Local observables in ${\mathrm{SU}}_q(2)$ lattice gauge theory

We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\mathrm{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities that live on the vertices of the lattice and are labeled by pairs of incident edges. Any function on the classical phase space, e.g., Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$ matrix, which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$ deformation of ${\mathfrak{so}}^{*}(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself.

Figure 1

The ribbon graph associated to the ribbon constraint. The ribbon carries two pairs of variables $(\ell ,u)$ and $(\tilde{\ell },\tilde{u})$. The ribbon constraint is the trivialization of the ribbon loop $\ell u{\tilde{\ell }}^{-1}{\tilde{u}}^{-1}$.

Figure 2

A piece of a graph $\mathrm{\Gamma }$ on the left and the corresponding piece of the ribbon graph ${\mathrm{\Gamma }}_{\mathrm{rib}}$ on the right.

Figure 3

The conjugate ribbon structure defined in terms of upper triangular matrices.

Figure 4

SU(2)-covariant spinors in the ribbon picture. Note that we can replace $\ell$ and $\tilde{\ell }$, respectively, by ${\ell }^{-1†}$ and ${\tilde{\ell }}^{-1†}$.

Figure 5

A node with three incident edges $e_1$, $e_2$, $e_3$ (in gray) and the correspondent ribbon graph. The four possible orientations for $e_1$ and $e_2$ with a fix orientation $o_3=-1$ for $e_3$ are illustrated separately. The spinors defining the scalar product $E_{12}^{{\epsilon }_1,{\epsilon }_2}$ can be read at the common vertex (in red) of the ribbons associated to $e_1$ and $e_2$.

Figure 6

The reference ribbon. The spinor operators ${\mathbf{t}}^{\epsilon }$ and ${\tilde{\mathbf{t}}}^{\epsilon }$ are ${\mathcal{U}}_q(\mathfrak{s}\mathfrak{u}(2))$ quantum spinors, while ${\mathit{\tau }}^{\epsilon }$ and ${\tilde{\mathit{\tau }}}^{\epsilon }$ are ${\mathcal{U}}_{q^{-1}}(\mathfrak{s}\mathfrak{u}(2))$ quantum spinors.

Figure 7

Flipping the reference ribbon due to the change of orientation of the edge $e$ is equivalent to the spinor flip $\tilde{\mathit{\tau }}\to \mathit{\tau },\tilde{\mathbf{t}}\to \mathbf{t}$.

Figure 8

The choice of cilium is given by the red bullet. In (a), the orientation is anticlockwise, while in (b), the orientation is clockwise. This choice matters since we usually order the tensor product from left to right.

Figure 9

The choice of cilium, the right end point of ribbon 1, is given by the red bullet. In each case, we transport the relevant spinor living on the right end point of ribbon 2 using the holonomy in ribbon 1. We recover the same spinor in each case as in the unflipped case.