# Twistors to twisted geometries

@article{Freidel2010TwistorsTT, title={Twistors to twisted geometries}, author={Laurent Freidel and Simone Speziale}, journal={Physical Review D}, year={2010}, volume={82} }

In a previous paper we showed that the phase space of loop quantum gravity on a fixed graph can be parametrized in terms of twisted geometries, quantities describing the intrinsic and extrinsic discrete geometry of a cellular decomposition dual to the graph. Here we unravel the origin of the phase space from a geometric interpretation of twistors.

## 104 Citations

Geometry of loop quantum gravity on a graph

- Physics
- 2010

We discuss the meaning of geometrical constructions associated to loop quantum gravity states on a graph. In particular, we discuss the "twisted geometries" and derive a simple relation between these…

Discrete Gravity Models and Loop Quantum Gravity: a Short Review

- Physics
- 2012

We review the relation between Loop Quantum Gravity on a fixed graph and discrete models of gravity. We compare Regge and twisted geometries, and discuss discrete actions based on twisted geometries…

Twistor Networks and Covariant Twisted Geometries

- Mathematics
- 2012

We study the symplectic reduction of the phase space of two twistors to the cotangent bundle of the Lorentz group. We provide expressions for the Lorentz generators and group elements in terms of the…

Spin connection of twisted geometry

- Mathematics
- 2013

Twisted geometry is a piecewise-flat geometry less rigid than Regge geometry. In Loop Gravity, it provides the classical limit for each step of the truncation utilized in the definition of the…

Spinors and Twistors in Loop Gravity and Spin Foams

- Physics
- 2011

Spinorial tools have recently come back to fashion in loop gravity and spin foams. They provide an elegant tool relating the standard holonomy-flux algebra to the twisted geometry picture of the…

Introductory lectures to loop quantum gravity

- Physics
- 2010

We give a standard introduction to loop quantum gravity, from the ADM variables to spin network states. We include a discussion on quantum geometry on a fixed graph and its relation to a discrete…

Octahedron of complex null rays and conformal symmetry breaking

- PhysicsPhysical Review D
- 2019

We show how the manifold T*SU(2,2) arises as a symplectic reduction from eight copies of the twistor space. Some of the constraints in the twistor space correspond to an octahedral configuration of…

A new action for simplicial gravity in four dimensions

- Physics
- 2015

We develop a proposal for a theory of simplicial gravity with spinors as the fundamental configuration variables. The underlying action describes a mechanical system with finitely many degrees of…

Loop gravity in terms of spinors

- Physics, Mathematics
- 2011

We show that loop gravity can equally well be formulated in in terms of spinorial variables (instead of the group variables which are commonly used), which have recently been shown to provide a…

Dynamics for a simple graph using the U(N) framework for loop quantum gravity

- Mathematics, Physics
- 2012

The implementation of the dynamics in loop quantum gravity (LQG) is still an open problem. Here, we discuss a tentative dynamics for the simplest class of graphs in LQG: Two vertices linked with an…

## References

SHOWING 1-10 OF 11 REFERENCES

Geometry of loop quantum gravity on a graph

- Physics
- 2010

We discuss the meaning of geometrical constructions associated to loop quantum gravity states on a graph. In particular, we discuss the "twisted geometries" and derive a simple relation between these…

U(N) coherent states for loop quantum gravity

- Mathematics
- 2011

We investigate the geometry of the space of N-valent SU(2) intertwiners. We propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under…

Twisted geometries: A geometric parametrisation of SU(2) phase space

- Mathematics
- 2010

A cornerstone of the loop quantum gravity program is the fact that the phase space of general relativity on a fixed graph can be described by a product of SU(2) cotangent bundles per edge. In this…

Triangulated surfaces in twistor space: a kinematical set up for open/closed string duality

- Mathematics
- 2006

We exploit the properties of the hyperbolic space 3 to discuss a simplicial setting for open/closed string duality based on (random) Regge triangulations decorated with null twistorial fields. We…

Quantum geometry from phase space reduction

- Mathematics
- 2009

In this work, we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes…

Area?angle variables for general relativity

- Mathematics, Physics
- 2008

We introduce a modified Regge calculus for general relativity on a triangulated four-dimensional Riemannian manifold where the fundamental variables are areas and a certain class of angles. These…

The symplectic geometry of polygons in Euclidean space

- Mathematics
- 1996

We study the symplectic geometry of moduli spaces M r of polygons with xed side lengths in Euclidean space. We show that M r has a natural structure of a complex analytic space and is…

Phase space descriptions for simplicial 4D geometries

- Mathematics
- 2008

Starting from the canonical phase space for discretised (4d) BF–theory, we implement a canonical version of the simplicity constraints and construct phase spaces for simplicial geometries. Our…

Geometric quantization and multiplicities of group representations

- Mathematics
- 1982

The Heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum-mechanical particle. This can be rephrased as follows: the smallest…

Holomorphic Factorization for a Quantum Tetrahedron

- Mathematics
- 2010

We provide a holomorphic description of the Hilbert space $${\mathcal{H}_{j_1,\ldots,j_n}}$$ of SU(2)-invariant tensors (intertwiners) and establish a holomorphically factorized formula for the…