Matrix model path integral vs Hilbert space
Assume a Hamiltonian $H$ with $N$ orthonormal eigenstates $\{\vert n\rangle\}$ of energies $\epsilon_n$. One can define a density of states,
\begin{align}
\rho(E)&=\mathrm{tr}\,\hat{\delta}(E-\hat{...
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Jackiw-Teitelboim (JT) gravity as a matrix integral
I am reading https://arxiv.org/abs/1903.11115 by Saad, Shenker and Stanford. They relate an (averaged) $n$-point function in a random double-scaled matrix model to a path integral genus expansion in ...
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Random matrix theory and the singularities of the Weingarten function
In the random matrix theory literature, one often encounters identities associated with averages over ensembles of random unitaries. For a simple example let's say we're interested exclusively in $2\...
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How to understand Heisenberg time in random matrix theory?
Recently, from few papers, I have encountered the word 'Heisenberg time' $t_{\text{H}}$ which is an inverse of a mean level spacing $\Delta(\hat{\mathcal{H}})$ of a finite system Hamiltonian $\hat{\...
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D0, D1, D2... DN branes consist of D0 branes?
In BFSS Matrix theory, D0-branes connect the ends of open strings. Do other D-branes (D1, D2... DN) consist of D0-branes?
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Trace identity for $SU(N)$ matrix integral
I would like to know if there's a nice way to compute the following:
$$ \int_{SU(N)} \underbrace{ dU}_{\text{Haar Measure}} \mathrm{tr} \left(U^n \right)~?$$
The following is necessary:
$U \in SU(N)$
$...
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Integral relation on Lie (Super)algebra
I have the following integral relation where $X$ is an element of a Lie (super)algebra, $Y_i$ are functions on the respective Lie group and $M$ is an element of the Lie group:
$$Y_1(M)=\int{D}Xe^{-\...
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Feynman Rules for the BFSS Model. Large $N$ Super Matrix Quantum Mechanics
BFSS model is a theory of super-symmetric matrix quantum mechanics describing $N$ coincident $D0$-branes, defined by the action
$$S=\frac{1}{g^2}\int dt\ \text{Tr}\left\{ \frac{1}{2}(D_t X^I)^2 + \...
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Gaussian Matrix Integral
I couldn't quite understand how we calculate the Gaussian Matrix Integral
$$\mathcal{Z}=\int dM\ e^{-N\text{tr}\left(\frac{1}{2}M^2+JM\right)},$$
where the integration measure $dM$ over the $N$ by $N$ ...
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Difference between resolvent and degeneracy
I am studying https://arxiv.org/abs/1903.11115. In equation (62), the resolvent is defined as the integral transform of partition function as
$$
R(E) = -\int_0^\infty d\beta\ e^{\beta E} Z(\beta)
$$
...
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Analytical expression for density of random matrix level ratios
Consider a hermitian matrix $H$ with eigenvalues $E_{i-1}<E_i$. The level spacings are defined as $s_i=E_i-E_{i-1}$ and the level ratios as $r_i = s_i/s_{i-1}$. To make the support of an underlying ...
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Matrix Integrals, Riemann Surfaces & Black Holes. A question regarding one of J.M. Maldacena's talks
I was watching this presentation of Juan Martin Maldacena at Princeton: https://www.youtube.com/watch?v=OMb_P5qPpMc&ab_channel=GraduatePhysics. In one slide he shows an interesting integral. (I ...
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How can I show that $1/N$ expansion for large $N$ matrix models have a string theoretical perturbation expansion?
While surfing through some further reading suggestions on string theory, I stumbled upon this slide from a talk by Nathan Seiberg. I wanted to derive the main argument by applying a perturbation ...
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Why is a static potential between supergravitons problematic?
When deriving the one and two-loop result for the effective potential between two scattering supergravitons, for example from here, we see that it is always a velocity dependent potential. ...
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Validity of DLCQ Matrix Theory near the Big Bang
In this paper,section 3.2, Craps, Sethi and Verlinde claim that DLCQ matrix theory is valid near the big bang if
The open string oscillators decouple and
Gravity decouples from the matrix description
...
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Why is there an additional $NI$ term in this ${\rm SU}(N)$ generator, from Matrix Quantum Mechanics?
This question refers to equation (11) in the latest preprint of the following paper:
X. Han, S. A. Hartnoll and J. Kruthoff, "Bootstrapping Matrix Quantum Mechanics", Phys. Rev. Lett. 125, ...
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Applications of the Gaussian unitary ensemble
It is well known that the pair correlation function of the zeros of the Riemann zeta function reproduces the correlation function of the random matrices from the Gaussian unitary ensemble (GUE). ...
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How are the saddle-point equations derived in the single random matrix model?
In this question, I'm referring to a specific step in https://arxiv.org/abs/hep-th/9306153. I want to reproduce equation (2.4) on page 15. I think I lack the experience required for dealing with ...
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Equivalence of 2d quantum and topological gravity
I have heard of the statement that 2d quantum gravity (defined as minimal models of CFT coupled to Liouville theory) and 2d topological gravity are equivalent. The former is described by the continuum ...
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Good starting point for quantum Hall matrix models
I am a recent Masters in theoretical condensed matter physics and have experience in working on topological insulators and Weyl semimetals. I have also dabbled a bit in the fractional quantum Hall ...
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Can we generalize matrix model theory?
As in the question, can matrix model theory be generalized to a tensor model theory? Will the results be different or useful in describing real world phenomena?
Details: in matrix model theory we ...
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$\delta^{(2)}$ convention
In this note: https://arxiv.org/abs/hep-th/0410165 at page 12 there is a delta-function constraint written as:
\begin{align}
\delta \left( ^ { U } M \right) = \prod _ { i < j } \delta ^ { ( 2 ) } \...
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Simultaneous shifted diagonalization of bunch of operators
I have the following Lie algebra which is generated by $\{L_n|n\geq 0\}.$ It satisfies the following commutation rule
$$ \Big[ L_i ,L_j \Big]= (i-j)(L_{i+j}-\frac14 L_{i+j-1}).....*$$
My question is ...
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Singular behavior of pure gravity
Can anyone plese explain what means singular part of partition function for pure gravity?
Let me specify my question. I am dealing with 2D quantum gravity and starts from path integral formulation of ...
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How to relate random matrix theory with Quantum mechanics approach
In Quantum mechanics we first built Hamiltonian and then find its eigenvalues and eigenvectors. How I can relate this with random matrix theory?
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How to calculate correlators in a 1D Conformally invariant Matrix Model?
I am working on a 1D Conformally invariant Matrix Model with the following Partition function:
$$
Z(g) = \int \mathcal{D}M(t) \exp \left[ -\text{tr}\int dt \left( \frac{1}{2} \dot{M}^2(t)+V(M) \...
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How to get the eigenvalue density contribution $\rho_1(x)$?
I'm studying the $1/N$ expansion beyond the planar limit in matrix models. Currently I'm trying to understand and reproduce the results of:
Antisymmetric Wilson loops in $\mathcal N \geq 4$ SYM ...
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Basis of eigenvectors common to H and B
Considering a three-dimensional state space spanned by the orthonormal basis formed by the three kets $|u_1\rangle,|u_2\rangle,|u_3\rangle $. In the basis of these three vectors, taken in order, are ...
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Hamiltonian matrix for a delta potential with periodic boundary condition
I'm trying to find the energy eigenvalues of a Dirac delta potential:
$$V(x)=-\alpha\delta(x)$$
with periodic boundary condition over some length $L$:
$$\psi(x+L)=\psi(x)$$
and only even ...
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Massless limit of Matrix Quantum Mechanics
I am working on a Matrix Quantum Mechanics model that is related to 2d string theory as defined here: http://arxiv.org/abs/hep-th/0311273 §Chapter III
The action is defined as $$
S = \text{Tr} \int ...
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Lumped mechanical system - Base Change and asymmetric matrices
Let's assume I have a system which can be wrote as:
$$ \underbrace{\begin{bmatrix}
m_{1} & 0 & \dots & 0 \\
0 & m_{2} & \...
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Gaussian beam propagation with ABCD matrix through a grin lens
I am currently trying to simulate a Gaussian beam that has a transverse offset of around 20um from the optic axis where the Gaussian beam travels through a grin lens using the ABCD matrix method. I ...
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Generalization of Itzykson-Zuber Formula to Path Integrals?
Some Background: In mathematical physics (matrix quantum mechanics in particular), one often runs into path integrals of the form:
$$ Z = \int \prod_{ab} DM^1_{ab}(\tau) DM^2_{ab}(\tau) e^{-NS(M^1,M^2,...
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References to Random Matrix Theory
I am looking for some good references - books/lecture notes/articles which contains Random Matrix Theory for Physicists. I am not particularly looking for mathematical rigor in derivations. I am more ...
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Relation between $1/N$ and perturbative expansions in QFT
I heard many times phrases like "large $N$ is a clever way to organize the diagrammatic expansion" or "each diagram in the large $N$ expansion contains an infinite number of usual perturbative ...
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Matching measured eigenvalue of a matrix [closed]
Given a Matrix of the form below, we have 8 variables, $\omega_{i}$ and $J_{ij}$. we want to diagonalise the Matrix to obtain values to match the observed Eigenvalue in an Experiment. i.e. $\bar{\...
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