Explore the fascinating world of matrix regularization in theoretical physics to preserve symmetries like space-time symmetry, supersymmetry, and internal rotations. Dive into the challenges of constructing theories of Quantum Gravity using matrix models, with a focus on momentum cutoff regularization and maintaining algebraic structures. Discover how matrix quantum mechanics with a finite number of degrees of freedom can lead to a unified description of topology in a lattice theory for M-theory. Unravel the difficulties in recovering the classical geometry of membranes, D-branes, and strings in path integrals of matrix models, paving the way for understanding shape recovery mechanisms and new observables in quantum physics.
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Presentation 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 matrices !!! In the large matrix size limit , 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-symand 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? ??? Geometry Matrix configuration (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.
coherent states general states Coherent states : quantum analogue of points on classical space Can be defined as the ground states of
Classical geometry = Set of all coherent states Nice property of It contains geometric information Connection, Curvature Vanishing on Metric on It is computable from given matrices (observable in matrix models)
We proposed a new observables in matrix models, which characterize geometric properties of matrices. Geometry Matrix configuration (Shape of membranes) 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 0.04 around here Also useful for numerical work 0.03 [Kim-Nishimura-Tsuchiya, Anagnostopoulos- Hanada-Nishimura-Takeuchi, Catterall-Wiseman] 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 0.8 0.9 1.0 1.1 1.2
