Matrix models of noncommutative geometry and string theory

Feynman path integrals and phase transitions in quantum field theory.

The Ising model and commutative scalar phi-four theory.

String theory in 2 dimensions as discretized surfaces and random matrices.

The large N saddle point and orthogonal polynomial methods.

The $\theta=0$ and $\theta=\infty$ limits of noncommutative field theory.

The multitrace expansion of noncommutative scalar field theory.

Monte Carlo algorithms for matrix models.

The phase structure of noncommutative phi-four.

The IKKT matrix model and non-perturbative superstring theory.

Yang-Mills matrix  models of noncommutative gauge theory. 

BFSS matrix quantum mechanics, M-(atrix) theory and the gauge/gravity duality.

Yang-Mills matrix quantum mechanical models.

Large $d$ expansion of Yang-Mills quantum mechanics.

Emergent geometry and noncommutative gauge theory  in Yang-Mills matrix models.

AdS/CFT correspondence in two dimensions and noncommutative geometry.

Wilson renormalization group equations for matrix and noncommutative models.

Noncommutative quantum black holes.

Emergent gravity and quantized symplectic geometry.

Matrix cosmology and emergent time in IKKT and BFSS matrix models.

Quantum Gravity: Noncommutative Geometry, The Gauge/Gravity Correspondence And Matrix Models

 Noncommutative geometry (and its matrix models) presents a distinct solution to the problem of quantum gravity whereas the gauge/gravity correspondence is currently the most successful proposal for quantum gravity. The two approaches intersect within the quantum mechanics of the BFSS (or M-(atrix)) theory and also within the IKKT matrix model which should be viewed as providing the starting unifying framework.
The BFSS-type Yang-Mills quantum mechanics and the IKKT-type Yang-Mills matrix models provide a non-perturbative formulation of superstring theory and its underlying eleven-dimensional M-theory. But they also provide a quantum gravitational formulation (gravitational Feynman path integral) of Connes' noncommutative geometry (classical phase space). In other words, we should think of noncommutative geometry as a "first quantization of geometry" (classical gravity) and think of the corresponding matrix models as a "second quantization" of geometry " (quantum gravity).
Poisson manifolds play therefore the fundamental role of "curved spacetime", the Darboux theorem plays the role of the "equaivalence principle" while Moyal-Weyl spaces are what defines our "flat spacetime".
The nature of quantum geometry can also be probed by means of multitrace matrix models where both the renormalization group equation, the large N saddle point analysis and the Monte Carlo method come together in a symphony of mathematical and computational methods applied to the same theoretical problem (which is quite rare). The multitrace matrix models is in fact an alternative to Yang-Mills matrix models which allow for emergent geometry (quantum geometry), emergent gravity (quantum gravity) and emergent time (quantum cosmology).
Another important gauge/gravity duality (besides the BFSS-type Yang-Mills quantum mechanics) is the AdS/CFT correspondence. The case of two dimensions is the most mysterious and is the most important for quantum black holes as well as it is the case most closely related to noncommutative geometry which is very intriguing indeed.

Question 1: Towards "computational physics of string theory"!
Answer 1: Preliminary results are reported in

Question 2: What is the relation between multitrace matrix models and quantum geometry?
Answer 2: The discussion of the fixed points of a cubic multitrace matrix model (which is important to a very important case of emergent noncommutative geometry in two dimensions, i.e. the fuzzy sphere, the noncommutative torus and the Moyal-Weyl plane) is discussed in

Question 3: Can we reformulate a noncommutative theory of the AdS/CFT correspondence and black hole evaporation problem?
Answer 3 (Sep 2021): Update number 9 (The AdS^2_θ/CFT_1 Correspondence and Noncommutative Geometry). See also

Symmetric Spaces

We start off with a brief discussion of the differential geometry of maximally symmetric spaces in two dimensions such as $\mathbb{S}^2$, ${\rm dS}^2$, $\mathbb{H}^2$ and ${\rm AdS}^2$ which play a prominent role in the near-horizon geometry of black holes, in Euclidean quantum field theory, in noncommutative geometry and in the  ${\rm AdS}^{d+1}/{\rm CFT}_d$ correspondence.


First we note that maximally symmetric spaces are essential ingredient in quantum gravity theories and cosmological models. These homogeneous and isotropic spaces enjoy the largest possible amount of spactime symmetries (isometries) and in Lorentzian signature they are exhausted with the three maximally symmetric spaces \cite{Bengtsson}:

The de Sitter spacetime ${\rm dS}^d$ (positive scalar curvature, repulsive cosmological constant, topology $\mathbb{S}^{d-1}\times \mathbb{R}$) which is relevant to cosmology.  The de Sitter spacetime ${\rm dS}^d$ as embedded in $\mathbb{M}^{1,d}$ is given by the ambiant metric and the quadric form

Minkowski spacetime $\mathbb{M}^d$ (zero scalar curvature, zero cosmological constant, topology $\mathbb{R}^d$) which can be viewed as a zero cosmological constant limit of de Sitter spacetime  $\mathbb{dS}^d$.

The anti-de Sitter spacetime ${\rm AdS}^d$ (negative scalar curvature, attractive cosmological constant,  topology $\mathbb{R}^{d-1}\times \mathbb{S}^1$) which is relevant to quantum gravity. The anti-de Sitter spacetime ${\rm AdS}^d$ as embedded in $\mathbb{M}^{2,d-1}$ is given by the ambiant metric and the quadric form

However, Wick rotation to Euclidean signature remains crucial to both quantum field theory and noncommutative geometry where quantization of fields and geometries makes strict sense only in Euclidean setting. In Euclidean signature, the maximally symmetric spaces are then given by the three spaces \cite{Bengtsson}:

The sphere $\mathbb{S}^d$ (positive curvature). The sphere $\mathbb{S}^d$ as embedded in $\mathbb{R}^{d+1}$ is given by the ambiant metric and the quadric form
The Killing vectors fields which leave both the ambiant metric and the quadric form invariant are
These $d(d+1)/2$ isometries generate the group of rotations $SO(d+1)$. This is to be contrasted with the isometry group of de Sitter spacetime is $SO(1,d)$.

Euclidean space $\mathbb{R}^d$ (zero curvature).

The pseudo-sphere $\mathbb{H}^d$ (negative curvature).  The pseudo-sphere (Hyperboloic space) $\mathbb{H}^d$  as embedded in $\mathbb{M}^{1,d}$ is given by the ambiant metric and the quadric form
The Hyperboloic space $\mathbb{H}^d$ is defined as the upper sheet of the two-sheeted hyperboloid $-X_1^2+X_2^2...+X_{d+1}^2=-R^2$. The Killing vectors fields which leave both the ambiant metric and the quadric form invariant are still given by as before but the underlying symmetry group is now given by $SO(1,d)$. This is to be contrasted with the isometry group of anti-de Sitter spacetime which is given by $SO(2,d-1)$.

It is intriguing to note that in two dimensions the spaces $\mathbb{S}^2$, ${\rm dS}^2$, $\mathbb{H}^2$ and ${\rm AdS}^2$ are simply related. For example, we can go from  ${\rm AdS}^2$ (closed timelike curves with isometry group $SO(2,1)$) to  ${\rm dS}^2$ (closed spacelike curves with isometry group $SO(1,2)$) and vice versa by switching the meaning of timelike and spacelike. While both ${\rm dS}^2$ and $\mathbb{H}^2$ share precisely the same isometry group $SO(1,2)$. And we can go from ${\rm dS}^2$ to $\mathbb{S}^2$ by an ordinary Wick rotation. We can also go from ${\rm AdS}^2$ to $\mathbb{H}^2$ by a Wick rotation.

Representation theory of the Lorentz groups $SO(1,2)$ and $SO(2,1)$ can be found for example in \cite{barg,bns}. See also \cite{Mukunda:1974gb,Girelli:2015ija,Basu:1981ju}.

In this article we will focus on the case of two dimensions with Euclidean signature where the positive curvature space is given by a sphere $\mathbb{S}^2$ with isometry group $SO(3)$ and the negative curvature space is given by a pseudo-sphere $\mathbb{H}^2$  with isometry group $SO(1,2)$. We will be mostly interested in the case of the pseudo-sphere $\mathbb{H}^2$ which we will simply denote by ${\rm AdS}^2$.

The quantization of these two spaces yields the fuzzy sphere $\mathbb{S}^2_N$  \cite{Hoppe,Madore:1991bw} and the noncommutative pseudo-sphere ${\rm AdS}^2_{\theta}$ \cite{Ho:2000fy,Ho:2000br,Jurman:2013ota,Pinzul:2017wch} respectively which enjoy the same isometry groups $SO(3)$ and $SO(1,2)$ as their commutative counterparts. The fuzzy sphere is unstable and suffers collapse in a phase transition to Yang-Mills matrix models (topology change or geometric transition) whereas the noncommutative pseudo-sphere can sustain black hole configurations (by including a dilaton field) and also suffers collapse in the form of the information loss process (quantum gravity transition).

In fact, the product space $\mathbb{S}^2\times {\rm AdS}^2$ is the near-horizon geometry of extremal black holes in general relativity and string theory, e.g. the four-dimensional Reissner-Nordstrom black hole. It is then observed that the information loss problem in  four dimensions on  $\mathbb{S}^2_N\times {\rm AdS}^2_{\theta}$ reduces to the information loss problem in two dimensions on  noncommutative ${\rm AdS}^2_{\theta}$.

As we have said we will be mostly interested here in the case of the pseudo-sphere $\mathbb{H}^2$ or the Euclidean ${\rm AdS}^2$. The goal naturally is to construct a consistent ${\rm AdS}^2$/${\rm CFT}_1$ correspondence.

Author biography

Badis Ydri —currently a professor of theoretical particle physics, teaching at the Department of Physics, Badji-Mokhtar Annaba University, Algeria—received his PhD from Syracuse University, New York, USA in 2001 and his Habilitation from Annaba University, Annaba, Algeria in 2011. His doctoral work, titled ‘Fuzzy Physics’, was supervised by Professor A P Balachandran. 


Professor Ydri is an Adjunct Professor at the Dublin Institute for Advanced Studies, Dublin, Ireland, and a research associate (regular ICTP associate) at the Abdus Salam Center for Theoretical Physics, Trieste, Italy.

His post-doctoral experience comprises a Marie Curie fellowship at Humboldt
University Berlin, Germany, and a Hamilton fellowship at the Dublin Institute for
Advanced Studies, Ireland.

His current research directions include: noncommutative geometry; the gauge/gravity duality; computational physics of string theory; renormalization group and Monte Carlo methods in matrix models and noncommutative field theories;  emergent geometry, gravity and cosmology from matrix models; and foundations of quantum mechanics.

Other related interests include string theory;  quantum information; causal dynamical triangulation; Horava–Lifshitz gravity; supersymmetric and noncommutative standard models; and supersymmetric gauge theory in four dimensions.

He has recently published six books. His other intellectual  interests include philosophy of physics and existential  philosophy.