^{1,a)}and Etera R. Livine

^{1,2,b)}

### Abstract

We describe fundamental equations which define the topological ground states in the lattice realization of the BF phase. We introduce a new scalar Hamiltonian, based on recent works in quantum gravity and topological models, which is different from the plaquette operator. Its gauge-theoretical content at the classical level is formulated in terms of spinors. The quantization is performed with Schwinger's bosonic operators on the links of the lattice. In the spin network basis, the quantum Hamiltonian yields a difference equation based on the spin 1/2. In the simplest case, it is identified as a recursion on Wigner 6j-symbols. We also study it in different coherent states representations, and compare with other equations which capture some aspects of this topological phase.

The authors also thank Laurent Freidel, who shared with us his ideas to extract the spin 1/2 recursions, which were basically the same as those we have used. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

I. INTRODUCTION

A. A practical guide through the equations of the topological BF phase

B. A new Hamiltonian for the magnetic part

II. INTRODUCTION TO THE BF PHASE

III. KINEMATICS OF LATTICE SU(2) GAUGE THEORY

A. The lattice spinor variables

B. Quantization: Intertwiners and Schwinger's boson operators

C. Graspings with spin 1/2 and holonomy operator

IV. DYNAMICS OF BF THEORY

A. The plaquette operator: Tent move evolution

B. The spin 1 Hamiltonian

C. The spin 1/2 Hamiltonian

1. The classical spinor Hamiltonian

2. The quantum spinor Hamiltonian

D. Geometric interpretation: Quantum Euclidean geometry

E. Equivalence with the flatness constraint

V. NEW REPRESENTATIONS OF THE TOPOLOGICAL EQUATION

A. The recursion formula in the coherent SU(2) basis

B. The Schwinger's generating function for 6j-symbols

VI. COMPARISONS WITH OTHER EQUATIONS FOR THE BF PHASE

A. Back to the plaquette operator, with spinors

B. Different types of recursions and differential equations

C. Generalization of the new scalar Hamiltonians to arbitrary lattices with 3-valent vertices

VII. CONCLUSION

### Key Topics

- SU2 group
- 67.0
- Holonomy
- 46.0
- Quantum optics
- 26.0
- Gauge field theory
- 21.0
- Lattice gauge theory
- 21.0

## Figures

The tetrahedral graph we consider throughout the paper. The orientations are the same as those of the graphical representation of the 6j-symbol.^{51} The three fat lines form the cycle (126) we will consider in order to explain the action of the new Hamiltonian.

The tetrahedral graph we consider throughout the paper. The orientations are the same as those of the graphical representation of the 6j-symbol.^{51} The three fat lines form the cycle (126) we will consider in order to explain the action of the new Hamiltonian.

The 3-valent nodes represent 3jm-symbols, where legs carry the spins. A magnetic index is summed when there is a link joining two nodes. The action of is a grasping between *e* _{1} and *e* _{2}, denoted by a dashed line which carries the spin 1/2. The final result is proportional to the 3jm-symbol with the spins .

The 3-valent nodes represent 3jm-symbols, where legs carry the spins. A magnetic index is summed when there is a link joining two nodes. The action of is a grasping between *e* _{1} and *e* _{2}, denoted by a dashed line which carries the spin 1/2. The final result is proportional to the 3jm-symbol with the spins .

A pictorial representation of (55). The character χ_{ j } along the closed loop acts on the left. On the right, we have depicted the situation after re-coupling. A specific 6j-symbol is extracted on each node, and one has to sum over the colorings *k* _{1}, … , *k* _{ n }. The dashed lines correspond to the dual 2D triangulation to the plaquette if we think of the latter as embedded in flat 3-space. The vertex *s* of the 2D triangulation is then dual to the plaquette.

A pictorial representation of (55). The character χ_{ j } along the closed loop acts on the left. On the right, we have depicted the situation after re-coupling. A specific 6j-symbol is extracted on each node, and one has to sum over the colorings *k* _{1}, … , *k* _{ n }. The dashed lines correspond to the dual 2D triangulation to the plaquette if we think of the latter as embedded in flat 3-space. The vertex *s* of the 2D triangulation is then dual to the plaquette.

Here, we have displayed the geometric interpretation of the character operator on the plaquette as a tent move. The vertex *s* is evolved to a new vertex *s* ^{′}, with an edge of length , the tent pole. Between the initial and the final triangulations we have a piece of 3D triangulation. The character operator then generates the evaluation of the Ponzano-Regge amplitude on this triangulation.

Here, we have displayed the geometric interpretation of the character operator on the plaquette as a tent move. The vertex *s* is evolved to a new vertex *s* ^{′}, with an edge of length , the tent pole. Between the initial and the final triangulations we have a piece of 3D triangulation. The character operator then generates the evaluation of the Ponzano-Regge amplitude on this triangulation.

A graphical representation of the action of the new Hamiltonian. The basic idea is that the holonomy around a closed loop in the topological sector only depends on its homotopy type, so that we can deform the grasping on the left to that on the right, picking up this way some holonomy which must be trivial.

A graphical representation of the action of the new Hamiltonian. The basic idea is that the holonomy around a closed loop in the topological sector only depends on its homotopy type, so that we can deform the grasping on the left to that on the right, picking up this way some holonomy which must be trivial.

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