Download Matrix Geometry And Coherent states

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