# Cosmological Constant from Condensation of Defect Excitations

## Abstract

**:**

## 1. Introduction

**Outline of the paper:**In the next Section 2 we will give a definition of the Turaev-Viro (TV) model and a related family of models. We will then follow up with a Hamiltonian description in Section 3. We will first introduce the string net framework that defines a physical Hilbert space associated to the TV model. We then define a kinematical Hilbert space by introducing punctures that can carry defect excitations and thus allow for a dynamics different from the TV model. The kinematical Hilbert spaces still carry a $\mathrm{k}$-dependent structure, but we will define an embedding of the $\tilde{\mathrm{k}}$-Hilbert space into the $\mathrm{k}$-Hilbert space for $\tilde{\mathrm{k}}<\mathrm{k}$. Using the simplest example, namely the two-punctured sphere, we then give a detailed description of the framework in Section 4. In Section 5 we will consider general triangulations and show how to construct the various Hamiltonians, ribbon operators and $\mathrm{k}$- and $\tilde{\mathrm{k}}$-vacua in this case. This will also allow us to analyze the $\mathrm{k}$-vacua in terms of the $\mathrm{k}$ excitations. We close with a discussion and outlook in Section 6. The Appendix A, Appendix B and Appendix C collect various more technical background material, in particular some essential definitions related to $\mathrm{SU}{\left(2\right)}_{\mathrm{k}}$.

## 2. The Turaev-Viro Partition Function and Related Models

## 3. Hamiltonian Description

#### 3.1. Physical Hilbert Space

- The strands of a graph can be deformed isotopically within $\mathsf{\Sigma}$:
- Strands with representation $j=0$ can be added or removed:
- The local connectivity of a graph can be changed by 2–2 (also referred to as F-move) and 3–1 moves and their inverses:

- We will allow for the crossing of two strands but have to denote which strand is over-crossing and which is under-crossing. Certain types of double crossings can be resolved by isotopic deformation of the strands, e.g.,
- We furthermore define a short-hand notation for a certain weighted sum over strands labelled by admissible irreps. We will refer to this combination as a vacuum strand and it is defined as

#### 3.2. Kinematical Hilbert Space Via Introduction of Punctures

#### Embedding the Kinematical $\tilde{\mathrm{k}}$-Hilbert Space Into the Kinematical $\mathrm{k}$-Hilbert Space

## 4. Example: The Two-Punctured Sphere

**Remark.**

## 5. General Triangulations

#### 5.1. Local Considerations

#### 5.2. Ribbon Operators

#### 5.3. Global Solutions

## 6. Discussion

**Quantum groups in quantum gravity:**As mentioned in the introduction, it would be highly interesting to derive a quantum group structure, and in particular the braiding in a quantum group, by imposing the dynamics of 3D gravity with a cosmological constant. To this end, one uses a first order formulation of 3D gravity with triads e and a ($\mathfrak{su}\left(2\right)$ valued) spin connection A. The constraints are then given by the curvature constraints $F-\mathsf{\Lambda}e\wedge e=0$ where F is the curvature of the spin connection. One also has the Gauß constraint (or no-torsion condition) ${d}_{A}e=0$. The curvature constraints are relatively simple to impose for $\mathsf{\Lambda}=0$ as in this case the constraints only involve the connection and one can work in a connection polarization. For non-vanishing $\mathsf{\Lambda}$ however the constraints involve canonically conjugated variables. The constraints can be however rewritten into $C={\textstyle \frac{1}{2}}({F}_{+}+{F}_{-})$ and $G={\textstyle \frac{1}{2}}({F}_{+}-{F}_{-})$ for the curvature and Gauß constraints respectively. This uses the curvatures ${F}_{\pm}$ of two Poincare connections ${A}_{\pm}\left(\mathsf{\Lambda}\right)=A\pm \sqrt{\mathsf{\Lambda}}e$. The price to pay is that the components of the redefined connections become non-commutative.

**Quantum cosmology:**One strategy for deriving a cosmological dynamics from quantum gravity is to consider states which are in a certain sense homogeneous. This approach has been taken in loop quantum cosmology [83,84,85,86,87] based on a notion of homogeneous connection [88,89,90]. Although there are a number of works which made this notion more precise in the quantum realm [91,92,93,94,95,96], the construction of loop quantum cosmology does rather start with a symmetry reduced classical phase space and does not consider a notion of quantum homogeneous states inside the full theory. A first step in this direction was taken in [97,98], where the homogeneity is based on conditions on the triads. The eigenvalues of the triads characterize the size of a fundamental cell and this can be seen as a discretization ambiguity, see also the discussion in [99,100]. An alternative framework is group field theory cosmology [101,102,103]. Group field theory is based on similar constructions as loop quantum gravity [104,105] but in a certain sense includes a sum over discretizations.

**Condensed matter:**Being able to separate kinematics and dynamics we can define $\tilde{\mathrm{k}}$-Hamiltonians ${\mathbf{H}}_{\tilde{\mathrm{k}}}$ whose ground states are given by the $\tilde{\mathrm{k}}$-vacua. We can thus consider linear combinations of two such Hamiltonians $\mathbf{H}\left(\alpha \right)=\alpha {\mathbf{H}}_{\mathrm{k}}+(1-\alpha ){\mathbf{H}}_{\tilde{\mathrm{k}}}$. Varying the coupling constant $\alpha $ we will obtain quantum phase transitions between the two phases characterized by $\tilde{\mathrm{k}}$ and $\mathrm{k}$. That is we expect to see that in the limit of large lattices, the gap between the energies for the ground state(s) and the first excited state(s) vanishes for a certain $\alpha $. A phase transition has to occur as the degeneracy for non-trivial topologies changes if we change the level, e.g., for the torus we have a degeneracy ${(\mathrm{k}+1)}^{2}$ and ${(\tilde{\mathrm{k}}+1)}^{2}$ respectively. We have furthermore order parameters—e.g., the $\mathrm{k}$ or $\tilde{\mathrm{k}}$ Wilson loops—which indicate such transitions by taking on a vanishing value in one phase and a non-vanishing value in the other phase.

**Lattice gauge theory and tensor networks:**The kinematical Hilbert space we construct can be also interpreted as a Hilbert space for $(2+1)$-dimensional lattice gauge theory, where the structure group is q-deformed to $\mathrm{SU}{\left(2\right)}_{\mathrm{k}}$. The $\tilde{\mathrm{k}}$ vacua can in this sense also be understood as phases of lattice gauge theory, which are characterized by non-vanishing expectations values of the curvature.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Essentials on SU (2) k

## Appendix B. Diagonalization of a Sum of Two Projectors onto One–Dimensional Subspaces

## Appendix C. Transformation for a Tetrahedral Lattice

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**Table 1.**Probabilities for (k = 3) anyon excitations (a,a)in the ($\tilde{\mathrm{k}}$ = 2)-vacuum state.

$a$ | $a=\frac{1}{2}$ | $a$ | $a=\frac{3}{2}$ |
---|---|---|---|

0.599 | 0.383 | 0.017 | 0.001 |

**Table 2.**Probabilities to find an (k = 3) anyon ($a,{a}^{\prime}$) for the fusion product of two plaquettes in the ($\tilde{\mathrm{k}}$ = 2)-vacuum state.

$a\u29f9{a}^{\prime}$ | 0 | $\frac{1}{2}$ | 1 | $\frac{3}{2}$ |
---|---|---|---|---|

0 | 0.524 | 0 | 0.0365 | 0 |

$\frac{1}{2}$ | 0 | 0.378 | 0 | 0.00261 |

1 | 0.0365 | 0 | 0.0189 | 0 |

$\frac{3}{2}$ | 0 | 0.00261 | 0 | 0.000693 |

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Dittrich, B.
Cosmological Constant from Condensation of Defect Excitations. *Universe* **2018**, *4*, 81.
https://doi.org/10.3390/universe4070081

**AMA Style**

Dittrich B.
Cosmological Constant from Condensation of Defect Excitations. *Universe*. 2018; 4(7):81.
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**Chicago/Turabian Style**

Dittrich, Bianca.
2018. "Cosmological Constant from Condensation of Defect Excitations" *Universe* 4, no. 7: 81.
https://doi.org/10.3390/universe4070081