# Group field theory on quantum groups

Florian Girelli<EMAIL_ADDRESS>Department of Applied
Mathematics, University of Waterloo, 200 University Avenue West, Waterloo,
Ontario, Canada, N2L 3G1 Matteo Laudonio<EMAIL_ADDRESS>Univ.
Bordeaux, LABRI, 351 Cours de la Libération, 33400 Talence, France  Department
of Applied Mathematics, University of Waterloo, 200 University Avenue West,
Waterloo, Ontario, Canada, N2L 3G1

###### Abstract

We introduce the framework of Hopf algebra field theory (HAFT) which
generalizes the notion of group field theory to the quantum group (Hopf
algebra) case. We focus in particular on the 3d case and show how the HAFT we
considered is topological. The highlight of the construction is the notion of
plane-wave which leads, in the specific example of $\mathrm{SU}_{q}(2)$ with
$q$ real, to a discretization of the Euclidian $BF$ action with a negative
cosmological constant. This work can be viewed as the generalization of the
seminal work by Baratin and Oriti [1]: we have a (non-commutative) formulation
of simplicial 3d gravity in the presence of a non-zero cosmological constant.

###### Contents

  1. 1 Introduction
  2. 2 Conventions and notations
    1. 2.1 Hopf algebra notations
    2. 2.2 Quantum double
    3. 2.3 Integral and Fourier transform
  3. 3 Hopf algebra field theory
    1. 3.1 Field and closure constraint
    2. 3.2 Action
    3. 3.3 Feynman diagram amplitude
    4. 3.4 Topological invariance
  4. 4 Example: $q$-deformed group field theory
    1. 4.1 $q$-deformed group field theory
      1. 4.1.1 Building blocks
      2. 4.1.2 Plane-wave
      3. 4.1.3 Closure constraint and Feynman diagram amplitude
      4. 4.1.4 Relation with Turaev-Viro model (with $q$ real)
    2. 4.2 Plane-wave and discretization of the $BF$ action
      1. 4.2.1 $BF$ action and its limits
      2. 4.2.2 Discretized $BF$ actions
      3. 4.2.3 Recovering the discretized $BF$ actions from the plane-wave
  5. 5 Conclusion
  6. A Further mathematical details
    1. A.1 Hopf algebra proofs
    2. A.2 Geometric computations
  7. B Further details on the Hopf algebra field theory
    1. B.1 Gauge projector
    2. B.2 Tetrahedron amplitude
    3. B.3 Feynman diagram amplitude
    4. B.4 Topological invariance
  8. C Derivation of skew symmetric bi-algebras
  9. D Further examples
    1. D.1 Finite Groups
    2. D.2 $\mathrm{SU}(2)$ group field theory

##  1 Introduction

The Turaev-Viro-Barrett-Westbury (TV) model is a topological state-sum model
defined in terms of the category of representations of a (weak) Hopf
algebra/quantum group (or even a spherical category [2, 3, 4]). Given a 3d
manifold ${\cal M}$, it consists in decorating the edges of a triangulation of
${\cal M}$ in terms of the representation labels. The key ingredient is the
(q-deformed) $6j$-symbol. From a field theory perspective, it is related to
the Chern-Simons theory or 3d gravity with a non-zero cosmological constant.
Indeed, it can be seen as providing a discretization of the amplitude of the
$BF$ action (with gauge symmetry given by the Lie algebra $\mathfrak{g}$) with
a cosmological constant.

${\cal Z}_{BF}=\int[\mathrm{d}B][\mathrm{d}A]\,e^{i\int_{{\cal
M}}\,B^{I}\wedge F_{I}+\frac{\Lambda}{6}\epsilon_{IJK}B^{I}\wedge B^{J}\wedge
B^{K}}\,,$ (1)

where $B$ is a 1-form with value in the Lie algebra $\mathfrak{g}^{*}$, $A$ is
the 1-form connection with value in the Lie algebra111The Lie algebras
$\mathfrak{g}$ and $\mathfrak{g}^{*}$ are dual to each other, and equipped
with a co-cycle structure inherit form the classical Drinfeld double.
$\mathfrak{g}$ and $F(A)$ is the associated curvature. As a 3d quantum gravity
model, the TV model is a spinfoam and it provides the full information about
the 3d quantum gravity regime (with no matter fields, but particles, as
topological defects can be introduced).

The TV model has no divergence when one deals with, for example, the $q$ root
of unity deformation of $\mathrm{SU}(2)$ (a quasi-Hopf algebra), which
corresponds in the (3d) gravity language to the Euclidian case with a positive
cosmological constant. In the other cases of signature and sign of the
cosmological constant, it is divergent, just as the Ponzano-Regge (PR)
model[5, 6]. The PR model corresponds to the state sum model based on the
representations of the gauge group of gravity with a zero cosmological
constant.

The appearance of a quantum group/deformed symmetry structure can be derived
from the Hamiltonian formulation of Chern-Simons or gravity, through some
discretization procedure of the fields [7, 8, 9]. In particular from the
gravity picture, one has to choose the proper variables to actually have the
gauge transformations depending on the cosmological constant, which fits the
quantum theory defined in terms a quantum group symmetry as gauge symmetry
[9], with the deformation parameter depending on the cosmological constant.

One can also derive the PR model from a direct discretization of the amplitude
(1), with $\Lambda=0$. Indeed, when the cosmological constant is zero, this
amplitude can be essentially seen as a plane-wave and therefore as a Dirac
delta function implementing that the curvature has to be zero (the $B$ field
can be seen as some kind of Lagrange multiplier to implement the zero
curvature condition).

$\displaystyle{\cal Z}^{\Lambda=0}_{BF}$
$\displaystyle=\int[dB][dA]\,e^{i\int_{{\cal M}}\,B^{I}\wedge
F_{I}}\sim\int[dA]\,\delta(F)$ (2) $\displaystyle\rightarrow\,\,\,{\cal
Z}^{\Lambda=0}_{d}=\int[dX][dg]\,\prod_{e}\,e^{i\langle
X_{e}\,,\,g_{e}\rangle}=\int[dg]\,\prod_{e}\,\delta(g_{e})\,,$ (3)

where $X_{e}\in\mathbb{R}^{3}$ is associated to the edge $e$ of a
triangulation and $g_{e}$ is the closed holonomy on the loop spanning the face
$e^{*}$ dual to the edge $e$. The notation $\langle X_{e}\,,\,g_{e}\rangle$
means that we consider a specific coordinate choice $k(g_{e})$ for $g_{e}$
(bearing in mind that multiple coordinate patches are typically needed) so
that

$e^{i\langle X_{e}\,,\,g_{e}\rangle}\equiv e^{iX^{a}k_{a}(g_{e})}\,,$ (4)

and we recover indeed a plane-wave. We note that as a function of $X$, the
plane-wave is a non-commutative function with star product $\star$ (which
depends on the choice of coordinates), since we must have

$e^{i\langle X_{e}\,,\,g_{e}\rangle}=e^{i\langle X_{e}\,,\,\prod_{l\in\partial
e^{*}}\,g_{l}\rangle}=\star_{l}\,\sigma_{l}\,\,,\quad\sigma_{l\in\partial
e^{*}}\equiv e^{i\langle X_{e}\,,\,g_{l}\rangle}\,,$ (5)

where the $g_{l}$ is the holonomy associated to the link $l$ of the dual
complex [1, 10, 11]. No analogue of such discretization really exists when the
cosmological constant is not zero. Asymptotic analysis of the TV amplitude
shows that one can recover the Regge action with a cosmological constant [12].
One can nevertheless wonder whether a similar derivation as the PR model could
exist for the TV model and in particular whether there would exist a
generalized notion of plane-wave which would be used to encode a discretized
$BF$ action. One should notice that the $BF$ amplitude with a non-zero
cosmological constant should be seen as some modified plane-wave. Indeed,
similarly to (2), we have

$\int\mathrm{d}^{3}B\,e^{i\int\,\big{(}B\cdot
F+\frac{\Lambda}{6}\,B\cdot(B\times
B)\big{)}}=\frac{2}{|\Lambda|}\,K_{0}\Big{(}\sqrt{\frac{4\,F\cdot(F\times
F)}{\Lambda}}\Big{)}\,,$ (6)

where $K_{0}(x)$ is the modified Bessel function of the second kind, which can
be seen as a smearing of the Dirac delta function.

We intend to show here that even in the TV case, there is a notion of plane-
wave which gives a discretized version of the $BF$ action with a non-zero
cosmological constant. Importantly, the recovered action is actually not the
standard one given in (1) since it is not defined in terms of the fields which
give rise to the quantum group structure upon discretization [9].

The framework of this construction will be the group field theory (GFT)
approach [13, 14], which we will upgrade to a general Hopf algebra context.
The reason is that group field theories have a configuration and a momentum
formulation which correspond to formulating the model in the triangulation
picture or in its dual complex. The two descriptions are related by a Fourier
transform and hence in terms of a notion of plane-wave. Such plane-wave will
appear as the building block to characterize the discretized $BF$ action
action with or without cosmological constant.

The group field theory approach on top of conveniently generating the
different triangulations (or dual complex) as Feynman diagrams, emphasizes the
fact that a pair of dual Hopf algebras are actually at play, one decorating
the triangulation, the other one decorating the dual complex. The pair of Hopf
algebras forms the symmetries of the system, namely the (generalized) Drinfeld
double and the duality is encoded in the canonical element of $H\otimes A$
[15]. On a practical manner, it is already known that when the cosmological
constant is zero, we recover a formulation of simplicial gravity with the
metric degrees of freedom, decorating the edges of the triangulation, which
are non-commutative [1]. Introducing an homogeneous curvature will deform
further the metric degrees of freedom and will render the decorations of the
dual complex non-commutative.

We will provide the general framework to construct group field theories for
any type of Hopf algebra $H$. The combinatorics we chose will focus on
generating 3d triangulations, but it can be generalized for any dimension. Our
construction emphasizes that the main structure at play is actually not the
Hopf algebra $H$, but rather its generalized Drinfeld double $H\bowtie A$. Our
work shares some similarities in the formulation with Krasnov’s construction
which also sets up the GFT in the Hopf algebra language to introduce matter
(or topological excitations) [16].

In section 2, we introduce the notations and conventions we will use
extensively in the paper. We introduce in particular the notion Fourier
transform and hence the notion of plane-wave that we will use to characterise
the duality between the Hopf algebras which constitute the Drinfeld double.

In section 3, we introduce the main ingredients of a Hopf algebra field theory
(HAFT), which generalizes the notion of group field theory. We consider a
specific action and show that the Feynman diagrams can be related by the
Pachner moves, so that their amplitude are actually topological invariants.

In section 4, we construct the main example, namely the case of the Drinfeld
double of $\mathrm{SU}_{q}(2)$, with $q$ real. We recall how the HAFT Feynman
diagrams in this specific example can be related to TV model, as Boulatov
originally discussed. We then discuss how the plane-wave we have can be
related to the discretization of the $BF$ action with a cosmological constant.

We have several appendices which contain the proofs of many of the
propositions given in the main text. We have also provided two more examples,
the case of the finite group, which is relevant to discuss spinfoams with
finite groups [17] and the undeformed $\mathrm{SU}(2)$ case.

##  2 Conventions and notations

In this section we recall the key Hopf algebra properties that we need and
introduce the relevant notations and conventions. The expert reader can skip
this section.

###  2.1 Hopf algebra notations

We note as usual $\Delta$ the co-product, $S$ the antipode, $\varepsilon$ the
co-unit, $m$ (or sometimes for short $\cdot$) the multiplication, $\eta$ (or,
for simplicity, $1$) the unit and $\tau$ the permutation map. Use use the
symbol $\circ$ for the composition of maps and $\textrm{id}^{n}$ for the
tensor product of $n$ identity maps. A bi-algebra $(A,m,\eta,\Delta,\epsilon)$
over the field $\mathbb{K}$ is a vector space over $\mathbb{K}$ which is both
an algebra and a co-algebra in a compatible way. An Hopf algebra
$(A,m,\eta,\Delta,\epsilon,S)$ over $\mathbb{K}$ is a bi-algebra over
$\mathbb{K}$ equipped with an antipode $S$.
Given the Hopf algebra $A$, the co-unit and antipode satisfy the axioms

$\displaystyle(\textrm{id}\otimes\varepsilon)\circ\Delta=(\varepsilon\otimes\textrm{id})\circ\Delta=\textrm{id}\,,$
(7) $\displaystyle
m\circ(S\otimes\textrm{id})\circ\Delta=m\circ(\textrm{id}\otimes
S)\circ\Delta=\eta\,\varepsilon\,.$ (8)

Given the Hopf algebra $A$, we denote $A^{n}=\bigotimes_{i}^{n}A_{i}$ its
associated tensor product and consider the following maps.

$\displaystyle\Delta^{n}$ $\displaystyle:A\rightarrow A^{n}$
$\displaystyle\text{with}\quad\Delta^{n}\equiv(\textrm{id}\otimes\dots\otimes\textrm{id}\otimes\Delta)\circ\dots\circ\Delta\,,\quad
n\geq 2$ (9) $\displaystyle m^{n}$ $\displaystyle:A^{n}\rightarrow A$
$\displaystyle\text{with}\quad m^{n}\equiv
m\circ\dots\circ(\textrm{id}\otimes\dots\otimes\textrm{id}\otimes m)\,,\quad
n\geq 2$ (10) $\displaystyle\tau^{n}$ $\displaystyle:A^{n}\rightarrow A^{n}$
$\displaystyle\text{with}\quad\tau^{n}(a_{1}\otimes\dots\otimes
a_{n})=a_{n}\otimes\dots\otimes a_{1}$ (11) $\displaystyle\varepsilon^{n}$
$\displaystyle:A^{n}\rightarrow\mathbb{K}$ (12) $\displaystyle\eta^{n}$
$\displaystyle:\mathbb{K}\rightarrow A^{n}$ (13)

We note that, since the co-product (resp. the multiplication) is co-
associative (resp. associative), the position of the co-product in (9) (resp.
the multiplication in (10)) does not matter. Furthermore, the tensor product
$A^{n}$ is in turn an Hopf algebra with associated maps

$\displaystyle\Delta^{(n)}$ $\displaystyle:A^{n}\rightarrow A^{n}\otimes
A^{n}$ $\displaystyle\text{with}\quad\Delta^{(n)}(a_{1}\otimes\dots\otimes
a_{n})=({a_{1}}_{(1)}\otimes\dots
a_{n}{}_{(1)})\otimes({a_{1}}_{(2)}\otimes\dots\otimes{a_{n}}_{(2)})$ (14)
$\displaystyle m^{(n)}$ $\displaystyle:A^{n}\otimes A^{n}\rightarrow A^{n}$
$\displaystyle\text{with}\quad m^{(n)}\big{(}(a_{1}\otimes\dots\otimes
a_{n})\otimes(b_{1}\otimes\dots\otimes b_{n})\big{)}=m(a_{1}\otimes
b_{1})\otimes\dots\otimes m(a_{n}\otimes b_{n})$ (15)
$\displaystyle\tau^{(n)}$ $\displaystyle:A^{n}\otimes A^{n}\rightarrow
A^{n}\otimes A^{n}$
$\displaystyle\text{with}\quad\tau^{(n)}\big{(}(a_{1}\otimes\dots\otimes
a_{n})\otimes(b_{1}\otimes\dots\otimes b_{n})\big{)}=(b_{1}\otimes\dots\otimes
b_{n})\otimes(a_{1}\otimes\dots\otimes a_{n})$ (16) $\displaystyle S^{(n)}$
$\displaystyle:A^{n}\rightarrow A^{n}$ $\displaystyle\text{with}\quad
S^{(n)}(a_{1}\otimes\dots\otimes a_{n})=(Sa_{1}\otimes\dots\otimes Sa_{n})$
(17)

Here we used the Sweedler notation for the co-product: $\Delta a=\sum
a_{(1)}\otimes a_{(2)}$. Given a tensor product of Hopf-algebras, we will
later use the notation

$\displaystyle m_{ij}(1\otimes\dots\otimes a_{1}\otimes\dots\otimes
a_{2}\otimes\dots\otimes 1)\equiv 1\otimes\dots\otimes m(a_{1}\otimes
a_{2})\otimes\dots\otimes 1\otimes\dots\otimes 1\,,$ (18)
$\displaystyle\Delta_{ij}a\equiv 1\otimes\dots\otimes
a_{(1)}\otimes\dots\otimes a_{(2)}\otimes\dots\otimes 1\,,\qquad a\in A\,.$
(19)

The map $m_{ij}$ stands for a product of two elements embedded in the tensor
product of Hopf algebras, respectively living in the $i^{th}$ and $j^{th}$
places, and the result lives in the $i^{th}$ space, for $i<j$. The map
$\Delta_{ij}$ stands for usual co-product embedded in a tensor product of Hopf
algebras and its two components live respectively in the $i^{th}$ and $j^{th}$
copies of the algebra. We use a minus sign as notation to imply the antipode
on one of the tensor spaces, for instance

$\Delta_{-i\,j}a\equiv 1\otimes\dots\otimes Sa_{(1)}\otimes\dots\otimes
a_{(2)}\otimes\dots\otimes 1\,,\qquad a\in A\,.$ (20)

These notations can be generalized to the $n$-dimensional product (10) and co-
product (9), which we denote $m^{n}_{i_{1}\dots i_{n}}$ and
$\Delta^{n}_{i_{1}\dots i_{n}}$. Similarly, they can be generalized to the
$n$-dimensional product (15) and co-product (14), which we denote
$m^{(n)}_{i_{1}\dots i_{n}\,\,\,j_{1}\dots j_{n}}$ and
$\Delta^{(n)}_{i_{1}\dots i_{n}\,\,\,j_{1}\dots j_{n}}$.

######  Definition 1 (Simplex maps).

Consider the maps

$\underline{m}:A^{12}\to A^{6}\,\,,\quad\underline{\Delta}:A^{6}\to
A^{12}\,\,,\quad\Delta_{{\cal V}}:A^{6}\to A^{16}\,,$ (21)

that stand for a combination of products or co-products that have the same
pattern of a tetrahedron. The maps are defined as

$\displaystyle\underline{m}(a_{1}\otimes\cdots\otimes a_{12})\equiv\,$
$\displaystyle(a_{1}\cdot a_{9})\otimes(a_{2}\cdot a_{12})\otimes(a_{3}\cdot
a_{4})\otimes(a_{5}\cdot a_{11})\otimes(a_{6}\cdot a_{7})\otimes(a_{8}\cdot
a_{10})\,,$ (22) $\displaystyle\underline{\Delta}(a_{1}\otimes\cdots\otimes
a_{6})\equiv\,$
$\displaystyle\Delta_{1\,9}a_{1}\otimes\Delta_{2\,12}a_{2}\otimes\Delta_{3\,4}a_{3}\otimes\Delta_{5\,11}a_{4}\otimes\Delta_{6\,7}a_{5}\otimes\Delta_{8\,10}a_{6}\,,$
(23) $\displaystyle\Delta_{{\cal V}}(a_{1}\otimes\cdots\otimes a_{6})\equiv\,$
$\displaystyle\Delta^{4}_{-16\,4\,-1\,6}\,a_{1}\cdot\Delta^{4}_{-15\,4\,2\,9}\,a_{2}\cdot\Delta^{4}_{-14\,4\,-3\,12}\,a_{3}\cdot\Delta^{4}_{-11\,3\,2\,10}\,a_{4}\cdot\Delta^{4}_{-13\,3\,-1\,5}\,a_{5}\cdot\Delta^{4}_{-8\,-2\,-1\,7}\,a_{6}\,,$
(24)

where $\Delta_{ij}$ (and similarly its generalization
$\Delta^{4}_{i_{1}\,i_{2}\,i_{3}\,i_{4}}$) is given in (19).

###  2.2 Quantum double

######  Definition 2 (Matched pair bi-algebras).

[18] Two bi-algebras $H$ and $A$ form a matched pair if there exist a pair of
actions

$\triangleright:A\otimes H\to H\,\,,\quad\triangleleft:A\otimes H\to A$ (25)

that satisfy the compatibility relations

$\displaystyle(ab)\triangleleft h=(a\triangleleft(b_{(1)}\triangleright
h_{(1)}))(b_{(2)}\triangleleft h_{(2)})\,\,,\qquad 1\triangleleft
h=\varepsilon(h)\,,$ (26) $\displaystyle
a\triangleright(hg)=(a_{(1)}\triangleright h_{(1)})((a_{(2)}\triangleleft
h_{(2)})\triangleright g)\,\,,\qquad a\triangleright 1=\varepsilon(a)\,,$
$\displaystyle\qquad a_{(1)}\triangleleft h_{(1)}\otimes a_{(2)}\triangleright
h_{(2)}=a_{(2)}\triangleleft h_{(2)}\otimes a_{(1)}\triangleright h_{(1)}\,.$

The following definition introduces a key-object of our construction.

######  Definition 3 (Skew paired bi-algebras.).

[18] Two bi-algebras $H$ and $A$ are skew paired if there exists a map called
skew pairing $\sigma:A\otimes H\to\mathbb{K}$ such that

$\displaystyle\sigma(a\cdot b\otimes h)=\sigma(a\otimes
h_{(1)})\,\sigma(b\otimes h_{(2)})\,,$ (27) $\displaystyle\sigma(a\otimes
h\cdot g)=\sigma(a_{(1)}\otimes g)\,\sigma(a_{(2)}\otimes h)\,,$

and

$\sigma(1\otimes h)=\varepsilon(h)\,\,,\quad\sigma(a\otimes
1)=\varepsilon(a)\,,$ (28)

for all $a,b\in A$ and $h,g\in H$. We have used the symbol $1$ as a shorthand
to encode the unit in both the bi-algebras $H$ and $A$. If either $A$ has an
antipode or $H$ has an inverse-antipode, then there exists a ”convolution
inverse” $\sigma^{-1}$ that satisfies [15]

$\sigma^{-1}(a\otimes h)\equiv\sigma(Sa\otimes h)=\sigma(a\otimes S^{-1}h)\,.$
(29)

######  Definition 4 (Generalized quantum double).

[18] Let $H$ and $A$ skew co-paired bi-algebras with skew pairing $\sigma$
which is convolution-invertible. The generalized quantum double is the double
cross product bi-algebra $D(H,A,\sigma)\equiv H\bowtie A$ built on $H\otimes
A$, equipped with mutual actions

$\displaystyle a\triangleleft h=a_{(2)}\,\sigma^{-1}(a_{(1)}\otimes
h_{(1)})\,\sigma(a_{(3)}\otimes h_{(2)})\,,$ (30) $\displaystyle
a\triangleright h=h_{(2)}\,\sigma^{-1}(a_{(1)}\otimes
h_{(1)})\,\sigma(a_{(2)}\otimes h_{(3)})\,,$

that make the bi-algebras $H$ and $A$ a matched pair. The generalized quantum
double $D(H,A,\sigma)$ is a bi-algebra with product

$(h\otimes a)\cdot(g\otimes b)=h(a_{(1)}\triangleright
g_{(1)})\otimes(a_{(2)}\triangleleft g_{(2)})b=hg_{(2)}\otimes
a_{(2)}b\,\sigma^{-1}(a_{(1)}\otimes g_{(1)})\,\sigma(a_{(3)}\otimes
g_{(3)})\,,$ (31)

co-product $\Delta=(\Delta_{H}\otimes\Delta_{A})$, tensor product, unit and
co-unit.

######  Definition 5 (Matched co-pair bi-algebras).

[18] Two bi-algebras $H$ and $A$ form a matched co-pair if there exist a pair
of co-actions

$\alpha:H\rightarrow H\otimes A\,\,,\quad\beta:A\rightarrow H\otimes A$ (32)

that satisfy the compatibility relations

$\displaystyle(\Delta\otimes\textrm{id})\circ\alpha(h)=((\textrm{id}\otimes\beta)\circ\alpha(h_{(1)}))(1\otimes\alpha(h_{(2)}))\,,$
(33)
$\displaystyle(\textrm{id}\otimes\Delta)\circ\beta(a)=((\beta(a_{(1)})\otimes
1)((\alpha\otimes\textrm{id})\circ\beta(a_{(2)}))\,,$
$\displaystyle\qquad\qquad\qquad\alpha(h)\beta(a)=\beta(a)\alpha(h)\,.$

######  Definition 6 (Skew co-paired bi-algebras.).

[18] Two bi-algebras $H$ and $A$ are skew co-paired if there exists an element
called skew co-pairing $\sigma\in H\otimes A$ such that222In (34), (35), (37)
and (38), we used for clarity the notation $\Delta_{H}$ and $\Delta_{A}$ to
encode the co-products on $H$ and $A$. In the following, for simplicity, we
will drop the indices $H$ and $A$.

$\displaystyle(\textrm{id}\otimes\Delta_{A})\,\sigma=\sigma_{13}\sigma_{12}\,,$
(34)
$\displaystyle(\Delta_{H}\otimes\textrm{id})\,\sigma=\sigma_{13}\sigma_{23}\,.$
(35)

If either $A$ admits an antipode or $H$ admits an inverse-antipode, then the
skew co-pairing is invertible, with inverse

$\sigma^{-1}\equiv(\textrm{id}\otimes
S)\,\sigma=(S^{-1}\otimes\textrm{id})\,\sigma\,,$ (36)

that satisfies the axioms

$\displaystyle(\textrm{id}\otimes\Delta_{A})\,\sigma^{-1}=\sigma^{-1}_{12}\sigma^{-1}_{13}\,,$
(37)
$\displaystyle(\Delta_{H}\otimes\textrm{id})\,\sigma^{-1}=\sigma^{-1}_{23}\sigma^{-1}_{13}\,.$
(38)

Using the co-unit axiom (7) and the co-pairing properties (34) and (35), one
derives the identities

$(\varepsilon\otimes\textrm{id})\,\sigma=(\textrm{id}\otimes\varepsilon)\,\sigma=1\,,$
(39)

that we call co-unit properties of the skew co-pairing element.

######  Definition 7 (Dual of the generalized quantum double).

[18] Let $H$ and $A$ skew co-paired bi-algebras with invertible skew co-
pairing. The dual of the generalized quantum double is the double cross co-
product bi-algebra
$D^{*}(A,H,\sigma)=A\mathrel{\scalerel*{\blacktriangleright\joinrel\blacktriangleleft}{x}}H$
built on $A\otimes H$, equipped with mutual co-actions $\alpha:H\to H\otimes
A$ and $\beta:A\to H\otimes A$ given by

$\displaystyle\alpha(h)=\sigma^{-1}(h\otimes 1)\sigma\,,$ (40)
$\displaystyle\beta(a)=\sigma^{-1}(1\otimes a)\sigma\,,$

that make the bi-algebras $H$ and $A$ a matched co-pair. The dual of the
generalized quantum double $D^{*}(A,H,\sigma)$ is a bi-algebra with co-product

$\Delta(a\otimes
h)=\big{(}(\textrm{id}\otimes\alpha\otimes\textrm{id})\circ(1\otimes\Delta_{H}h)\big{)}\cdot\big{(}(\textrm{id}\otimes\beta\otimes\textrm{id})\circ(\Delta_{A}a\otimes
1)\big{)}=\sigma_{23}^{-1}(\Delta_{A}a\otimes\Delta_{H}h)\sigma_{23}\,,$ (41)

product $(a\otimes h)\cdot(b\otimes g)=ab\otimes hg$, tensor product, unit and
co-unit.

We will later need to extend this construction in the context of tensor
product of bi-algebras. We consider two tensor product bi-algebras
$H^{n}=\bigotimes_{i}^{n}H_{i}$ and $A^{n}=\bigotimes_{i}^{n}A_{i}$ such that
the sub bi-algebras $H_{i}$ and $A_{i}$ are skew paired (resp. skew co-
paired), that is for each pair $H_{i},A_{i}$ we have a map
$\sigma_{i}:A_{i}\otimes H_{i}\rightarrow\mathbb{K}$ (resp. an element
$\sigma_{i}\in H_{i}\otimes A_{i}$). Then the tensor product bi-algebras
$H^{n},A^{n}$ are skew paired by the map

$\Sigma^{n}:A^{n}\otimes H^{n}\rightarrow\mathbb{K}$ (42)

or skew co-paired by the element

$\Sigma^{n}\in H^{n}\otimes A^{n}\,.$ (43)

In particular, since the bi-algebras $H^{n}$ and $A^{n}$ are tensor products
of independent bi-algebras, the $n$ dimensional skew co-pairing can be written
in the tensor product notation as

$\Sigma^{n}=\sigma_{1\,n+1}\,\sigma_{2\,n+2}\,\cdots\,\sigma_{n\,2n}\,.$ (44)

######  Proposition 1 (Properties of the multidimensional skew co-pairing
element).

Let $H^{n}$ and $A^{n}$ be skew co-paired tensor product bi-algebras with
invertible skew co-pairing $\Sigma^{n}$ and convolution inverse
$\Sigma^{n\,-1}$, the identities below are satisfied.

$\begin{aligned} &(m^{n}\otimes\textrm{id}^{\otimes
n})\,\Sigma^{n}=(\textrm{id}\otimes\tau^{n}\circ\Delta^{n})\,\sigma\,,\\\
&(m^{n}\otimes\textrm{id}^{\otimes
n})\,\Sigma^{n\,-1}=(\textrm{id}\otimes\Delta^{n})\,\sigma^{-1}\,,\end{aligned}\qquad\begin{aligned}
&(\textrm{id}^{\otimes n}\otimes
m^{n})\,\Sigma^{n}=(\Delta^{n}\otimes\textrm{id})\,\sigma\,,\\\
&(\textrm{id}^{\otimes n}\otimes
m^{n})\,\Sigma^{n\,-1}=(\tau^{n}\circ\Delta^{n}\otimes\textrm{id})\,\sigma^{-1}\,.\end{aligned}$
(45)

The proof of this proposition follows directly from the definition of
$\sigma$.

###  2.3 Integral and Fourier transform

In this section we recall the definition of the Fourier transform between Hopf
algebras. Hence we work at a purely algebraic level, we are not concerned
about analytical issues, which would matter if we want to analyze the possible
divergences of the theory.
First we define the notion of integral in the Hopf algebra setting. This
notion differs from the usual notion of integral one uses in calculus. This
latter is actually what will be called the co-integral.

######  Definition 8 (Integral and co-integral).

[15] A left (resp. right) integral in $A$ is an element $\ell_{L}$, (resp.
$\ell_{R}$) in $A$ such that

$a\cdot\ell_{L}=\varepsilon(a)\ell_{L}\,\,,\quad\ell_{R}\cdot
a=\varepsilon(a)\ell_{R}\,,\qquad\forall a\in A\,.$ (46)

The integral $\ell$ is normalized if $\varepsilon(\ell)=1$.
A left (resp. right) co-integral on $A$ is a map
$\int_{A}^{L}:A\rightarrow\mathbb{K}$ (resp.
$\int_{A}^{R}:A\rightarrow\mathbb{K}$) that satisfies the left (resp. right)
invariance condition

$\left(\text{id}\otimes\int_{A}^{L}\right)\,\Delta
a=1\otimes\int_{A}^{L}a\,\,,\qquad\left(\int_{A}^{R}\otimes\text{id}\right)\,\Delta
a=\int_{A}^{R}a\otimes 1\,,\qquad\forall a\in A\,.$ (47)

The co-integral $\int_{A}$ is normalized if $\int_{A}\,1=1$.

In order to keep track of divergences, in the following we consider non-
normalized co-integrals on $H$ and $A$

$\int_{H}\,1\equiv V_{H}\,\,,\quad\int_{A}\,1\equiv V_{A}\,.$ (48)

Using $\sigma$, we can now define the notion of Fourier transform and its
inverse.

######  Definition 9 (Fourier transform).

Let $H$ and $A$ be skew co-paired bi-algebras. The Fourier transform from $H$
to $A$ is a map $\mathcal{F}:H\rightarrow A$ defined as333Note that we used
left co-integrals for both the Fourier transform (49) and its inverse (50). As
an alternative convention, we could have used right co-integrals. In this case
the inverse skew co-pairing element $\sigma^{-1}$ would appear in the
definition (49) and the skew co-pairing element $\sigma$ in (50).

$\mathcal{F}[h]\equiv\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\,\textrm{id}\right)\,\big{(}\sigma\cdot(h\otimes
1)\big{)}\,,$ (49)

with inverse map $\mathcal{F}^{-1}:A\rightarrow H$

$\mathcal{F}^{-1}[a]\equiv\frac{1}{\sqrt{\mu}}\left(\textrm{id}\otimes\int_{A}^{L}\right)\,\big{(}\sigma^{-1}\cdot(1\otimes
a)\big{)}\,,$ (50)

with $h\in H$, $a\in A$, and normalization factor

$\mu\equiv\left(\int_{H}^{L}\otimes\int_{A}^{L}\right)\,\sigma=\left(\int_{H}^{L}\otimes\int_{A}^{L}\right)\,\sigma^{-1}\,.$
(51)

######  Proposition 2.

The Fourier transform (49) and the inverse Fourier transform (50) are inverse
maps in the sense that

$(\mathcal{F}\circ\mathcal{F}^{-1})=(\mathcal{F}^{-1}\circ\mathcal{F})=\textrm{id}\,.$
(52)

The analogue of the Dirac delta function will be a very important object in
our construction since it allows to identify the decorations of the geometric
structures.

######  Definition 10 (Delta function).

Let $H$ and $A$ be skew co-paired bi-algebras. The Dirac delta functions of
$H$ (noted $\delta_{H}$) and $A$ (noted $\hat{\delta}_{A}$), are respectively
defined as the Fourier transform and the inverse Fourier transform of the
unit:

$\hat{\delta}_{A}\equiv{\cal
F}[1]=\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\,\textrm{id}\right)\,\sigma\,\,,\quad\delta_{H}\equiv{\cal
F}^{-1}[1]=\frac{1}{\sqrt{\mu}}\left(\textrm{id}\otimes\int_{A}^{L}\right)\,\sigma^{-1}\,.$
(53)

Similarly, we call opposite delta functions the elements

$\hat{\delta}_{A}^{-1}=\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\,\textrm{id}\right)\,\sigma^{-1}\,\,,\quad\delta_{H}^{-1}=\frac{1}{\sqrt{\mu}}\left(\textrm{id}\otimes\int_{A}^{L}\right)\,\sigma\,.$
(54)

######  Proposition 3 (Properties of the delta function).

Let $H$ and $A$ be skew co-paired Hopf algebras. The delta functions satisfy
the identities below.

$\begin{aligned} &(a\otimes 1)\cdot\Delta\hat{\delta}_{A}=(1\otimes
Sa)\cdot\Delta\hat{\delta}_{A}\,,\\\ &(h\otimes
1)\cdot\Delta\delta_{H}=(1\otimes
S^{-1}h)\cdot\Delta\delta_{H}\,,\end{aligned}\qquad\quad\begin{aligned}
&\Delta\hat{\delta}_{A}^{-1}\cdot(a\otimes
1)=\Delta\hat{\delta}_{A}^{-1}\cdot(1\otimes Sa)\,,\\\
&\Delta\delta_{H}^{-1}\cdot(h\otimes 1)=\Delta\delta_{H}^{-1}\cdot(1\otimes
S^{-1}h)\,.\end{aligned}$ (55)

Moreover, the delta functions are normalized in the sense that

$\frac{1}{\sqrt{\mu}}\int_{A}^{L}\,\hat{\delta}_{A}=\frac{1}{\sqrt{\mu}}\int_{A}^{L}\,\hat{\delta}_{A}^{-1}=1\,\,,\quad\frac{1}{\sqrt{\mu}}\int_{H}^{L}\,\delta_{H}=\frac{1}{\sqrt{\mu}}\int_{H}^{L}\,\delta_{H}^{-1}=1\,.$
(56)

######  Proposition 4 (Delta function as integral in the Hopf algebra).

Consider the skew co-paired Hopf algebras $H,A$. The delta functions (53) of
the Hopf algebras $H$ and $A$ are resp. left integrals of $A$ and $H$:

$a\cdot\hat{\delta}_{A}=\hat{\delta}_{A}\,\varepsilon(a)\,\,,\quad
h\cdot\delta_{H}=\delta_{H}\,\varepsilon(h)\,.$ (57)

The opposite delta functions (54) are resp. right integrals of $A$ and $H$:

$\hat{\delta}_{A}^{-1}\cdot
a=\hat{\delta}_{A}^{-1}\,\varepsilon(a)\,\,,\quad\delta_{H}^{-1}\cdot
h=\delta_{H}^{-1}\,\varepsilon(h)\,.$ (58)

For the sake of clarity, here we specified whether the co-integral are left or
right by using the indices $L,R$. In the following we will omit such index,
implying that all the co-integrals are left invariant ones.
Since the skew co-pairing element $\sigma$ is the kernel of the Fourier
transform, we will call it plane wave.
In App. A we give some proofs of the propositions and statements given in this
section.

##  3 Hopf algebra field theory

In this section we describe what we will call a Hopf algebra field theory
(HAFT). This is a field theory based on Hopf algebras which can be seen as a
generalization of ordinary group field theories [19, 20]. We provide an action
specified by a kinetic and an interaction term, and we will derive the
amplitude of a general cellular decomposition, expanded as a ”sum” over the
building blocks of a triangulation (or its dual complex).
We call three dimensional HAFT the field theory which Feynman diagrams can be
seen as dual of a three dimensional triangulation. This will be the focus of
our discussion. It can be generalized in a direct manner to any dimensions.

HAFT is meant to be a model about discrete geometries. Hence, throughout its
construction, we will be driven by our geometric intuition. In particular, we
consider a three dimensional triangulation and its dual complex. We use
elements of the Hopf algebra $A$ to decorate objects in the triangulation and
elements of $H$ to decorate objects in the dual complex. This is analogue to
what happens for the Kitaev model defined in terms of Drinfeld doubles [21,
22], which is using a quadrangulation that is then self-dual.

We introduce the nomenclature node and links resp. for 0$d$ and 1$d$ objects
living in the dual complex, while vertices and edges are resp. 0$d$ and 1$d$
objects in the triangulation. In 3d nodes and links are respectively dual to
3- and 2-simplices (tetrahedra and triangles).

###  3.1 Field and closure constraint

We note $\Phi$ the field which is an element in the tensor product of three
copies of $H$, $\Phi\in H^{3}$. On the other hand, $\hat{\Phi}$ is the dual
field and is an element in three copies of $A$, $\hat{\Phi}\in A^{3}$. The
fields $\Phi$ and $\hat{\Phi}$ are related by a Fourier transform (49) and its
inverse (50), with kernel $\Sigma$.

Geometrically, the dual field $\hat{\Phi}\in A^{3}$ is associated to a
triangle (2-simplex), where each of its sub-components (elements of $A$)
decorates one of the edges that compose its boundary. Dually, the field
$\Phi\in H^{3}$ is associated to the graph dual to such triangle and its sub-
components (elements of $H$) are the links, that share a single node, dual to
the edges.
The fields $\Phi$ and $\hat{\Phi}$ are represented in Fig. 1. In this context,
the skew co-pairing $\sigma$ encodes the information on both the triangulation
and the dual complex.

$\hat{\phi}$$\phi$ Figure 1: The dual field $\hat{\Phi}$ is represented in
black as a triangle; the field $\Phi$ is its dual graph represented in blue,
made of a central node and three links.

We impose that the field $\Phi$ should be invariant under some gauge symmetry,
which we enforce through a projector.

######  Definition 11 (Gauge projector).

Consider the (left444We may alternatively introduce a right projector as
$({\cal P}_{R}\,\Phi)=\frac{1}{V_{H}}\left(\textrm{id}^{\otimes
3}\otimes\int_{H}\right)\,\big{(}(\textrm{id}^{\otimes 3}\otimes
m^{3})\circ\Delta^{(3)}\Phi\big{)}\,,$ (59) defined through a right co-
integral.) projector ${\cal P}_{L}:H^{3}\to H^{3}$ whose action on the field
is called gauge averaging

$({\cal P}_{L}\,\Phi)=\frac{1}{V_{H}}\left(\int_{H}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}(m^{3}\otimes\textrm{id}^{\otimes
3})\circ\Delta^{(3)}\Phi\big{)}\,.$ (60)

We recall that $\int_{H}$ is the left co-integral on $H$, while $m^{3}$ and
$\Delta^{(3)}$ are the maps resp. defined in (10) and (14).

In App. B.1 we prove that the operator ${\cal P}_{L}$ defined above is a
projector. The field $\Phi$ we defined is invariant under gauge averaging:

$({\cal P}_{L}\,\Phi)=\Phi\,.$ (61)

The element $(m^{3}\otimes\textrm{id}^{\otimes 3})\circ\Delta^{(3)}\,\Phi$ in
(60) belongs to the Hopf algebra $H^{4}$: this can be understood as the graph
dual to the triangle (represented by the field) plus an extra link. Such extra
link is interpreted as a parallel transport in the dual complex, and (61)
enforces the invariance of the field under any possible translation of this
type. In analogy with ordinary group field theory, we call it gauge symmetry
and (61) encodes the gauge invariance of the field. We represent the gauge
symmetry in Fig. 2.

$\phi$ Figure 2: The gauge symmetry is interpreted as a translation in the
dual complex. As the field $\Phi$ is given by three links dual to a triangle
(in blue), the (gauge) translation is an extra variable (the dashed blue
line).

Dually (upon Fourier transform), the gauge symmetry translates into the
closure constraint.

######  Proposition 5 (Closure constraint).

The Fourier transform of the gauged projected field (60) gives

${\cal F}[({\cal P}_{L}\,\Phi)]=\hat{{\cal C}}\cdot\hat{\Phi}\,,$ (62)

where the element $\hat{{\cal C}}\in A^{3}$ is

$\hat{{\cal C}}=\frac{\sqrt{\mu}}{V_{H}}\,\Delta^{3}\hat{\delta}_{A}^{-1}\,.$
(63)

###### Proof.

We show here that the Fourier transform of the projected field (60) gives the
closure constraint (63). The Fourier transform of the gauge projected field
(60) is

$\displaystyle{\cal F}[({\cal P}_{L}\,\Phi)]$
$\displaystyle=\frac{1}{\sqrt{\mu^{3}}}\left(\int_{H^{3}}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}\Sigma\cdot(({\cal P}_{L}\,\Phi)\otimes 1)$
$\displaystyle=\frac{1}{\sqrt{\mu^{3}}\,V_{H}}\left(\int_{H}\otimes\int_{H^{3}}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}(m^{3}\otimes\textrm{id}^{\otimes
3}\otimes\textrm{id}^{\otimes 3})\,\Sigma_{23}\cdot(\Delta^{(3)}\Phi\otimes
1)\big{)}$
$\displaystyle=\frac{1}{\sqrt{\mu^{3}}\,V_{H}}\left(\int_{H}\otimes\int_{H^{3}}\otimes\textrm{id}^{\otimes
3}\right)\,(m^{3}\otimes\textrm{id}^{\otimes 3}\otimes\textrm{id}^{\otimes
3})\,\big{(}\Sigma_{13}^{-1}\cdot\Sigma_{13}\cdot\Sigma_{23}\cdot(\Delta^{(3)}\Phi\otimes
1)\big{)}\,.$ (64)

Then, using the property (35) for the multi dimensional skew co-pairing
element555We will often simply use $\Sigma$ as a short hand notion for
$\Sigma^{n}$ in order to not clutter the notations. $\Sigma$ and left
invariance of the co-integral on $H$, the expression reduces to

$\displaystyle{\cal F}[({\cal P}_{L}\,\Phi)]$
$\displaystyle=\frac{1}{\sqrt{\mu^{3}}\,V_{H}}\left(\int_{H}\otimes\int_{H^{3}}\otimes\textrm{id}^{\otimes
3}\right)\,(m^{3}\otimes\textrm{id}^{\otimes 3}\otimes\textrm{id}^{\otimes
3})\,\big{(}\Sigma_{13}^{-1}\cdot(\Delta^{(3)}\otimes\textrm{id}^{\otimes
3})\,(\Sigma\cdot(\Phi\otimes 1)\big{)}$
$\displaystyle=\frac{1}{\sqrt{\mu^{3}}\,V_{H}}\left(\int_{H}\otimes\int_{H^{3}}\otimes\textrm{id}^{\otimes
3}\right)\,(m^{3}\otimes\textrm{id}^{\otimes 3}\otimes\textrm{id}^{\otimes
3})\,\big{(}\Sigma_{13}^{-1}\cdot\Sigma_{23}\cdot(1\otimes\Phi\otimes
1)\big{)}\,.$ (65)

Last, using the property (45) for the three dimensional convolution inverse
$\Sigma^{-1}$, together with the definition of Fourier transform and opposite
delta function, we get the expression of the closure constraint (63).

$\displaystyle{\cal F}[({\cal P}_{L}\,\Phi)]$
$\displaystyle=\frac{1}{\sqrt{\mu^{3}}\,V_{H}}\left(\int_{H}\otimes\int_{H^{3}}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}(\textrm{id}\otimes\textrm{id}^{\otimes
3}\otimes\Delta^{3})\,\sigma^{-1}\cdot\Sigma_{23}\cdot(1\otimes\Phi\otimes
1)\big{)}$
$\displaystyle=\frac{1}{V_{H}}\left(\int_{H}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}(\textrm{id}\otimes\Delta^{3})\,\sigma^{-1}\cdot(1\otimes\hat{\Phi})\big{)}=\frac{\sqrt{\mu}}{V_{H}}\,\Delta^{3}\hat{\delta}_{A}^{-1}\cdot\hat{\Phi}\,.$
(66)

∎

Performing a Fourier transformation of the gauge invariance condition (61)
gives the identity

$(\hat{{\cal C}}\cdot\hat{\Phi})=\hat{\Phi}\,.$ (67)

$\hat{{\cal C}}$ is called the closure constraint as it encodes the closure of
the triangle [10]. Indeed, as an element of the tensor product algebra
$A^{3}$, the closure constraint is interpreted as the combination of three
edges in the triangulation. Being given as the co-product of the delta
function, these edges are naturally interpreted as part of a discrete closed
path. Therefore, (62) implements the closure of the boundary (the three sub-
components) of the (triangle) dual field $\hat{\Phi}$.

Note that we could also demand to implement a symmetry under permutations of
the arguments $H$ in $H^{3}$. We will not impose this and follow instead [23]
where it was shown that in 3d, if the proper interaction term is chosen, there
is no need to use such permutation symmetry.

###  3.2 Action

The HAFT action is composed by a kinetic plus an interaction term (with
coupling constant set to $1$ for simplicity)

${\cal S}={\cal S}_{{\cal K}}+{\cal S}_{{\cal V}}\,.$ (68)

The interaction term is defined as the product of fields with the
combinatorics of the tetrahedron.

######  Definition 12 (Interaction term).

The interaction term of three dimensional HAFT is

${\cal S}_{{\cal V}}=\int_{H^{6}}\,\big{(}\underline{m}\,({\cal
P}_{L}\,\Phi\otimes{\cal P}_{L}\,\Phi\otimes{\cal P}_{L}\,\Phi\otimes{\cal
P}_{L}\,\Phi)\big{)}\,.$ (69)

The map $\underline{m}$ was introduced in Def. 1 and it is explicitly given in
(22).

Going to the dual picture, namely the triangulation picture, allows a better
interpretation of such interaction term.

######  Proposition 6 (Tetrahedron).

The interaction term (69) can be expressed as an integral operator in the two
forms below

${\cal S}_{{\cal V}}=\int_{H^{12}}\,\big{(}{\cal
V}\cdot(\Phi\otimes\Phi\otimes\Phi\otimes\Phi)\big{)}=\int_{A^{12}}\,\big{(}\hat{{\cal
V}}\cdot(\hat{\Phi}\otimes\hat{\Phi}\otimes\hat{\Phi}\otimes\hat{\Phi})\big{)}\,,$
(70)

where

$\displaystyle{\cal V}$
$\displaystyle=\frac{1}{\mu^{6}\,V_{H}^{4}}\left(\int_{H^{4}}\otimes\textrm{id}^{12}\right)\,\Delta_{{\cal
V}}\delta_{H}^{-1}\,,$ (71) $\displaystyle\hat{{\cal V}}$
$\displaystyle=\frac{1}{\mu^{6}}\,\underline{\Delta}\hat{\delta}_{A}^{-1}\,\cdot\,(\hat{{\cal
C}}\otimes\hat{{\cal C}}\otimes\hat{{\cal C}}\otimes\hat{{\cal C}})\,,$

are the kernels of the tetrahedron amplitudes, resp. in the $H$ and $A$
representations. The co-product $\Delta_{{\cal V}}$ was introduced in Def. 1,
moreover the opposite delta functions $\delta_{H}^{-1}$ and
$\hat{\delta}_{A}^{-1}$ in both the amplitudes (71) are six dimensional delta
functions.

Due to the length and complexity of the proof of Prop. 6, we provide it in
App. B.2.
Each dual field is associated to a triangle, by construction, the interaction
term represents the composition of four triangles, with the combinatorics of a
tetrahedron specified by the co-product (24). The associated amplitude is thus
interpreted as the boundary of a tetrahedron or the bulk and the boundary of
its dual graph and it is illustrated in Fig. 3(b). In analogy with ordinary
field theory, the tetrahedron amplitude encodes the smallest information (we
call it building block) of any Feynman diagram of Hopf algebra field theory.
As a consequence, the HAFT Feynman diagrams are (dual to) three dimensional
triangulations.

(a) The propagator amplitude enforces the identification of a pair of
triangles (in black) in the $A$ representation or their dual graphs (in blue)
in the $H$ representation. We used arrows to emphasize that edges are
identified with opposite orientations and links with the same orientation.
This is reflected by the antipode in (74).

(b) The tetrahedron amplitude is given by the combination of four closure
constraints (in black) representing the triangles that compose the boundary of
a tetrahedron. The graph dual to the bulk of the tetrahedron is in red and the
graph dual to its boundary is in blue. The graph dual to a tetrahedron is
given as a combination of six loops made of two bulk links (solid red) and two
boundary links (solid blue). Each of these loops is given by one of the six
co-products in $\Delta_{{\cal V}}$.

Figure 3: Representation of the propagator and vertex/tetrahedron amplitude.

######  Definition 13 (Kinetic term).

The kinetic term of three dimensional Hopf algebra field theory is

${\cal S}_{{\cal K}}=\int_{H^{3}}\,\Phi\cdot(\tau\Phi)\,.$ (72)

The associated amplitude, called propagator, thus represents the
identification of the dual graph of two triangles. As in ordinary in ordinary
field theories, the propagator amplitudes are used to glue pairs of amplitudes
associated to the interaction term. In this case, two tetrahedra are merged by
identifying two triangles (or the associated boundary graph). We represent
such identification in Fig. 3(a). Such composition allows to construct
arbitrary simplicial three dimensional triangulations or dual complexes (with
or without boundaries).
In the triangulation picture, the propagator consists in identifying the edges
of the triangles.

######  Proposition 7 (Propagator).

The kinetic term (72) can be expressed as an integral operator in two
different ways,

${\cal S}_{{\cal
K}}=\left(\int_{H^{3}}\otimes\int_{H^{3}}\right)\,\big{(}{\cal
K}\cdot(\Phi\otimes\Phi)\big{)}=\left(\int_{A^{3}}\otimes\int_{A^{3}}\right)\,\big{(}\hat{{\cal
K}}\cdot(\hat{\Phi}\otimes\hat{\Phi})\big{)}\,,$ (73)

where

${\cal
K}=\frac{1}{\sqrt{\mu^{3}}}\,(S^{(3)}\otimes\tau^{3})\circ\Delta^{(3)}\delta_{H}\,\,,\quad\hat{{\cal
K}}=\frac{1}{\sqrt{\mu^{3}}}\,(\textrm{id}^{\otimes
3}\otimes\tau^{3})\circ\Delta^{(3)}\hat{\delta}_{A}^{-1}\,,$ (74)

are the kernels of the propagator amplitudes, resp. in the $H$ and $A$
representations.

###### Proof.

We show that the kinetic term (72) can be expressed as in (73) with kernels
(74). Let us first focus on the kernel in the $A$ representation. We use the
definition of the inverse Fourier transform, the property (37) and the
definition of opposite delta function.

$\displaystyle{\cal S}_{{\cal K}}$
$\displaystyle=\int_{H^{3}}\,\Phi\cdot(\tau^{3}\Phi)=\frac{1}{\mu^{3}}\left(\int_{H^{3}}\otimes\int_{A^{3}}\otimes\int_{A^{3}}\right)\,\big{(}\Sigma_{12}^{-1}\cdot(\textrm{id}\otimes\tau^{3})\Sigma_{13}^{-1}\cdot(1\otimes\hat{\Phi}\otimes\hat{\Phi})\big{)}$
$\displaystyle=\frac{1}{\mu^{3}}\left(\int_{H^{3}}\otimes\int_{A^{3}}\otimes\int_{A^{3}}\right)\,\big{(}(\textrm{id}\otimes\textrm{id}\otimes\tau^{3})\circ(\textrm{id}\otimes\Delta^{(3)})\Sigma^{-1}\cdot(1\otimes\hat{\Phi}\otimes\hat{\Phi})\big{)}$
$\displaystyle=\frac{1}{\sqrt{\mu^{3}}}\left(\int_{A^{3}}\otimes\int_{A^{3}}\right)\,\big{(}(\textrm{id}\otimes\tau^{3})\circ\Delta^{(3)}\hat{\delta}_{A}^{-1}\cdot(\hat{\Phi}\otimes\hat{\Phi})\big{)}\,.$
(75)

From this, we get the expression of $\hat{{\cal K}}$ in (74). Now, to derive
the kernel ${\cal K}$ in the $H$ representation, use the definition of the
Fourier transform (49) for both the fields $\hat{\Phi}$, and the property (55)
of the delta function.

$\displaystyle{\cal S}_{{\cal K}}$
$\displaystyle=\frac{1}{\sqrt{\mu^{6}}}\left(\int_{H^{6}}\otimes\int_{A^{6}}\right)\,\big{(}(1\otimes
1\otimes(\textrm{id}\otimes\tau^{3})\circ\Delta^{(3)}\hat{\delta}_{A}^{-1})\cdot\Sigma_{13}\cdot\Sigma_{24}\cdot(\Phi\otimes\Phi\otimes
1\otimes 1)\big{)}$
$\displaystyle=\frac{1}{\sqrt{\mu^{6}}}\left(\int_{H^{6}}\otimes\int_{A^{6}}\right)\,(\textrm{id}\otimes\textrm{id}\otimes\textrm{id}\otimes\tau^{3})\big{(}(1\otimes
1\otimes\Delta^{(3)}\hat{\delta}_{A}^{-1})\cdot\Sigma_{13}\cdot(\textrm{id}\otimes\tau^{3}\otimes\textrm{id}\otimes\textrm{id})\,\Sigma_{24}\cdot(\Phi\otimes\Phi\otimes
1\otimes 1)\big{)}$
$\displaystyle=\frac{1}{\sqrt{\mu^{6}}}\left(\int_{H^{6}}\otimes\int_{A^{6}}\right)\,\big{(}(1\otimes
1\otimes(\textrm{id}\otimes\tau^{3})\circ\Delta^{(3)}\hat{\delta}_{A}^{-1})\cdot(\textrm{id}\otimes\tau^{3}\otimes\textrm{id}\otimes\textrm{id})\,\Sigma_{23}^{-1}\cdot\Sigma_{13}\cdot(\Phi\otimes\Phi\otimes
1\otimes 1)\big{)}\,.$ (76)

We get rid of the delta function by using the left invariance of the co-
integral on $A$, plus the normalization property (56).

$\displaystyle{\cal S}_{{\cal K}}$
$\displaystyle=\frac{1}{\sqrt{\mu^{6}}}\left(\int_{H^{6}}\otimes\int_{A^{6}}\right)\,\big{(}(1\otimes
1\otimes
1\otimes\tau^{3}\hat{\delta}_{A}^{-1})\cdot(\textrm{id}\otimes\tau^{3}\otimes\textrm{id}\otimes\textrm{id})\Sigma_{23}^{-1}\cdot\Sigma_{13}\cdot(\Phi\otimes\Phi\otimes
1\otimes 1)\big{)}$
$\displaystyle=\frac{1}{\mu^{3}}\left(\int_{H^{6}}\otimes\int_{A^{3}}\right)\,\big{(}(\textrm{id}\otimes\tau^{3}\otimes\textrm{id}\otimes\textrm{id})\,\Sigma_{23}^{-1}\cdot\Sigma_{13}\cdot(\Phi\otimes\Phi\otimes
1)\big{)}\,.$ (77)

Last, using the definition of convolution inverse, its property (34) and the
definition of delta function we derive the expression of the kernel ${\cal K}$
in (74).

$\displaystyle{\cal S}_{{\cal K}}$
$\displaystyle=\frac{1}{\mu^{3}}\left(\int_{H^{3}}\otimes\int_{H^{3}}\otimes\int_{A^{3}}\right)\,\big{(}(S^{(3)}\otimes\tau^{3}\otimes\textrm{id})\circ(\Delta^{(3)}\otimes\textrm{id})\,\Sigma^{-1}\cdot(\Phi\otimes\Phi\otimes
1)\big{)}$
$\displaystyle=\frac{1}{\sqrt{\mu^{3}}}\left(\int_{H^{3}}\otimes\int_{H^{3}}\right)\,\big{(}(S^{(3)}\otimes\tau^{3})\circ\Delta^{(3)}\delta_{H}\cdot(\Phi\otimes\Phi)\big{)}\,.$
(78)

∎

###  3.3 Feynman diagram amplitude

The HAFT Feynman diagrams represent decorated three dimensional triangulations
or their dual complexes. The strength of GFT’s or HAFT’s is that their Feynman
diagrams can be seen as the triangulations (or their dual complexes) of a
manifold ${\cal M}$. The interaction term is chosen to encode the $d$-simplex
if the manifold ${\cal M}$ has dimension $d$. As argued before we focused here
on the case $d=3$. The Feynman diagrams amplitudes of the HAFT in the $A$
polarization defined earlier can be expanded in terms of diagrams $\Gamma$
which correspond to the triangulations of manifolds. In particular, the
amplitude of a triangulation is expressed as a combination of tetrahedron
amplitudes (71) glued by the proper propagator amplitudes (74).

Dually, in the $H$ polarization, the Feynman diagrams $\Gamma^{*}$ correspond
to the dual complexes of the triangulation $\Gamma$. In particular, the
amplitude of a general dual complex can be expressed as a combination of loops
made of an arbitrary number $N$ of bulk links (part of the bulk graph of a
tetrahedron). Each of such loops is the closed path spanning a face dual to an
edge (shared by a number $N$ of tetrahedra) of the triangulation.
In line with the usual interpretation of curvature in models of discrete
geometries, each of these loops can be interpreted as probing the local
curvature around this edge.

We intend now to give the expression of the Feynman diagram amplitude in the
different polarizations. We will show that it can actually be expressed in
terms of the plane-wave $\sigma$ (which we recall is the canonical element of
the dual of the generalized quantum double $D^{*}(H,A,\sigma)$). As we will
show in Sec. 4, this expression is associated to the discretization of a $BF$
theory.

######  Proposition 8 (Amplitude in the triangulation).

Let $\Gamma$ be the triangulation of a manifold built as the combination of
$M$ tetrahedra $\tau$. As a Feynman diagram of the HAFT, it is associated to
the amplitude

${\cal
A}_{\Gamma}=\frac{1}{\mu^{6M}}\,\int_{A^{12M}}\,\prod_{\tau}\,\big{(}\underline{\Delta}^{(3)}\hat{\delta}_{A}^{-1}\,\cdot\,(\hat{{\cal
C}}\otimes\hat{{\cal C}}\otimes\hat{{\cal C}}\otimes\hat{{\cal C}})\big{)}\,.$
(79)

######  Proposition 9 (Amplitude in the dual complex).

Consider $N$ tetrahedra sharing a single edge and let ${\cal L}_{N}$ be the
closed loop made of $2N$ half bulk links. The closed loop spans a face dual to
the edge. Let $\Gamma^{*}$ be the complex dual to a three dimensional
triangulation $\Gamma$, built as the combination of such loops ${\cal L}_{N}$.
Each loop is made of some number $N$ links (in general different for each
loop). The HAFT amplitude for the graph $\Gamma^{*}$ in the $H$ polarization
is

${\cal A}_{\Gamma^{*}}=\prod_{\\{{\cal
L}_{N}\\}}\,\frac{1}{\mu^{N}\,V_{H}^{N}}\,\int_{H^{2N}}\,\Delta^{2N}_{1\,-2\,3\,-4\,\dots\,(2N-1)\,-2N}\,\delta_{H}^{-1}\,.$
(80)

######  Proposition 10 (Amplitude in terms of the plane-wave).

The amplitude of the Feynman diagram $\Gamma^{*}$ can be represented in terms
of the plane wave $\sigma$, as

${\cal A}_{\Gamma^{*}}=\prod_{\\{{\cal
L}_{N}\\}}\,\frac{1}{\sqrt{\mu^{2N+1}}\,V_{H}^{N}}\,\left(\int_{H^{2N}}\otimes\int_{A}\right)\,\big{(}\Delta^{2N}_{1\,-2\,3\,-4\,\dots\,(2N-1)\,-2N}\otimes\textrm{id}\big{)}\,\sigma\,.$
(81)

In the appendix B.3, we show how to derive these three amplitudes.

###  3.4 Topological invariance

Given two simplicial decompositions of a manifold ${\cal M}$, one can find a
finite set of transformations, called Pachner moves, that maps one simplicial
decomposition into the other [24]. If the associated amplitudes are invariant
under the action of these Pachner moves (up to some possible constant re-
scaling), then we say that the amplitude is a topological invariant.

######  Proposition 11.

The amplitude (81) is a topological invariant.

In three dimensions there exist two Pachner moves, denoted $P_{(1,4)}$ and
$P_{(2,3)}$. We provide below the precise relation between the different
relevant amplitudes, showing the topological invariance of the model. In App.
B.4 we provide the details of the proofs.

##### Pachner move $P_{(1,4)}$.

The Pachner move $P_{(1,4)}$ takes the amplitude of one tetrahedron (denoted
${\cal A}_{{\cal V}}$) into the amplitude of four tetrahedra (denoted ${\cal
A}_{{\cal V}^{4}}$). The (reduced) amplitudes of one and four tetrahedra are

$\displaystyle{\cal A}_{{\cal V}}$
$\displaystyle=\frac{1}{\mu^{3}}\,\int_{A^{6}}\,\big{(}\hat{{\cal
C}}_{-6\,-4\,-2}\cdot\hat{{\cal C}}_{-5\,6\,-1}\cdot\hat{{\cal
C}}_{-3\,4\,5}\cdot\hat{{\cal C}}_{1\,2\,3}\big{)}\,,$ (82)
$\displaystyle{\cal A}_{{\cal V}^{4}}$
$\displaystyle=\frac{1}{\mu^{12}}\,\int_{A^{6}}\,\big{(}\hat{{\cal
C}}_{-1\,-3\,-2}\cdot\hat{{\cal C}}_{-5\,1\,-4}\cdot\hat{{\cal
C}}_{-6\,4\,2}\cdot\hat{{\cal C}}_{3\,5\,6}\big{)}\,.$

The action of the $P_{(1,4)}$ move gives

$(P_{(1,4)}\,{\cal A}_{{\cal V}})={\cal A}_{{\cal
V}^{4}}=\frac{1}{\mu^{9}}\,{\cal A}_{{\cal V}}\,.$ (83)

##### Pachner move $P_{(2,3)}$.

The Pachner move $P_{(2,3)}$ takes the amplitude of two tetrahedra (denoted
${\cal A}_{{\cal V}^{2}}$) into the one of three (denoted ${\cal A}_{{\cal
V}^{3}}$). The (reduced) amplitudes of two and three tetrahedra are

$\displaystyle{\cal A}_{{\cal V}^{2}}$
$\displaystyle=\frac{1}{\mu^{6}}\,\int_{A^{9}}\,\big{(}\hat{{\cal
C}}_{-6\,-4\,-2}\cdot\hat{{\cal C}}_{-5\,6\,-1}\cdot\hat{{\cal
C}}_{-3\,4\,5}\cdot\hat{{\cal C}}_{-9\,2\,-7}\cdot\hat{{\cal
C}}_{-8\,1\,9}\cdot\hat{{\cal C}}_{3\,8\,7}\big{)}\,,$ (84)
$\displaystyle{\cal A}_{{\cal V}^{3}}$
$\displaystyle=\frac{1}{\mu^{9}}\,\int_{A^{9}}\,\big{(}\hat{{\cal
C}}_{-5\,-3\,-2}\cdot\hat{{\cal C}}_{-4\,5\,-1}\cdot\hat{{\cal
C}}_{-8\,2\,-6}\cdot\hat{{\cal C}}_{-7\,1\,8}\cdot\hat{{\cal
C}}_{-9\,6\,3}\cdot\hat{{\cal C}}_{4\,7\,9}\big{)}\,.$

The relation between the two amplitudes, encoded by the Pachner move
$P_{(2,3)}$ is

$(P_{(2,3)}\,{\cal A}_{{\cal V}^{2}})={\cal A}_{{\cal
V}^{3}}=\frac{1}{\mu^{3}}\,{\cal A}_{{\cal V}^{2}}\,.$ (85)

In Fig. 4 and 5 we give the representation of the Pachner moves $P_{(1,4)}$
and $P_{(2,3)}$ in the triangulation picture. The amplitudes associated to the
building blocks of any Feynman diagram of the HAFT are invariant (up to some
constant factors) under the action of the Pachner moves. Hence, the HAFT we
proposed is a topological model.

$P_{(1,4)}$ Figure 4: The Pachner move $P_{(1,4)}$ takes one tetrahedron to
the combination of four. In the triangulation it is realized by connecting the
center of the tetrahedron with its four vertices. The surfaces ranging between
the center of the tetrahedron and any pair of vertices are internal triangles.
The four external faces of the initial tetrahedron thus become the external
faces of the four different tetrahedra that share the six internal triangles.
In the dual complex, the move is realized by taking four nodes (the colored
ones on the right) at the place of the central node (in blue on the left).
Each node is connected to one of the four external links and to each of the
other nodes. $P_{(2,3)}$ Figure 5: The Pachner move $P_{(2,3)}$ takes two
tetrahedra to the combination of three. In the triangulation it is realized by
connecting the vertices above and below through an internal edge. The surfaces
ranging between the internal edge and one of the three vertices shared by the
two initial tetrahedra become internal triangles. The three faces above, that
initially belonged to a single tetrahedron, now belong to three different
tetrahedra. The same for the faces below. In the dual complex, the move is
realized by taking three nodes (the colored ones on the right) at the place of
the two initial ones (in blue on the left). Each node is thus connected to
each other by two of its links, plus to one external link above and one below.

Note that, in the standard analysis of the three dimensional Pachner moves,
the proportionality constant for the move $P_{(1,4)}$ amounts to the cube of
the volume of the group (or Hopf algebra in our case) that decorates the dual
complex, whereas, the proportionality constant for the Pachner move
$P_{(2,3)}$ is a single volume term. Geometrically, these terms amount to the
difference between the number of (independent) internal edges from the initial
and to the final amplitudes of a given move. In (83) and (85) we do not
encounter the (usually) expected proportionality constants. One can retrieve
them by setting the normalization factor (51) to the identity, $\mu=1$ (as in
the standard case), and by removing the volume term $V_{H}$ in the definition
of gauge projector (60). This would prevent the operator ${\cal P}_{L}$ in
(60) to be a projector, as it would satisfy the identity ${\cal
P}_{L}^{2}=V_{H}\,{\cal P}_{L}$ instead of ${\cal P}_{L}^{2}={\cal P}_{L}$,
but this choice would lead to the standard proportionality constants in the
moves (83) and (85),

$(P_{(1,4)}\,{\cal A}_{{\cal V}}):={\cal A}_{{\cal V}^{4}}=V_{H}^{3}\,{\cal
A}_{{\cal V}}\,\,,\quad(P_{(2,3)}\,{\cal A}_{{\cal V}^{2}}):={\cal A}_{{\cal
V}^{3}}=V_{H}\,{\cal A}_{{\cal V}^{2}}\,.$ (86)

##  4 Example: $q$-deformed group field theory

We use our formalism to construct the HAFT when dealing with the quantum group
which would generate the Turaev-Viro amplitude [2], which is Boulatov’s
original group field theory [19]. We will show how the plane-wave $\sigma$
allows to recover the discretization of (Euclidian) BF theory in the presence
of a (negative) cosmological constant.
A quantum group is typically seen as a deformation of the usual notion of
group. Therefore the HAFT we will construct can be seen as a deformation of
the usual notion of group field theory. In appendix D, we discuss the cases of
a finite group and the undeformed $\mathrm{SU}(2)$ case.

###  4.1 $q$-deformed group field theory

We would like to illustrate our construction with the Hopf algebras $H$ and
$A$ taken to be the $q$-deformations of $F(\mathrm{SU}(2))$ and ${\cal
U}(\mathfrak{su}(2))$ respectively. Such standard deformations [15] are the
non commutative and non co-commutative Hopf algebras denoted
$\mathcal{U}_{q}(\mathfrak{an}(2))\cong F(\mathrm{SU}_{q}(2))$ and
$\mathcal{U}_{q}(\mathfrak{su}(2))\cong F(\mathrm{AN}_{q}(2))$.

####  4.1.1 Building blocks

Let the real deformation parameter be $q=e^{\ell\lambda}$, where $\ell$ has
the dimension of a length and is the characteristic scale of $\mathrm{SU}(2)$,
while $\lambda$ has the dimension of an inverse length and is the
characteristic scale of $\mathrm{AN}(2)$. Let $H,X_{\pm}$ be the generators of
$\mathcal{U}_{q}(\mathfrak{su}(2))$ with the dimension of a length ($\ell$)
and obeying Hopf algebra structure

Product: $\displaystyle\quad[H,X_{\pm}]={\pm}2\ell X_{\pm}\qquad\qquad\to\quad
e^{\lambda H}X_{\pm}=q^{\pm 2}\,X_{\pm}e^{\lambda H}\,,$ (87)
$\displaystyle\quad[X_{+},X_{-}]=\ell^{2}q^{-1}\frac{\sinh(\lambda
H)}{\sinh(\ell\lambda)}\,,$ Co-product: $\displaystyle\quad\Delta H=H\otimes
1+1\otimes H$ $\displaystyle\quad\Delta X_{+}=X_{+}\otimes 1+e^{-\lambda
H}\otimes X_{+}$ $\displaystyle\quad\Delta X_{-}=X_{-}\otimes e^{\lambda
H}+1\otimes X_{-}\,,$ Co-unit:
$\displaystyle\quad\varepsilon(H)=\varepsilon(X_{\pm})=0\,,$ Antipode:
$\displaystyle\quad S(H)=-H\,,$ $\displaystyle\quad S(X_{+})=-e^{\lambda
H}X_{+}\,,$ $\displaystyle\quad S(X_{-})=-X_{-}e^{-\lambda H}\,.$

Let $\phi,\varphi_{\pm}$ be the generators of $F(SU_{q}(2))$ with the
dimension of an inverse length ($\lambda$) and with Hopf algebra structure
specified by

Product:
$\displaystyle\quad[\phi,\varphi_{\pm}]=-i\lambda\varphi_{\pm}\qquad\qquad\to\quad
e^{i\ell\phi}\varphi_{\pm}=q\,\varphi_{\pm}e^{i\ell\phi}\,,$ (88)
$\displaystyle\quad[\varphi_{+},\varphi_{-}]=0\,,$ Co-product:
$\displaystyle\quad\Delta\phi=\frac{i}{\ell}\log\bigg{(}\frac{1}{\Delta\varphi_{0}}\big{(}\varphi_{0}e^{-i\ell\phi}\otimes\varphi_{0}e^{-i\ell\phi}-\ell^{2}\varphi_{-}\otimes\varphi_{+}\big{)}\bigg{)}\,,$
$\displaystyle\quad\Delta\varphi_{+}=\varphi_{+}\otimes\varphi_{0}e^{-i\ell\phi}+e^{i\ell\phi}\varphi_{0}\otimes\varphi_{+}\,,$
$\displaystyle\quad\Delta\varphi_{-}=\varphi_{-}\otimes
e^{i\ell\phi}\varphi_{0}+\varphi_{0}e^{-i\ell\phi}\otimes\varphi_{-}\,,$ Co-
unit: $\displaystyle\quad\varepsilon(\phi)=\varepsilon(\varphi_{\pm})=0\,,$
Antipode: $\displaystyle\quad S(\phi)=-\phi\,,$ $\displaystyle\quad
S(\varphi_{\pm})=-q^{\mp}\varphi_{\pm}\,.$

with $\varphi_{0}=\sqrt{1-q^{-1}\ell^{2}\varphi_{-}\varphi_{+}}$. We express
the two Hopf algebras through their PBW basis [25]:

$\\{X_{+}^{b}H^{a}X_{-}^{c}\\}_{abc=0}^{\infty}\in\mathcal{U}_{q}(\mathfrak{su}(2))\,\,,\quad\\{\varphi_{-}^{b}\phi^{a}\varphi_{+}^{c}\\}_{abc=0}^{\infty}\in\mathcal{U}_{q}(\mathfrak{an}(2))\,.$
(89)

The above ${\cal U}_{q}(\mathfrak{su}(2))$ basis is obtained from the standard
symmetric deformation (with generators $\\{J_{\pm},J_{3}\\}$) presented in
[15], with the re-scaling

$J_{\pm}\,\,\,\to\,\,\,e^{\mp\frac{1}{2}\lambda H}\,X_{\pm}\,\,,\quad
J_{3}\,\,\,\to\,\,\,H\,.$ (90)

The $F(\mathrm{SU}_{q}(2))$ basis given above is obtained by the following
change of coordinates on the standard $\mathrm{SU}_{q}(2)$ matrix element [15]

$g=\begin{pmatrix}a&b\\\
-qb^{*}&a^{*}\end{pmatrix}\,\,\,\to\,\,\,g=\begin{pmatrix}\varphi_{0}e^{-i\ell\phi}&i\ell\varphi_{-}\\\
i\ell\varphi_{+}&e^{i\ell\phi}\varphi_{0}\end{pmatrix}\quad\Rightarrow\quad\left\\{\,\,\begin{aligned}
&a=\varphi_{0}e^{-i\ell\phi}\,,\\\ &b=i\ell\varphi_{-}\,.\end{aligned}\right.$
(91)

The Haar measures on $F(\mathrm{SU}_{q}(2))$ and $F(\mathrm{AN}_{q}(2))$ are
resp. given by the standard $q$-deformation of the Haar measure on
$\mathrm{SU}(2)$ [15] and by the $q$-deformation of the Haar measure on
$AN(2)$, in the coordinate basis used above.

####  4.1.2 Plane-wave

The $q$-deformed group field theory can be seen as an example of a HAFT
presented in Sec. 3, specified by $H=F_{q}(\mathrm{SU}(2))\cong{\cal
U}_{q}(\mathfrak{an}(2))$ and $A=\mathcal{U}_{q}(\mathfrak{su}(2))\cong
F_{q}(AN(2))$.

######  Proposition 12 (Generalized quantum double of $q$-deformed group field
theory).

The generalized quantum double (Def. 4) of the $q$-deformation of group field
theory is the Drinfeld double of $\mathcal{U}_{q}(\mathfrak{su}(2))$:

${\cal D}({\cal U}_{q}(\mathfrak{su}(2)),{\cal
U}_{q}(\mathfrak{an}(2)),\sigma)={\cal D}({\cal
U}_{q}(\mathfrak{su}(2))\cong{\cal
U}_{q}(\mathfrak{su}(2))\bowtie_{\sigma}{\cal U}_{q}(\mathfrak{an}(2))\,,$
(92)

with (diagonal or canonical) skew pairing

$\sigma\big{(}X_{+}^{j}H^{i}X_{-}^{k}\,,\,\varphi_{-}^{b}\phi^{a}\varphi_{+}^{c}\big{)}=i^{a+b+c}\,\delta_{ai}\delta_{bj}\delta_{ck}\,a![b]_{q^{2}}![c]_{q^{-2}}!\,,$
(93)

and mutual actions

$\begin{aligned} &H\triangleright\phi=0\,,\\\
&X_{\pm}\triangleright\phi=\mp\ell q^{2B}\varphi_{\pm}\,,\\\
&H\triangleright\varphi_{\pm}=\pm\ell\varphi_{\pm}\,,\\\
&X_{\pm}\triangleright\varphi_{\pm}=0\,,\\\
&X_{\pm}\triangleright\varphi_{\mp}=\pm
i(\varphi_{0}e^{-i\ell\phi}-e^{i\ell\phi}\varphi_{0})\,,\end{aligned}\qquad\begin{aligned}
&H\triangleleft\phi=0\,,\\\ &X_{\pm}\triangleleft\phi=i\lambda X_{\pm}\,,\\\
&H\triangleleft\varphi_{\pm}=0\,,\\\
&X_{\pm}\triangleright\varphi_{\pm}=0\,,\\\
&X_{\pm}\triangleright\varphi_{\mp}=\mp i(1-qe^{\pm\lambda
H})\,.\end{aligned}$ (94)

Here

$[n]_{f}=\sum_{i=0}^{n}\,f^{i}=\frac{1-f^{n+1}}{1-f}\,\,,\quad[n]_{f}!=\prod_{i=1}^{n}\,[i]_{f}\,,$
(95)

are respectively the $q$-number and the $q$-factorial.

###### Proof.

Let us prove that the skew pairing

$\sigma\big{(}X_{+}^{j}H^{i}X_{-}^{k}\,,\,\varphi_{-}^{b}\phi^{a}\varphi_{+}^{c}\big{)}=i^{a+b+c}\,\delta_{ai}\delta_{bj}\delta_{ck}\,a![b]_{q^{2}}![c]_{q^{-2}}!$
(96)

satisfies the relations (27). We carry out the proof just using the bi-algebra
structures (87) of ${\cal U}_{q}(\mathfrak{su}(2))$ and (88) of
$F(\mathrm{SU}_{q}(2))$. To compute the $n^{th}$ power of the co-product of
$X_{\pm}$ we will make use of the $q$-deformed binomial theorem (see [15]).
Given two elements $A,B$ with obeying the relation $AB=q^{\alpha}BA$, the
deformed binomial theorem states that

$(A+B)^{n}=\sum_{i=0}^{n}\,\frac{A^{i}B^{n-i}}{[n]_{q^{\alpha}}}\,.$ (97)

Consider first

$\displaystyle\sigma\big{(}\Delta(X_{+}^{j}H^{i}X_{-}^{k})$
$\displaystyle\,,\,\varphi_{-}^{b}\phi^{a}\varphi_{+}^{c}\otimes\varphi_{-}^{r}\phi^{p}\varphi_{+}^{s}\big{)}:=\sum_{u,v,w=0}^{i,j,k}\,\binom{i}{u}\begin{bmatrix}j\\\
v\end{bmatrix}_{q^{2}}\begin{bmatrix}k\\\ w\end{bmatrix}_{q^{-2}}\,$
$\displaystyle\qquad\sigma\big{(}X_{+}^{j-v}e^{-v\lambda
H}H^{i-u}X_{-}^{w}\otimes X_{+}^{v}H^{u}e^{w\lambda
H}X_{-}^{k-w}\,,\,\varphi_{-}^{b}\phi^{a}\varphi_{+}^{c}\otimes\varphi_{-}^{r}\phi^{p}\varphi_{+}^{s}\big{)}$
$\displaystyle=\sum_{u,v,w=0}^{i,j,k}\sum_{\alpha,\beta=0}^{\infty}\,\binom{i}{u}\begin{bmatrix}j\\\
v\end{bmatrix}_{q^{2}}\begin{bmatrix}k\\\
w\end{bmatrix}_{q^{-2}}\,\frac{\lambda^{\alpha+\beta}(-v)^{\alpha}w^{\beta}}{\alpha!\beta!}$
$\displaystyle\qquad\sigma\big{(}X_{+}^{j-v}H^{i-u+\alpha}X_{-}^{w}\otimes
X_{+}^{v}H^{u+\beta}X_{-}^{k-w}\,,\,\varphi_{-}^{b}\phi^{a}\varphi_{+}^{c}\otimes\varphi_{-}^{r}\phi^{p}\varphi_{+}^{s}\big{)}$
$\displaystyle=i^{a+p+j+k}\,\delta_{j,b+r}\delta_{k,c+s}[b+r]_{q^{2}}[c+s]_{q^{-2}}\,\lambda^{a+p-i}\,\sum_{u=0}^{i}\,\frac{i!a!p!\,(-r)^{a-i+u}c^{p-u}}{u!(i-u)!(p-u)!(a-i+u)!}\,,$
(98) $\displaystyle\sigma\big{(}X_{+}^{j}H^{i}X_{-}^{k}$
$\displaystyle\,,\,\varphi_{-}^{b}\phi^{a}\varphi_{+}^{c}\cdot\varphi_{-}^{r}\phi^{p}\varphi_{+}^{s}\big{)}=\sigma\big{(}X_{+}^{j}H^{i}X_{-}^{k}\,,\,\varphi_{-}^{b}\varphi_{-}^{r}(\phi-i\lambda
r)^{a}(\phi+i\lambda c)^{p}\varphi_{+}^{c}\varphi_{+}^{s}\big{)}$
$\displaystyle=\sum_{u,v=0}^{a,p}\,\binom{a}{u}\binom{p}{v}\,(i\lambda)^{a+p-u-v}(-r)^{a-u}c^{p-v}\,\sigma\big{(}X_{-}^{j}H^{i}X_{+}^{k}\,,\,\varphi_{+}^{b+r}\phi^{u+v}\varphi_{-}^{c+s}\big{)}$
$\displaystyle=i^{a+p+j+k}\,\delta_{j,b+r}\delta_{k,c+s}\,[b+r]_{q^{2}}[c+s]_{q^{-2}}\,\lambda^{a+p-i}\,\sum_{u=0}^{p}\,\frac{i!a!p!\,(-r)^{a-i+u}c^{p-u}}{u!(p-u)!(i-u)!(p-i+u)!}\,.$
(99)

Here

$\begin{bmatrix}n\\\
m\end{bmatrix}_{f}=\frac{[n]_{f}!}{[m]_{f}!\,[n-m]_{f}!}\,,$ (100)

is the $q$-binomial. The second identity is more involved, as in the flat
case, it requires the star product on ${\cal U}_{q}(\mathfrak{su}(2))$. We
refer to App. C which provides the general setting for such type of
computation. Note that also the unit relations are trivially satisfied. ∎

######  Proposition 13 (Dual of the generalized quantum double of $q$-deformed
group field theory).

The dual of generalized quantum double (Def. 7) of the $q$-deformation of
group field theory is

$\mathcal{D}(\mathcal{U}_{q}(\mathfrak{an}(2)),\mathcal{U}_{q}(\mathfrak{su}(2)),\sigma)=\mathcal{U}_{q}(\mathfrak{an}(2))\mathrel{\scalerel*{\blacktriangleright\joinrel\blacktriangleleft}{x}}_{\sigma}\mathcal{U}_{q}(\mathfrak{su}(2))\,,$
(101)

with skew co-pairing element given by the $q$-star exponential

$\sigma=e_{\star\,q^{2}}^{i\varphi_{+}\otimes X_{-}}\,e_{\star}^{i\phi\otimes
H}\,e_{\star\,q^{-2}}^{i\varphi_{-}\otimes X_{+}}\,,$ (102)

and co-actions (40)

$\displaystyle\alpha(\phi)$ $\displaystyle=(\phi\otimes
1)-i\lambda\sum_{n=1}^{\infty}\,\frac{(1-q^{2})^{n}}{1-q^{2n}}\,\Big{(}q^{2n}(e^{2ni\ell\phi}\otimes
e^{-n\lambda H})\,e_{\star\,q^{2}}^{-i\varphi_{-}\otimes
X_{+}}\,(i\varphi_{+}\otimes X_{-})^{n}e_{\star\,q^{-2}}^{i\varphi_{-}\otimes
X_{+}}-(-i\varphi_{-}\otimes X_{+})^{n}\Big{)}$
$\displaystyle\approx\phi\otimes 1+\lambda(\varphi_{+}\otimes
X_{-}+\varphi_{-}\otimes X_{+})+{\cal O}(\lambda^{2})+{\cal O}(\ell)+{\cal
O}(\ell\lambda)\,,$ (103) $\displaystyle\alpha(\varphi_{\pm})$
$\displaystyle=e_{\star\,q^{2}}^{-i\varphi_{-}\otimes
X_{+}}\,e_{\star\,q^{-2}}^{i\varphi_{-}\otimes
X_{+}\,q^{-2}}\,(\varphi_{\pm}\otimes e^{-\lambda
H})\approx\varphi_{\pm}\otimes 1-\lambda\varphi_{\pm}\otimes H+{\cal
O}(\lambda^{2})+{\cal O}(\ell)+{\cal O}(\ell\lambda)\,,$
$\displaystyle\beta(H)$ $\displaystyle=(1\otimes
H)-2\ell\sum_{n=1}^{\infty}\,\frac{(1-q^{2})^{n}}{1-q^{2n}}\,\Big{(}q^{2n}(e^{2ni\ell\phi}\otimes
e^{-n\lambda H})\,e_{\star\,q^{2}}^{-i\varphi_{-}\otimes
X_{+}}\,(i\varphi_{+}\otimes X_{-})^{n}e_{\star\,q^{-2}}^{i\varphi_{-}\otimes
X_{+}}-(-i\varphi_{-}\otimes X_{+})^{n}\Big{)}$ $\displaystyle\approx 1\otimes
H-2i\ell(\varphi_{+}\otimes X_{-}+\varphi_{-}\otimes X_{+})+{\cal
O}(\lambda)+{\cal O}(\ell^{2})+{\cal O}(\ell\lambda)\,,$ (104)
$\displaystyle\beta(X_{\pm})$ $\displaystyle\approx 1\otimes X_{\pm}\pm
i\ell(\varphi_{\pm}\otimes H-2\phi\otimes X_{\pm})+{\cal O}(\lambda)+{\cal
O}(\ell^{2})+{\cal O}(\ell\lambda)\,.$ (105)

Since the explicit expression of the mutual co-actions is very involved as it
require the use of the star products, we provided the first terms of the
expansions wither in the parameter $\ell$ or $\lambda$. The $q$-exponential is
defined as

$e_{q}^{x}=\sum_{n=0}^{\infty}\,\frac{x^{n}}{[n]_{q}!}=\exp\bigg{(}\sum_{n=1}^{\infty}\,\frac{x^{n}}{n}\frac{(1-q)^{n}}{1-q^{n}}\bigg{)}\,.$
(106)

###### Proof.

Let us prove that the skew co-pairing (quantum plane wave)

$\sigma=e_{\star\,q^{2}}^{i\varphi_{+}\otimes X_{-}}\,e_{\star}^{i\phi\otimes
H}\,e_{\star\,q^{-2}}^{i\varphi_{-}\otimes X_{+}}\,,$ (107)

satisfies the relations (34),(35). We carry out the proof just using the bi-
algebra structures (87) of ${\cal U}_{q}(\mathfrak{su}(2))$ and (88) of
$F(\mathrm{SU}_{q}(2))$. In order to split the $q$-$\star$ exponentials we
used their property related to the $q$-deformed binomial theorem (see [15]).
Given two elements $A,B$ with obeying the relation $AB=q^{\alpha}BA$, the
$q$-exponential satisfies the relation

$e_{q^{\alpha}}^{A}\,e_{q^{\alpha}}^{B}=e_{q^{\alpha}}^{A+B}\,.$ (108)

Consider first

$\displaystyle(\textrm{id}\otimes\Delta)\,\sigma:=$ $\displaystyle
e_{\star\,q^{2}}^{i\varphi_{+}\otimes\Delta
X_{-}}\,e_{\star}^{i\phi\otimes\Delta
H}\,e_{\star\,q^{-2}}^{i\varphi_{-}\otimes\Delta
X_{+}}=e_{\star\,q^{2}}^{i\varphi_{+}\otimes(1\otimes X_{-}+X_{-}\otimes
e^{\lambda H})}\,e_{\star}^{i\phi\otimes(1\otimes H+H\otimes
1)}\,e_{\star\,q^{-2}}^{i\varphi_{-}\otimes(e^{-\lambda H}\otimes
X_{+}+X_{+}\otimes 1)}$ $\displaystyle=$ $\displaystyle
e_{\star\,q^{2}}^{i\varphi_{+}\otimes 1\otimes
X_{-}}e_{\star\,q^{2}}^{i\varphi_{+}\otimes X_{-}\otimes e^{\lambda
H}}\,e_{\star}^{i\phi\otimes 1\otimes H}e_{\star}^{i\phi\otimes H\otimes
1}\,e_{\star\,q^{-2}}^{i\varphi_{-}\otimes e^{-\lambda H}\otimes
X_{+}}e_{\star\,q^{-2}}^{i\varphi_{-}\otimes X_{+}\otimes 1}$ $\displaystyle=$
$\displaystyle e_{\star\,q^{2}}^{i\varphi_{+}\otimes 1\otimes
X_{-}}e_{\star}^{i\phi\otimes 1\otimes
H}\,e_{\star\,q^{2}}^{i\varphi_{+}\otimes X_{-}\otimes
1}e_{\star\,q^{-2}}^{i\varphi_{-}\otimes 1\otimes
X_{+}}\,e_{\star}^{i\phi\otimes H\otimes
1}e_{\star\,q^{-2}}^{i\varphi_{-}\otimes X_{+}\otimes 1}$ $\displaystyle=$
$\displaystyle e_{\star\,q^{2}}^{i\varphi_{+}\otimes 1\otimes
X_{-}}e_{\star}^{i\phi\otimes 1\otimes
H}e_{\star\,q^{-2}}^{i\varphi_{-}\otimes 1\otimes
X_{+}}\,e_{\star\,q^{2}}^{i\varphi_{+}\otimes X_{-}\otimes
1}e_{\star}^{i\phi\otimes H\otimes 1}e_{\star\,q^{-2}}^{i\varphi_{-}\otimes
X_{+}\otimes 1}:=\sigma_{13}\,\sigma_{12}\,.$ (109)

The second identity is more involved, as it requires the star product on
${\cal U}_{q}(\mathfrak{su}(2))$. We refer to App. C which provides the
general setting for such type of computation. Note that also the co-unit
relations are trivially satisfied. ∎

We note that the plane wave (102) is similar to the exponential map obtained
in [26].

####  4.1.3 Closure constraint and Feynman diagram amplitude

Let us now derive the fundamental ingredients of the $q$-deformation of three
dimensional group field theory, using the HAFT based on the dual of the
generalized quantum double of ${\cal U}_{q}(\mathfrak{su}(2))$ above. We
review the expressions of the closure constraint and the amplitude of a given
graph.
The closure constraint (63) in the triangulation picture is given by the delta
function on ${\cal U}_{q}(\mathfrak{su}(2))$, which enforces the following co-
products to vanish

$\displaystyle H\otimes 1\otimes 1+1\otimes H\otimes 1+1\otimes 1\otimes
H=0\,,$ (110) $\displaystyle X_{+}\otimes 1\otimes 1+e^{-\lambda H}\otimes
X_{+}\otimes 1+e^{-\lambda H}\otimes e^{-\lambda H}\otimes X_{+}=0\,,$
$\displaystyle X_{-}\otimes e^{\lambda H}\otimes e^{\lambda H}+1\otimes
X_{-}\otimes e^{\lambda H}+1\otimes 1\otimes X_{-}=0\,.$

Such closure condition can be re-packaged as a product of three $AN_{q}(2)$
group elements, see [9]. This is the quantum Gauss constraint associated to a
triangle [27].
The amplitude (81) for a graph $\Gamma^{*}$ is

${\cal A}_{\Gamma^{*}}=\prod_{\\{{\cal
L}_{N}\\}}\,\frac{1}{V_{\mathrm{SU}_{q}(2)}^{N}}\,\int[\mathrm{d}\phi\mathrm{d}\varphi_{\pm}]^{N}[\mathrm{d}H\mathrm{d}X_{\pm}]\,e_{\star\,q^{2}}^{i\Delta^{2N}\varphi_{+}\otimes
X_{-}}\,e_{\star}^{i\Delta^{N}\phi\otimes
H}\,e_{\star\,q^{-2}}^{i\Delta^{N}\varphi_{-}\otimes X_{+}}\,,$ (111)

where $V_{\mathrm{SU}_{q}(2)}$ is the volume of $F(\mathrm{SU}_{q}(2))$, and
for simplicity, we assumed666We recall that $\mu$ comes from the normalization
of the integration of the delta function (or of the double integration of the
plane-wave), see (51). $\mu=1$.

####  4.1.4 Relation with Turaev-Viro model (with $q$ real)

We are going to show in the next section that the Feynman amplitude (111) is
naturally associated to the discretized action of a $BF$ model with non-
vanishing cosmological constant. However before that, we are going to discuss
how it is related to the Turaev-Viro model. To this aim, it is enough to
explain how to recover the original Boulatov model [19]. In his work, Boulatov
defines a field theory based on the representations of a given group $G$,
where the fundamental field is expanded in the Fourier decomposition

$\displaystyle\Phi(x,y,z)=\sum_{i_{1},j_{2},j_{3}}\sum_{\\{m,n,k\\}}$
$\displaystyle\Phi^{m_{1}\,m_{2}\,m_{3};k_{1}\,k_{2}\,k_{3}}_{j_{1}\,j_{2}\,j_{3}}\,D^{j_{1}}_{m_{1},n_{1}}(x)\,D^{j_{2}}_{m_{2},n_{2}}(y)\,D^{j_{3}}_{m_{3},n_{3}}(z)$
(112) $\displaystyle\int\mathrm{d}\omega$ $\displaystyle
D^{j_{1}}_{n_{1},k_{1}}(\omega)\,D^{j_{2}}_{n_{2},k_{2}}(\omega)D^{j_{3}}_{n_{3},k_{3}}(\omega)\,,$

where $x,y,z,\omega\in G$ and $D^{j}_{m,n}(x)$ are matrix elements obeying the
orthogonality condition

$\int\mathrm{d}x\,D^{j_{1}}_{m_{1},n_{1}}(x)\,D^{j_{2}}_{m_{2},n_{2}}(x)=\frac{1}{d_{j}}\,\delta_{j_{1},j_{2}}\delta_{m_{1},m_{2}}\delta_{n_{1},n_{2}}\,,$
(113)

with $d_{j}$ being the dimension of the irreducible representation associated
to $j$. Taking $G=\mathrm{SU}(2)$, the matrix elements $D^{j}_{m,n}(x)$ are
the standard Wigner $D$-matrices, whereas – as Boulatov explains – for
$G=SU_{q}(2)$ the matrix elements $D^{j}_{m,n}(x)$ become the $q$-deformed
Wigner matrices [15]. In Hopf algebra field theory, the fundamental field
$\phi$ is an element of the tensor product Hopf algebra $H^{3}$. We claimed
that the $q$-deformed group field theory is obtained by choosing
$H=F(\mathrm{SU}_{q}(2))$, for which we used the parametrization in terms of
the coordinates $\\{\varphi_{+},\phi,\varphi_{-}\\}$ of the
$\mathrm{SU}_{q}(2)$ group element (91). Each field $\Phi$ in Hopf algebra
field theory is thus given by the tensor product of three copies of the linear
combination of monomials $\varphi_{+}^{j}\,\phi^{i}\,\varphi_{-}^{k}$.
Therefore, to make contact between Hopf algebra field theory and the original
Boulatov model, we define the Fourier expansion for the fields $\Phi\in
F(\mathrm{SU}_{q}(2)^{\times 3})$

$\displaystyle\Phi=\sum_{i_{1},j_{2},j_{3}}\sum_{\\{m,n,k\\}}$
$\displaystyle\Phi^{m_{1}\,m_{2}\,m_{3};k_{1}\,k_{2}\,k_{3}}_{j_{1}\,j_{2}\,j_{3}}\,D^{j_{1}}_{m_{1},n_{1}}(\\{\varphi_{\pm},\phi\\}_{1})\,D^{j_{2}}_{m_{2},n_{2}}(\\{\varphi_{\pm},\phi\\}_{2})\,D^{j_{3}}_{m_{3},n_{3}}(\\{\varphi_{\pm},\phi\\}_{3})$
(114)
$\displaystyle\int\mathrm{d}\varphi_{+}^{\prime}\mathrm{d}\phi^{\prime}\mathrm{d}\varphi_{-}^{\prime}$
$\displaystyle
D^{j_{1}}_{n_{1},k_{1}}(\\{\varphi_{\pm}^{\prime},\phi^{\prime}\\})\,D^{j_{2}}_{n_{2},k_{2}}(\\{\varphi_{\pm}^{\prime},\phi^{\prime}\\})\,D^{j_{3}}_{n_{3},k_{3}}(\\{\varphi_{\pm}^{\prime},\phi^{\prime}\\})\,,$

where we used the symbol $\\{\varphi_{\pm},\phi\\}_{i}$ for the monomial in
the coordinates $\varphi_{+},\phi,\varphi_{-}$ in the $i^{th}$ tensor space,
and once again the $D^{j}_{m,n}$ are matrix elements obeying the same
orthogonality condition. The HAFT defined for $H=\mathrm{SU}_{q}(2)$ we
proposed is thus equivalent to the Boulatov model. In particular, the Feynman
amplitude of the Boulatov model, and thus the one of HAFT (79), can be given
in terms of the $q$-deformed 6-$j$ symbols and thus it allows to recover the
Turaev-Viro invariant [2] with $q$ real.

###  4.2 Plane-wave and discretization of the $BF$ action

The amplitudes we calculated are expressed in terms of the plane-wave. Since
the model is topological we expect that it should be related to the
$BF$-action. More concretely, we expect that the amplitude (111) provides a
regularization/discretization of $e^{i{\cal S}^{\lambda}_{BF}}$, where ${\cal
S}^{\lambda}_{BF}$ is the $BF$ action with a cosmological constant as given in
[9].

####  4.2.1 $BF$ action and its limits

Let us focus on the Euclidian case with a negative or null cosmological
constant $\Lambda\leq 0$, as we know this corresponds to the the deformation
with $q$ real [9]. The fundamental fields are the frame field $e$ and the
connection $A$, which are respectively 1-forms with values in the boosts $K$
and the Lie algebra $\mathfrak{su}(2)$. Noting $\langle\,,\,\rangle$ the
Killing form of $\mathfrak{sl}(2,C)$, the $BF$ model is governed by the action

${\cal S}^{\Lambda}_{BF}[A,e]=\int_{{\cal M}}\,\langle
e\,,\,\Big{(}F+\frac{\Lambda}{6}\,(e\times e)\Big{)}\rangle\,.$ (115)

The frame field variables are difficult to quantize since they are valued in
the boosts. It was shown in [9] that it is convenient to do a canonical
transformation

$\omega_{I}=A_{I}+\epsilon_{IJK}n^{J}e_{K}\,\,,\quad
n^{I}=(0,0,\lambda)\,\,,\quad n^{2}=\lambda^{2}=-\Lambda\,,$ (116)

which makes the frame field discretizable with values in the Lie algebra
$\mathfrak{an}(2)$. With this new connection the action becomes, up to a
boundary term,

${\cal S}^{\lambda}_{BF}[\omega,e]=\int_{{\cal M}}\,\big{(}e^{I}\wedge
F_{I}+\lambda\,(\omega_{-}\wedge e_{3}\wedge e_{+}-\omega_{+}\wedge
e_{-}\wedge e_{3})\big{)}\,.$ (117)

Note that this action is also valid if we have $\Lambda=\lambda^{2}=0$ but
$\lambda\neq 0$. On the other hand if $\lambda=0$, we just recover the
standard $BF$ action ${\cal S}_{BF}^{\Lambda=0}={\cal S}_{BF}^{\lambda=0}$.
Let us emphasize that this action depends on two constants, Newton’s constant
$G$ (hence the Planck length $\ell$) and the cosmological constant (hence
$\lambda$). The constant $\ell$ typically encodes the curvature (radius) of
$\mathrm{SU}(2)$, which is why the fluxes (Poisson) non-commutativity is
encoded by $\ell$, the Planck length. The cosmological constant or $\lambda$
encodes the curvature of the group $\mathrm{AN}(2)$. It encodes then the
(Poisson) non-commutativity of the holonomy components.

When looking at the action (117), we can therefore take two limits, one where
$\lambda\rightarrow 0$ and another one where $\ell\rightarrow 0$. This latter
one is less common from the (quantum) gravity side since in a sense it amounts
to switch off gravity.

In terms of the underlying symmetry structure, the actions (115) or (117) are
built from the classical Drinfeld double
$\mathfrak{sl}(2,C)\cong\mathfrak{su}(2)\bowtie\mathfrak{an}(2)$. The two
limits we just discussed become then

$\displaystyle\mathfrak{su}(2)\bowtie\mathfrak{an}(2)\,\,\,\rightarrow\,\,\,\left\\{\begin{array}[]{c}\mathfrak{su}(2)\ltimes\mathbb{R}^{3},\quad\lambda=0\\\
\mathbb{R}^{3}\rtimes\mathfrak{an}(2),\quad\ell=0\end{array}\right.$ (120)

In terms of the action, we can write (117) highlighting the different brackets
which would be turned to zero,

$\displaystyle{\cal S}^{\lambda}_{BF}[\omega,e]=\int_{\cal M}\,\Big{(}\langle
e\,,\,d\omega\rangle+\frac{1}{2}\langle
e\,,\,[\omega\,,\,\omega]_{\mathfrak{su}(2)}\rangle+\frac{1}{2}\langle[e\,,\,e]_{\mathfrak{an}(2)}\,,\,\omega\rangle\Big{)}\,.$
(121)

We have therefore the two types of limiting actions

${\cal S}^{\lambda}_{BF}[\omega,e]\,\,\,\to\,\,\,\left\\{\,\,\begin{aligned}
&{\cal S}^{\mathfrak{su}(2)}=\int_{{\cal M}}\,\Big{(}\langle
e\,,\,d\omega+\frac{1}{2}[\omega\,,\,\omega]_{\mathfrak{su}(2)}\rangle\Big{)}\,\,\,,\quad\lambda=0\\\
&{\cal S}^{\mathfrak{an}(2)}=\int_{{\cal M}}\,\Big{(}\langle
de+\frac{1}{2}[e\,,\,e]_{\mathfrak{an}(2)}\,,\,\omega\rangle\Big{)}\,\,\,,\quad\ell=0\end{aligned}\right.$
(122)

where in the last case, we omitted the boundary term and we recall that in
this case, $\omega$ is with value in the abelian Lie algebra $\mathbb{R}^{3}$.
In the first case, it is $e$ which is with value in $\mathbb{R}^{3}$. The
first action is $\mathfrak{su}(2)$ $BF$ theory, while the second one is a
$\mathfrak{an}(2)$ $BF$ theory.

####  4.2.2 Discretized $BF$ actions

Let us recall the standard discretization procedure of a $\mathfrak{g}$ $BF$
theory, with no cosmological constant.

$\displaystyle{\cal S}=\int_{\cal M}\langle B,F(A)\rangle\,.$ (123)

The field $B$ is a $\mathfrak{g}^{*}$ valued one-form and $A$ is a
$\mathfrak{g}$-valued connection. The 3$d$ manifold ${\cal M}$ is discretized
on a cellular decomposition $\Gamma$, noting $\Gamma^{*}$ its dual complex.
According to the Poincaré duality, we discretize $B$ and $F$ on dual
structures. The curvature $F(A)$ is chosen to be discretized along an holonomy
$g_{e}$ in $G$ forming a closed loop, spanning a face $e^{*}$ dual to an edge
$e$. In this construction, we can attribute to each link $l$ an holonomy
$g_{l}\in G$, such that $g_{e}=\prod_{l}\,g_{l}$. Upon quantization, such
product is actually characterized by the co-product

$g_{e}=\prod_{l}g_{l}\,\,\,\rightarrow\,\,\,\Delta^{n}g\,.$ (124)

The closed loop defines the boundary of a surface dual to an edge. We
discretize $B$ on such edge as a 3$d$ vector777We mean that that the
coordinates are $\star$ non-commutative coordinates, so that
$\mathbf{x}_{e}\in F_{\star}(\mathbb{R}^{d})$.
$\mathbf{x}_{e}\in\mathbb{R}^{d}_{\star}$, where $d$ is the dimension of
$\mathfrak{g}$. The natural discretization of the three dimensional $BF$
action can be represented with the help of the co-product or the $\star$
product representation.

$e^{i{\cal S}_{BF}[A,B]}\approx e^{\langle
g_{e}\,\mathbf{x}\rangle}\,\,\,\to\,\,\,\left\\{\begin{array}[]{l}(\Delta^{N}\otimes\textrm{id})e^{g\cdot\mathbf{x}_{e}}=e^{\Delta^{N}g\cdot\mathbf{x}_{e}}\\\
e_{\star}^{\langle
g\,,\,\mathbf{x}_{e}\rangle}=e_{\star}^{\langle\prod_{l}g_{l}\,,\,\mathbf{x}_{e}\rangle}=\star_{l}e_{\star}^{\langle
g_{l}\,,\,\mathbf{x}_{e}\rangle}\end{array}\right.\,,$ (125)

where $\Delta^{N}g\cdot\mathbf{x}_{e}$ is the quantization of $\langle
g_{e}\,,\,\mathbf{x}\rangle$. We emphasize that there is a freedom in the
choice of coordinates for the element $g_{e}$. As operators we will have
clearly different operator coordinates, and dually different $\mathbf{x}_{e}$.
We will have then different co-product for the coordinate operators which is
equivalent to different $\star$ products.

For instance, taking the specific example of $\mathrm{SU}(2)$, we can use the
Euler angles $(\varphi_{\pm},\phi)$ as a parametrization of an element of
$\mathrm{SU}(2)$, or the ”standard” parameterization $(p_{\pm},p_{3},p_{0})$
[1])

$g=\begin{pmatrix}e^{i\phi}&ie^{i\phi}\varphi_{-}\\\
ie^{i\phi}\varphi_{+}&e^{-i\phi}-\varphi_{-}\varphi_{+}e^{i\phi}\\\
\end{pmatrix}=\begin{pmatrix}p_{0}+ip_{3}&ip_{-}\\\
ip_{+}&p_{0}-ip_{3}\end{pmatrix}\,.$ (126)

where $p_{0}$ depends on the other coordinates. The two sets of coordinates
are obviously related as explicitly seen in (126).

$\displaystyle p_{\pm}=e^{i\phi}\varphi_{\pm}\,\,,\quad
p_{3}=\sin\phi+\frac{i}{2}\varphi_{+}\varphi_{-}e^{i\phi}\,\,,\quad
p_{0}=\cos\phi-\frac{1}{2}\varphi_{+}\varphi_{-}e^{i\phi}.$ (127)

We also always have to bear in mind that one might need more than one
coordinate patch so one needs to be cautious [28].

The dual variables to the coordinates $(\varphi_{\pm},\phi)$ and
$(p_{\pm},p_{3},p_{0})$ are different. They can be seen as the vectors fields
generating the translations on the sphere $S^{3}\cong\mathrm{SU}(2)$.
Depending on the choice of coordinates, they take different shapes. The co-
product in the $p$-variables is different than the one in the
$\varphi$-variables, or equivalently the $\star$ product for the dual
variables of the $p$-variables is different than the one of the dual of the
$\varphi$-variables. Nevertheless they are related by the change of
coordinates (127).

Finally, we emphasize that while we chose the group $\mathrm{SU}(2)$ to
illustrate our discussion, this is true for any group, in particular
$\mathrm{AN}(2)$.

We have seen that we can have two actions ${\cal S}^{\mathfrak{su}(2)}$,
${\cal S}^{\mathfrak{an}(2)}$ obtained as different limits of a single action
${\cal S}^{\lambda}$. The limiting actions in (122) can be discretized along
the same procedure. Since we use the same fields $(e,\omega)$ in both cases,
the fields $(e,\omega)$ should be discretized on the same structure in each
case. Following the loop quantum gravity picture, we choose to discretize the
frame field $e$ on the edges of $\Gamma$, while $\omega$ is discretized on the
links of $\Gamma^{*}$.

The fundamental element in both discretization takes the same general shape
but with different locations according to the different cases.

$\left\\{\begin{array}[]{c}e^{\langle g_{l}\textbf{x}_{e}\rangle},\textrm{
with }g_{l}\in\mathrm{SU}(2)\textrm{ on
}\Gamma^{*},\quad\textbf{x}_{e}\in\mathbb{R}^{3}\textrm{ on }\Gamma\\\
e^{\langle\textbf{x}_{l}g_{e}\rangle},\textrm{ with
}g_{e}\in\mathrm{AN}(2)\textrm{ on
}\Gamma,\quad\textbf{x}_{l}\in\mathbb{R}^{3}\textrm{ on
}\Gamma^{*}\end{array}\right.$ (128)

Using the co-product, we can construct the discretized actions. We note the
choice of coordinates $k$, $q$ on respectively $\mathrm{SU}(2)$ and
$\mathrm{AN}(2)$ and $X$, x on respectively
$\mathbb{R}^{3}_{\star_{\mathfrak{su}}}$ and
$\mathbb{R}^{3}_{\star_{\mathfrak{an}}}$. We have therefore the discretization
of the actions (122) given by

$\displaystyle\,{\cal S}^{\mathfrak{su}(2)}_{d}\approx\,{\cal
S}_{d}^{\lambda=0}$ $\displaystyle=\Delta^{n}k^{a}\otimes X_{a}$ (129)
$\displaystyle\,{\cal S}^{\mathfrak{an}(2)}_{d}\approx\,{\cal S}_{d}^{\ell=0}$
$\displaystyle=\Delta^{n}\textbf{x}^{a}\otimes q_{a}$ (130)

To obtain the discretization of the general case $e^{i{\cal
S}^{\lambda}_{BF}[\omega,e]}$, we need to consider the HAFT and its associated
plane-wave.

####  4.2.3 Recovering the discretized $BF$ actions from the plane-wave

As we alluded earlier, we expect that the plane-wave (102) would provide a
discretization of (117) since this formulation is the proper one leading to
the quantum group variables [9]. The fact that the triangulation is decorated
by quantum group elements indicates that we are dealing with the quantum
version of a homogeneously curved discrete geometry [29].

Following the previous section, if we consider an edge dual to a loop formed
of $n$ links, we would have the discretization scheme,

$(\Delta^{n}\otimes\textrm{id})\,\sigma\,\,\,\leftrightarrow\,\,\,e^{i{\cal
S}^{\lambda}_{BF}[\omega,e]}\,.$ (131)

We note that the main complication relies on the fact that the amplitude
expressed in terms of the fields (131) is expressed in terms of the fields all
included in a single exponential, contrary to the plane-wave formula (102),
where each component has its own exponential (similarly to the Euler angles
parametrization). Putting them together under a common exponential generally
requires to use the Baker-Campbell-Hausdorff formula

$e^{A}e^{B}\approx e^{A+B+\frac{1}{2}[A,B]}\,,$ (132)

which we will truncate at first order to identify the relationship between the
continuum actions and their discrete versions.

As recalled in the table 1, one can obtain different bi-algebras as limiting
cases of the deformed case. We recover the bi-algebras associated to
$\mathrm{AN}(2)$ in the limit $\ell\to 0$ and those associated to
$\mathrm{SU}(2)$ when $\lambda\to 0$. This is consistent with the different
limits we discussed in the previous section.

$q=e^{\ell\lambda}$ | $\lambda\rightarrow 0$ : ${\cal U}(\mathfrak{su}(2))$ | $\ell\rightarrow 0$ : $F(AN(2))$
---|---|---
$[H,X_{\pm}]={\pm}2\ell J_{\pm}$ | $[H,X_{\pm}]={\pm}2\ell J_{\pm}$ | $[H,X_{\pm}]=0$
$[X_{+},X_{-}]=\ell^{2}\frac{\sinh(\lambda H)}{\sinh(\ell\lambda)}$ | $[X_{+},X_{-}]=\ell H$ | $[X_{+},X_{-}]=0$
$\Delta H=H\otimes 1+1\otimes H$ | $\Delta H=H\otimes 1+1\otimes H$ | $\Delta H=H\otimes 1+1\otimes H$
$\Delta X_{+}=X_{+}\otimes 1+e^{-\lambda H}\otimes X_{+}$ | $\Delta X_{+}=X_{+}\otimes 1+1\otimes X_{+}$ | $\Delta X_{+}=X_{+}\otimes 1+e^{-\lambda H}\otimes X_{+}$
$\Delta X_{-}=X_{-}\otimes e^{\lambda H}+1\otimes X_{-}$ | $\Delta X_{-}=X_{-}\otimes 1+1\otimes X_{-}$ | $\Delta X_{-}=X_{-}\otimes e^{\lambda H}+1\otimes X_{-}$
| $\lambda\rightarrow 0$ : $F(SU(2))$ | $\ell\rightarrow 0$ : ${\cal U}(\mathfrak{an}(2))$
$\varphi_{0}^{2}=1-\ell^{2}\varphi_{+}\varphi_{-}$ | $\varphi_{0}^{2}=1-\ell^{2}\varphi_{+}\varphi_{-}$ | $\varphi_{0}^{2}=1$
$[\phi,\varphi_{\pm}]=-i\lambda\varphi_{\pm}$ | $[\phi,\varphi_{\pm}]=0$ | $[\phi,\varphi_{\pm}]=-i\lambda\varphi_{\pm}$
$[\varphi_{+},\varphi_{-}]=0$ | $[\varphi_{+},\varphi_{-}]=0$ | $[\varphi_{+},\varphi_{-}]=0$
$\Delta(\varphi_{0}e^{-i\ell\phi})=\big{(}\varphi_{0}e^{-i\ell\phi}\otimes\varphi_{0}e^{-i\ell\phi}-q\ell^{2}\varphi_{-}\otimes\varphi_{+}\big{)}$ | $\Delta(\varphi_{0}e^{-i\ell\phi})=\big{(}\varphi_{0}e^{-i\ell\phi}\otimes\varphi_{0}e^{-i\ell\phi}-q\ell^{2}\varphi_{-}\otimes\varphi_{+}\big{)}$ | $\Delta\phi=\phi\otimes 1+1\otimes\phi$
$\Delta\varphi_{+}=\varphi_{+}\otimes\varphi_{0}e^{-i\ell\phi}+e^{i\ell\phi}\varphi_{0}\otimes\varphi_{+}$ | $\Delta\varphi_{+}=\varphi_{+}\otimes\varphi_{0}e^{-i\ell\phi}+e^{i\ell\phi}\varphi_{0}\otimes\varphi_{+}$ | $\Delta\varphi_{+}=\varphi_{+}\otimes 1+1\otimes\varphi_{+}$
$\Delta\varphi_{-}=\varphi_{-}\otimes e^{i\ell\phi}\varphi_{0}+\varphi_{0}e^{-i\ell\phi}\otimes\varphi_{-}$ | $\Delta\varphi_{-}=\varphi_{-}\otimes e^{i\ell\phi}\varphi_{0}+\varphi_{0}e^{-i\ell\phi}\otimes\varphi_{-}$ | $\Delta\varphi_{-}=\varphi_{-}\otimes 1+1\otimes\varphi_{-}$
Table 1: The different limiting cases.

One of our main statements is that the plane wave (102) provides the
discretization of the $BF$ amplitude with non-vanishing cosmological constant

$e^{i{\cal S}_{BF}^{\lambda}[\omega,e]}\approx e^{i{\cal
S}_{d}^{\Lambda}}=(\Delta^{n}\otimes\textrm{id})\,\big{(}e_{\star\,q^{2}}^{i\varphi_{+}\otimes
X_{-}}\,e_{\star}^{i\phi\otimes H}\,e_{\star\,q^{-2}}^{i\varphi_{-}\otimes
X_{+}}\big{)}\,.$ (133)

Using the full BCH formula would provide us the exact discretization of the
$BF$ action with or without cosmological constant. However since it is very
complicated, we will truncate the formula at the first order in $\ell$ and
$\lambda$.

Since we can recover the actions (122) from the main action ${\cal
S}^{\lambda}$, we should be recovering the discretized actions (129) and (130)
from the full plane-wave. To this aim, we use the BCH formula on the
$q$-deformed plane wave (102) to express it as a single exponential and take
the different limits.

$\displaystyle\sigma$ $\displaystyle=e_{\star\,q^{2}}^{i\varphi_{+}\otimes
X_{-}}\,e_{\star}^{i\phi\otimes H}\,e_{\star\,q^{-2}}^{i\varphi_{-}\otimes
X_{+}}\approx e_{\star}^{i\varphi_{+}\otimes
X_{-}+\frac{1}{2}\ell\lambda\varphi_{+}^{2}\otimes
X_{-}^{2}}\,e_{\star}^{i\phi\otimes H}\,e_{\star}^{i\varphi_{-}\otimes
X_{+}-\frac{1}{2}\ell\lambda\varphi_{-}^{2}\otimes X_{+}^{2}}$
$\displaystyle\approx e_{\star}^{i\varphi_{+}\otimes X_{-}+i\phi\otimes
H+i\varphi_{-}\otimes X_{+}+\frac{1}{2}\ell\lambda\varphi_{+}^{2}\otimes
X_{-}^{2}-\frac{1}{2}\ell\lambda\varphi_{-}^{2}\otimes
X_{+}^{2}+\frac{1}{2}[\phi\otimes H\,,\,\varphi_{+}\otimes
X_{-}]-\frac{1}{2}[\phi\otimes H\,,\,\varphi_{-}\otimes
X_{+}]+\frac{1}{2}[\varphi_{-}\otimes X_{+}\,,\,\varphi_{+}\otimes X_{-}]}$
$\displaystyle\approx e_{\star}^{i\varphi_{+}(1+i\ell\phi)\otimes
X_{-}+i(1+i\ell\phi)\varphi_{-}\otimes
X_{+}+i(\phi-\frac{i}{2}\ell\varphi_{+}\varphi_{-})\otimes
H+\frac{i\lambda}{2}\,(\varphi_{-}\otimes HX_{+}-\varphi_{+}\otimes
X_{-}H)}\,.$ (134)

In the case where $\lambda=0$, we are looking at the standard GFT with group
$\mathrm{SU}(2)$, and in this case $H=F(\mathrm{SU}(2))$ and
$A=\mathcal{U}(\mathfrak{su}(2))\cong F_{\star}(\mathbb{R}^{3})$. At first
order in $\ell$, we have the contribution in the exponent is given by

$\displaystyle\varphi_{+}(1+\ell\phi)\otimes
X_{-}+\varphi_{-}(1+\ell\phi)\otimes
X_{+}H+(\phi-i\ell/2\,\varphi_{-}\varphi_{+})\otimes H\,,$ (135)

which allows us to identify the coordinates $k$ in the discretization (129),
at first order in $\ell$,

$\displaystyle k_{+}=\varphi_{+}(1+\ell\phi),\quad
k_{-}=\varphi_{-}(1+\ell\phi),\quad
k_{3}=(\phi-i\ell/2\,\varphi_{-}\varphi_{+})\,.$ (136)

This means that the discretized $BF$ action with $\lambda=0$ can be written as

$\displaystyle\,{\cal S}_{\mathfrak{su}(2)}\approx\,{\cal S}_{d}^{\lambda=0}$
$\displaystyle=\Delta^{n}k^{a}\otimes X_{a}$
$\displaystyle\approx(\Delta^{n}\otimes\textrm{id})\,\big{(}\varphi_{+}(1+\ell\phi)\otimes
X_{-}+\varphi_{-}(1+\ell\phi)\otimes
X_{+}+(\phi-i\ell/2\,\varphi_{-}\varphi_{+})\otimes H\big{)}\,.$ (137)

We are now interested at looking at the case where
$H=F_{\star}(\mathbb{R}^{3})\cong{\cal U}(\mathfrak{an}(2))$ and $A=F(AN(2))$.
The contribution in the exponent is

$\displaystyle\varphi_{+}\otimes
X_{-}(1-\frac{\lambda}{2}H)+\varphi_{-}\otimes(1+\frac{\lambda}{2}H)X_{+}+\phi\otimes
H\,.$ (138)

The coordinates $(\varphi_{\pm},\phi)=(x_{\pm},x)=\textbf{x}$ are non-
commutative variables of the $\mathfrak{an}(2)$ Lie algebra type, while the
coordinates $H,X_{\pm}$ are commutative according to table 1. Noting the
coordinate $q$ on the group $\mathrm{AN}(2)$, we have then at first order in
$\lambda$

$\displaystyle q_{-}=X_{-}(1-\frac{\lambda}{2}H),\quad
q_{+}=(1+\frac{\lambda}{2}H)X_{+},\quad q_{3}=H.$ (139)

The discretized action (130) in the $\ell=0$ case becomes then

$\displaystyle\,{\cal S}_{\mathfrak{an}(2)}\approx\,{\cal S}_{d}^{\ell=0}$
$\displaystyle=(\Delta^{n}\textbf{x})^{a}\otimes q_{a}$
$\displaystyle\approx(\Delta^{n}\otimes\textrm{id})\,\big{(}\varphi_{+}\otimes
X_{-}(1-\frac{\lambda}{2}H)+\varphi_{-}\otimes(1+\frac{\lambda}{2}H)X_{+}+\phi\otimes
H\big{)}\,.$ (140)

Finally, the case where both $\ell$ and $\lambda$ are not zero corresponds to
the case specified by $H=F_{q}(\mathrm{SU}(2))\cong{\cal
U}_{q}(\mathfrak{an}(2))$ and $A=\mathcal{U}_{q}(\mathfrak{su}(2))\cong
F_{q}(AN(2))$. The approximation (134) provides therefore the basic building
block to construct the discretization of the $BF$ action ${\cal
S}^{\lambda}_{BF}$. The extra term is naturally identified with the
cosmological constant contribution in (117). Therefore, working at the first
order in $\ell$ and $\lambda$, we obtain the discretization of the the $BF$
action with cosmological constant.

$\displaystyle{\cal S}_{BF}^{\lambda}[\omega,e]$ $\displaystyle={\cal
S}_{BF}^{\lambda=0}[\omega,e]+\lambda\,\big{(}\omega_{-}\wedge e_{3}\wedge
e_{+}-\omega_{+}\wedge e_{-}\wedge e_{3}\big{)}$ $\displaystyle\approx{\cal
S}_{d}^{\lambda}$ $\displaystyle={\cal
S}_{d}^{\lambda=0}+\frac{\lambda}{2}\,\big{(}\Delta^{n}\varphi_{-}\otimes
HX_{+}-\Delta^{n}\varphi_{+}\otimes X_{-}H\big{)}\,.$ (141)

This corresponds to the generalization of the non-commutative simplicial
gravity picture introduced by Baratin and Oriti [1]. We have non-commutative
coordinates both on the triangulation side (as they did) and in the dual
complex. On the triangulation side the non-commutativity is further modified
with respect to [1] due to the presence of the cosmological constant.

##  5 Conclusion

We proposed a model based on Hopf algebras that provides a (3d) topological
state sum. The formulation of the model is a generalization of the notion
group field theories, where the fundamental degrees of freedom are encoded in
a gauge invariant field and the Feynman diagram amplitudes are expressed as a
sum over the amplitudes associated to three dimensional geometries.

The field theory is discussed both in the configuration and momentum spaces,
and a map (generalized Fourier transform) between the two is provided. The
object used as the kernel of such transformation is a generalized notion of
plane-wave given by the canonical element of $D^{*}(A,H,\sigma)$. It is the
first time, to the best of our knowledge, that such canonical element takes a
central place in the definition of of a topological model and is used to
provide a discretization of the $BF$ action.

The main scope of this work was to construct a field theory generating curved
discrete geometries as Feynman diagrams, in such a way that the amplitudes
would provide a discretization of the $BF$ action with non-vanishing
cosmological constant. In this sense, HAFT provides a non-commutative
formulation of simplicial 3d gravity. The non-commutativity appears due to the
homogeneous curvature induced by the cosmological constant (and also due to
the Planck length [1]). We emphasize that the discretization is not done at
the level of the action but instead at the level of the amplitude $e^{i{\cal
S}^{\lambda}_{BF}}$.

Let us mention several directions that we would find interesting to explore.

##### Relation with integrable systems.

The canonical element/plane-wave associated to the dual Hopf algebras also
appeared in the context of integrable systems under the shape of the
transfer/transport matrix, or T-matrix [30, 31, 32]. In this context, we
consider a lattice which sites are associated with a T-matrix $\sigma$. It can
be written as

$\sigma=e^{x\cdot k}\,,$ (142)

where $L=x\cdot k$ originates from a Lax pair and the $x^{i}$ are the
dynamical variables, which can be seen as generating a Lie algebra
$\mathfrak{g}_{x}$, while the $k_{i}$ are the generators of the Lie algebra
$\mathfrak{g}_{k}$. We can multiply the T-matrices sitting at different sites,
with dynamical variables $x$ and $x^{\prime}$ to obtain a new one
$x^{\prime\prime}=x\oplus x^{\prime}$. This is precisely the structure we
would expect from a plane-wave. The dynamical variables are equipped with a
Poisson bracket which induces a Poisson structure for the T-matrix. We expect
then that the Poisson bracket (associated with the dynamical variables) should
be compatible with the product of the T-matrices. Since there is an obvious
symmetry in (142), we can make the same type of construction for the $k$
sector. The main features of the T-matrix are captured by the expected plane-
wave properties

$\displaystyle\sigma_{x,k_{1}}\sigma_{x,k_{2}}=e^{ix\cdot k_{1}}e^{ix\cdot
k_{2}}=e^{ix\cdot(k_{1}\oplus k_{2})}\equiv\sigma_{x,(k_{1}\oplus k_{2})}\,,$
(143) $\displaystyle\sigma_{x_{1},k}\sigma_{x_{2},k}=e^{ix_{1}\cdot
k}e^{ix_{2}\cdot k}=e^{i(x_{1}\oplus x_{2})\cdot k}\equiv\sigma_{(x_{1}\oplus
x_{2})\cdot k}\,,$

such that the product is a Poisson map. Note that, at the quantum level, by
performing some projection in terms of representations, we can recover from
the T-matrix the notion of $R$-matrix [15] which also play a fundamental role
in the study of integrable systems. It would be interesting to explore whether
the fact that this common structure appears both for integrable systems and 3d
gravity could clarify some interesting questions, such as holography for
example.

##### Generalization to quasi Hopf algebras.

Our construction relied on (co-associative) Hopf algebras. The original TV
model is defined for quasi Hopf algebras, such as for the deformation at $q$
root of unity of $\mathrm{SU}(2)$. It would be interesting to explore how our
proposed framework extends to this quasi case. As a starting point we could
explore how twisting the Drinfeld double and weakening the co-associativity
would modify our construction.

##### Introducing matter.

Krasnov has argued that to introduce matter, one should not only consider a
field on some copies of $H=F(\mathrm{SU}(2))$ as we did, but instead on the
full Drinfeld double $H=F(\mathrm{SU}(2))\rtimes{\cal U}(\mathfrak{su}(2))$
[16]. It would be interesting to see how this is implemented in our general
framework so that we could deal with particles coupled to gravity in the case
where the cosmological constant is not zero. Along a similar line, it is well
understood now that the Kitaev model is equivalent to a $BF$ theory with
particle excitations, in particular the fundamental symmetry is given by the
Drinfeld double. In [22], some abstract Fourier transform was discussed, for
the specific case of a quadrangulation which is therefore self-dual. It would
be interesting to check how our proposal of Fourier transform could be applied
in this set up.

##### Four dimensional model: quantum 2-groups.

Despite the HAFT can be easily generalized to the $d$ dimensional case, we
discussed the three dimensional version of the model for two reasons. First,
in 3$d$ it is easier to visualize the underlying geometric objects. Second, if
we were to consider the 4d case, instead of using Hopf algebras we
could/should use categorified Hopf algebras according to the categorical
ladder [33]. In particular, in 4d we expect to have triangulation faces, as
dual to the links of the dual complex to be decorated. Without any edge
decoration, it is actually not possible to decorate faces by non-abelian group
elements (eg at the classical level) due to the Eckmann-Hilton argument [34].
This therefore prevents the description of a discrete curved geometry unless
we also decorate the edges. A 2-group or crossed module or their deformation
appears therefore as the natural structure to describe discrete curved
geometries in 4d. Even though there already exists a proposal for the notion
quantum 2-group (see [35]) the relevant structures and duality properties
(such as a Fourier transform for 2-groups) to extend our construction are not
clear yet. Note that some recent work has extended the notion of GFT when
using 2-groups instead of groups [36]. It is therefore an interesting question
to determine what is the notion of plane-wave when dealing with 2-groups.

## Acknowledgment

FG would like to thank A. Baratin for discussions at the early stage of this
work, a long long time ago.

##  Appendix A Further mathematical details

###  A.1 Hopf algebra proofs

Here we provide some relevant proofs regarding the bi-algebra notions
introduced in Sec. 2.

Inverse of the Fourier transform: Prop. 2.
We prove Prop. 2, which states that the map (50) is the inverse of the Fourier
transform (49). Here we first use both the definitions of Fourier transform
and inverse

$(\mathcal{F}\circ\mathcal{F}^{-1})[\hat{\Phi}]=\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\,\textrm{id}\right)\,\big{(}\sigma\cdot(\mathcal{F}^{-1}[\hat{\Phi}]\otimes
1)\big{)}=\frac{1}{\mu}\left(\int_{H}^{L}\otimes\int_{A}^{L}\otimes\textrm{id}\right)\,\big{(}\sigma_{13}\cdot\sigma_{12}^{-1}\cdot(1\otimes\hat{\Phi}\otimes
1)\big{)}\,.$ (144)

Then, using the definition of inverse skew co-pairing (36), the expression
becomes

$(\mathcal{F}\circ\mathcal{F}^{-1})[\hat{\Phi}]=\frac{1}{\mu}\left(\int_{H}^{L}\otimes\int_{A}^{L}\otimes\textrm{id}\right)\,(\textrm{id}\otimes
S\otimes\textrm{id})\,\big{(}(1\otimes S^{-1}\hat{\Phi}\otimes
1)\cdot(\sigma_{13}\cdot\sigma_{12})\big{)}\,.$ (145)

Since the co-integral of $H$ of the product of co-pairing
$\sigma_{13}\cdot\sigma_{12}$ can be written as the co-product of the delta
function of $A$,

$\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\textrm{id}\otimes\textrm{id}\right)\,(\sigma_{13}\cdot\sigma_{12})=\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\Delta\right)\,\sigma=\Delta\hat{\delta}_{A}\,,$
(146)

one can use the first of the identities (55) to reduce the expression as

$(\mathcal{F}\circ\mathcal{F}^{-1})[\hat{\Phi}]=\frac{1}{\sqrt{\mu}}\left(\int_{A}^{L}\otimes\textrm{id}\right)\,(S\otimes\textrm{id})\,\big{(}(S^{-1}\hat{\Phi}\otimes
1)\cdot\Delta\hat{\delta}_{A}\big{)}=\frac{1}{\sqrt{\mu}}\left(\int_{A}^{L}\otimes\textrm{id}\right)\,(S\otimes\textrm{id})\,\big{(}(1\otimes\hat{\Phi})\cdot\Delta\hat{\delta}_{A}\big{)}\,.$
(147)

Last, since the antipode is an anti-homomorphism map, is passes through the
co-product, which becomes $\Delta_{21}$. This allows us to use the left
invariance property of the co-integral and compute it using the normalization
property of the delta function (56). This closes the proof of the proposition.

$(\mathcal{F}\circ\mathcal{F}^{-1})[\hat{\Phi}]=\frac{1}{\sqrt{\mu}}\left(\int_{A}^{L}\otimes\textrm{id}\right)\,\big{(}\Delta_{-21}(S\hat{\delta}_{A})\cdot(1\otimes\hat{\Phi})\big{)}=\frac{1}{\sqrt{\mu}}\left(\int_{A}^{L}\otimes\textrm{id}\right)\,\big{(}(S\hat{\delta}_{A}\otimes
1)\cdot(1\otimes\hat{\Phi})\big{)}=\hat{\Phi}\,.$ (148)

Similarly, we can also prove the identity
$(\mathcal{F}^{-1}\circ\mathcal{F})=\textrm{id}$.

Delta function of an Hopf algebra: Prop. 3.
Let us prove that the delta function of the algebra $A$, defined in (53),
satisfies the identity (55) of Prop. 3. We use first the definition of delta
function and that of Fourier transform, plus the property (34) of the skew co-
pairing element.

$\displaystyle(\hat{\Phi}\otimes 1)\cdot\Delta\hat{\delta}_{A}$
$\displaystyle=\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\textrm{id}\otimes\textrm{id}\right)\,\big{(}(1\otimes\hat{\Phi}\otimes
1)\cdot(\textrm{id}\otimes\Delta)\,\sigma\big{)}=\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\textrm{id}\otimes\textrm{id}\right)\,\big{(}(1\otimes\hat{\Phi}\otimes
1)\cdot\sigma_{13}\cdot\sigma_{12}\big{)}$
$\displaystyle=\frac{1}{\mu}\left(\int_{H}^{L}\otimes\int_{H}^{L}\otimes\textrm{id}\otimes\textrm{id}\right)\,\big{(}\sigma_{13}\cdot\sigma_{24}\cdot\sigma_{23}\cdot(\Phi\otimes
1\otimes 1\otimes 1)\big{)}\,.$ (149)

Now multiply the skew co-pairings from the left by
$\sigma_{14}^{-1}\cdot\sigma_{14}$ and use the property (35), plus the left
invariance of the co-integral on $H$.

$\displaystyle(\hat{\Phi}\otimes 1)\cdot\Delta\hat{\delta}_{A}$
$\displaystyle=\frac{1}{\mu}\left(\int_{H}^{L}\otimes\int_{H}^{L}\otimes\textrm{id}\otimes\textrm{id}\right)\,\big{(}\sigma_{14}^{-1}\cdot\sigma_{14}\cdot\sigma_{24}\cdot\sigma_{13}\cdot\sigma_{23}\cdot(\Phi\otimes
1\otimes 1\otimes 1)\big{)}$
$\displaystyle=\frac{1}{\mu}\left(\int_{H}^{L}\otimes\int_{H}^{L}\otimes\textrm{id}\otimes\textrm{id}\right)\,\big{(}\sigma_{14}^{-1}\cdot(\Delta\otimes\textrm{id}\otimes\textrm{id})\,(\sigma_{12}\cdot\sigma_{13})\cdot(\Phi\otimes
1\otimes 1\otimes 1)\big{)}$
$\displaystyle=\frac{1}{\mu}\left(\int_{H}^{L}\otimes\int_{H}^{L}\otimes\textrm{id}\otimes\textrm{id}\right)\,\big{(}\sigma_{14}^{-1}\cdot\sigma_{23}\cdot\sigma_{24}\cdot(\Phi\otimes
1\otimes 1\otimes 1)\big{)}\,.$ (150)

Last, using the definitions of inverse skew co-pairing (36), that of Fourier
transform and delta function, we prove the first part of the proposition.

$\displaystyle(\hat{\Phi}\otimes 1)\cdot\Delta\hat{\delta}_{A}$
$\displaystyle=\frac{1}{\mu}\left(\int_{H}^{L}\otimes\int_{H}^{L}\otimes\textrm{id}\otimes\textrm{id}\right)\,\big{(}(\textrm{id}\otimes\textrm{id}\otimes\textrm{id}\otimes
S)\,\sigma_{14}\cdot(\Phi\otimes 1\otimes 1\otimes
1)\cdot(\textrm{id}\otimes\textrm{id}\otimes\Delta)\,\sigma_{23}\big{)}$
$\displaystyle=\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\textrm{id}\otimes\textrm{id}\right)\,\big{(}(\textrm{id}\otimes\textrm{id}\otimes
S)\,\sigma_{13}\cdot(\Phi\otimes 1\otimes
1)\big{)}\,\cdot\,(\textrm{id}\otimes\Delta)\,\hat{\delta}_{A}$
$\displaystyle=(1\otimes S\hat{\Phi})\cdot\Delta\hat{\delta}_{A}\,.$ (151)

The proofs of the other three identities of the first part of Prop. 3 are
similar.
The proof of normalization of the delta function (56), in the second part of
the proposition, is straightforward.

Delta function of an Hopf algebra as integral in it: Prop. 4.
We show that the delta function in the Hopf algebra $A$ is a left integral in
it, as stated in Prop. 4. First note that the co-unit of an element of $A$
coincides with the co-integral of its dual:

$\varepsilon\hat{\Phi}=\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\varepsilon\right)\,\big{(}\sigma\cdot(\Phi\otimes
1)\big{)}=\frac{1}{\sqrt{\mu}}\,\int_{H}^{L}\,\Phi\,.$ (152)

Here we simply used the definition of Fourier transform (49) and the unit
property (34) of the skew co-pairing element. Now take the product
$\hat{\Phi}\cdot\hat{\delta}_{A}$, use the definition of delta function (53)
and that of Fourier transform.

$\hat{\Phi}\cdot\hat{\delta}_{A}=\frac{1}{\sqrt{\mu}}\left(\int_{H}^{L}\otimes\textrm{id}\right)\,\big{(}(1\otimes\hat{\Phi})\cdot\sigma\big{)}=\frac{1}{\mu}\left(\int_{H}^{L}\otimes\int_{H}^{L}\otimes\textrm{id}\right)\,\big{(}\sigma_{13}\cdot\sigma_{23}\cdot(\Phi\otimes
1\otimes 1)\big{)}\,.$ (153)

Then, using the property (35) and the left invariance of the co-integral on
$H$, we prove the identity.

$\displaystyle\hat{\Phi}\cdot\hat{\delta}_{A}$
$\displaystyle=\frac{1}{\mu}\left(\int_{H}^{L}\otimes\int_{H}^{L}\otimes\textrm{id}\right)\,\big{(}(\Delta\otimes\textrm{id})\sigma\cdot(\Phi\otimes
1\otimes
1)\big{)}=\frac{1}{\mu}\left(\int_{H}^{L}\otimes\int_{H}^{L}\otimes\textrm{id}\right)\,\big{(}\sigma_{23}\cdot(\Phi\otimes
1\otimes 1)\big{)}$
$\displaystyle=\frac{1}{\sqrt{\mu}}\,\hat{\delta}_{A}\int_{H}\Phi=\hat{\delta}_{A}\,\varepsilon(\hat{\Phi}).$
(154)

Similarly, one can prove that $\delta_{H}$ is the left integral in $H$ and
that $\hat{\delta}_{A}^{-1}$ and $\delta_{H}^{-1}$ defined in (54) are the
right integrals respectively in $A$ and $H$.
Let us provide a further property of the skew co-pairing element.

######  Proposition 14 (Integral property of the skew co-pairing element).

Let $H$ and $A$ be skew co-paired bi-algebras with invertible skew co-pairing.
It satisfies the identities below.

$\begin{aligned}
&\left(\int_{H}\otimes\textrm{id}\right)\,\sigma=\left(\int_{H}\otimes\textrm{id}\otimes\textrm{id}\right)\,(\sigma^{\pm
1}_{23}\cdot\sigma_{13})\,,\\\
&\left(\int_{H}\otimes\textrm{id}\right)\,\sigma^{-1}=\left(\int_{H}\otimes\textrm{id}\otimes\textrm{id}\right)\,(\sigma^{-1}_{13}\cdot\sigma^{\pm
1}_{23})\,,\end{aligned}\qquad\begin{aligned}
&\left(\textrm{id}\otimes\int_{A}\right)\,\sigma=\left(\textrm{id}\otimes\textrm{id}\otimes\int_{A}\right)\,(\sigma_{13}\cdot\sigma^{\pm
1}_{12})\,,\\\
&\left(\textrm{id}\otimes\int_{A}\right)\,\sigma^{-1}=\left(\textrm{id}\otimes\textrm{id}\otimes\int_{A}\right)\,(\sigma^{\pm
1}_{12}\cdot\sigma^{-1}_{13})\,.\end{aligned}$ (155)

###### Proof.

Consider the first of the identities (155). We prove it simply using the
property (35) and the left invariance of the co-integral on $H$.

$\displaystyle\left(\int_{H}\otimes\textrm{id}\right)\,\sigma$
$\displaystyle=\left(\int_{H}\otimes\textrm{id}\otimes\textrm{id}\right)\,(\sigma_{23}^{-1}\cdot\sigma_{23}\cdot\sigma_{13})$
$\displaystyle=\left(\int_{H}\otimes\textrm{id}\otimes\textrm{id}\right)\,(\sigma_{23}^{-1}\cdot(\Delta_{21}\otimes\textrm{id})\,\sigma)=\left(\int_{H}\otimes\textrm{id}\otimes\textrm{id}\right)\,(\sigma_{23}^{-1}\cdot\sigma_{13})\,.$
(156)

Alternatively, using the same properties, we have

$\displaystyle\left(\int_{H}\otimes\textrm{id}\right)\,\sigma$
$\displaystyle=\left(\int_{H}\otimes\textrm{id}\otimes\textrm{id}\right)\,(\sigma_{23}\cdot\sigma^{-1}_{23}\cdot\sigma_{13})=\left(\int_{H}\otimes\textrm{id}\otimes\textrm{id}\right)\,(\sigma_{23}\cdot(\textrm{id}\otimes
S^{-1}\otimes\textrm{id})\,(\sigma_{23}\cdot\sigma_{13}))$
$\displaystyle=\left(\int_{H}\otimes\textrm{id}\otimes\textrm{id}\right)\,(\sigma_{23}\cdot(\textrm{id}\otimes
S^{-1}\otimes\textrm{id})\circ(\Delta_{21}\otimes\textrm{id})\,\sigma)=\left(\int_{H}\otimes\textrm{id}\otimes\textrm{id}\right)\,(\sigma_{23}\cdot\sigma_{13})\,.$
(157)

The other three identities in (155) can be proved in a similar way. ∎

Note that, by direct application of the above proposition, it is
straightforward to derive the further identities

$\left(\int_{H}\otimes\textrm{id}\right)\,\sigma=\left(\int_{H}\otimes\textrm{id}\right)\,\sigma^{-1}\,\,,\quad\left(\textrm{id}\otimes\int_{A}\right)\,\sigma=\left(\textrm{id}\otimes\int_{A}\right)\,\sigma^{-1}\,.$
(158)

###  A.2 Geometric computations

Let us provide here some useful results which will be used later in this
section. The computations below are strictly related to the geometric
interpretation of Hopf algebra field theory, as a path integral formulation of
a model of discrete curved geometries.
We remind that the closure constraint (63) represents the closure of the
boundary of a triangle. Let us use the tensor notation $\hat{{\cal
C}}=\hat{{\cal C}}_{1\,2\,3}\in A^{3}$. We list below some propositions
involving the geometric operations on the closure constraint that we will need
in Hopf algebra field theory, such as the cyclic permutation of the edges,
inversion or gluing of two triangles.

######  Proposition 15 (Cyclic permutation and inversion properties of a the
closure constraint).

The closure constraint (63) is invariant under cyclic permutation of its
elements and under inversion.

$\displaystyle\hat{{\cal C}}_{1\,2\,3}=\hat{{\cal C}}_{-3\,-2\,-1}\,,$ (159)
$\displaystyle\hat{{\cal C}}_{1\,2\,3}=\hat{{\cal C}}_{2\,3\,1}\,.$

Moreover, the closure constraint is a projector,

$\hat{{\cal C}}\cdot\hat{{\cal C}}=\hat{{\cal C}}\,.$ (160)

###### Proof.

Let us prove first the invariance under the inversion. We carry out the
computation just using the definition of closure constraint (63) and that of
opposite delta function (54), plus the left invariance of the co-integral on
$H$.

$\displaystyle\hat{{\cal C}}$
$\displaystyle=\frac{1}{V_{H}}\left(\int_{H}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}(\textrm{id}\otimes\Delta^{3})\,\sigma^{-1}\big{)}=\frac{1}{V_{H}}\left(\int_{H}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}\sigma^{-1}_{12}\sigma^{-1}_{13}\sigma^{-1}_{14}\big{)}$
$\displaystyle=\frac{1}{V_{H}^{2}}\left(\int_{H^{2}}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}(\sigma^{-1}_{13}\sigma^{-1}_{14}\sigma^{-1}_{15})\cdot(\sigma^{-1}_{23}\sigma^{-1}_{24}\sigma^{-1}_{25})\cdot(\sigma_{25}\sigma_{24}\sigma_{23})\big{)}\qquad\text{\tiny{use
\eqref{SkewEl_CoProduct-id} and the left invariance}}$
$\displaystyle=\frac{1}{V_{H}^{2}}\left(\int_{H^{2}}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}(\sigma^{-1}_{13}\sigma^{-1}_{14}\sigma^{-1}_{15})\cdot(\sigma_{25}\sigma_{24}\sigma_{23})\big{)}\qquad\text{\tiny{use
\eqref{SkewEl_CoProduct-id} with the antipode and the left invariance}}$
$\displaystyle=\frac{1}{V_{H}^{2}}\left(\int_{H^{2}}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}\sigma_{25}\sigma_{24}\sigma_{23}\big{)}=\frac{1}{V_{H}}\left(\int_{H}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}\sigma_{14}\sigma_{13}\sigma_{12}\big{)}=S^{3}\circ\tau^{3}\,\hat{{\cal
C}}=\hat{{\cal C}}_{-3\,-2\,-1}\,.$ (161)

We can prove in a similar way the invariance under cyclic permutations by
multiplying the closure constraint by the properly chosen identity.

The fact that the product of two closure constraints reduces to a single one,
eq. (160), is expected since the closure constraint is the Fourier transform
of a projector. We carry out the proof writing the closure constraints in
terms of skew co-pairing elements and then using the left invariance of the
co-integral on $H$.

$\displaystyle\hat{{\cal C}}\cdot\hat{{\cal C}}$
$\displaystyle=\frac{\mu}{V_{H}^{2}}\big{(}\Delta^{3}\hat{\delta}_{A}^{-1}\cdot\Delta^{3}\hat{\delta}_{A}^{-1}\big{)}$
$\displaystyle=\frac{1}{V_{H}^{2}}\left(\int_{H^{2}}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}(\sigma^{-1}_{13}\sigma^{-1}_{14}\sigma^{-1}_{15})\cdot(\sigma^{-1}_{23}\sigma^{-1}_{24}\sigma^{-1}_{25})\big{)}\qquad\text{\tiny{use
\eqref{SkewEl_CoProduct-id_inv} and the left invariance}}$
$\displaystyle=\frac{1}{V_{H}^{2}}\left(\int_{H^{2}}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}\sigma^{-1}_{13}\sigma^{-1}_{14}\sigma^{-1}_{15}\big{)}=\frac{1}{V_{H}}\left(\int_{H}\otimes\textrm{id}^{\otimes
3}\right)\,\big{(}\sigma^{-1}_{12}\sigma^{-1}_{14}\sigma^{-1}_{15}\big{)}=\hat{{\cal
C}}\,.$ (162)

∎

We now consider the kernel of the vertex amplitude (71) associated to a
tetrahedron and illustrate how to reduce its expression. The reduction
consists in gluing the closure constraints in it, by evaluating the co-product
$\underline{\Delta}$. Geometrically, it amounts to identifying the edges of
the four triangles pairwise.

######  Proposition 16 (Gluing triangles).

Consider the kernel (71) and two closure constraints (63) in it. Consider also
the co-product $\underline{\Delta}$, given in (23) (which is a combination of
six co-products), and take the co-product $\Delta_{ij}$ shared by the two
closure constraints. The gluing of the two triangles is performed by taking
the co-integral on the $j^{th}$ tensor space:

$\int_{A_{j_{1}}}\,\big{(}\Delta_{i_{1}\,j_{1}}\hat{\delta}_{A}^{-1}\cdot{\cal
C}_{i_{1}\,i_{2}\,i_{3}}\cdot{\cal
C}_{j_{1}\,j_{2}\,j_{3}}\big{)}=\sqrt{\mu}\,{\cal
C}_{-i_{1}\,j_{2}\,j_{3}}\cdot{\cal C}_{i_{1}\,i_{2}\,i_{3}}\,.$ (163)

Similarly, the following identities hold.

$\displaystyle\int_{A_{j_{1}}}\,\big{(}\Delta_{-i_{1}\,j_{1}}\hat{\delta}_{A}^{-1}\cdot{\cal
C}_{i_{1}\,i_{2}\,i_{3}}\cdot{\cal
C}_{j_{1}\,j_{2}\,j_{3}}\big{)}=\sqrt{\mu}\,{\cal
C}_{i_{1}\,i_{2}\,i_{3}}\cdot{\cal C}_{i_{1}\,j_{2}\,j_{3}}\,,$ (164)
$\displaystyle\int_{A_{j_{1}}}\,\big{(}{\cal
C}_{i_{1}\,i_{2}\,i_{3}}\cdot{\cal
C}_{j_{1}\,j_{2}\,j_{3}}\cdot\Delta_{i_{1}\,-j_{1}}\hat{\delta}_{A}^{-1}\big{)}=\sqrt{\mu}\,{\cal
C}_{i_{1}\,i_{2}\,i_{3}}\cdot{\cal C}_{i_{1}\,j_{2}\,j_{3}}\,,$
$\displaystyle\int_{A_{j_{1}}}\,\big{(}{\cal
C}_{i_{1}\,i_{2}\,i_{3}}\cdot{\cal
C}_{j_{1}\,j_{2}\,j_{3}}\cdot\Delta_{-i_{1}\,-j_{1}}\hat{\delta}_{A}^{-1}\big{)}=\sqrt{\mu}\,{\cal
C}_{-i_{1}\,j_{2}\,j_{3}}\cdot{\cal C}_{i_{1}\,i_{2}\,i_{3}}\,.$

###### Proof.

In order to prove the proposition we first use the property (55) for the co-
product of the opposite delta function, moving the first component of the
second closure constraint from the $j_{1}^{th}$ to the $i_{1}^{th}$ tensor
space.

$\int_{A_{j_{1}}}\,\big{(}\Delta_{i_{1}\,j_{1}}\hat{\delta}_{A}^{-1}\cdot{\cal
C}_{i_{1}\,i_{2}\,i_{3}}\cdot{\cal
C}_{j_{1}\,j_{2}\,j_{3}}\big{)}=\int_{A_{j_{1}}}\,\big{(}\Delta_{i_{1}\,j_{1}}\hat{\delta}_{A}^{-1}\cdot{\cal
C}_{-i_{1}\,j_{2}\,j_{3}}\cdot{\cal C}_{i_{1}\,i_{2}\,i_{3}}\big{)}\,.$ (165)

Then use the left invariance of the co-integral and the normalization property
(56) to remove the opposite delta function.

$\int_{A_{j_{1}}}\,\big{(}\Delta_{i_{1}\,j_{1}}\hat{\delta}_{A}^{-1}\cdot{\cal
C}_{i_{1}\,i_{2}\,i_{3}}\cdot{\cal
C}_{j_{1}\,j_{2}\,j_{3}}\big{)}=\left(\int_{A}\,\hat{\delta}_{A}^{-1}\right)\,\big{(}{\cal
C}_{-i_{1}\,j_{2}\,j_{3}}\cdot{\cal
C}_{i_{1}\,i_{2}\,i_{3}}\big{)}=\sqrt{\mu}\,{\cal
C}_{-i_{1}\,j_{2}\,j_{3}}\cdot{\cal C}_{i_{1}\,i_{2}\,i_{3}}\,.$ (166)

The proof of the other three identities (164) is completely analogous. ∎

Let us now illustrate how to glue two tetrahedra (amplitude (71)) by
identifying their faces (one closure constraint for each amplitude).

######  Proposition 17 (Gluing two tetrahedra).

Consider the tensor product of a pair of independent tetrahedron amplitudes
(196) obtained by the use of Prop. 16; take their product with a propagator
amplitude (74). The gluing of two tetrahedra is performed by taking the co-
integral of such product:

$\displaystyle\int_{A^{3}_{j_{2}\,j_{4}\,j_{6}}}\,$ $\displaystyle(\hat{{\cal
C}}_{-i_{6}-i_{4}-i_{2}}\cdot\hat{{\cal C}}_{-i_{5}i_{6}-i_{1}}\cdot\hat{{\cal
C}}_{-i_{3}i_{4}i_{5}}\cdot\hat{{\cal
C}}_{i_{1}i_{2}i_{3}})\cdot\Delta^{(3)}_{i_{1}i_{2}i_{3}\,\,j_{4}j_{6}j_{2}}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-j_{6}-j_{4}-j_{2}}\cdot\hat{{\cal C}}_{-j_{5}j_{6}-j_{1}}\cdot\hat{{\cal
C}}_{-j_{3}j_{4}j_{5}}\cdot\hat{{\cal C}}_{j_{1}j_{2}j_{3}})$
$\displaystyle=\sqrt{\mu^{3}}\,(\hat{{\cal
C}}_{-i_{6}-i_{4}-i_{2}}\cdot\hat{{\cal C}}_{-i_{5}i_{6}-i_{1}}\cdot\hat{{\cal
C}}_{-i_{3}i_{4}i_{5}}\cdot\hat{{\cal C}}_{i_{1}i_{2}i_{3}})\cdot(\hat{{\cal
C}}_{-j_{5}-i_{2}-j_{1}}\cdot\hat{{\cal C}}_{-j_{3}-i_{1}j_{5}}\cdot\hat{{\cal
C}}_{-i_{3}j_{3}j_{1}})\,.$ (167)

As in Prop. 16, there are other identities similar to (167), involving
antipodes or different combinatorics, whose derivation is completely analogous
to that of the above identity.

###### Proof.

The proof of the proposition follows the same pattern of Prop. 16: we use the
property (55) to move the components of the second amplitude from the
$j_{1}j_{2}j_{3}$ to the $i_{1}i_{2}i_{3}$ tensor spaces, so that we can use
the left invariance of the co-integral and the normalization property (56) to
remove the propagator amplitude. Note that, for non-commutativity reasons, in
order to correctly use the property (55), some of the closure constraints of
the second tetrahedron amplitude have to be permuted or inverted using Prop.
15.

$\displaystyle\int_{A^{3}_{j_{2}\,j_{4}\,j_{6}}}\,$ $\displaystyle(\hat{{\cal
C}}_{-i_{6}-i_{4}-i_{2}}\cdot\hat{{\cal C}}_{-i_{5}i_{6}-i_{1}}\cdot\hat{{\cal
C}}_{-i_{3}i_{4}i_{5}}\cdot\hat{{\cal
C}}_{i_{1}i_{2}i_{3}})\cdot\Delta^{(3)}_{i_{1}i_{2}i_{3}\,\,j_{4}j_{6}j_{2}}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-j_{6}-j_{4}-j_{2}}\cdot\hat{{\cal C}}_{-j_{5}j_{6}-j_{1}}\cdot\hat{{\cal
C}}_{-j_{3}j_{4}j_{5}}\cdot\hat{{\cal C}}_{j_{1}j_{2}j_{3}})$
$\displaystyle=\int_{A^{3}_{j_{2}\,j_{4}\,j_{6}}}\,(\hat{{\cal
C}}_{-i_{6}-i_{4}-i_{2}}\cdot\hat{{\cal C}}_{-i_{5}i_{6}-i_{1}}\cdot\hat{{\cal
C}}_{-i_{3}i_{4}i_{5}}\cdot\hat{{\cal
C}}_{i_{1}i_{2}i_{3}})\cdot\Delta^{(3)}_{i_{1}i_{2}i_{3}\,\,j_{4}j_{6}j_{2}}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{j_{4}j_{6}j_{2}}\cdot\hat{{\cal C}}_{-j_{5}j_{6}-j_{1}}\cdot\hat{{\cal
C}}_{-j_{3}j_{4}j_{5}}\cdot\hat{{\cal C}}_{j_{2}j_{3}j_{1}})$
$\displaystyle=\int_{A^{3}_{j_{2}\,j_{4}\,j_{6}}}\,(\hat{{\cal
C}}_{-i_{6}-i_{4}-i_{2}}\cdot\hat{{\cal C}}_{-i_{5}i_{6}-i_{1}}\cdot\hat{{\cal
C}}_{-i_{3}i_{4}i_{5}}\cdot\hat{{\cal
C}}_{i_{1}i_{2}i_{3}})\cdot\Delta^{(3)}_{i_{1}i_{2}i_{3}\,\,j_{4}j_{6}j_{2}}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-i_{1}-i_{2}-i_{3}}\cdot\hat{{\cal
C}}_{-j_{5}-i_{2}-j_{1}}\cdot\hat{{\cal C}}_{-j_{3}-i_{1}j_{5}}\cdot\hat{{\cal
C}}_{-i_{3}j_{3}j_{1}})$
$\displaystyle=\Big{(}\int_{A^{3}_{j_{4}\,j_{4}\,j_{2}}}\,\hat{\delta}_{A}^{-1}\Big{)}\,(\hat{{\cal
C}}_{-i_{6}-i_{4}-i_{2}}\cdot\hat{{\cal C}}_{-i_{5}i_{6}-i_{1}}\cdot\hat{{\cal
C}}_{-i_{3}i_{4}i_{5}}\cdot\hat{{\cal C}}_{i_{1}i_{2}i_{3}})\cdot(\hat{{\cal
C}}_{-i_{1}-i_{2}-i_{3}}\cdot\hat{{\cal
C}}_{-j_{5}-i_{2}-j_{1}}\cdot\hat{{\cal C}}_{-j_{3}-i_{1}j_{5}}\cdot\hat{{\cal
C}}_{-i_{3}j_{3}j_{1}})$ $\displaystyle=\sqrt{\mu^{3}}\,(\hat{{\cal
C}}_{-i_{6}-i_{4}-i_{2}}\cdot\hat{{\cal C}}_{-i_{5}i_{6}-i_{1}}\cdot\hat{{\cal
C}}_{-i_{3}i_{4}i_{5}}\cdot\hat{{\cal C}}_{i_{1}i_{2}i_{3}})\cdot(\hat{{\cal
C}}_{-i_{1}-i_{2}-i_{3}}\cdot\hat{{\cal
C}}_{-j_{5}-i_{2}-j_{1}}\cdot\hat{{\cal C}}_{-j_{3}-i_{1}j_{5}}\cdot\hat{{\cal
C}}_{-i_{3}j_{3}j_{1}})$ $\displaystyle=\sqrt{\mu^{3}}\,(\hat{{\cal
C}}_{-i_{6}-i_{4}-i_{2}}\cdot\hat{{\cal C}}_{-i_{5}i_{6}-i_{1}}\cdot\hat{{\cal
C}}_{-i_{3}i_{4}i_{5}}\cdot\hat{{\cal C}}_{i_{1}i_{2}i_{3}})\cdot(\hat{{\cal
C}}_{-j_{5}-i_{2}-j_{1}}\cdot\hat{{\cal C}}_{-j_{3}-i_{1}j_{5}}\cdot\hat{{\cal
C}}_{-i_{3}j_{3}j_{1}})\,.$ (168)

In the last step we used a the identity (160) to remove the closure constraint
$\hat{{\cal C}}_{-i_{1}-i_{2}-i_{3}}$. ∎

Finally, we give the details of the gluing of three tetrahedra amplitudes (71)
sharing a single internal edge.

######  Proposition 18 (Gluing three tetrahedra: internal edge).

Consider the tensor product of three independent tetrahedron amplitudes (196).
Take their product with three propagators as in Prop. 17, in such a way that
the three pairs of closure constraints identified (multiplied by the
propagators) share a single tensor space (a common edge). Focus on the six
closure constraints involved in the product. The pairwise identification of
the six triangles (closure constraints) sharing one edge is performed by
taking the co-integral of such product:

$\displaystyle\int_{A^{3}_{i,j,k}}$ $\displaystyle(\hat{{\cal
C}}_{-i\,i_{3}\,i_{4}}\cdot\hat{{\cal
C}}_{i\,i_{1}\,i_{2}})\cdot\Delta_{-i\,j}\hat{\delta}_{A}^{-1}\cdot\Delta_{-i\,k}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-j\,j_{3}\,j_{4}}\cdot\hat{{\cal
C}}_{j\,j_{1}\,j_{2}})\cdot\Delta_{-j\,k}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-k\,k_{3}\,k_{4}}\cdot\hat{{\cal C}}_{k\,k_{1}\,k_{2}})$ (169)
$\displaystyle=\sqrt{\mu^{3}}\,\big{(}\hat{{\cal
C}}_{i_{4}\,i_{3}\,i_{1}\,i_{2}}\cdot\hat{{\cal
C}}_{-i_{3}\,-i_{4}\,j_{3}\,j_{4}}\cdot\hat{{\cal
C}}_{i_{4}\,i_{3}\,j_{1}\,j_{2}}\cdot\hat{{\cal
C}}_{-i_{3}\,-i_{4}\,k_{3}\,k_{4}}\cdot\hat{{\cal
C}}_{i_{4}\,i_{3}\,k_{1}\,k_{2}}\big{)}\,.$

Here the co-integral is on the Hopf algebras related to the internal edge:
tensor spaces $i,j,k$.

###### Proof.

The proof follows a pattern similar to that of Prop. 17. We first use the
identity (55) on the co-product $\Delta_{-i\,k}$ and then the left invariance
of the co-integral on the $k^{th}$ tensor space and the normalization property
of the delta function (56) to remove this co-product.

$\displaystyle\int_{A^{3}_{i,j,k}}\,$ $\displaystyle(\hat{{\cal
C}}_{-i\,i_{3}\,i_{4}}\cdot\hat{{\cal
C}}_{i\,i_{1}\,i_{2}})\cdot\Delta_{-i\,j}\hat{\delta}_{A}^{-1}\cdot\Delta_{-i\,k}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-j\,j_{3}\,j_{4}}\cdot\hat{{\cal
C}}_{j\,j_{1}\,j_{2}})\cdot\Delta_{-j\,k}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-k\,k_{3}\,k_{4}}\cdot\hat{{\cal C}}_{k\,k_{1}\,k_{2}})$
$\displaystyle=\int_{A^{3}_{i,j,k}}\,(\hat{{\cal
C}}_{-i\,i_{3}\,i_{4}}\cdot\hat{{\cal
C}}_{i\,i_{1}\,i_{2}})\cdot\Delta_{-i\,j}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-j\,j_{3}\,j_{4}}\cdot\hat{{\cal
C}}_{j\,j_{1}\,j_{2}})\cdot\Delta_{-j\,i}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-i\,k_{3}\,k_{4}}\cdot\hat{{\cal
C}}_{i\,k_{1}\,k_{2}})\cdot\Delta_{-i\,k}\hat{\delta}_{A}^{-1}$
$\displaystyle=\int_{A^{2}_{i,j}}\,(\hat{{\cal
C}}_{-i\,i_{3}\,i_{4}}\cdot\hat{{\cal
C}}_{i\,i_{1}\,i_{2}})\cdot\Delta_{-i\,j}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-j\,j_{3}\,j_{4}}\cdot\hat{{\cal
C}}_{j\,j_{1}\,j_{2}})\cdot\Delta_{-j\,i}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-i\,k_{3}\,k_{4}}\cdot\hat{{\cal
C}}_{i\,k_{1}\,k_{2}})\,\Big{(}\int_{A_{k}}\,\hat{\delta}_{A}^{-1}\Big{)}$
$\displaystyle=\sqrt{\mu}\,\int_{A^{2}_{i,j}}\,(\hat{{\cal
C}}_{-i\,i_{3}\,i_{4}}\cdot\hat{{\cal
C}}_{i\,i_{1}\,i_{2}})\cdot\Delta_{-i\,j}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-j\,j_{3}\,j_{4}}\cdot\hat{{\cal
C}}_{j\,j_{1}\,j_{2}})\cdot\Delta_{-j\,i}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-i\,k_{3}\,k_{4}}\cdot\hat{{\cal C}}_{i\,k_{1}\,k_{2}})\,.$ (170)

Then repeating the same procedure with the co-product $\Delta_{-i\,j}$, we get

$\displaystyle\int_{A^{3}_{i,j,k}}\,$ $\displaystyle(\hat{{\cal
C}}_{-i\,i_{3}\,i_{4}}\cdot\hat{{\cal
C}}_{i\,i_{1}\,i_{2}})\cdot\Delta_{-i\,j}\hat{\delta}_{A}^{-1}\cdot\Delta_{-i\,k}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-j\,j_{3}\,j_{4}}\cdot\hat{{\cal
C}}_{j\,j_{1}\,j_{2}})\cdot\Delta_{-j\,k}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-k\,k_{3}\,k_{4}}\cdot\hat{{\cal C}}_{k\,k_{1}\,k_{2}})$
$\displaystyle=\mu\,\int_{A_{i}}\,(\hat{{\cal
C}}_{-i\,i_{3}\,i_{4}}\cdot\hat{{\cal C}}_{i\,i_{1}\,i_{2}})\cdot(\hat{{\cal
C}}_{-i\,j_{3}\,j_{4}}\cdot\hat{{\cal
C}}_{i\,j_{1}\,j_{2}})\cdot\Delta_{-i\,i}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-i\,k_{3}\,k_{4}}\cdot\hat{{\cal C}}_{i\,k_{1}\,k_{2}})$
$\displaystyle=V_{H}\sqrt{\mu}\,\int_{A_{i}}\,(\hat{{\cal
C}}_{-i\,i_{3}\,i_{4}}\cdot\hat{{\cal C}}_{i\,i_{1}\,i_{2}})\cdot(\hat{{\cal
C}}_{-i\,j_{3}\,j_{4}}\cdot\hat{{\cal C}}_{i\,j_{1}\,j_{2}})\cdot(\hat{{\cal
C}}_{-i\,k_{3}\,k_{4}}\cdot\hat{{\cal C}}_{i\,k_{1}\,k_{2}})\,.$ (171)

Here in the last step we used that
$\sqrt{\mu}\,\Delta_{-i\,i}\hat{\delta}_{A}^{-1}=V_{H}$, which is
straightforward to verify. The last step to prove the proposition consists in
using the definition of closure constraint (63) for the first constraint
$\hat{{\cal C}}_{-i\,i_{1}\,i_{2}}$, together with the inverse property (159).
Use the property (55) on this co-product, then the left invariance of the co-
integral on the $i^{th}$ tensor space and the normalization property of the
delta to remove it.

$\displaystyle\int_{A^{3}_{i,j,k}}\,$ $\displaystyle(\hat{{\cal
C}}_{-i\,i_{3}\,i_{4}}\cdot\hat{{\cal
C}}_{i\,i_{1}\,i_{2}})\cdot\Delta_{-i\,j}\hat{\delta}_{A}^{-1}\cdot\Delta_{-i\,k}\hat{\delta}_{A}^{-1}\cdot(\hat{{\cal
C}}_{-j\,j_{3}\,j_{4}}\cdot\hat{{\cal