Quantum (matrix) geometry and quasi-coherent states

Abstract

A general framework is described which associates geometrical structures to any set of D finite-dimensional Hermitian matrices X a , a = 1, …, D. This framework generalizes and systematizes the well-known examples of fuzzy spaces, and allows to extract the underlying classical space without requiring the limit of large matrices or representation theory. The approach is based on the previously introduced concept of quasi-coherent states. In particular, a concept of quantum Kähler geometry arises naturally, which includes the well-known quantized coadjoint orbits such as the fuzzy sphere S N 2 and fuzzy C P N n . A quantization map for quantum Kähler geometries is established. Some examples of quantum geometries which are not Kähler are identified, including the minimal fuzzy torus.

Exploring the gravity sector of emergent higher-spin gravity: effective action and a solution

“…[6,27]. Finally, (quasi-) coherent states on quantum spaces are related to yet another metric [25,26].…”

A Twistorial Description of the IKKT-Matrix Model

“…Hence, the fuzzy S 4 N can be understood as projection of fuzzy twistor space CP 3 N . We note that in the non-commutative case, the incident relation (50) does not have a well-defined inverse. Moreover, since there is not a geometry in the usual sense, differential forms and complex structures are not defined a priori.…”

“…For our purpose, the most important feature of IKKT-type matrix models is that they define a gauge theory on suitable matrix backgrounds. Such a background is defined by a set of 10 "almost-commutative" matrices Ȳ I , and typically defines a noncommutative or quantized space(time) [50]. This means that the matrix algebra EndpHq generated by the Ȳ I can be interpreted as quantized algebra of functions on some symplectic space M, and Ȳ I is interpreted as quantized coordinate function.…”

“…Therefore, the incidence relation (50) of the twistor construction corresponds precisely to the Hopf map followed by a stereographic projection. By denoting x 2 " x µ x µ , and x 2 " x αα 1 x αα 1 , we see that…”

Matrix regularization for tensor fields

“…18) where, d n are integration constants. The equation (B.14) imposes the recursion relation,d n+N = d n e −π(N +2n) (B.19)for any integer n. By writing the integer n asn = Nk + I k ∈ Z, I ∈ {1, 2, • • • , N}, (B.20)we can solve the recursion relation (B 19…”