Relationship between classical $q$-deformed General Relativity and the cosmological constant

Question

The frame-connection formulation of pure General Relativity in 4 dimensions is given by the action $$ S_{4d}[e, \omega] = \frac{1}{2 \kappa} \int \varepsilon_{IJKL} e^I \wedge e^J \wedge F^{KL}, $$ where $e^I = e^I_{\mu} dx^{\mu}$ is the frame field, $\omega^I_{\;J} = \omega^I_{\;J\;\mu} dx^{\mu}$ is the spin connection and $F^I_{\;J} = d\omega^I_{\;J} + \omega^{IK} \wedge \omega_{KJ}$ is its curvature tensor. In my notation, $I, J, \dots$ are internal flat space indices and $\mu, \nu, \dots$ are coordinate indices on the space-time manifold, though I use differential forms whenever possible for bookkeeping.

The analogous action in 3 dimensions takes an even simpler form $$ S_{3d}[e, \omega] = \frac{1}{2 \kappa} \int \varepsilon_{IJK} e^I \wedge F^{JK}. $$ This is a special case of the topological $BF$ theory action for the group $SL(2,\mathbb{R})$, the non-compact form of $SU(2)$. In Euclidean signature, the group is just the compact $SU(2)$.

A well known toy model of quantum gravity in $3d$ is the Ponzano-Regge model, which is also the result of applying the covariant Loop Quantum Gravity programme in Euclidean signature in $3d$.

The Ponzano-Regge model has a peculiar "twin" called the Turaev-Viro model defined for $q \in \mathbb{C}$ a root of unity. The definition mirrors the definition of the Ponzano-Regge model that makes use of the representation theory of $SU(2)$, except it employs the representation theory of the corresponding $q$-deformed Hopf algebra $SU_q (2)$.

It has many interesting features, like finiteness of its amplitudes (in fact, Turaev-Viro is frequently used to regularize the formal infinite Ponzano-Regge amplitudes), triangulation independence (which essentially means Turaev-Viro is a TQFT, even though it uses triangulations in its definition). But probably the most unexpected consequence is that Turaev-Viro turns out to give $3d$ General Relativity with a non-zero cosmological constant $\Lambda$ that depends on $q$, and vanishes in the $q \rightarrow 1$ limit when Turaev-Viro becomes Ponzano-Regge.

This relationship between $q$-deformed Lie groups (aka quantum groups, but I try to avoid this terminology, because physicists have a very different notion of "quantum" which is not to be confused) and the cosmological constant fascinates me.

I am curious if it also exists on a purely classical level. To be more precise, my hypothesis is that a similar action with the gauge group $SL(2,\mathbb{R})$ or $SU(2)$ replaced by the corresponding $q$-deformed Hopf algebra, whatever that means, somehow automatically contains a cosmological constant term without it being explicitly present in the theory.

Similarly, one may also hope that a similar mysterious relationship between $q$-deformed groups and the cosmological constant exists in $4d$.

I'm looking for references that pursued this direction.

Answer 1

In 4D case, if you q-deform the Lorentz group (or more precisely $Spin(1,3)$) $$SO(1,3)$$ to arrive at the de Sitter group $$SO_q(1,3) = SO(1,4),$$ then the curvature $$ F^I_{\;J} = d\omega^I_{\;J} + \omega^{IK} \wedge \omega_{KJ} $$ will be q-deformed to (given that the q-deformed Lorentz connection has two parts $\omega^I_{\;J} + qe^I$) $$ F^I_{\;Jq} = (d\omega^I_{\;J} + \omega^{IK} \wedge \omega_{KJ}) + q^2(e^I\wedge e_J). $$ And the Lagrangian $$ S_{4d}[e, \omega] = \frac{1}{2 \kappa} \int \varepsilon_{IJKL} e^I \wedge e^J \wedge F^{KL} $$ will be q-deformed to $$ S_{4dq}[e, \omega] = \frac{1}{2 \kappa} \int \varepsilon_{IJKL} e^I \wedge e^J \wedge F^{KL}_q. $$ Therefore the $q^2$ part will contribute a Lagrangian term $$\frac{q^2}{2 \kappa} \int \varepsilon_{IJKL} e^I \wedge e^J \wedge e^K \wedge e^L, $$ which amounts to a cosmological constant.

BTW, there is also a torsion part of the q-deformed curvature too $$ F^I_{q} = q(de^I + \omega^{IJ} \wedge e_J), $$ which will not contribute to the q-deformed Lagrangian.