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Transcript

Goro Ishiki (University of Tsukuba) arXiv: 1503. 01230 [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. Known examples → Fuzzy sphere, torus, CPn, etc We need deeper understanding ・Consider momentum cutoff regularization on sphere spherical harmonics Of course, functions with a cutoff do not form a closed algebra (ring). Exceeds cutoff In most physical theories , this breaks symmetries… ・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 In the large matrix size limit matrices !!! , the algebra becomes isomorphic to the original. [DeWitt-Hoppe-Nicoli, BFSS] Nambu-Goto action for membranes ( After some gauge fixing ) ・When the world volume has spherical topology, Matrix regularization Matrix Quantum Mechanics with finite number of DOF ・ Matrix regularization preserves rotaion, R-sym and SUSY (with some fermions) ・ The case of torus leads the same matrix model ⇒ Unified description of topology ・ This model (+ fermions) is conjectured to be a correct “Lattice theory” for M-theory, [BFSS] i.e. Non-perturbative formulation for Quantum Gravity ・ Difficulty in matrix models In the path-integral of matrix models, the geometry has become invisible. 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 ] ・ 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. general states coherent states Coherent states : quantum analogue of points on classical space Can be defined as the ground states of ・We are given a one parameter family of D Hermitian matrices Hermitian ・We define Hamiltonian ・ Eigenstates ・ Coherent states ⇔ with Coherent state ⇔ Wave packet which shrinks to a point at ◇ Classical geometry = Set of all coherent states ◇ Nice property of ・ It contains geometric information Vanishing on Metric on Connection, Curvature ・It is computable from given matrices (observable in matrix models) ◆ Metric on ◆ Levi-Civita connection ◆ Curvature ◆ Poisson tensor Assumption: is manifold & dim rep of SU(2) generators ・Classical space ・Metric ・ Poisson Tensor Represented by Clock-Shift matrices “Fuzzy Clifford Torus” ・Classical space ・Metric ・We proposed a new observables in matrix models, which characterize geometric properties of matrices. Geometry (Shape of membranes) Matrix configuration via coherent states ・Checked for fuzzy sphere and torus ・Dimension of ・The geometric objects we defined are gauge invariant ⇒ Observables in MM. ・Geometric interpretation of matrix models Emergent space time in AdS/CFT Fuzzy sphere with N=50 0.05 [cf. Berentsin, Aoki-san’s talk] is around here 0.04 ・Also useful for numerical work 0.03 [Kim-Nishimura-Tsuchiya, AnagnostopoulosHanada-Nishimura-Takeuchi, Catterall-Wiseman] 0.8 0.9 1.0 0.8 0.9 1.0 1.1 1.2 2.8 2.6 2.4 2.2 Dimension is 2 2.0 1.8 1.1 1.2