Quantum deformation of two four-dimensional spin foam models
We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the EPRL model. The q-deformed models are based on the representation theory of two copies of at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models we give a definition of the quantum EPRL intertwiners, study their convergence and braiding properties and construct an amplitude for the four-simplexes. We find that both of the resulting models are convergent.
Background and motivation
Manifold invariants of representation theoretic origin such as the Reshetikhin-Turaev invariant  or the Turaev-Viro invariant  play an important role in mathematical physics. In particular, they are of interest to quantum gravity, where models of the Turaev-Viro type are known under the name of spin foam models. The basic principle in the construction of a dimensional spin foam model is to assign irreducible representations to the -simplexes of a triangulated manifold and intertwining operators to the -simplexes which intertwine the representations associated to their boundaries. Higher-dimensional simplexes in the triangulation are then decorated with representation theoretical data constructed from these representations and intertwiners, often referred to as “amplitudes” in the literature. After a summation over all such assignments of representations and intertwiners the product of the amplitudes assigned to each higher-dimensional simplex define a number, the partition function of the model. In some three- and higher-dimensional models, one can show that the partition function converges and is independent of the choice of the triangulation. It thus defines a manifold invariant.
Among the spin foam models in the literature, one distinguishes models which are based on the representation theory of Lie groups (in the following referred to as “classical models”) and models which are based on the representation theory of quantum groups. Compared to the classical models, the latter have several advantages. The most important one is that they exhibit an improved convergence behaviour. When the relevant quantum group is a -deformed universal enveloping algebra of a simple Lie algebra at a root of unity, its tilting modules form a non-degenerate finite semisimple spherical category [3, 4]. In the corresponding spin foam model, this has the effect of replacing infinite sums in the classical models by finite sums in the -deformed models.
In the case where is not a root of unity, the representation theory of the -deformed universal enveloping algebras is more complicated, but the associated spin foam models still often show an improved convergence behaviour compared to the classical ones. From the physics perspective, the -deformed models can be viewed as regularisations of the corresponding classical models. They remove divergences that occur when the representation labels grow large. Since these labels carry an interpretation in terms of lengths or areas of the simplicial complex, these divergences are often referred to as infrared.
Examples of this effect in three dimensions are the Turaev-Viro model , which is based on the tilting modules for at a root of unity. It can be viewed as a regularisation of the (divergent) Ponzano-Regge model , which is based on the representation theory of . A similar example in four dimensions is the relation between the Crane-Yetter model [6, 7] and the Ooguri model , which are based on the representation theory of, respectively, at a root of unity and . The former defines a topological invariant of -manifolds which can be expressed in terms of the signature and the Euler characteristic of the manifold .
Quantum groups also appear in constrained topological models of four-dimensional quantum gravity. The idea underlying the construction of such models is the fact that gravity in dimensions greater than three can be written as a constrained BF theory333The name BF theory stems from the letters and that were used for the field variables in the first formulation of the theory. . Constrained topological models are an attempt of incorporating these constraints into a spin foam model by imposing appropriate restrictions on the representations and intertwiners that are assigned to the faces and tetrahedra of the triangulation.
The first such constrained topological model is the BC model due to Barrett and Crane [11, 12]. Similarly to the topological models, this model diverges for large values of the representation labels. Yetter  and Noui and Roche  constructed -deformed analogues of the Euclidean and Lorentzian versions of the model based on, respectively, the representation category of and . Both -deformed models exhibit enhanced convergence properties compared to their classical counterparts.
Recently, the implementation of the constraints reducing BF theory to gravity in 4d spin foam models was refined and improved. This led to the formulation of new constrained topological models by Engle, Pereira, Rovelli and Livine (EPRL)  and Freidel and Krasnov (FK) . These models incorporate a free parameter , called the Immirzi parameter. The limits and yield respectively the BC and the flipped EPR model [17, 18, 19].
These recent models exhibit interesting physical properties, such as an asymptotic behaviour related to the Regge action [20, 21, 22, 23, 24, 25], but suffer from divergences for large area variables (see  for an analysis of these divergences). As originally suggested by Rovelli , this indicates the need for -deformed versions of these models, which still have the relevant physical properties but have a better chance of convergence. Furthermore, there is evidence that such models could be related to gravity with a positive cosmological constant. The construction of these models is the aim of this article.
In this article, we construct a -deformed version of both the Euclidean and Lorentzian EPRL model . The formulation of the Lorentzian model uses techniques from the representation theory of non-compact quantum groups and applies them to the quantum Lorentz group with a real deformation parameter. Using the results by Buffenoir and Roche [28, 29] on the harmonic analysis of the quantum Lorentz group, we construct a generalisation of the integral definition of the classical EPRL intertwiner to the quantum Lorentz group by means of a Haar measure on the latter. This is followed by a detailed analysis of the properties of the resulting intertwiner, including a proof of its convergence. We also derive explicit expressions for its transformations under braiding, under which the quantum EPRL intertwiner turns out not to be invariant. We then define an amplitude for the -simplexes of a triangulated four-manifold and construct the associated quantum spin foam model. Remarkably, the model converges although it is based on representations of an infinite-dimensional Hopf algebra.
The Euclidean model is formulated in terms of representation categories rather than the language of Hopf algebras. This structural difference with the Lorentzian model is explained by the fact that the Euclidean model is based on the representation theory of (two copies of) at a root of unity. While the representation theory simplifies considerably in this case - the tilting modules [3, 4] of at a root of unity define a non-degenerate, finite semisimple spherical category- the corresponding Hopf algebra structures become complicated, and one is forced to enter the framework of weak, quasi-Hopf algebras. The -deformed generalisation of the Euclidean EPRL intertwiner is therefore formulated in terms of the representation theory of rather than in terms of Haar measures on Hopf algebras. In this case, there are no convergence issues since there is only a finite number of representations with non-vanishing -dimension. As in the Lorentzian case, the quantum EPRL intertwiner does not appear to be invariant under braiding. After our discussion of the EPRL intertwiner, we then define the amplitude for a four simplex in the resulting model and show that it factorises into two quantum symbols with fusion coefficients. This mirrors the behaviour of the amplitude in the classical Euclidean EPRL model . The resulting -deformed spin foam model is again finite.
Outline of the paper
The paper is organised as follows. Section is dedicated to the construction of the Lorentzian model. We start by reviewing essential notions on the quantum Lorentz group, its representation theory and Harmonic analysis in Section 2.1. In Section 2.2 we show how the Lorentzian EPRL intertwiner can be generalised to the quantum Lorentz group by means of a Haar measure on the quantum Lorentz group. We then prove an important convergence theorem that ensures that the quantum EPRL intertwiner is well-defined and investigate its behaviour of under braiding. In Section 2.3 we define an amplitude for the four-simplexes of a closed oriented triangulated four-manifold. This definition is given via a graphical calculus defined by means of an invariant bilinear form on the representation spaces of the quantum Lorentz group. This naturally leads to the definition of the quantum spin foam model which is shown to converge for all triangulated manifolds.
In Section , we construct the -deformed EPRL model for Euclidean signature. We start by summarising the relevant aspects of the quantum group at a root of unity and of its representation theory in Section 3.1. In Section 3.1.3 we give a brief description of the “quantum rotation group” and its irreducible representations. This background is then used in the construction of the Euclidean quantum EPRL intertwiner in Section 3.2. We analyse the properties of this intertwiner and then define the associated four-simplex amplitude in Section 3.3. We conclude the discussion by proving that this amplitude factorises into a quantum symbol and that the associated spin foam model converges for all triangulated closed four-manifolds. Section contains a discussion of the physical interpretation and properties of the two -deformed EPRL models as well as our outlook and conclusions.
2 The Lorentzian model
2.1 The quantum Lorentz group
In this subsection, we summarise the relevant definitions and results about the quantum Lorentz group following [30, 28, 29]. The quantum Lorentz group is an infinite-dimensional ribbon Hopf algebra, which is obtained as the quantum double or Drinfel’d double of the -deformed universal enveloping algebra , where is a real deformation parameter.
2.1.1 Hopf algebra structure
The Hopf algebra .
We start by introducing the Hopf algebra , adopting the conventions from . The star Hopf algebra is the associative algebra generated multiplicatively by the elements , , subject to the relations
The comultiplication, counit and antipode are given by
and the star structure takes the form
The star Hopf algebra is a ribbon algebra with universal -matrix
where is the -exponential:
The ribbon element is defined by the identity , where is the invertible element and denotes the multiplication map. The group-like element is given by
Representation theory of
The representation theory of with a deformation parameter closely resembles the representation theory of the Lie group . Irreducible finite-dimensional -representations are labelled by “spins” and by an additional parameter . In the following we will restrict attention to equivalence classes of unitarizable representations which are characterised by the condition . As in the case of the Lie group , the representation space of the irreducible representation is -dimensional. There exists an orthonormal basis of the complex vector space in which the generators act according to
where is defined as in (7). The fusion rules for the tensor products resemble the ones for the representations of . We have
where the isomorphism is given by the Clebsch-Gordan intertwining operators
As all multiplicities in (10) are equal to one, these intertwiners are unique up to normalisation. They are non-zero if and only if , and are non-negative integers. Their coefficients with respect to the bases are the Clebsch-Gordan coefficients
To fix the phase of the Clebsch-Gordan coefficients, we impose reality conditions
and we use Wigner’s convention to fix the remaining sign ambiguity. The Clebsch-Gordan coefficients satisfy numerous relations. In the following we will frequently use their first orthogonality property and permutation symmetry
In some parts of the paper, we will identify the representation spaces with their duals by means of an invariant bilinear form on . We denote by be the dual vector space of the representation space in (2.1.1) and by be the basis dual to the basis :
The antipode (4) associates to each representation on a representation on via
The representations and are equivalent. The equivalence is given by a bijective intertwiner . Its matrix elements and those of the dual map with respect to the bases , are defined by
Here and throughout the paper, we use Einstein summation convention: repeated upper and lower indices are summed over unless stated otherwise. A short calculation using expression (4) and (2.1.1) shows that the matrix elements of and its dual are given up to normalisation by
where the constant is fixed to and is given by the action of the ribbon element (96) on . The matrix elements satisfy the identities
The intertwiner defines an invariant bilinear form on the vector space via
The Hopf algebra .
The star Hopf algebra is the dual of the Hopf algebra and can be viewed as a quantum deformation of the algebra of polynomial functions on . A basis of is given by the matrix elements in the unitary irreducible representations (2.1.1) of
The pairing between and takes the form
The Hopf algebra structure of induced by the one on via the pairing (20). In terms of the matrix elements , its algebra structure is characterised by the relations
Its comultiplication, counit and antipode take the form
where is the Kronecker symbol for the representation labelled by and the coefficients are given by (16). The star structure is given by
As any irreducible representation of can be obtained by tensoring the fundamental representation labelled by , a set of multiplicative generators of is given by the matrix elements in the fundamental representation
They generate multiplicatively subject to the relations
In terms of these generators, the comultiplication, counit and antipode are given by
and the pairing takes the form
The quantum Lorentz group .
The quantum Lorentz group is the quantum double of . It is given as the star Hopf algebra
where is the Hopf algebra with opposite coproduct, and the symbol ‘’ indicates that the Hopf subalgebras and do not commute inside . The algebra structure is given by (1), (21) together with mixed relations, that are most easily given in terms of the multiplicative generators of , see for instance the appendix of . In terms of these variables and the standard generators of they take the form
The comultiplication, counit and antipode for are given by equations (2), (3), (4) for the Hopf subalgebra and by the opposite of the comultiplication in (22), the counit in (23) and the inverse444The antipode of a Hopf algebra with the opposite coproduct is the inverse of the antipode of the Hopf algebra . of the antipode (24) for the Hopf subalgebra . The star structure is given by (5) and by (25), where the antipode in (25) is replaced by its inverse. As it is a quantum double, the quantum Lorentz group is a braided Hopf algebra. We will describe its universal -matrix after introducing its double dual in Section 2.1.2 below.
The quantum Lorentz group is a quantum deformation of the universal enveloping algebra of the real Lie algebra . This is a direct consequence of the quantum duality principle, see for instance [28, 31, 32, 33]. Recall that the coalgebra structure on the deformed enveloping algebra of a Lie algebra induces a Lie algebra structure of the dual vector space . The principle of quantum duality states that the -deformed universal enveloping algebra of is given by . On the level of quantised function spaces, one obtains that the Hopf algebra of quantum deformations of polynomial functions on the group is given by , where .
In the case at hand, this yields , where is the Lie algebra of the group . Here, denotes the group of diagonal positive -matrices of determinant one and is the nilpotent group of lower triangular two by two matrices with diagonal elements equal to one. The quantum double construction is therefore the quantum analogue of the Iwasawa decomposition of the classical Lorentz algebra , and we will use the notation .
2.1.2 Irreducible representations, duals and -matrix
Irreducible representations of the quantum Lorentz group
The irreducible unitary representations of were first classified by Pusz . In this paper, we will only consider the representations of the principal series. These representations are labelled by a couple with and or with and . We denote by the representation of labelled by . It is a Harish-Chandra representation which decomposes into representations of as follows
where is the left -module (2.1.1). A basis of the infinite dimensional vector space is given by where, for fixed , is the basis of defined before equation (2.1.1). In terms of this basis, the action of on the representation space is given by equation (2.1.1) for the action of and the following action of
where are complex numbers defined in terms of analytic continuations of symbols for . As their expressions are lengthy and complicated, we will not give them here but refer the reader to , where they are derived explicitly, and to  where their properties are studied in depth.
By introducing a hermitian form for which the basis is orthonormal
the representation spaces can be given a pre-Hilbert space structure. Its completion with respect to the associated norm is separable Hilbert space with Hilbert basis . The representations of the principal series are unitary in the sense that for all in and for all in , , that is, .
Note that the finite-dimensional unitary representations of the quantum Lorentz group do not form a ribbon category. As the representation spaces are infinite-dimensional, there is no notion of (quantum) trace. As in the classical case, this poses a considerable obstacle for the definition of the amplitude for the four-simplexes of the EPRL model and the definition of a consistent graphical calculus for the quantum Lorentz group. We will come back to this point in the sequel.
Algebra of functions on the quantum Lorentz group.
Matrix elements of the principal representations of the quantum Lorentz group are linear forms on and hence elements of its dual . In the following, we will therefore consider the Hopf algebra of functions on the quantum Lorentz group. The algebra decomposes as , where the Hopf algebra is interpreted as the space of (compactly supported) functions on . It is the dual of the Hopf algebra and hence can be identified with the Hopf algebra with the opposite multiplication.
Denoting by the basis of introduced in (19) and by the elements of its dual basis given by
one finds that a convenient basis of is provided by the elements . The star Hopf algebra structure of follows immediately from the one of via the duality principle. In particular, one finds that the multiplication on takes the form
Universal -matrix and braiding.
To obtain a simple expression for the universal- matrix of , it is convenient to work with the Hopf algebra introduced in , which can be viewed as the double dual of . As the Hopf algebra is infinite-dimensional, this is not identical to but contains as a Hopf subalgebra . It factorises as . A convenient basis of is given by the basis dual to . In terms of this basis, the universal -matrix of takes a particularly simple form, namely
To derive an explicit expression for the braiding, we note that if is a principal representation of on , there is a unique representation  of on , which will also be denoted by . Its action on the basis elements of the preHilbert space is given by
where are the coefficients from (33). It is then immediate to obtain explicit expressions for the action of the universal -matrix in these representations:
where and . Note that although these sums are infinite, there is only a finite number of non-zero terms . Consequently, there are no issues with convergence.
2.2 The quantum EPRL intertwiner
We are now ready to construct the generalised EPRL model associated with the quantum Lorentz group. The first step is to define the state space associated to the three-simplexes of a triangulated -manifold . This is the vector space of EPRL intertwiners between the four EPRL representations associated with its boundary triangles and the trivial EPRL representation on .
2.2.1 Quantum EPRL representations
A central ingredient in the construction is the generalisation of the notion of an EPRL representation to the quantum Lorentz group. In the classical model, an EPRL representation assigns a representation of to each finite-dimensional representation of . This assignment can be viewed as a lift from the representation category of to the representation category of , and is given by the prescription
where labels the irreducible representations of and is the Immirzi parameter. In the quantum Lorentz group, the -representations in the classical model correspond to irreducible representations of the Hopf algebra , which are labelled by a parameter . The inclusion , is replaced by the inclusion , . It is thus natural to require that that quantum EPRL representations are given by a tensor functor between the category of representations of and the representation category of the quantum Lorentz group, which is compatible with this inclusion of into . The image of the functor is a subset of representations of called EPRL representations. The analogy with the classical case suggests that this functor should act on the irreducible representations of according to
where labels an irreducible unitary principal series representation of the quantum Lorentz group and is the Immirzi parameter. Note, however, that the parameter in the representation labels of the principal series is restricted to the interval . To obtain a consistent definition, it is thus necessary to restrict the preimage of this functor, i. e. the representation label . Remark also that there is no loss of generality in considering only the case where is positive since the representations and are equivalent. We are now led to the following definition.
(EPRL representations) Let be a fixed real positive parameter (the Immirzi parameter) and consider the subset of irreducible representations of labelled by
The Lorentzian EPRL representation of spin is the principal representation of labelled by
The restriction of the representation labels of to the label set ensures that the Lorentzian EPRL representation is a principal representation of the quantum Lorentz group, which would not be the case otherwise. It decomposes into irreducible representations of as follows
2.2.2 Quantum EPRL intertwiners
We are now ready to generalise the notion of quantum EPRL intertwiner to the quantum Lorentz group. This requires a generalisation of integral expressions of the type
where is a compact, unimodular Lie group with Haar measure , are unitary irreducible representations of and and . The generalisation of such integrals to -deformed universal enveloping algebras and the associated -deformed function spaces requires the notion of a Haar measure or biinvariant normalised integral on . This is a non-degenerate linear form that satisfies the identities
For a pedagogical introduction, see for instance . The identities (40) imply that a Haar measure on is invariant under the left- and right-action of on . For given , the left- and right- action are defined via the pairing between and :
The invariance property (40) then implies
The Haar measure is thus invariant under the left- and right-action of the Hopf algebra on its dual , and this invariance can be viewed as a a generalisation of the left- and right-invariance of the Haar measure on a unimodular Lie group.
Given a Haar measure on , it is straightforward to generalise expression (39) to the representations of , . For this, one introduces a basis of and denotes by the associated dual basis of . Given irreducible representations , , of , one obtains a representation of on the tensor product of the associated representation spaces