For matrix models and QFT, we discuss how holographic emergent geometry appears from matrix degrees of freedom (specifically, adjoint scalars in super Yang-Mills theory) and how operator algebra that describes an arbitrary region of the bulk geometry can be constructed. We pay attention to the subtle difference between the notions of wave packets that describe low-energy excitations: QFT wave packet associated with the spatial dimensions of QFT, matrix wave packet associated with the emergent dimensions from matrix degrees of freedom, and bulk wave packet which is a combination of QFT and matrix wave packets. In QFT, there is an intriguing interplay between QFT wave packet and matrix wave packet that connects quantum entanglement and emergent geometry. We propose that the bulk wave packet is the physical object in QFT that describes the emergent geometry from entanglement. This proposal sets a unified view on two seemingly different mechanisms of holographic emergent geometry: one based on matrix eigenvalues and the other based on quantum entanglement. Further intuition comes from the similarity to traversable wormholes.

We will give an intuitive explanation of why and how matrices (or, more precisely, large-N gauge theories) can describe a black hole, without assuming knowledge of quantum mechanics and holographic duality. Firstly, we explain an intuitive picture inroduced by Witten: diagonal entries of matrices describe particles and off-diagonal entries describe strings connecting particles. When many strings are excited, a lot of energy and entropy are packed in a small region and form black hole. Next, we consider classical dynamics of matrix model. Specifically, we colide two black holes. Using the energy conservation, equipartition law of energy and elementary school math, we show that black hole becomes colder after the merger. Matrices know black hole's negative heat capacity! To gain a little bit more intuition, we will look at ants. Collective behavior of ants has a striking similarity to black hole. The mapping rule is ant -> particle, pheromone -> string, and ant trail -> black hole. Tuning parameters such as temperature or each ant's laziness, we can obtain three kinds of phase diagrams. Each of them has a counterpart in large-N gauge theories. If time permits, I will explain the mechanism applicable to strongly-coupled and highly quantum regime needed for quantitative agreement with Einstein gravity. (This part requires a good understanding of undergraduate-level quantum mechanics.)

In holography, the geometry of gravitational theory should be encoded in its non-gravitational dual. In particular, in gauge/gravity duality, the geometry should be encoded into the color degrees of freedom (matrices) in U(N) supersymmetric Yang-Mills theory. When the Yang-Mills theory is regarded as the low-energy effective theory of a system of N D-branes and open strings between them. Transverse spatial directions emerge from scalar fields, which are N*N matrices with color indices; roughly speaking, the eigenvalues are the locations of D-branes. In the past, it was argued that this simple 'emergent space' picture cannot be used in the context of gauge/gravity duality, because the ground-state wave function delocalizes at large N, leading to a conflict with the locality in the bulk geometry. We show that this conventional wisdom is not correct: the ground-state wave function does not delocalize, and there is no conflict with the locality of the bulk geometry. This conclusion is obtained by clarifying the meaning of the 'diagonalization of a matrix' in Yang-Mills theory, which is not as obvious as one might think. This observation opens up the prospect of characterizing the bulk geometry via the color degrees of freedom in Yang-Mills theory, all the way down to the center of the bulk. [please email a.held@imperial.ac.uk for zoom link or password]

The D0-brane matrix model (the BFSS matrix model and the BMN matrix model) can describe various objects including type IIA black zero-brane, M-theory black hole, M2-branes and M5-branes. We study this theory from several different angles. We put the emphasis on the importance of the dynamics of eigenvalues of matrices, and more generally, color degrees of freedom. Furthermore we explain how the Euclidean theory can be studied by using the Monte Carlo method, and discuss the future directions. If you have a good idea we can test it on computer together! Part of the Black Hole Information Paradox Journal Club. Please email damian.galante@kcl.ac.uk for link to the meeting.

We propose a unified description of two important phenomena: color confinement in large-N gauge theory, and Bose-Einstein condensation (BEC). We focus on the confinement/deconfinement transition characterized by the increase of the entropy from N^0 to N^2, which persists in the weak coupling region. Indistinguishability associated with the symmetry group --- SU(N) or O(N) in gauge theory, and S_N permutations in the system of identical bosons --- is crucial for the formation of the condensed (confined) phase. We relate standard criteria, based on off-diagonal long range order (ODLRO) for BEC and the Polyakov loop for gauge theory. The constant offset of the distribution of the phases of the Polyakov loop corresponds to ODLRO, and gives the order parameter for the partially-(de)confined phase at finite coupling. Furthermore we show the numerical evidence for this phenomenon at strong coupling, by using the Yang-Mills matrix model as a concrete example and solving it numerical via lattice simulation. This talk is based on a series of papers, especially "Color Confinement and Bose-Einstein Condensation" by Hanada, Shimada and Wintergerst, 2001.10459 [hep-th] and "Partial Deconfinement at Strong Coupling on a Lattice' by Bergner, Bodendorfer, Funai, Hanada, Rinaldi, Schaefer, Vranas and Watanabe to appear (should be in hep-th by the talk).

We propose a unified description of two important phenomena: color confinement in large-$N$ gauge theory, and Bose-Einstein condensation (BEC). The key lies in relating standard criteria, based on off-diagonal long range order (ODLRO) for BEC and the Polyakov loop for gauge theory: the constant offset of the distribution of the phases of the Polyakov loop corresponds to ODLRO. Indistinguishability associated with the symmetry group --- SU(N) or O(N) in gauge theory, and S_N permutations in the system of identical bosons --- is crucial in either case. This viewpoint may have implications for confinement at finite N, and for quantum gravity via gauge/gravity duality. As a byproduct, we obtain a characterization of the partially-confined/partially-deconfined phase at finite coupling: the constant offset of the distribution of the phases of the Polyakov loop is the order parameter.