Four-Dimensional Spin Foam Perturbation Theory

We define a four-dimensional spin-foam perturbation theory for the BF-theory with a B∧B potential term defined for a compact semi-simple Lie group G on a compact orientable 4-manifold M. This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. W...

Full description

Saved in:

Bibliographic Details
Date:	2011
Main Authors:	Martins, J.F., Mikovic, A.
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2011
Series:	Symmetry, Integrability and Geometry: Methods and Applications
Online Access:	http://dspace.nbuv.gov.ua/handle/123456789/147406
Tags:	Add Tag No Tags, Be the first to tag this record!
Journal Title:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:	Four-Dimensional Spin Foam Perturbation Theory / J.F. Martins, A. Mikovic // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 27 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-147406
record_format	dspace
spelling	irk-123456789-1474062019-02-15T01:24:46Z Four-Dimensional Spin Foam Perturbation Theory Martins, J.F. Mikovic, A. We define a four-dimensional spin-foam perturbation theory for the BF-theory with a B∧B potential term defined for a compact semi-simple Lie group G on a compact orientable 4-manifold M. This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group Uq(g) where g is the Lie algebra of G and q is a root of unity. The Chain-Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners Λ⊗Λ→A, where A is the adjoint representation of g, is 1-dimensional for each irrep Λ. We calculate the partition function Z in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold M. We prove that the first-order perturbative contribution vanishes for finite triangulations, so that we define a dilute-gas limit by using the second-order contribution. We show that Z is an analytic continuation of the Crane-Yetter partition function. Furthermore, we relate Z to the partition function for the F∧F theory. 2011 Article Four-Dimensional Spin Foam Perturbation Theory / J.F. Martins, A. Mikovic // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81T25; 81T45; 57R56 DOI: http://dx.doi.org/10.3842/SIGMA.2011.094 http://dspace.nbuv.gov.ua/handle/123456789/147406 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We define a four-dimensional spin-foam perturbation theory for the BF-theory with a B∧B potential term defined for a compact semi-simple Lie group G on a compact orientable 4-manifold M. This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group Uq(g) where g is the Lie algebra of G and q is a root of unity. The Chain-Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners Λ⊗Λ→A, where A is the adjoint representation of g, is 1-dimensional for each irrep Λ. We calculate the partition function Z in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold M. We prove that the first-order perturbative contribution vanishes for finite triangulations, so that we define a dilute-gas limit by using the second-order contribution. We show that Z is an analytic continuation of the Crane-Yetter partition function. Furthermore, we relate Z to the partition function for the F∧F theory.
format	Article
author	Martins, J.F. Mikovic, A.
spellingShingle	Martins, J.F. Mikovic, A. Four-Dimensional Spin Foam Perturbation Theory Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Martins, J.F. Mikovic, A.
author_sort	Martins, J.F.
title	Four-Dimensional Spin Foam Perturbation Theory
title_short	Four-Dimensional Spin Foam Perturbation Theory
title_full	Four-Dimensional Spin Foam Perturbation Theory
title_fullStr	Four-Dimensional Spin Foam Perturbation Theory
title_full_unstemmed	Four-Dimensional Spin Foam Perturbation Theory
title_sort	four-dimensional spin foam perturbation theory
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/147406
citation_txt	Four-Dimensional Spin Foam Perturbation Theory / J.F. Martins, A. Mikovic // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 27 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT martinsjf fourdimensionalspinfoamperturbationtheory AT mikovica fourdimensionalspinfoamperturbationtheory
first_indexed	2025-07-11T02:17:37Z
last_indexed	2025-07-11T02:17:37Z
_version_	1837315405682573312
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 094, 22 pages Four-Dimensional Spin Foam Perturbation Theory João FARIA MARTINS † and Aleksandar MIKOVIĆ ‡§ † Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal E-mail: jn.martins@fct.unl.pt URL: http://ferrari.dmat.fct.unl.pt/personal/jnm/ ‡ Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologia, Av do Campo Grande, 376, 1749-024 Lisboa, Portugal E-mail: amikovic@ulusofona.pt § Grupo de F́ısica Matemática da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal Received June 03, 2011, in final form September 23, 2011; Published online October 11, 2011 http://dx.doi.org/10.3842/SIGMA.2011.094 Abstract. We define a four-dimensional spin-foam perturbation theory for the BF-theory with a B ∧ B potential term defined for a compact semi-simple Lie group G on a compact orientable 4-manifold M . This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group Uq(g) where g is the Lie algebra of G and q is a root of unity. The Chain–Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners Λ⊗ Λ→ A, where A is the adjoint representation of g, is 1-dimensional for each irrep Λ. We calculate the partition function Z in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold M . We prove that the first- order perturbative contribution vanishes for finite triangulations, so that we define a dilute- gas limit by using the second-order contribution. We show that Z is an analytic continuation of the Crane–Yetter partition function. Furthermore, we relate Z to the partition function for the F ∧ F theory. Key words: spin foam models; BF-theory; spin networks; dilute-gas limit; Crane–Yetter invariant; spin-foam perturbation theory 2010 Mathematics Subject Classification: 81T25; 81T45; 57R56 1 Introduction Spin foam models are state-sum representations of the path integrals for BF theories on sim- plicial complexes. Spin foam models are used to define topological quantum field theories and quantum gravity theories, see [1]. However, there are also perturbed BF theories in various di- mensions, whose potential terms are powers of the B field, see [10]. The corresponding spin-foam perturbation theory generating functional was formulated in [10], but further progress was hin- dered by the lack of the regularization procedure for the corresponding perturbative expansion and the problem of implementation of the triangulation independence. The problem of implementation of the triangulation independence for general spin foam per- turbation theory was studied in [2], and a solution was proposed, in the form of calculating the perturbation series in a special limit. This limit was called the dilute-gas limit, and it was given by λ → 0, N → ∞, such that g = λN is a fixed constant, where λ is the perturbation theory parameter, also called the coupling constant, N is the number of d-simplices in a simpli- cal decomposition of a d-dimensional compact manifold M and g is the effective perturbation mailto:jn.martins@fct.unl.pt http://ferrari.dmat.fct.unl.pt/personal/jnm/ mailto:amikovic@ulusofona.pt http://dx.doi.org/10.3842/SIGMA.2011.094 2 J. Faria Martins and A. Miković parameter, also called the renormalized coupling constant. However, the dilute-gas limit could be used in a concrete example only if one knew how to regularize the perturbative contributions. The regularization problem has been solved recently in the case of three-dimensional (3d) Euclidean quantum gravity with a cosmological constant [9], following the approach of [3, 8]. The 3d Euclidean classical gravity theory is equivalent to the SU(2) BF-theory with a B3 perturbation, and the corresponding spin foam perturbation expansion can be written by using the Ponzano–Regge model. The terms in this series can be regularized by replacing all the spin-network evaluations with the corresponding quantum spin-network evaluations at a root of unity. By using the Chain–Mail formalism [23] one can calculate the quantum group perturbative corrections, and show that the first-order correction vanishes [9]. Consequently, the dilute-gas limit has to be modified so that g = λ2N is the effective perturbation parameter [9]. Another result of [9] was to show that the dilute gas limit cannot be defined for an arbitrary class of triangulations of the manifold. One needs a restricted class of triangulations such that the number of possible isotopy classes of a graph defined from the perturbative insertions is bounded. In 3d this can be achieved by using the triangulations coming from the barycentric subdivisions of a regular cubulation of the manifold [9]. In this paper we are going to define the four-dimensional (4d) spin-foam perturbation theory by using the same approach and the techniques as in the 3d case. We start from a BF-theory with a B ∧ B potential term defined for a compact semi-simple Lie group G on a compact 4-manifold M . In Section 2 we define the formal spin foam perturbative series by using the spin-foam generating functional method. We then regularize the terms in the series by passing to the category of representations for the quantum group Uq(g) where g is the Lie algebra of G and q is a root of unity. In Sections 4 and 5 we then use the Chain–Mail formalism to calculate the perturbative contributions. The first-order perturbative contribution vanishes, so that we define the dilute-gas limit in Section 6 by using the second-order contribution. We calculate the partition function Z in the dilute-gas limit for a class of triangulations of a 4-dimensional manifold which are arbitrarily fine and have a controllable local complexity. We conjecture that such a class of triangulations always exists for any 4-dimensional manifold, and can be given by the triangulations corresponding to the barycentric subdivisions of a fixed cubulation of the manifold. We then show that Z is given as an analytic continuation of the Crane–Yetter partition function. In Section 7 we relate the path-integral for the F ∧ F theory with the spin foam partition function and in Section 8 we present our conclusions. 2 Spin foam perturbative expansion Let g be the Lie algebra of a semisimple compact Lie group G. The action for a perturbed BF-theory in 4d can be written as S = ∫ M ( BI ∧ FI + λ gIJ B I ∧BJ ) , (2.1) where B = BILI is a g-valued two-form, LI is a basis of g, F = dA + 1 2 [A,A] is the curvature 2-form for the G-connection A on a principal G-bundle over M , XI = XI and gIJ is a symmetric G-invariant tensor. Here if X and Y are vector fields in the manifold M then [A,A](X,Y ) = [A(X), A(Y )]. We will consider the case when gIJ ∝ δIJ , where δIJ is the Kronecker delta symbol. In the case of a simple Lie group, this is the only possibility, while in the case of a semisimple Lie group one can also have gIJ which are not proportional to δIJ . For example, in the case of the SO(4) group one can put gab,cd = εabcd, where ε is the totally antisymmetric tensor and 1 ≤ a, . . . , d ≤ 4. We will also use the notation Tr (XY ) = XIYI and 〈XY 〉 = gIJX IY J . Four-Dimensional Spin Foam Perturbation Theory 3 Consider the path integral Z(λ,M) = ∫ DADBei ∫ M(BI∧FI+λ〈B∧B〉). (2.2) It can be evaluated perturbatively in λ by using the generating functional Z0(J ,M) = ∫ DADBei ∫ M(BI∧FI+BI∧JI), (2.3) (where JILI is an arbitrary 2-form valued in g) and the formula Z(λ,M) = exp ( −iλ ∫ M gIJ δ δJI ∧ δ δJJ ) Z0(J ,M) ∣∣∣ J=0 . (2.4) The path integrals (2.3) and (2.4) can be represented as spin foam state sums by discretizing the 4-manifold M , see [10]. This is done by using a simplicial decomposition (triangulation) of M , T (M). It is useful to introduce the dual cell complex T ∗(M) [24] (a cell decomposition of M), and we will denote the vertices, edges and faces of T ∗(M) as v, l and f , respectively. A vertex v of T ∗(M) is dual to a 4-simplex σ of T (M), an edge l of T ∗(M) is dual to a tetra- hedron τ of T (M) and a face f of T ∗(M) is dual to a triangle ∆ of T (M). The action (2.1) then takes the following form on T (M) S = ∑ ∆ Tr (B∆Ff ) + λ 5 ∑ σ ∑ ∆′,∆′′∈σ 〈B∆′B∆′′〉, where ∆′ and ∆′′ are pairs of triangles in a four-simplex σ whose intersection is a single vertex of σ and B∆ = ∫ ∆B. The variable Ff is defined as eFf = ∏ l∈∂f gl, where f is the face dual to a triangle ∆, l’s are the edges of the polygon boundary of f and gl are the dual edge holonomies. One can then show that Z0(J ,M) = ∑ Λf ,ιl ∏ f dim Λf ∏ v A5(Λf(v), ιl(v),Jf(v)), (2.5) where the amplitude A5(Λf(v), ιl(v),Jf(v)), also called the weight for the 4-simplex σ, is given by the evaluation of the four-simplex spin network whose edges are colored by ten Λf(v) irreps and five ιl(v) intertwiners, while each edge has a D(Λ)(eJ ) insertion. Here D(Λ)(eJ ) is the repre- sentation matrix for a group element eJ in the irreducible representation (irrep) Λ, see [10, 18]. Note that a vertex v is dual to a 4-simplex σ, so that the set of faces f(v) intersecting at v is dual to the set of ten triangles of σ. Similarly, the set of five dual edges l(v) intersecting at v is dual to the set of five tetrahedrons of σ. The sum in (2.5) is over all colorings Λf of the set of faces f of T ∗(M) by the irreps Λf of G, as well as over the corresponding intertwiners ιl for the dual complex edges l. Equivalently, Λf label the triangles of T (M), while ιl label the tetrahedrons of T (M). In the case of the SU(2) group and J = 0, the amplitude A5 gives the 15j symbol, see [1, 13]. For the general definition of 15j-symbols A5(Λf(v), ιl(v)) see [16, 17]. Then Z0(J = 0,M) can be written as Z0(M) = ∑ Λf ,ιl ∏ f dim Λf ∏ v A5(Λf(v), ιl(v)), (2.6) 4 J. Faria Martins and A. Miković which after quantum group regularization (by passing to a root of unity) becomes a manifold invariant known as the Crane–Yetter invariant [6]. The formula (2.4) is now given by the discretized expression Z(λ,M, T ) = exp − iλ 5 ∑ σ ∑ ∆,∆′∈σ gIJ ∂2 ∂J If ∂J Jf ′ Z0(J ,M) ∣∣∣ J=0 , (2.7) where T denotes the triangulation of M . The equation (2.7) can be rewritten as Z(λ,M, T ) = ∑ Λf ,ιl ∏ f dim Λf exp ( −iλ ∑ σ V̂σ )∏ v A5(Λf(v), ιl(v)), (2.8) where the operator V̂σ is given by V̂σ = 1 5 ∑ ∆,∆′∈σ gIJL (Λ) I ⊗ L(Λ′) J ≡ 1 5 ∑ f,f ′;v∈f∩f ′ gIJL (Λf ) I ⊗ L(Λf ′ ) J . (2.9) This operator acts on the σ-spin network evaluation A5 by inserting the Lie algebra basis ele- ment L(Λ) for an irrep Λ into the spin network edge carrying the irrep Λ. The expression (2.9) follows from (2.5), (2.7) and the relation ∂D(Λ)(eJ ) ∂J I ∣∣∣ J=0 = L (Λ) I . Following (2.9), let us define a g-edge in a 4-simplex spin network as a line connecting the middle points of two edges of the spin network, such that this line is labelled by the tensor gIJ . We associate to a g-edge the linear map∑ IJ gIJL (Λf ) I ⊗ L(Λf ′ ) J , where Λf and Λf ′ are the labels of the spin network edges connected by the g-edge and L (Λf ) I denotes the action of the basis element LI of g in the representation Λf . The action of the operator gIJL (Λf ) I ⊗ L(Λf ′ ) J in a single 4-simplex of the J = 0 spin foam state sum (2.5) can be represented as the evaluation of a spin network Γσ,g obtained from the 4-simplex spin network Γσ by adding a g-edge between the two edges of Γσ labeled by Λf and Λf ′ . When gIJ ∝ δIJ and the intertwiners CΛΛ′A, from Λ⊗Λ′ to A, where A is the adjoint repre- sentation, form a one-dimensional vector space, Γσ,g becomes the 4-simplex spin network with an insertion of an edge labeled by the adjoint irrep, see Fig. 1. This simplification happens because the matrix elements of L (Λ) I can be identified with the components of the intertwiner CΛΛ′A, since these intertwiners are one-dimensional vector spaces, i.e.( L (Λ) I ) αβ = CΛΛA αβ I , (2.10) so that∑ I ( L (Λ) I ) αβ ( L (Λ′) I ) α′β′ = ∑ I CΛΛA αβ IC AΛ′Λ′ I α′ β′ . (2.11) Then the right-hand side of the equation (2.11) represents the evaluation of the spin network in Fig. 2. The condition (2.10) is not too restrictive since it includes the SU(2) and SO(4) groups. We need to consider this particular case in order to be able to use the Chain–Mail techniques. Four-Dimensional Spin Foam Perturbation Theory 5 Figure 1. A 15j symbol (4-simplex spin network) with a g-edge insertion (dashed line). Here A is the adjoint representation. Figure 2. Spin network form of equation (2.11). The action of (V̂σ)n in A5 is given by the evaluation of a Γσ,n spin network which is obtained from the Γσ spin network by inserting n g-edges labeled by the adjoint irrep. These additional edges connect the edges of Γσ which correspond to the triangles of the 4-simplex σ where the operators L (Λ) I ⊗ L(Λ′) I from V̂σ act. Let Z(M,T ) = ∞∑ n=0 inλnZn(M,λ, T ), then Z0(M) = Z0(J ,M) ∣∣∣ J=0 . The state sum Z0 is infinite, unless it is regularized. The usual way of regularization is by using the representations of the quantum group Uq(g) at a root of unity, which, by passing to a finite-dimensional quotient, yields a modular Hopf algebra [26]. There are only finitely many irreps with non-zero quantum dimension in this case, and the corresponding state sum Z0 has the same form as in the Lie group case, except that the usual spin network evaluation used for the spin-foam amplitudes has to be replaced by the quantum spin network evaluation. In this way one obtains a finite and triangulation independent Z0, usually known as the Crane–Yetter invariant [6]. This 4-manifold invariant is determined by the signature of the manifold [23, 26]. The same procedure of passing to the quantum group at a root of unity can be applied to the perturbative corrections Zn, but in order to obtain triangulation independent results, the dilute gas limit has to be implemented [2, 9]. 6 J. Faria Martins and A. Miković 2.1 The Chain–Mail formalism and observables of the Crane–Yetter invariant The Chain–Mail formalism for defining the Turaev–Viro invariant and the Crane–Yetter inva- riant was introduced by Roberts in [23]. In the four-dimensional case, the construction of the related manifold invariant Z0(M) had already been implemented by Broda in [4]. However, the equality with the Crane–Yetter invariant, as well as the relation of Z0(M) with the signature of M appears only in the work of Roberts [23]. We will follow the conventions of [3]. Let M be a four-dimensional manifold. Suppose we have a handle decomposition [12, 14, 24] of M , with a unique 0-handle, and with hi handles of order i (where i = 1, 2, 3, 4). This gives rise to the link CHLM in the three-sphere S3, with h2 + h1 components (the “Chain–Mail link”), which is the Kirby diagram of the handle decomposition [12, 14]. Namely, we have a dotted unknotted (and 0-framed) circle for each 1-handle of M , determining the attaching of the 1-handle along the disjoint union of two balls, and we also have a framed knot for each 2-handle, along which we attach it. This is the four- dimensional counterpart of the three-dimensional Chain–Mail link of Roberts, see [23, 3]. The Crane–Yetter invariant Z0(M), which coincides with the invariant Z0(J ,M)J=0, defined in the introduction, see equation (2.6), can be represented as a multiple of the spin-network eva- luation of the chain mail link CHLM , colored with the following linear combination of quantum group irreps (the Ω-element): Ω = ∑ Λ (dimq Λ)Λ, see [23]. Explicitly, by using the normalizations of [3]: Z0(M) = η− 1 2 (h2+h1+h3−h4+1)〈CHLM ,Ω h2+h1〉, (2.12) where η = ∑ Λ (dimq Λ)2. Roberts also proved in [23] that Z0(M) = κs(M), where κ is the evaluation of a 1-framed unknot colored with the Ω-element and s(M) denotes the signature of M . Given a triangulated manifold (M,T ), consider the natural handle decomposition of M ob- tained from thickening the dual cell decomposition of M ; see [23, 24]. Then a handle de- composition of M (with a single 0-handle), such that (2.12) explicitly gives the formula for Z0(M) = Z0(J ,M)J=0, appearing in equation (2.5), is obtained from this handle decomposi- tion by canceling pairs of 0- and 1-handles [12, 14, 24], until a single 0-handle is left; in this case, in the vicinity of each 4-simplex, the Chain–Mail link has the form given in Fig. 3. This explicit calculation appears in [23, 3] and essentially follows from the Lickorish encircling lemma [13, 15]: the spin-network evaluation of a graph containing a unique strand (colored with the represen- tation Λ) passing through a zero framed unknot colored with Ω vanishes unless Λ is the trivial representation. A variant of the Crane–Yetter model Z0 in (2.12) is achieved by inspecting its observables, addressed in [3]. Consider a triangulated 4-manifold M . Consider the handle decomposition of M obtained from thickening the dual complex of the triangulation, and eliminating pairs of 0- and 1-handles until a single 0-handle is left. Any triangle of the triangulation of M therefore yields a 2-handle of M . Now choose a set S with nS triangles of M , which will span a (possibly singular) surface Σ2 of M . Color each t ∈ S with a representation Λt. The associated observable of the Crane–Yetter invariant is: Z0(M,S,ΛS) = η− 1 2 (h2+h1+h3−h4+1)〈CHLM ; Ωh2+h1−nS ,ΛS〉 ∏ t∈S dimq(Λt), (2.13) Four-Dimensional Spin Foam Perturbation Theory 7 Figure 3. Portion of the chain-mail link corresponding to a 4-simplex; this may have additional meridian circles (corresponding to 1-handles) since we also eliminate pairs of 0- and 1-handles, until a single 0- handle is left. where 〈CHLM ; Ωh2+h1−nS ,ΛS〉 denotes the spin-network evaluation of the Chain–Mail link CHLM , where the components associated with the triangles t ∈ S are colored by Λs and the remaining components with Ω. We can see CHLM as a pair (LS ,KS), where KS denotes the components of the Chain–Mail link given by the triangles of S and LS the remaining components of the Chain–Mail link. We thus have CHLM = LS ∪KS . Let ZWRT(N,Γ) denote the Witten–Reshetikhin–Turaev invariant of the colored graph Γ embedded in the 3-manifold N , in the normalization of [3]. Then Theorem 2 of [3] says that: Z0(M,S,ΛS) = ZWRT ( ∂(M \ Σ̂2),K ′S ) κs(M\Σ̂2) η χ(Σ2) 2 −nS ∏ t∈S dimq(Λt). (2.14) Here Σ̂2 is an open regular neighborhood of Σ2 in M , s denotes the signature of the manifold and χ denotes the Euler characteristic. The link K ′S is the link in ∂(M \ Σ̂2) along which the 2- handles associated to the triangles of Σ2 would be attached, in order to obtain M . This theorem of [3] essentially follows from the fact that the pair (LS ,KS) is a surgery presentation of the pair ( ∂(M \ Σ̂2),K ′S ) , a link embedded in a manifold, apart from connected sums with S1×S2. 3 The first-order correction Recall that there are two possible ways of representing the Crane–Yetter invariant Z0: as a state sum invariant (2.6) and as the evaluation of a Chain–Mail link (2.12). It follows from (2.8) that Z1 can be written as NZ ′0 where N is the number of 4-simplices of T (M) and Z ′0 is the state sum given by a modification of the state sum Z0 where a single 4-simplex is perturbed by the operator V̂σ. In order to calculate Z ′0 consider a 4-manifold M with a triangulation T whose dual complex is T ∗. Given a 4-simplex σ ∈ T we define an insertion I as being a choice of a pair of triangles of σ which do not belong to the same tetrahedron of σ and have therefore a single vertex in common (following (2.7) we will distinguish the order in which the triangles appear). Given the colorings Λf of the triangles of σ (or of the dual faces f) and the colorings ιl of the tetrahedrons of σ (or of the dual edges l), then A5(Λf , ιl, I) is the evaluation of the spin network of Fig. 1. 8 J. Faria Martins and A. Miković Figure 4. Portion of the graph CMLI M corresponding to a 4-simplex with an insertion. All strands are to be colored with Ω, unless they intersect the insertion, as indicated. We then have: Z1(M,T ) = 1 5 ∑ σ ∑ Iσ ∑ Λf ,ιl A5(Λf(σ), ιl(σ), Iσ) ∏ f dim Λf ∏ v 6=v(σ) A5(Λf(v), ιl(v)), (3.1) where v(σ) is the vertex of T ∗ corresponding to σ. This sum is over the set of all 4-simplices σ of T (M), as well as over the set Iσ of all insertions of σ and over the set of all colorings (Λf , ιl) of the faces and the edges of T ∗(M) (or equivalently, a sum over the colorings of the triangles and the tetrahedrons of T (M).) The infinite sum in (3.1) is regularized by passing to the category of representations of the quantum group Uq(g), where q is a root of unity. In order to calculate Z1(M,T ) in this case, let us represent it as an evaluation of the Chain–Mail link CHL(M,T ) [23] in the three-sphere S3. As explained in Subsection 2.1, the invariant Z0(M) can be represented as a multiple of the evaluation of the chain mail link CHLM colored with the linear combination of the quan- tum group irreps Ω = ∑ Λ(dimq Λ)Λ, see equation (2.12). Analogously, by extending the 3- dimensional approach of [9], a chain-mail formulation for the equation (3.1) can be given. Con- sider the handle decomposition of M obtained by thickening the dual cell decomposition of M associated to the triangulation T of M . One can cancel pairs of 0- and 1-handles, until a single 0-handle is left. Let CHLM be the associated chain-mail link (the Kirby diagram of the handle decomposition). We then have Z1(M,T ) = 1 5 ∑ I ∑ Λ,Λ′ η− 1 2 (h2+h1+h3−h4+1) dimq Λ dimq Λ′〈CHLIM ,Ω h2+h1−2,Λ,Λ′〉, (3.2) where, as before, an insertion I is the choice of a pair of triangles t1 and t2 in some 4-simplex of M , such that t1 and t2 have only one vertex in common. Given an insertion I, the graph CHLIM is obtained from the chain-mail link CHLM by inserting a single edge (colored with the adjoint representation of g) connecting the components of CHLM (colored with Λ and Λ′) corresponding to t1 and t2; see Fig. 4. CHLIM can be considered as a pair (LI ,ΓI) where LI denotes the components of CHLIM not incident to the insertion I (which are exactly h2 +h2−2) and ΓI denotes the component of CHLIM containing the insertion I. Hence we use the notation 〈CHLIM ,Ω h2+h1−2,Λ,Λ′〉 to mean the evaluation of the pair (LI ,ΓI) where all components Four-Dimensional Spin Foam Perturbation Theory 9 Figure 5. Spin network Γ1(Λ,Λ′). Here A is the adjoint representation. of LI are colored with Ω and the two circles of ΓI are colored with Λ and Λ′, with an extra edge connecting them, colored with the adjoint representation A. Consider an insertion I connecting the triangles t1 and t2, which intersect at a single vertex. Equation (3.2) coincides, apart from the inclusion of the single insertion, with the observables of the Crane–Yetter invariant [3] (Subsection 2.1) for the pair of triangles colored with Λ and Λ′; see equation (2.13). By using the discussion in Section 6 and Theorem 2 of [3] (see equation (2.14)) one therefore obtains, for each insertion I and each pair of irreps Λ, Λ′: η− 1 2 (h2+h1+h3−h4+1)〈CHLIM ,Ω h2+h1−2,Λ,Λ′〉 = ZWRT(S3,Γ(Λ,Λ′))η− 3 2Z0(M), (3.3) where Γ(Λ,Λ′) is the colored link of Fig. 5. In addition ZWRT denotes the Witten–Reshetikhin– Turaev invariant [27, 22] of graphs in manifolds, in the normalization of [3, 8]. Note that in the notation of equation (2.14), Σ2 = t1 ∪ t2 is two triangles which intersect at a vertex, thus χ(Σ2) = 1 and also its regular neighborhood Σ̂2 is homeomorphic to the 4-disk, thus s(M) = s(M \ Σ̂2). Equation (3.3) follows essentially from the fact that the pair CHLIM = (LI ,ΓI) is a surgery presentation [14, 12] of the pair (S3,Γ(Λ,Λ′)), apart from connected sums with S1 × S2; c.f. Theorem 3, below. To see this, note that (after turning the circles associated with the 1-handles of M into dotted circles) the link LI is a Kirby diagram for the manifoldM minus an open regular neighborhood Σ′ of the 2-complex Σ made from the vertices and edges of the triangulation of M , together with the triangles t1 and t2. Since t1 and t2 intersect at a single vertex, any regular neighborhood Σ̂2 of the (singular) surface Σ2 spanned by t1 and t2 is homeomorphic to the 4-disk D4. Therefore Σ′ is certainly homeomorphic to the boundary connected sum ( \ki=1(D3×S1) ) \D4, whose boundary is( #k i=1(S2×S1) ) #S3, for some positive integer k. Here # denotes the connected sum of manifolds and \ denotes the boundary connected sum of manifolds. The circles c1, c2 ⊂ ΓI associated with the triangles t1 and t2 define a link which lives in ∂Σ̂2 ∼= S3 ⊂ ( #k i=1(S2 × S1) ) #S3. The two circles c1 and c2 define a 0-framed unlink in S3, with each individual component being unknotted. Let us see why this is the case. We will turn the underlying handle decomposition of M upside down, by passing to the dual handle decomposition of M , where each i-simplex of the triangulation of M yields an i-handle of M ; see [12, p. 107]. Consider the bit P ⊂ M of the handle-body yielded by the 2-complex Σ, thus P is (like Σ′) a regular neighborhood of Σ. Maintaining the 0-handle generated by the vertex t1∩t2, eliminate some pairs of 0- and 1-handles, in the usual way, until a single 0-handle of P is left. Clearly ∂P ∗ = ∂(M \Σ′), where ∗ denotes the orientation reversal. The circles c1 and c2, in ∂P ∗, correspond now (since we considered the dual handle decomposition) to the belt-spheres of the 2-handles of P (attached along ct1 and ct2) and associated with the triangles t1 and t2. Since c1 and c2 are 0-framed meridians going around ct1 and ct2 (see [12, Example 1.6.3]) it therefore follows that these circles are unlinked and are also, individually, unknotted; see Fig. 6. Given this and the fact that the insertion I colored with A also lives in S3, it follows that CHLIM = (LI ,ΓI) is a surgery presentation of the pair (S3,Γ(Λ,Λ′)), apart from the connected sums, distant from Γ(Λ,Λ′), with S1 × S2. Since the evaluation of the tadpole spin network is zero, it follows that ZWRT(S3,Γ(Λ,Λ′))= 0, and consequently 10 J. Faria Martins and A. Miković Figure 6. The Kirby diagram for P in the vicinity of the triangles t1 and t2. We show the belt-spheres c1 and c2 of the 2-handles of P (attaching along ct1 and ct2) associated with the triangles t1 and t2. Theorem 1. For any triangulation T of M we have Z1(M,T ) = 0. 4 The second-order correction Since Z1 = 0, we have to calculate Z2 in an appropriate limit such that the partition function Z is different from Z0 and such that Z is independent of the triangulation [9]. Let N be the number of 4-simplices. From (2.8) we obtain Z2(M) = 1 2 ∑ Λf ,ιl ∏ f dim Λf ∑ σ V̂ 2 σ + ∑ σ 6=σ′ V̂σV̂σ′ ∏ v A5(Λf(v), ιl(v)), (4.1) where V̂σA5(Λf(v), ιl(v)) = 1 5 ∑ insertions I of σ A5(Λf(v), ιl(v), I), (4.2) if σ is dual to v, see Fig. 1. On the other hand, V̂σA5(Λf(v), ιl(v)) = A5(Λf(v), ιl(v)) if v is not dual to σ. In order to to solve the possible framing and crossing ambiguities arising from the equation (4.1), a method analogous to the one used in [9] can be employed. Note that there are exactly 30 insertions in a 4-simplex σ, corresponding to pairs of triangles of σ with a single vertex in common. This is because there are exactly three triangles of σ having only one vertex in common with a given triangle of σ. Analogously to the first-order correction, Z2 can be written as Z2(M,T ) = 1 2 η− 1 2 (h2+h1+h3−h4+1) N∑ k=1 〈CHLM ,Ω h2+h1 , V̂ 2 k 〉 + 1 2 η− 1 2 (h2+h1+h3−h4+1) ∑ 1≤k 6=l≤N 〈CHLM ,Ω h2+h2 , V̂k, V̂l〉, (4.3) where the first sum denotes the contributions from two insertions V̂ in the same 4-simplex σk and the second sum represents the contributions when the two insertions V̂ act in different 4-simplices σk and σl. As in the previous section, we will use the handle decomposition of M with an unique 0-handle naturally obtained from the thickening of T ∗(M). Note that each 〈CHLM ,Ω h2+h1 , V̂ 2 k 〉 corresponds to a sum over all the possible choices of pairs of insertions in the 4-simplex σk. The value of 〈CHLM ,Ω h2+h1 , V̂ 2 k 〉 is obtained from the evaluation of the chain-mail link CHLM colored with Ω, which contains g-edges carrying the adjoint representation, as in the calculation of the first-order correction. Four-Dimensional Spin Foam Perturbation Theory 11 Figure 7. Colored graph Γ′ 2(Λ,Λ′,Λ′′), a wedge graph. Here A is the adjoint representation. A configuration C is, by definition, a choice of insertions distributed along a set of 4-simplices of M . Given a positive integer n and a set R of 4-simplices of M , we denote by CnR the set of configurations with n insertions distributed along R. By expanding each V into a sum of insertions, the equation (4.3) can be written as: Z2(M,T ) = 1 50 η− 1 2 (h2+h1+h3−h4+1) N∑ k=1 ∑ C∈C2 {σk} 〈CHLCM ,Ω h2+h1〉 + 1 50 η− 1 2 (h2+h1+h3−h4+1) ∑ 1≤k 6=l≤N ∑ C∈C2 {σk,σl} 〈CHLCM ,Ω h2+h2〉. (4.4) Note that each graph CHLCM splits naturally as (LC ,ΓC), where the first component contains the circles non incident to any insertion of C. The second sum in equation (4.4) vanishes, because it is the sum of terms proportional to:∑ Λ,Λ′ 〈Γ1(Λ,Λ′)〉 2 Z0(M) (4.5) and to ∑ Λ,Λ′,Λ′′ 〈Γ′2(Λ,Λ′,Λ′′)〉Z0(M), (4.6) where Γ1 is the dumbbell spin network of Fig. 5, and Γ′2 is a three-loop spin network, see Fig. 7. These spin networks arise from the cases when the pair (LC ,ΓC) is a surgery presentation of the disjoint union ( S3,Γ1 ) t ( S3,Γ1 ) and of ( S3,Γ′2 ) , respectively, apart from connected sums with S1 × S2; see Theorem 3 below. The former case corresponds to a situation where the two insertions act in pairs of triangles without a common triangle, and the latter corresponds to a situation where the two pairs of triangles have a triangle in common, which necessarily is a triangle in the intersection σk ∩ σl of the 4-simplices σk and σl. The evaluations in (4.5) and (4.6) vanish since the corresponding spin networks have tadpole subdiagrams. The first sum in (4.4) also gives the terms proportional to the ones in equations (4.5) and (4.6). These terms correspond to two insertions connecting two pairs made from four distinct triangles of σk and to two insertions connecting two pairs of triangles made from three distinct triangles of σk, respectively. All these terms vanish. The non-vanishing terms in equation (4.4) arise from a pair of insertions connecting the same two triangles in a 4-simplex. There are exactly 30 of these. Therefore, by using Theorem 3 of Section 5, we obtain: Z2(M,T ) = 3N 5 η−2 ∑ Λ,Λ′ dimq Λ dimq Λ′〈Γ2(Λ,Λ′)〉Z0(M), where Γ2 is a two-handle dumbbell spin network, see Fig. 8. We thus have: 12 J. Faria Martins and A. Miković Figure 8. Colored graph Γ2(Λ,Λ′), a double dumbbell graph. As usual A is the adjoint representation. Theorem 2. The second-order perturbative correction Z2(M,T ) divided by the number of N of 4-simplices of the manifold is triangulation independent. In fact: Z2(M,T ) N = 3 5 η−2 ∑ Λ,Λ′ dimq(Λ) dimq(Λ ′)〈Γ2(Λ,Λ′)〉Z0(M). Here 〈Γ2(Λ,Λ′)〉 denotes the spin-network evaluation of the colored graph Γ2(Λ,Λ′). Note that 〈Γ2(Λ,Λ′)〉 = θ(A,Λ,Λ)θ(A,Λ′,Λ′) dimq A , which is is obviously non-zero, and therefore Z2(M,T ) 6= 0. 5 Higher-order corrections For n > 2, the contributions to the partition function will be of the form η− 1 2 (h2+h1+h3−h4+1)〈CHLM ,Ω h1+h2 , (V̂1)k1 · · · (V̂N )kN 〉, where k1 + · · · + kN = n. By using the equation (4.2), each of these terms splits as a sum of terms of the form: 1 5n η− 1 2 (h2+h1+h3−h4+1)〈CHLCM ,Ω h1+h2〉 = 〈C〉, (5.1) where C is a set of n insertions (a “configuration”) distributed among the N 4-simplices of the chosen triangulation of M , such that the 4-simplex σi has ki insertions. Insertions are added to the chain-mail link CHLM as in Fig. 4, forming a graph CHLCM (for framing and crossing ambiguities we refer to [9]). Note that each graph CHLCM splits as (LC ,ΓC) where LC contains the components of CHLM not incident to any insertion. As in the n = 1 and n = 2 cases, equation (5.1) coincides, apart from the extra insertions, with the observables of the Crane–Yetter invariant defined in [3]; see Subsection 2.1. Therefore, by using the same argument that proves Theorem 2 of [3] we have: Theorem 3. Given a configuration C consider the 2-complex ΣC spanned by the k triangles of M incident to the insertions of C. Let Σ′C be a regular neighborhood of ΣC in M . Then M can be obtained from M \ Σ′C by adding the 2-handles corresponding to the faces of Σ (and some further 3- and 4-handles, corresponding to the edges and vertices of Σ). These 2-handles attach along a framed link K in ∂(M \Σ′), a manifold diffeomorphic to ∂(Σ′C) with the reverse orientation. The insertions of C can be transported to this link K defining a graph KC in ∂(Σ′∗C) = ∂(M \ Σ′C). We have: 〈C〉 = ∑ Λ1,...,Λk ZWRT ( ∂(Σ′∗C),KC ; Λ1, . . . ,Λk ) κs(M\Σ ′)η χ(Σ) 2 −k dimq Λ1 · · · dimq Λk, where s(M \ Σ′) denotes the signature of the manifold M \ Σ′ and χ denotes the Euler charac- teristic. Four-Dimensional Spin Foam Perturbation Theory 13 Note that, up to connected sums with S1 × S2, the pair CHLCM = (LC ,ΓC) is a surgery presentation of (∂(M \ Σ′C),KC) = (∂(Σ′∗C),KC). Unlike the n = 1 and n = 2 cases, it is not possible to determine the pair (∂(Σ′∗C),KC) for n ≥ 3 without having an additional information about the configuration. In fact, considering the set of all triangulations of M , an infinite number of diffeomorphism classes for (M \Σ′,KC) is in general possible for a fixed n; see [9] for the three dimensional case. This makes it complicated to analyze the triangulation independence of the formula for Zn(M,T ) for n ≥ 3. Since Z(M,T ) = ∑ n λnZn(M,T ), where Zn(M,T ) = ∑ k1+···+kN=n 1 k1! · · · kN ! η− 1 2 (h2+h1+h3−h4+1)〈CHLM ,Ω h1+h2 , (V̂1)k1 · · · (V̂N )kN 〉, in order to resolve the triangulation dependence of Zn, let us introduce the quantities zn = lim N→∞ Zn(M,T ) N n 2Z0(M) ; (5.2) see [9, 2]. The limit is to be extended to the set of all triangulations T of M , with N being the number of 4-simplices of M , in a sense to be made precise; see [9]. From Sections 3, 4 and Theorem 2 it follows that z1 = 0, z2 = 3η−2 5 dimq A (∑ Λ dimq Λθ(A,Λ,Λ) )2 . Note that the values of z1 and z2 are universal for all compact 4-manifolds. The expression for z2 is finite because there are only finitely many irreps for the quantum group Uq(g), of non-zero quantum dimension, when q is a root of unity. 6 Dilute-gas limit We will now show how to define and calculate the limit in the equation (5.2). Let M be a 4- manifold and let us consider a set S of triangulations of M , such that for any given ε > 0 there exists a triangulation T ∈ S such that the diameter of the biggest 4-simplex is smaller than ε, i.e. the triangulations in S can be chosen to be arbitrarily fine. We want to calculate the limit in equation (5.2) only for triangulations belonging to the set S. Furthermore, we suppose that S is such that (c.f. [9]): Restriction 1 (Control of local complexity-I). Together with the fact that the triangulations in S are arbitrarily fine we suppose that: There exists a positive integer L such that any 4-simplex of any triangulation T ∈ S intersects at most L 4-simplices of T . Let us fix n and consider Z2n(M,T ) when N →∞. The value of Z2n will be given as a sum of contributions of configurations C such that n1 insertions of V̂ act in a 4-simplex σ1, n2 of insertions of V̂ act in the 4-simplex σ2 6= σ1 and so on, such that n1 + n2 + · · ·+ nN = 2n and nk ≥ 0. 14 J. Faria Martins and A. Miković A configuration for which any 4-simplex has either zero or two insertions, with all 4-simplices which have insertions being disjoint will be called a dilute-gas configuration. There will be 15n N ! n!(N − n)! − δ(N,T ), dilute-gas configurations, where δ is the number of pairs of 4-simplices in T with non-empty intersection. From Restriction 1 it follows that δ(N,T ) = O(N) as N →∞. Each dilute-gas configuration contributes zn2Z0(M) to Z2n(M,T ) and we can write Z2n Z0 = ( N ! n!(N − n)! −O(N) ) zn2 + ∑ non-dilute C 〈C〉 Z0 , (6.1) where 〈C〉 = 1 5n η − 1 2 (h2+h1+h3−h4+1)〈CHLCM ,Ω h1+h2〉, denotes the contribution of the configura- tion C. Let us describe the contribution of the non-dilute configurations C more precisely. Recall that a configuration C is given by a choice of n insertions connecting n pairs of triangles of M , where each pair belongs to the same 4-simplex of M and the triangles have only one common vertex. Given a configuration C with n insertions, consider a (combinatorial) graph γC with a vertex for each triangle appearing in C and for each insertion and edge connecting the corresponding vertices. The graph γC is obtained from ΓC by collapsing the circles of ΓC of it into vertices. However note that γC is merely a combinatorial graph, whereas ΓC is a graph in S3, which can have a complicated embedding. If γC has a connected component homeomorphic to the graph made from two vertices and an edge connecting them, then 〈C〉 vanishes, since in this case the embedded graph whose surgery presentation is given by (CHLCM ,ΓC) will have a tadpole. In fact, looking at Theorem 3, one of the connected components of (∂(Σ′∗),KC) will be (S3,Γ1), where Γ1 is the graph in Fig. 5. Consider a manifold with a triangulation with N 4-simplices, satisfying Restriction 1. The number of possible configurations C with l insertions with make a connected graph γC is bounded by N(10L)l−1(l−1)!. In particular the number of non-dilute configurations V with 2n insertions and yielding a non-zero contribution is bounded by max l1+···+l2n=2n li 6=1 ∃i : li≥3 b 2n∏ i=1 N(10L)lk−1(lk − 1)! = O ( Nn−1 ) . (6.2) This is simply the statement that if a graph γC has k connected components, then it has O(Nk) possible configurations. Since k ≤ n− 1 for a non-dilute configuration, the bound (6.2) follows. We now need to estimate the value of 〈C〉 for a non-dilute configuration C. We will need to make the following restriction on the set S. We refer to the notation introduced in Theorem 3. Restriction 2 (Control of local complexity-II). The set of S of triangulations of M is such that given a positive integer n then the number of possible diffeomorphism classes for the pair (∂(Σ′C),KC) is finite as we vary the triangulation T ∈ S and the configuration C with n inser- tions. In the three-dimensional case a class of triangulations S satisfying Restrictions 1 and 2 was constructed by using a particular class of cubulations of 3-manifolds (which always exist, see [5]) and their barycentric subdivisions, see [9]. These cubulations have a simple local structure, with only three possible types of local configurations, which permits a case-by-case analysis as the cubulations are refined through barycentric subdivisions. In the case of four-dimensional Four-Dimensional Spin Foam Perturbation Theory 15 cubulations, no such list is known, although it has been proven that a finite (and probably huge) list exists. Therefore the approach used in the three-dimensional case cannot be directly applied to the four-dimensional case. However, it is reasonable to assume that triangulations coming from the barycentric subdivisons of a cubulation of M satisfy Restriction 2. More precisely, given a cubulation � of M , let ∆� be the triangulation obtained from � by taking the cone of each i-face of each cube of �, starting with the 2-dimensional faces. Consider the class S = {∆�(n)}∞n=0, where �(n) is the barycentric subdivision of order n of �. Then we can see that Restriction 1 is satisfied by this example, and we conjecture that S also satisfies Restriction 2. The Restriction 2 combined with Theorem 3 implies that the value of 〈C〉 in equation (6.1) is bounded for a fixed n, considering the set of all triangulations in the set S and all possible configurations with 2n insertions. Since the number of non-dilute configurations C which have a non-zero contribution is of O(Nn−1), it follows that∑ non-dilute C 〈C〉 Z0 = O ( Nn−1 ) . Therefore lim N→∞ Z2n(M,T ) NnZ0(M) = zn2 n! , or Z2n(M,T ) Z0(M) ≈ zn2 n! Nn, (6.3) for large N . In the case of Z2n+3, the dominant configurations for triangulations with a large number N of 4-simplices consist of configurations C (as before called dilute) whose associated combinatorial graph γC has as connected components a connected closed graph with three edges and (n− 1) connected closed graphs with two edges. We can write: Z2n+3 Z0 = ∑ dilute C 〈C〉+ ∑ non-dilute C 〈C〉. Since the number of dilute configurations is of O(Nn), while the second sum is of O(Nn−1), due to the Restrictions 1 and 2, we then obtain for large N Z2n+3 Z0 = O(Nn). More precisely Z2n+3 Z0 ≈ z3(z2)nN (N − 1)! n!(N − 1− n)! , or Z2n+3 Z0 ≈ z3(z2)n Nn n! , (6.4) for large N , where z3 is the sum of two terms. The first term is 30 6× 53 ∑ Λ,Λ′ dimq Λ dimq Λ′〈Γ3(Λ,Λ′)〉, 16 J. Faria Martins and A. Miković Figure 9. Colored graph Γ3(Λ,Λ′), a triple dumbbell graph. As usual A is the adjoint representation. Figure 10. Colored graph Γ′ 3(Λ,Λ′,Λ′′). As usual A is the adjoint representation. where Γ3 is the triple dumbbell graph of Fig. 9, corresponding to three insertions connecting the same pair of triangles of the underlying 4-simplex (there are exactly 30 of these). The second term is 30 6× 53 ∑ Λ,Λ′,Λ′′ dimq Λ dimq Λ′ dimq Λ′′〈Γ′3(Λ,Λ′,Λ′′)〉, where Γ′3 appears in Fig. 10. This corresponds to three insertions making a chain of triangles, pairwise having only a vertex in common (there are exactly 30 insertions like these for each 4-simplex). 6.1 Large-N asymptotics Let us now study the asymptotics of Z(M,T ) for N →∞. We will denote Z(M,T ) as Z(λ,N) and Z0(M) as Z0(λ0), in order to highlight the fact that the Crane–Yetter state sum Z0(M) can be understood as a path integral for the BF -theory with a cosmological constant term λ0 ∫ M 〈B ∧ B〉, such that λ0 is a certain function of an integer k0, which specifies the quantum group at a root of unity whose representations are used to construct the Crane–Yetter state sum. In the case of a quantum SU(2) group it has been conjectured that λ0 = 4π/k0, see [25]. Consequently Z(λ) = ∫ DADB ei ∫ M 〈B∧F+λB∧B〉 = ∫ DADBei ∫ M 〈B∧F+λ0B∧B〉ei(λ−λ0) ∫ M 〈B∧B〉, which means that our perturbation parameter is λ− λ0 instead of λ. Let us consider the partial sums ZP (λ,N) = P∑ n=0 inZn(N)(λ− λ0)n, where P = √ N . In this way we ensure that each perturbative order n in ZP is much smaller than N when N is large. We can then use the estimates from the previous section, and from (6.3) Four-Dimensional Spin Foam Perturbation Theory 17 and (6.4) we obtain ZP (λ,N) Z0(λ0) ≈ P/2∑ m=0 i2m(λ− λ0)2mN m m! zm2 + (P−3)/2∑ m=0 i2m+3(λ− λ0)2m+3z3(z2)m Nm m! ≈ [ 1 + iz3(λ− λ0)3 ] ∞∑ m=0 (−1)m gm m! zm2 = [ 1 + iz3(λ− λ0)3 ] e−gz2 , where g = (λ− λ0)2N. (6.5) Given that Z(λ,N) = lim P→∞ ZP (λ,N) for \|λ− λ0\| < r, where r is the radius of convergence of Z(λ,N) = ∞∑ n=0 in(λ− λ0)nZn(N), (6.6) then Z(λ,N) ≈ Z√N (λ,N) for large N . Therefore Z(λ,N) Z0(λ0) ≈ [ 1 + iz3(λ− λ0)3 ] exp (−gz2) , (6.7) for λ ≈ λ0, where g = (λ− λ0)2N . In the limit N →∞, λ→ λ0 and g constant we obtain Z(λ,N) Z0(λ0) → exp(−gz2). We can rewrite this result as Z(M, g) = e−gz2Z0(M), (6.8) where Z(M, g) is the perturbed partition function in the dilute-gas limit. This value is triangu- lation independent and it depends on the renormalized coupling constant g. Note that (6.7) can be rewritten as Z∗(λ, g) Z0(λ0) ≈ [ 1 + iz3(λ− λ0)3 ] exp (−gz2) , (6.9) where we have changed the variable N to variable g = N(λ− λ0)2 and Z∗(λ, g) = Z ( λ, g (λ− λ0)2 ) . (6.10) The approximation (6.9) is valid for λ→ λ0 and g (λ− λ0)2 →∞. 18 J. Faria Martins and A. Miković The result (6.8) can be understood as the lowest order term in the asymptotic expansion (6.9) where g is a constant. However, one can have a more general situation where g = f(λ − λ0) such that f(λ− λ0) (λ− λ0)2 →∞, (6.11) for λ→ λ0. In this case Z∗(λ, g) ≈ e−z2f(λ−λ0) [ 1 + iz3(λ− λ0)3 ] Z0(λ0), (6.12) which opens the possibility that Z∗(λ, g) ≈ [Z0(k)]k=φ(λ) , (6.13) where Z0(k) is the real number extension of the Crane–Yetter state sum and k = φ(λ) is the relation between k and λ. In the case of a quantum SU(2) group at a root of unity Z0(M,k) = e−iπs(M)R(k), (6.14) where s(M) is the signature of M and R(k) = k(2k + 1)/4(k + 2), see [23]. The value of λ which corresponds to k is conjectured to be k = 4π/λ, see [25], and this is an example of the function φ. The relation λ ∝ 1/k could be checked by calculating the large-spin asymptotics of the quantum 15j-symbol for the case of the root of unity, in analogy with the three-dimensional case, where by computing the asymptotics of the quantum 6j-symbol one can find that λ = 8π2/k2, see [21]. The quantum 15j-symbol asymptotics is not known, but for our purposes it is sufficient to know that λ→ 0 as k →∞. Equations (6.12), (6.13) and (6.14) imply f(λ− λ0) ≈ iπs(M) z2 [h(λ)− h(λ0)] + i z3 z2 (λ− λ0)3, (6.15) where h(λ) = R(φ(λ)). The solution (6.15) is consistent with the condition (6.11), since h′(λ0) 6= 0. However, f has to be a complex function, although the original definition (6.5) suggests a real function. This means that λ has to take complex values in order for (6.13) to hold, i.e. we need to perform an analytic continuation of the function in (6.9). Also note that the definition (6.6) and the fact that Z1(N) = 0 (Theorem 1) imply that Z ′(λ0, N) = 0, but this does not imply that lim λ→λ0 dZ∗(λ, f(λ− λ0)) dλ = 0 since Z(λ,N) and Z∗(λ, g) are different functions of λ due to (6.10) and g = f(λ− λ0). There- fore the approximation (6.13) is consistent. This also implies that we can define a triangulation independent Z(λ,M) as the real number extension of the function Z0(k,M), see (6.14). There- fore Z(λ,M) = Z0(k,M)\|k=φ(λ) = Z̃0(λ,M). (6.16) Four-Dimensional Spin Foam Perturbation Theory 19 7 Relation to 〈F ∧ F 〉 theory It is not difficult to see that the equations of motion for the action (2.1) are equivalent to the equations of motion for the action S̃ = ∫ M 〈F ∧ F 〉, because the B field can be expressed algebraically as BI = −gIJF J/(2λ). At the path-integral level, this property is reflected by the following consideration. One can formally perform a Gaussian integration over the B field in the path integral (2.2), which gives the following path integral Z = D(λ,M) ∫ DA exp ( i 4λ ∫ M 〈F ∧ F 〉 ) , (7.1) where D(λ,M) is the factor coming from the determinant factor in the Gaussian integration formula. More precisely, if we discretize M by using a triangulation T with n triangles, then the path integral (2.2) becomes a finite-dimensional integral∫ ∏ l,I dAIl ∫ ∏ f,I dBI f exp ( i ∑ ∆ Tr (B∆Ff ) + iλ 5 ∑ σ ∑ ∆′,∆′′∈σ 〈B∆′B∆′′〉 ) . (7.2) The integral over the B variables in (7.2) can be written as∫ +∞ −∞ · · · ∫ +∞ −∞ ∏ k,l dBkle iλ(B,QB)+i(B,F), (7.3) where m = dimA, B = (B11, . . . , Bmn) and F = (F11, . . . , Fmn) are vectors in Rmn, (X,Y ) is the usual scalar product in Rmn and Q is an mn×mn matrix. The integral (7.3) can be defined as the analytic continuation λ→ iλ, F → iF of the formula∫ +∞ −∞ · · · ∫ +∞ −∞ ∏ k,l dBkle −λ(B,QB)+(F ,B) = √ πmn λmn detQ e(F ,Q−1F)/4λ, (7.4) so that when n→∞ such that the triangulations become arbitrarily fine, we can represent the limit as the path integral (7.1). Since ∫ M 〈F ∧ F 〉 is a topological invariant of M , which is the second characteristic class number c2(M), see [7], we can write Z(M,λ) = E(M,λ)eic2(M)/λ, where E(M,λ) = D(M,λ) ∫ DA and D(M,λ) denotes the (λmn detQ)−1/2 factor from (7.4). As we have shown in the previous section, Z(λ,M) = Z̃0(M,λ), so that in the case of SU(2) E(M,λ) = e−ic2(M)/λ−iπs(M)h(λ). (7.5) Therefore one can calculate the volume of the moduli space of connections on a principal bundle provided that the relation k = φ(λ) is known for the corresponding quantum group. 20 J. Faria Martins and A. Miković 8 Conclusions The techniques developed for the 3d spin foam perturbation theory in [9] can be extended to the 4d case, and hence the 4d partition function has the same form as the corresponding 3d partition function in the dilute gas limit, see (6.8). The constant z2 depends only on the group G and an integer, and z2 is related to the second-order perturbative contribution, see Section 5. The constant z2 appears because the constant z1 vanishes for the same reason as in the 3d case, which is the vanishing of the tadpole spin network evaluation. The result (6.8) implies that Z(M, g) is not a new manifold invariant, but it is propor- tional to the Crane–Yetter invariant. Given that the renormalized coupling constant g is an arbitrary number, a more usefull way of representing our result is the asymptotic for- mula (6.12). This formula allowed us to conclude that Z(M,λ) can be identified as the Crane– Yetter partition function evaluated at k = φ(λ), see (6.13) and (6.16). The formula (6.12) also applies to the spin foam perturbation expansion in 3d, where z2 and z3 are given as the state sums of the corresponding 3d graphs, see [9]. Therefore the formula (6.12) is the jus- tification for the conjecture made in [9], where the triangulation independent Z(λ,M) was identified with the Turaev–Viro partition function ZTV (M,k) for k = 4π2/λ2 in the SU(2) case. The relation (6.16) was useful for determining the volume of the moduli space of connections on the G-principal bundle for arbitrary values of λ, given that Z(M,λ) is related to the path integral of the 〈F ∧ F 〉 theory, see (7.5). However, it still remains to be proved the conjec- ture that k ∝ 1/λ for G = SU(2), while for the other groups the function k = φ(λ) is not known. Note that the result (6.8) depends on the existence of a class of triangulations of M which are arbitrarily fine, but having a finite degree of local complexity. As explained in Section 5 it is reasonable to assume that such a class exists, and can be constructed by considering the triangulations coming from the barycentric subdivisions of a fixed cubulation of M . Our approach applies to Lie groups whose vector space of intertwiners Λ ⊗ Λ → A is one- dimensional for each irreducible representation Λ. This is true for the SU(2) and SO(4) groups, but it is not true for the SU(3) group. This can probably be fixed by adding extra information to the chain-mail link with insertions at the 3-valent vertices. Also note that we only considered the gIJ ∝ δIJ case. This is sufficient for simple Lie groups, but in the case of semi-simple groups one can have non-trivial gIJ . Especially interesting is the SO(4) case, where gIJ ∝ εabcd. In the general case one will have to work with spin networks which will have L(Λ) and gIJ insertions, so that it would be very interesting to find out how to generalize the Chain–Mail formalism to this case. One of the original motivations for developing a four-dimensional spin-foam perturbation theory was a possibility to obtain a nonperturbative definition of the four-dimensional Euclidean quantum gravity theory, see [10] and also [19, 20]. The reason for this is that general relativity with a cosmological constant is equivalent to a perturbed BF -theory given by the action (2.1), where G = SO(4, 1) for a positive cosmological constant, while G = SO(3, 2) for a negative cosmological constant and gIJ = εabcd in both cases, see [19, 20]. However, the gIJ in the gravity case is not a G-invariant tensor, since it is only invariant under a subgroup of G, which is the Lorentz group. Consequently this perturbed BF theory is not topological. In the Euclidean gravity case one has G = SO(5), and the invariant subgroup is SO(4) since gIJ = εabcd. One can then formulate a spin foam perturbation theory along the lines of Section 3. However, the Chain–Mail techniques cannot be used, because gIJ is not a G-invariant tensor and therefore one lacks an efficient way of calculating the perturbative contributions. In order to make further progress, a generalization of the Chain–Mail calculus has to be found in order to accommodate the case when gIJ is invariant only under a subgroup of G. Four-Dimensional Spin Foam Perturbation Theory 21 Acknowledgments This work was partially supported FCT (Portugal) under the projects PTDC/MAT/099880/2008, PTDC/MAT/098770/2008, PTDC/MAT/101503/2008. This work was also partially supported by CMA/FCT/UNL, through the project PEst-OE/MAT/UI0297/2011. References [1] Baez J., An introduction to spin foam models of quantum gravity and BF theory, in Geometry and Quantum Physics (Schladming, 1999), Lecture Notes in Phys., Vol. 543, Springer, Berlin, 2000, 25–93, gr-qc/9905087. [2] Baez J., Spin foam perturbation theory, in Diagrammatic Morphisms and Applications (San Francisco, CA, 2000), Contemp. Math., Vol. 318, Amer. Math. Soc., Providence, RI, 2003, 9–21, gr-qc/9910050. [3] Barrett J.W., Faria Martins J., Garćıa-Islas J.M., Observables in the Turaev–Viro and Crane–Yetter models, J. Math. Phys. 48 (2007), 093508, 18 pages, math.QA/0411281. [4] Broda B., Surgical invariants of four-manifolds, hep-th/9302092. [5] Cooper D., Thurston W., Triangulating 3-manifolds using 5 vertex link types, Topology 27 (1988), 23–25. [6] Crane L., Yetter D.A., A categorical construction of 4D topological quantum field theories, in Quantum Topology, Ser. Knots Everything, Vol. 3, World Sci. Publ., River Edge, NJ, 1993, 120–130, hep-th/9301062. [7] Eguchi T., Gilkey P.B., Hanson A.J., Gravitation, gauge theories and differential geometry, Phys. Rep. 66 (1980), 213–393. [8] Faria Martins J., Miković A., Invariants of spin networks embedded in three-manifolds, Comm. Math. Phys. 279 (2008), 381–399, gr-qc/0612137. [9] Faria Martins J., Miković A., Spin foam perturbation theory for three-dimensional quantum gravity, Comm. Math. Phys. 288 (2009), 745–772, arXiv:0804.2811. [10] Freidel L., Krasnov K., Spin foam models and the classical action principle, Adv. Theor. Math. Phys. 2 (1999), 1183–1247, hep-th/9807092. [11] Freidel L., Starodubtsev A., Quantum gravity in terms of topological observables, hep-th/0501191. [12] Gompf R.E., Stipsicz A.I., 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, Vol. 20, American Mathematical Society, Providence, RI, 1999. [13] Kauffman L.H., Lins S.L., Temperley–Lieb recoupling theory and invariants of 3-manifolds, Annals of Math- ematics Studies, Vol. 134, Princeton University Press, Princeton, NJ, 1994. [14] Kirby R.C., The topology of 4-manifolds, Lecture Notes in Mathematics, Vol. 1374, Springer-Verlag, Berlin, 1989. [15] Lickorish W.B.R., The skein method for three-manifold invariants, J. Knot Theory Ramifications 2 (1993), 171–194. [16] Mackaay M., Spherical 2-categories and 4-manifold invariants, Adv. Math. 143 (1999), 288–348, math.QA/9805030. [17] Mackaay M., Finite groups, spherical 2-categories, and 4-manifold invariants, Adv. Math. 153 (2000), 353– 390, math.QA/9903003. [18] Miković A., Spin foam models of Yang–Mills theory coupled to gravity, Classical Quantum Gravity 20 (2003), 239–246, gr-qc/0210051. [19] Miković A., Quantum gravity as a deformed topological quantum field theory, J. Phys. Conf. Ser. 33 (2006), 266–270, gr-qc/0511077. [20] Miković A., Quantum gravity as a broken symmetry phase of a BF theory, SIGMA 2 (2006), 086, 5 pages, hep-th/0610194. [21] Mizoguchi S., Tada T., Three-dimensional gravity from the Turaev–Viro invariant, Phys. Rev. Lett. 68 (1992), 1795–1798, hep-th/9110057. [22] Reshetikhin N., Turaev V.G., Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547–597. http://dx.doi.org/10.1007/3-540-46552-9_2 http://arxiv.org/abs/gr-qc/9905087 http://arxiv.org/abs/gr-qc/9910050 http://dx.doi.org/10.1063/1.2759440 http://arxiv.org/abs/math.QA/0411281 http://arxiv.org/abs/hep-th/9302092 http://dx.doi.org/10.1016/0040-9383(88)90004-3 http://dx.doi.org/10.1142/9789812796387_0005 http://arxiv.org/abs/hep-th/9301062 http://dx.doi.org/10.1016/0370-1573(80)90130-1 http://dx.doi.org/10.1007/s00220-008-0422-8 http://arxiv.org/abs/gr-qc/0612137 http://dx.doi.org/10.1007/s00220-009-0776-6 http://dx.doi.org/10.1007/s00220-009-0776-6 http://arxiv.org/abs/0804.2811 http://arxiv.org/abs/hep-th/9807092 http://arxiv.org/abs/hep-th/0501191 http://dx.doi.org/10.1142/S0218216593000118 http://dx.doi.org/10.1006/aima.1998.1798 http://arxiv.org/abs/math.QA/9805030 http://dx.doi.org/10.1006/aima.1999.1909 http://arxiv.org/abs/math.QA/9903003 http://dx.doi.org/10.1088/0264-9381/20/1/317 http://arxiv.org/abs/gr-qc/0210051 http://dx.doi.org/10.1088/1742-6596/33/1/029 http://arxiv.org/abs/gr-qc/0511077 http://dx.doi.org/10.3842/SIGMA.2006.086 http://arxiv.org/abs/hep-th/0610194 http://dx.doi.org/10.1103/PhysRevLett.68.1795 http://arxiv.org/abs/hep-th/9110057 http://dx.doi.org/10.1007/BF01239527 http://dx.doi.org/10.1007/BF01239527 22 J. Faria Martins and A. Miković [23] Roberts J., Skein theory and Turaev–Viro invariants, Topology 34 (1995), 771–787. [24] Rourke C.P., Sanderson B.J., Introduction to piecewise-linear topology, Reprint, Springer Study Edition, Springer-Verlag, Berlin – New York, 1982. [25] Smolin L., Linking topological quantum field theory and nonperturbative quantum gravity, J. Math. Phys. 36 (1995), 6417–6455, gr-qc/9505028. [26] Turaev V.G., Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, Vol. 18, Walter de Gruyter & Co., Berlin, 1994. [27] Witten E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351–399. http://dx.doi.org/10.1016/0040-9383(94)00053-0 http://dx.doi.org/10.1063/1.531251 http://arxiv.org/abs/gr-qc/9505028 http://dx.doi.org/10.1007/BF01217730 1 Introduction 2 Spin foam perturbative expansion 2.1 The Chain-Mail formalism and observables of the Crane-Yetter invariant 3 The first-order correction 4 The second-order correction 5 Higher-order corrections 6 Dilute-gas limit 6.1 Large-N asymptotics 7 Relation to "426830A FF"526930B theory 8 Conclusions References

Four-Dimensional Spin Foam Perturbation Theory

Institution

Similar Items