Goro Ishiki (University of Tsukuba) arXiv: [hep-th]

◆ Matrix regularization ・ Can preserve a lot of symmetries Space-time symmetry, SUSY, Internal rotation etc. ・ Matrix models ⇔ ``Lattice theory’’ for string/M theory Candidates for theories of Quantum Gravity ・ Not easy to construct. We need deeper understanding Known examples → Fuzzy sphere, torus, CP n, etc

・ Consider momentum cutoff regularization on sphere Of course, functions with a cutoff do not form a closed algebra (ring). In most physical theories, this breaks symmetries… spherical harmonics Exceeds cutoff

・ More efficient momentum cutoff map : SU(2) generators in spin representation. Consider a set of ``Matrix Spherical Harmonics’’ Actually, they form a closed algebra of matrices !!! In the large matrix size limit, the algebra becomes isomorphic to the original.

Nambu-Goto action for membranes ( After some gauge fixing ) ・ When the world volume has spherical topology, Matrix regularization [DeWitt-Hoppe-Nicoli, BFSS]

・ This model (+ fermions) is conjectured to be a correct “Lattice theory” for M-theory, [BFSS] ・ Matrix regularization preserves rotaion, R-sym and SUSY (with some fermions) i.e. Non-perturbative formulation for Quantum Gravity Matrix Quantum Mechanics with finite number of DOF ・ The case of torus leads the same matrix model ⇒ Unified description of topology

・ Difficulty in matrix models In the path-integral of matrix models, How can we recover the shape of membranes, D-branes or strings? Is there any good observables in MM, which characterize the classical geometry (shape) of membranes? Matrix configuration Geometry (Shape of membranes) ??? [ Cf. Berenstein, Aoki-san’s talk ] the geometry has become invisible.

・ We generalized “coherent states” to matrix geometries ・ We defined classical geometry as a set of coherent states ・ We proposed a new set of observables in matrix models, which describe the classical geometry and geometric objects like metric, curvature and so on.

Coherent states : quantum analogue of points on classical space general states coherent states Can be defined as the ground states of

・ We are given a one parameter family of D Hermitian matrices Hermitian ・ We define Hamiltonian Coherent state ⇔ Wave packet which shrinks to a point at ・ Eigenstates ・ Coherent states ⇔ with

◇ Classical geometry = Set of all coherent states ◇ Nice property of Metric on Connection, Curvature Vanishing on ・ It contains geometric information ・ It is computable from given matrices (observable in matrix models)

◆ Levi-Civita connection ◆ Curvature ◆ Poisson tensor ◆ Metric on Assumption: is manifold &

dim rep of SU(2) generators ・ Classical space ・ Metric ・ Poisson Tensor

“Fuzzy Clifford Torus” ・ Classical space ・ Metric Represented by Clock-Shift matrices

・ We proposed a new observables in matrix models, which characterize geometric properties of matrices. ・ Dimension of via coherent states Matrix configuration Geometry (Shape of membranes) ・ Checked for fuzzy sphere and torus

・ The geometric objects we defined are gauge invariant ⇒ Observables in MM. Fuzzy sphere with N=50 is around here Dimension is 2 ・ Also useful for numerical work ・ Geometric interpretation of matrix models Emergent space time in AdS/CFT [Kim-Nishimura-Tsuchiya, Anagnostopoulos- Hanada-Nishimura-Takeuchi, Catterall-Wiseman] [cf. Berentsin, Aoki-san’s talk]