NUPHB14609S0550-3213(19)30089-610.1016/j.nuclphysb.2019.03.021The Author(s)High Energy Physics – TheoryAspects of higher-abelian gauge theories at zero and finite temperature: Topological Casimir effect, duality and Polyakov loopsGeraldKelnhofergerald.kelnhofer@gmail.comFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, AustriaFaculty of PhysicsUniversity of ViennaBoltzmanngasse 5ViennaA-1090AustriaEditor: Clay CórdovaAbstractHigher-abelian gauge theories associated with Cheeger-Simons differential characters are studied on compact manifolds without boundary. The paper consists of two parts: First the functional integral formulation based on zeta function regularization is revisited and extended in order to provide a general framework for further applications. A field theoretical model - called extended higher-abelian Maxwell theory - is introduced, which is a higher-abelian version of Maxwell theory of electromagnetism extended by a particular topological action. This action is parametrized by two non-dynamical harmonic forms and generalizes the θ-term in usual gauge theories. In the second part the general framework is applied to study the topological Casimir effect in higher-abelian gauge theories at finite temperature at equilibrium. The extended higher-abelian Maxwell theory is discussed in detail and an exact expression for the free energy is derived. A non-trivial topology of the background space-time modifies the spectrum of both the zero-point fluctuations and the occupied states forming the thermal ensemble. The vacuum (Casimir) energy has two contributions: one related to the propagating modes and the second one related to the topologically inequivalent configurations of higher-abelian gauge fields. In the high temperature limit the leading term is of Stefan-Boltzmann type and the topological contributions are suppressed. With a particular choice of parameters extended higher-abelian Maxwell theories of different degrees are shown to be dual. On the n-dimensional torus we provide explicit expressions for the thermodynamic functions in the low- and high temperature regimes, respectively. Finally, the impact of the background topology on the two-point correlation function of a higher-abelian variant of the Polyakov loop operator is analyzed.1Introduction and summary1.1Motivation and objectiveSince its discovery by Casimir in 1948 [1], the Casimir effect at zero and nonzero temperature has been a subject of intense research both theoretically and experimentally, ranging from string theory to condensed matter physics, nanotechnology and cosmology. For a general review see [2–7] and further references therein.In brief, the Casimir effect is caused by a modification of the vacuum energy of a quantum field subjected to boundary conditions as compared to the quantum field without any constraints. At finite temperature not only the vacuum energy but also the energy of the thermal excitations is affected giving rise to a modified free energy. These constraints on the quantum field may be imposed either by the presence of material boundaries, interfaces, domain walls or by a topologically non-trivial space-time background. The effect implied by the latter is usually called topological Casimir effect and indicates a deep relation between the global properties of underlying space-time and the quantum phenomenon. The topological Casimir effect at zero and finite temperature has been considered for many years in different contexts (see e.g. [8–42]).11The reviews [2–7] and the references therein cover also further aspects of the topological Casimir effect. Due to the vast amount of literature we are well aware that our selection may by no means be complete. These areas of applications include for example the impact of the topological Casimir effect in models of a compact universe with non-trivial topology, its role for the stabilization of moduli in multidimensional Kaluza-Klein type theories, its occurrence in brane-world cosmological scenarios or its consequences for compactified condensed matter systems. The interest in the topological Casimir effect at finite temperature is motivated in a large part by the standard cosmological model and inflationary models, where at very early stages the evolution of our universe undergoes a high temperature phase. In addition to the modifications of the partition function and the thermodynamic functions, a non-trivial topology of the background space-time also affects the (thermal) vacuum expectation values (VEVs) of physical observables.Most of the research has been devoted to scalar fields and to the electromagnetic field, but to a lesser extent to antisymmetric tensor fields. In fact, these fields are represented by differential forms A∈Ωp(M) of a certain degree p≤n on a n-dimensional manifold M and are subjected to gauge transformations A↦A+dξ, where ξ∈Ωp−1(M). The most prominent example is the so-called p-form Maxwell theory [43,44], which is governed by the classical action(1.1)S(A)=qp2∫MdA∧⋆dA, and whose dynamical (i.e. propagating) degrees of freedom are represented by co-exact p-forms on M (qp denotes the coupling constant). During the last years several aspects of the topological Casimir effect in p-form Maxwell theory were discussed on different manifolds at zero [45–50] and at finite temperature [51–53].However, antisymmetric tensor fields represent just a special class of higher-abelian gauge fields. Compared to the former which are topologically trivial, the latter possess topologically inequivalent configurations giving rise to quantized charges and fluxes. These higher-abelian fields play an important role in string theory and supergravity and nowadays gain importance also in other branches of physics. A suitable mathematical framework for their description is provided by differential cohomology, which, in brief, can be viewed as a specific combination of ordinary (e.g. singular) cohomology with the algebra of differential forms on smooth manifolds. Following [54], we may call a higher-abelian (or generalized abelian) gauge theory any field theory on a smooth manifold whose space of gauge inequivalent configurations (i.e. the gauge orbit space) is modeled by a certain differential cohomology group. A review of differential cohomology and its application in physics can be found in [54–56].For sake of clarity and due to the fact that differential cohomology is uniquely determined up to equivalence [57,58], we focus on higher-abelian gauge theories whose gauge orbit space is the abelian group of Cheeger-Simons differential characters Hˆp(M) [59] of some degree p. Let us recall that differential characters are U(1)-valued group homomorphisms on the abelian group Zp(M) of smooth singular p-cycles in M that satisfy a certain smoothness condition. To each differential character uˆ∈Hˆp(M) is associated a closed p+1 differential form δ1(uˆ) with integer periods, called curvature (or field strength), and a so-called characteristic class δ2(uˆ) which belongs to the integer cohomology group Hp+1(M;Z). As uˆ varies, the assigned characteristic classes label the different topological sectors or in physical terms the topologically inequivalent configurations of higher-abelian gauge fields.It is natural to raise the question which impact a non-trivial topology of the background space has on these topologically inequivalent configurations which are not present in theories of ordinary antisymmetric tensor fields. In regard to the applications mentioned above, particularly the combined effect of topology and temperature seems to deserve closer attention. However, up to now there does not exist exists an exhaustive analysis of the topological Casimir effect in higher-abelian gauge theories.Our main motivation is to fill this gap and to present a detailed study of higher-abelian gauge theories at finite temperature at equilibrium on a n-dimensional compact and closed spatial background. The aim is to shed some light on the impact of the topology of the background on both the vacuum (Casimir) energy and the free energy associated with the occupied states forming the thermal ensemble of higher-abelian gauge fields at equilibrium. The present paper is a continuation of previous work on abelian gauge theories at zero [60,61] and non-zero temperature [62].1.2Outline and summary of resultsWe perform our analysis in the functional integral formalism by using zeta function regularization. It is well known that especially in gauge theories at finite temperature particular care has to be taken to obtain a correct functional integral measure including all relevant temperature and volume dependent factors.22For a discussion on the necessity to retain the ghost-for-ghost term in order to get the correct number for the degrees of freedom in pure QED at finite temperature we refer to the seminal paper [63]. See also analogous statements regarding the need for the ghost-for-ghost factor in antisymmetric tensor field theories [64] or in supergravity [65]. For Maxwell theory on topologically non-trivial manifolds we refer again to [62] and references therein. This justifies to review the corresponding construction principle before discussing the topological Casimir effect.For this reason the present paper is split into two main parts. The first part which contains section 2 is devoted to the functional integration of higher-abelian gauge theories. In order to make the presentation self-contained we begin in section 2.1 with a short introduction to Cheeger-Simons differential characters. Since higher-abelian gauge theories are affected in general by Gribov ambiguities [61], the conventional Faddeev-Popov method to fix the gauge needs to be modified. By revisiting in section 2.2 the construction of the functional integral representation for the partition function [61] to resolve the Gibov problem we find, however, that higher-abelian gauge theories exhibit a larger global gauge symmetry than discussed previously. After extracting the volume of the corresponding gauge group, the evaluation of the functional integral along the gauge inequivalent configurations yields a field independent Jacobian which is formed by two factors. One part derives from the conventional - due to the existence of zero modes suitable regularized - ghost-for-ghost contribution of the propagating modes. The other contribution to the Jacobian now is related to the volumes of the subgroups of flat differential characters (i.e. differential characters with vanishing curvature). In the thermal case this topological factor depends on temperature and volume and hence - like the conventional ghost-for-ghost contribution - has to be retained in the partition function. In summary, we obtain a general formula from which the partition function can be derived once the higher-abelian gauge model is specified.In this paper we consider an extended version of higher-abelian Maxwell theory which we introduce in section 2.3. In its original form, higher-abelian Maxwell theory [54,56,61,66,67] is governed by the classical action(1.2)S0(uˆ)=qp2∫Mδ1(uˆ)∧⋆δ1(uˆ),uˆ∈Hˆp(M), and generalizes (1.1) by replacing the trivial field strength dA by δ1(uˆ). We will extend this action furthermore by adding a specific topological action Stop=Stop(uˆ,γ,θ) to S0, such that the classical equation of motion is unchanged. The topological action is parametrized by two arbitrary harmonic forms γ and θ on M representing two de Rham cohomology classes of degree p+1 and n−p, respectively. They can be regarded as two external, non-dynamical topological fields. Although the addition of Stop does not change the dynamics classically, it contributes non-trivially to the quantum partition function. This resembles the situation of the θ-term in ordinary gauge theory and is related to the fact that the quantization is not unique on topologically non-trivial backgrounds. For short we may call this model with total action Stot=S0+Stop extended higher-abelian Maxwell theory. We compute the corresponding partition function and show that the ratio of the partition functions of degrees p and n−p−1 becomes independent of the metric of M if certain conditions on the coupling constants and on the topological fields are met. Additionally, in odd dimensions and for topological fields with integer periods the corresponding partition functions are equal, meaning that the two extended higher-abelian Maxwell theories are exactly dual to each other. In the special case of vanishing topological fields we recover the result previously obtained in [68]. This closes the first part of the present paper.The second part which contains sections 3 and 4 is devoted to the application of the general framework of the first part to the thermal case. In section 3.1, we determine an exact formula for the regularized free energy and derive a corresponding asymptotic expansion for the high-temperature regime. These results are applied in section 3.2 to extended higher-abelian Maxwell theory. It is shown that the topologically inequivalent configurations of the higher-abelian gauge fields give rise to additional contributions in the thermodynamic functions. As a consequence the total vacuum (Casimir) energy is the sum of the vacuum energy of the propagating modes and the vacuum energy generated by the topological fields. The latter vanishes whenever the topological fields are absent or have integer periods. This generalizes our findings obtained previously in Maxwell theory at finite temperature [62], where the effect of these topological contributions was studied. This has initiated a lot of further investigations with interesting applications [69–72]. One of the main conclusions was that these topological contributions should be regarded as physically observable. They can be traced back to tunneling events between different topological sectors. In analogy we would suppose that the same arguments apply to the topological Casimir effect in extended higher-abelian Maxwell theory as well.At low temperatures the free energy is governed by the vacuum (Casimir) energy. For the high-temperature limit an asymptotic expansion for the free energy is computed in terms of the Seeley (heat kernel) coefficients. The leading contribution is the extensive Stefan-Boltzmann term in n dimensions generated by the propagating degrees of freedom, followed by subleading terms related to finite size and topological effects. Due to regularization ambiguities and a non-trivial topology of the spatial background the invariance of the free energy under constant scale transformations is violated leading to a modified equation of state compared to an ideal Bose gas in flat Euclidean space. As a consequence of the duality relation discussed in section 2 we infer that the thermodynamic functions of extended higher-abelian Maxwell theories of degrees p and n−p−1 are equal implying that the two theories are exactly dual to each other.As an explicit example we consider extended higher-abelian Maxwell theory on the n-dimensional torus Tn in section 3.3. Exact expressions for the thermodynamic functions are computed for both the low- and high temperature regime in terms of Epstein-zeta functions. If the topological field θ has half-integral periods, the entropy converges in the zero temperature limit to a non-vanishing universal term indicating that the ground state is degenerate. In the high-temperature limit each thermodynamic function is dominated by its respective Stefan-Boltzmann term followed by a subleading topological contribution. All other terms decrease exponentially with increasing temperature. Since the topological contributions are suppressed in the high-temperature limit we find that the leading term in the entropy to energy ratio becomes linear in the inverse temperature and depends only on the dimension of the n-torus.Of particular importance for thermal gauge theories is the so-called Polyakov loop operator, a variant of the Wilson loop operator, which measures the holonomy of the gauge fields along the periodic (thermal) time. In section 4 we investigate how the two-point correlation function of a higher-abelian version of the Polyakov loop operator is affected by the topology of the spatial background. By analogy with point-like static charges in gauge theory this two-point correlation function is interpreted as change in the free energy in the presence of a pair of oppositely charged, static and closed branes in a bath of thermal higher-abelian gauge fields. Apart from a temperature independent interaction term of Coulomb type the topological sectors are shown to give rise to both a temperature independent and temperature dependent contribution.Appendix A contains some information on Riemann Theta functions and Epstein zeta functions needed in this paper. We choose the conventions c=ħ=kB=1.2Higher-abelian gauge theoriesIn this section we want to introduce the geometrical setting for the formulation of higher-abelian gauge theories as well as the notation and conventions that will be used in this paper. In the second step the configuration space is studied and the functional integral over the group of Cheeger-Simons differential characters is introduced.2.1The configuration space - Cheeger-Simons differential charactersLet M be a (n+1)-dimensional compact, oriented and connected Riemannian manifold without boundary, (Ω•(M);dM) the de-Rham complex of smooth differential forms and Ωcl•(M)=kerd•M the space of closed differential forms. Furthermore, let (C•(M;Z);∂) be the complex of smooth singular chains in M with boundary map ∂•:C•(M;Z)→C•−1(M;Z). For an abelian group G, let C•(M;G)=C•(M;Z)⊗G denote the smooth singular complex with coefficients in G and let C•(M;G)=Hom(C•(M;Z),G) be the complex of abelian groups of G-valued cochains with coboundary δ•:C•(M;G)→C•+1(M;G). Furthermore, Z•(M;G)=ker(∂•⊗id) is the subcomplex of smooth singular G-valued cycles and Z•(M;G)=kerδ• is the subcomplex of smooth singular G-valued cocycles. The corresponding singular homology and cohomology groups with coefficients in G are denoted by H•(M;G) and H•(M;G), respectively.Let ip:Ωp(M)→Cp(M;U(1)) be the map ip(A)(Σ)=e2πi∫ΣA, where Σ∈Cp(M;Z). Moreover, let ΩZp(M) denote the abelian group of closed smooth differential p-forms with integer periods, i.e. for ω∈ΩZp(M) its integral ∫Σω over any p-cycle Σ∈Zp(M;Z) gives an integer.The abelian group of Cheeger-Simons differential characters [59] of degree p is defined by(2.1)Hˆp(M)={uˆ∈Hom(Zp(M;Z),U(1))|uˆ∘∂∈ip+1(ΩZp+1(M))}, with group multiplication (uˆ1⋅uˆ2)(Σ)=uˆ1(Σ)⋅uˆ2(Σ). By definition there exists a (p+1)-form δ1(uˆ)∈ΩZp+1(M) such that uˆ(∂Σ)=e2πi∫Σδ1(uˆ) for all Σ∈Cp+1(M;Z). This correspondence uˆ↦δ1(uˆ) defines a homomorphism δ1:Hˆp(M)→ΩZp+1(M), which is called the curvature (or field strength) of uˆ. On the other hand, since Zp(M;Z) is a free Z module, there exists a homomorphism u˜:Zp(M;R)→R such that uˆ(Σ)=e2πiu˜(Σ). But then the map u˘:Cp+1(M;Z)→R, defined by u˘(Σ)=∫Σδ1(uˆ)−u˜(∂Σ), belongs to Zp+1(M;Z) and gives rise to the cohomology class [u˘]∈Hp+1(M;Z). This assignment uˆ↦[u˘] generates a homomorphism δ2:Hˆp(M)→Hp+1(M,Z) and δ2(uˆ)=[u˘] is called the characteristic class of uˆ.Let Rp+1(M)={(ω,c)∈ΩZp+1(M)×Hp+1(M,Z)|[ω]=j⁎(c)}, where j⁎:Hp+1(M,Z)→Hp+1(M,R) is induced by the inclusion Z↪R. Each class [v]∈Hp(M,U(1)) defines a differential character j1([v])∈Hˆp(M) by j1([v])(Σ):=v|Zp(M;Z)(Σ)=<[v],[Σ]> where [Σ] denotes the homology class of the p-cycle Σ∈Zp(M;Z). On the other hand each class [A]∈Ωp(M)/ΩZp(M) defines a differential character j2([A])∈Hˆp(M) by setting j2([A])(Σ):=e2πi∫ΣA. The homomorphisms j1, j2, δ1 and δ2 fit into the following exact sequences of abelian groups [59](2.2)0→Hp(M,U(1))→j1Hˆp(M)→δ1ΩZp+1(M)→00→Ωp(M)/ΩZp(M)→j2Hˆp(M)→δ2Hp+1(M,Z)→00→Hp(M;R)/j⁎(Hp(M;Z))→Hˆp(M)→(δ1,δ2)Rp+1(M)→0. Furthermore δ1(j2([A]))=dA and δ2(j1(v))=−δ⁎([v]), where δ⁎:Hp(M,U(1))→Hp(M,Z) is the Bockstein operator [73] induced from the short exact sequence 0→Z→R→U(1)→0. Since δ2(j2([A]))=0, the differential characters induced by p-forms are called topologically trivial, whereas differential characters induced by classes in Hp(M,U(1)) are called flat, since δ1(j1([v]))=0. The group of flat differential characters is a compact abelian group and fits into the short exact sequence(2.3)0→Hp(M;R)/j⁎(Hp(M;Z))→Hp(M,U(1))→δ⁎Htorp+1(M,Z)→0, where Htorp+1(M,Z) is the torsion subgroup in Hp+1(M,Z).In fact, Hˆ•(.) is a graded functor from the category of smooth manifolds to the category of Z-graded abelian groups, equipped with natural transformations j1, j2, δ1, δ2. It provides a specific model for a differential cohomology theory.33While there exist many different models in addition, like Deligne cohomology [74], the de Rham-Federer model [75] or Hopkins-Singer differential cocycles [76], they were all shown [57,58] to be isomorphic to Hˆ•(.). Moreover Hˆp(M) can be given the structure of an infinite dimensional abelian Lie group, whose connected components are labeled by Hp+1(M,Z) (for a detailed account on its Frechet-Lie group structure we refer to [77]). In physical terms Hˆp(M) is interpreted as gauge orbit space of higher-abelian gauge theories. In this context a differential character uˆ∈Hˆp(M) represents an equivalence class of a higher abelian gauge field.44More generally, higher-abelian gauge fields can be naturally modeled as objects of an action groupoid whose morphisms are the local gauge transformations and whose automorphisms represent the global gauge transformations. The set of isomorphism classes of objects gives then the corresponding gauge orbit space. An explicit model is provided by the category of Hopkins-Singer differential cocycles [76]. This resembles the situation encountered in ordinary gauge theory where one works with gauge fields rather than gauge equivalence classes. The equivalence class induced by an antisymmetric tensor field corresponds to a differential character whose field strength is an exact form and whose characteristic class vanishes.A L2-inner product on Ωp(M) is defined by(2.4)<υ1,υ2>=∫Mυ1∧⋆υ2υ1,υ2∈Ωp(M), where ⋆ is the Hodge star operator on M with respect to a fixed metric g. On p-forms the Hodge star operator satisfies ⋆2=(−1)p(n+1−p). Together with the related co-differential operator (dpM)†=(−1)(n+1)(p+1)+1⋆dn+1−pM⋆:Ωp(M)→Ωp−1(M) the Laplace operator on p-forms is defined by ΔpM=(dp+1M)†dpM+dp−1M(dpM)†.Let Hp(M):=kerΔpM denote the space of harmonic p-forms on M and let Hp(M)⊥ be its orthogonal complement. The Green’s operator is given by(2.5)GpM:Ωp(M)→Hp(M)⊥,GpM:=(ΔpM|Hp(M)⊥)−1∘πHp⊥, where πHp⊥ is the projection onto Hp(M)⊥. By construction ΔpM∘GpM=GpM∘ΔpM=πHp⊥. The dimension of Hp(M) is given by the p-th Betti number bp(M) of M. The abelian subgroup of harmonic p-forms with integer periods is denoted by HZp(M):=Hp(M)∩ΩZp(M).Since the homology of M is finitely generated we can choose a set of p-cycles Σi(p)∈Zp(M), i=1,…,bp(M), whose corresponding homology classes [Σi(p)] provide a Betti basis of Hp(M;Z). Let (ρM(n+1−p))j∈HZn+1−p(M) be a basis associated to the Betti basis [Σi(p)] via Poincare duality, where j=1,…,bn+1−p(X). A dual basis (ρM(p))i∈HZp(M) can be adjusted such that ∫Σj(p)(ρM(p))i=∫M(ρM(p))i∧(ρM(n+1−p))j=δij holds, implying that ∫Σj(p)B=∫MB∧(ρM(n+1−p))j for any [B]∈Hp(M;R). Then(2.6)(hM(p))ij=<(ρM(p))i,(ρM(p))j>=∫M(ρM(p))i∧⋆(ρM(p))j is the induced metric on Hp(M). With respect to this basis, the projector onto the harmonic forms reads(2.7)πHp(A)=∑j,k=1bp(M)(hM(p))jk−1<A,(ρM(p))j>(ρM(p))k=∑j=1bp(M)(∫MA∧(ρM(n+1−p))j)(ρM(p))j. Two such bases of harmonic forms with integer periods are connected by a unimodular transformation.Since im(dp−1M)≅im(dpM)†, one has equivalently ΩZp(M)≅im(dpM)†×HZp(M). In the following we will utilize this isomorphic representation of ΩZp(M). Correspondingly, the action of ΩZp(M) on Ωp(M) has the form Ap⋅(τp−1,ηp):=Ap+dτp−1+ηp, where (τp−1,ηp)∈im(dpM)†×HZp(M).2.2The partition function - general caseIn this section we revisit the functional integral quantization of higher-abelian gauge theories in terms of differential characters extending [61]. Our aim is to provide a general framework for the functional integral formulation of these theories without referring to a particular action or observable.Let S=S(uˆp), uˆp∈Hˆp(M), be an Euclidean action for a higher-abelian gauge theory of degree p on the manifold M. Additionally, the action may depend on other external (i.e. non dynamical) fields. The vacuum expectation value (VEV) of an observable O=O(uˆp) is defined by(2.8)〈O〉:=ZM(p)(O)ZM(p), where(2.9)ZM(p)(O)=∫Hˆp(M)Duˆpe−S(uˆp)O(uˆp). Furthermore ZM(p):=ZM(p)(1) denotes the partition function of the theory. The quite formal functional integral (2.9) can be given a definite meaning as follows:Since the abelian group Ωp(M)/ΩZp(M) is divisible, the second exact sequence in (2.2) splits. Accordingly, we can assign to each cohomology class cM(p+1)∈Hp+1(M;Z) a differential character, denoted by cˆM(p+1)∈Hˆp(M), such that δ2(cˆM(p+1))=cM(p+1). We call cˆM(p+1) the background differential character associated to cM(p+1). Let Hˆp(M):={uˆp∈Hˆp(M)|δ1(uˆp)∈HZp(M)} denote the subgroup of harmonic differential characters [78]. By using the Hodge decomposition theorem one can always select a family of harmonic background differential characters cˆM(p+1)∈Hˆp(M) associated to cM(p+1).Since Hp+1(M;Z) is finitely generated it admits a (non-canonical) splitting of the form Hp+1(M;Z)≅Hfreep+1(M;Z)⊕Htorp+1(M;Z) into its free and torsion subgroup. For the free part we choose a Betti basis (fM(p+1))i, where i=1,…,bp+1(M). The torsion subgroup is generated by the basis (tM(p+1))j, where j=1,…,rp+1(M). Let wp+11,…,wp+1rp+1(M)∈N be such that wp+1j(tM(p+1))j=0 for each j=1,…,rp+1(M), then the order of the torsion subgroup is given by |Htorp+1(M;Z)|=∏j=1rp+1(M)wp+1j. With respect to these choices any class cM(p+1)∈Hp+1(M;Z) admits the following non-canonical decomposition(2.10)cM(p+1)=(cM(p+1))f+(cM(p+1))t=∑i=1bp+1(M)mp+1i(fM(p+1))i+∑j=1rp+1(M)m˜p+1j(tM(p+1))j, where mp+1i,m˜p+1j are integer coefficients. Let us choose (fˆM(p+1))i∈Hˆp(M), such that(2.11)δ1((fˆM(p+1))i)=(ρM(p+1))i,δ2((fˆM(p+1))i)=(fM(p+1))i,i=1,…,bp+1(M), and let us introduce (tˆM(p+1))j∈Hˆp(M) with j=1,…,rp+1(M), satisfying(2.12)δ1((tˆM(p+1))j)=0,δ2((tˆM(p+1))j)=(tM(p+1))j,((tˆM(p+1))j)wp+1j=1. In summary, the family of harmonic background differential characters is given by(2.13)cˆM(p+1)=∏i=1bp+1(M)(fˆM(p+1))imp+1i∏j=1rp+1(M)(tˆM(p+1))jm˜p+1j, which satisfies(2.14)δ1(cˆM(p+1))=∑i=1bp+1(M)mp+1i(ρM(p+1))i,δ2(cˆM(p+1))=cM(p+1),d⁎δ1(cˆM(p+1))=0. For each fixed cohomology class cM(p+1)∈Hp+1(M;Z), δ2−1(cM(p+1)) is a torsor for the abelian group Ωp(M)/ΩZp(M) of topologically trivial differential characters. Hence there exists a unique class [Ap]∈Ωp(M)/ΩZp(M) for each uˆp∈δ2−1(cM(p+1)) such that uˆp=cˆM(p+1)⋅j2([Ap]). However, the choice of a gauge field from the class [Ap] is not unique. From a field theoretical viewpoint this ambiguity can be interpreted as a gauge symmetry. Thus for fixed class cM(p+1) the higher-abelian gauge theory can be interpreted as an “ordinary” abelian gauge theory with gauge orbit space Ωp(M)/ΩZp(M), action ScˆM(p+1)([Ap]):=S(cˆM(p+1)⋅j2([Ap])) and observable OcˆM(p+1)([Ap])=O(cˆM(p+1)⋅j2([Ap])). This interpretation suggests to define the formal functional integral in (2.9) as functional integral over the gauge orbit space Ωp(M)/ΩZp(M) with respect to a yet to be defined measure D[Ap], followed by summing over all characteristic classes cM(p+1), namely(2.15)ZM(p)(O):=∑cM(p+1)∈Hp+1(M;Z)∫Ωp(M)/ΩZp(M)D[Ap]e−ScˆM(p+1)([Ap])OcˆM(p+1)([Ap]). To calculate the sum over the cohomology classes the recipe is to use the explicit expression (2.13) and to sum over the integers mp+1i and m˜p+1j, respectively.Due to the principle of locality we want to replace the integration over the gauge orbit space by an integration over the whole configuration space Ωp(M) subjected to a certain gauge fixing condition and followed by dividing by the volume of the gauge group in order to factor out redundant gauge degrees of freedom.Two questions arise: First, what is the total gauge group and second is it possible to fix the gauge globally? An answer to the first question is motivated by the following exact sequence which derives directly from (2.2):(2.16)0→Hp−1(M,U(1))→j1Hˆp−1(M)→δ1Ωp(M)→j2˜Hˆp(M)→δ2Hp+1(M,Z)→0, where j2˜(Ap):=j2([Ap]).For an interpretation of (2.16) let us first consider the case p=1: Recall that Maxwell theory of electromagnetism can be equivalently described in terms of differential cohomology. In fact, the holonomy associated to a gauge field, which is represented geometrically by a connection on a principal U(1)-bundle over M, defines a certain differential character whose field strength is the curvature of that connection and whose characteristic class is the first Chern class of the underlying principal U(1)-bundle. This correspondence induces a bijection between the set of isomorphism classes of principal U(1)-bundles with connections and Hˆ1(M). In physical terms, Hˆ1(M) represents the gauge orbit space of Maxwell theory. For p=1 the exact sequence (2.16) reduces to(2.17)0→U(1)→j1C∞(M;U(1))→δ1Ω1(M)→j2˜Hˆ1(M)→δ2H2(M,Z)→0, where we have used that H0(M,U(1))≅U(1) and Hˆ0(M)≅C∞(M;U(1)). Hence H0(M,U(1)) and Hˆ0(M) can be identified with the groups of global and local U(1) gauge transformations, respectively. Therefore the sequence (2.17) can be interpreted as follows: Since δ1(uˆ0)=12πiuˆ0−1duˆ0 holds for a gauge transformation uˆ0∈C∞(M;U(1)), the homomorphism δ1 generates apparently the gauge transformation A↦A+δ1(uˆ0) on the space of topologically trivial gauge fields. Evidently, the kernel of δ1 consists of the global gauge transformations. On the other hand the homomorphism j2˜ assigns to each η∈Ω1(M) the trivial U(1)-bundle on M with connection η modulo connection preserving isomorphisms. Finally, δ2 assigns to each equivalence class of principal U(1)-bundles the corresponding first Chern class.By analogy we interpret Hˆp−1(M) as the total gauge group of a higher-abelian gauge theory of degree p. According to (2.16) the curvature δ1(uˆp−1) of a gauge transformation uˆp−1∈Hˆp−1(M) gives a morphism connecting two topologically trivial p-form gauge fields Ap and Ap′ by Ap′−Ap=δ1(uˆp−1). The isotropy group of this action is the subgroup of flat differential characters of degree (p−1), namely Hp−1(M,U(1)), which represents the group of global gauge transformations. The quotient Hˆp−1(M)/Hp−1(M,U(1)) gives the reduced gauge group, which is isomorphic to ΩZp(M). Finally, the subgroup of topologically trivial differential characters Ωp(M)/ΩZp(M) is the corresponding gauge orbit space.Concerning the second question raised above the answer is negative. It was shown in [61] that in general the principal fiber bundle πΩp:Ωp(M)→Ωp(M)/ΩZp(M) is not trivializable implying that the theory suffers from a Gribov ambiguity. Therefore the conventional Faddeev-Popov method to fix the gauge does not longer apply.To overcome this drawback we follow an approach, which was introduced and studied in detail in [60,61,79]. The idea is to modify the gauge invariant measure DAp on Ωp(M), which is induced by the metric (2.4) in such a way that the resulting measure is damped along the gauge degrees of freedom yielding a finite value for the functional integral of gauge invariant observables. Since this modification is possible only locally, these modified but local measures are pasted together in such a way that the vacuum expectation values of gauge invariant observables become independent of the particular way the gluing is provided. The construction is as follows: Let us introduce a contractible open cover {Va|a∈J} of the 1-torus T (where J is an appropriate index set) and introduce local sections sa:Va→R of the universal cover e2πi(.):R→T. A contractible open cover of Tbp(M) is provided by(2.18)V(p)={Va(p):=Va1×⋯×Vaj×⋯×Vabp(X)|a:=(a1,…,aj,…,abp(M)),aj∈J,∀j}. The family of local sections sa=(sa1,…,sabp(M)):Va(p)→Rbp(M) provides a bundle atlas for Rbp(M)→Tbp(M). The projection π′:Ωp(M)/ΩZp(M)→Tbp(M), given by(2.19)π′([Ap])=(e2πi∫MAp∧(ρM(n+1−p))1,…,e2πi∫MAp∧(ρM(n+1−p))bp(M)) induces an open cover {Ua(p):=π′−1(Va(p))} of the gauge orbit space. For the principal ΩZp(M)-fiber bundle πΩp:Ωp(M)→Ωp(M)/ΩZp(M) a bundle atlas {Ua(p),φa(p)} is provided by the family of maps φˆa(p):πΩp−1(Ua(p))→ΩZp(M), φˆa(p)(Ap)=((dpM)†GpMAp,∑j=1bp(M)εaj(Ap)(ρM(p))j), where(2.20)εaj(Ap):=∫MAp∧(ρM(n+1−p))j−saj(e2πi∫MAp∧(ρM(n+1−p))j)∈Zj=1,…,bp(M). Let us introduce a (smooth) real-valued function Rp on ΩZp(M) in such a way that the regularized group volume(2.21)vol(ΩZp(M)):=∑ηp∈HZp(M)∫im(dpM)†Dτp−1e−Rp(τp−1,ηp), becomes finite. Dτp−1 is the measure on im(dpM)† induced by the inner product (2.4). Now we replace DAp by the product DAp⋅Ξ(Ap), where(2.22)Ξ(Ap):=(2π)−bp(M)2∑a℘a(p)([Ap])e−φˆa(p)⁎Rp(Ap). Here {℘a(p)} denotes a partition of unity subordinate to the open cover {Ua(p)} and the pre-factor is chosen for convenience. We will see, however, that the final result is independent of the concrete choice for Rp.55A natural choice for Rp is Rp(τp−1,ηp)=12‖Δp−1M|im(dpM)†τp−1‖2+12‖ηp‖2, which on πΩp−1(Ua(p)) becomes φˆa(p)⁎Rp(Ap)=12‖(dpM)†Ap‖2+12∑i,j=1bp(M)(hM(p))ijεai(Ap)εaj(Ap). The first term is the conventional gauge fixing term in the Lorentz gauge. The second term damps the harmonic degrees of freedom (i.e. zero modes) and due to the Gribov ambiguity is defined only locally. As intended we rewrite (2.15) in the form(2.23)ZM(p)(O)=∑cM(p+1)∈Hp+1(M;Z)1vol(Hˆp−1(M))×∫Ωp(M)DApΞ(Ap)πΩp⁎OcˆM(p+1)(Ap)e−πΩp⁎ScˆM(p+1)(Ap), where we have factored out the yet to be defined (regularized) volume of the total gauge group Hˆp−1(M) to account for all redundant gauge degrees of freedom. In order to compute the functional integral in (2.23) we notice that the subgroup of topologically trivial differential characters Ωp(M)/ΩZp(M) has the structure of a trivializable vector bundle over the harmonic torus Hp(M;R)/j⁎(Hp(M;Z)) with typical fiber im(dp+1M)†. Since Hp(M;R)≅Hp(M) and j⁎(Hp(M;Z))≅Hfreep(M;Z)≅HZp(M) one has(2.24)Hp(M;R)/j⁎(Hp(M;Z))≅Hp(M)/HZp(M)≅Tbp(M). Correspondingly, a global bundle trivialization is given by(2.25)φ_a(p)(z→p,τp)=[∑i=1bp(M)sai(zi)(ρM(p))i+τp],z→p=(z1,…,zbp(M))∈Tbp(M),τp∈im(dp+1M)†. With respect to the local diffeomorphisms ϕa(p):=φa(p)∘(φ_a(p)×idΩZp(M)) the volume form transforms as(2.26)ϕa(p)⁎DAp=(dethM(p))12(detΔp−1M|im(dpM)†)12dt1∧…∧dtbp(M)∧Dτp∧Dτp−1, where t→=(t1,…,tbp(M)) are local coordinates for Tbp(M). Notice that ϕa(p)⁎DAp is a globally defined volume form on Tbp(M)×im(dp+1M)†×ΩZp(M).Since φ_a(p)⁎OcˆM(p+1)=φ_b(p)⁎OcˆM(p+1) holds on each overlap Va(p)∩Vb(p)×im(dp+1M)†, the local contributions can be extended to a globally defined function on Tbp(M)×im(dp+1M)†, denoted by O_cˆM(p+1). For fixed τp∈im(dp+1M)†, z→p↦O_cˆM(p+1)(z→p,τp) is a function on Tbp(M), so it can be expanded in a Fourier series with coefficients(2.27)[O_cˆM(p+1)(τp)](m1,…,mbp(M)):==∫01dt1…∫01dtbp(M)O_cˆM(p+1)(e2πit1,…,e2πitbp(M),τp)e−2πi∑j=1bp(M)mjtj. We transform the integral in (2.23) into an integral over Tbp(M)×im(dp+1M)†×ΩpZ(M). By using (2.21), (2.26) and the Fourier series expansion (2.27) one obtains(2.28)ZM(p)(O)=(2π)−bp(M)2(dethM(p))12(detΔp−1M|im(dpM)†)12vol(ΩZp(M))vol(Hˆp−1(M))×∑cM(p+1)∈Hp+1(M;Z)∫im(dp+1M)†Dτp[e−S_cˆM(p+1)(τp)O_cˆM(p+1)(τp)](0,…,0). Notice that ZM(p)(O) is independent of the partition of unity and the local trivialization. Moreover, it does not depend on the specific choice of background differential characters either.66Let us consider a different family cˆM′(p+1) of background differential characters related to cM(p+1)∈Hp+1(M;Z). Then there exists a family of harmonic forms BcM(p+1)∈Hp(M), parametrized by cM(p+1), such that cˆM′(p+1)=cˆM(p+1)⋅j2([BcM(p+1)]). By (2.25) there exists for each cM(p+1) a w→∈Tbp(M) such that [BcM(p+1)]=φ_a(p)(w→,0). This gives a homomorphism cM(p+1)↦[BcM(p+1)] from Hp+1(M;Z) to Tbp(M). But then O_cˆM′(p+1)=lw→⁎O_cˆM(p+1), where lw→ denotes the (left) multiplication by w→. This gives [O_cˆM′(p+1)(τp)](0,…,0)=[O_cˆM(p+1)(τp)](0,…,0).Now it remains to determine the quotient of the gauge group volumes in (2.28). Let Rp be an arbitrary but fixed regularizing functional for ΩZp(M) and introduce the homomorphism δˆ1:Hˆp−1(M)→im(dpM)†×HZp(M) by uˆp−1↦((dpM)†GpMδ1(uˆp−1),πHp(δ1(uˆp−1))). Then we define the regularized volume of the total gauge group Hˆp−1(M) in a natural way by the formal functional integral(2.29)vol(Hˆp−1(M)):=∫Hˆp−1(M)Duˆp−1e−Rˆp−1(uˆp−1), where Rˆp−1:=δˆ1⁎Rp. In order to compute (2.29) and verify that it yields a finite volume we are now in exactly the same situation as we were before when we calculated (2.9), which finally led to (2.28). Thus we will use formula (2.28), but now in degree p−1 and applied to the integrand e−Rˆp−1 instead of e−SO as in (2.9). Let us begin with the computation of the corresponding integral in (2.28). Since Rˆ_p−1cˆM(p)(z→p−1,τp−1)=Rp(τp−1,δ1(cˆM(p))) and δ1(cˆM(p))=(cM(p))f, we obtain(2.30)∑cM(p)∈Hp(M;Z)∫im(dpM)†Dτp−1[e−Rˆ_p−1cˆM(p)(τp−1)](0,…,0)=vol(ΩZp(M))|Htorp(M;Z)|, where (2.21) was used. The group of global gauge transformations Hp−1(M;U(1)) can be assigned a volume in a canonical way by noting from (2.3) that its connected component is the harmonic torus Hp−1(M;R)/j⁎(Hp−1(M;Z)). With respect to the induced metric on Hp−1(M) the volume of the harmonic torus reads(2.31)vol(Hp−1(M;R)/j⁎(Hp−1(M;Z)))=(dethM(p−1))12. According to (2.3) this leads to(2.32)vol(Hp−1(M;U(1)))=(dethM(p−1))12|Htorp(M,Z)|. From formula (2.28) we obtain for the volume of the total gauge group(2.33)vol(Hˆp−1(M))==(2π)−bp−1(M)2(detΔp−2M|im(dp−1M)†)12×vol(Hp−1(M;U(1)))vol(ΩZp(M))vol(ΩZp−1(M))vol(Hˆp−2(M)). By induction this leads to(2.34)vol(ΩZp(M))vol(Hˆp−1(M))=∏r=0p−1(2π)br(M)2(−1)p+1−r∏r=0p−2(detΔrM|im(dr+1M)†)12(−1)p+1−r×∏r=0p−1vol(Hr(M;U(1)))(−1)p−r. We interpret this quotient as the ghost-for-ghost contribution for higher abelian gauge theories.77Let us remark that in [61] the groups Ωp−1(M)×HZp(M) and Ωclp−1(M) were considered as total gauge group and group of global gauge transformations, respectively. Here we derived Hˆp−1(M) and Hp−1(M;U(1)) as the corresponding gauge groups. The reduced gauge groups are in both cases ΩZp(M), the ghost-for-ghost terms differ by the order of the torsion subgroups of M, however. In the topologically trivial case the quotient (2.34) reduces to the alternating product of Laplace operators, hence recovering the well-known ghost-for-ghost contribution in the functional integral measure [45,64,80,81]. Substituting (2.34) into (2.28) finally yields(2.35)ZM(p)(O)=∏r=0p((det(12πhM(r)))12|Htorr(M;Z)|)(−1)p−r∏r=0p−1(detΔrM|im(dr+1M)†)12(−1)p+1−r×∑cM(p+1)∈Hp+1(M;Z)∫im(dp+1M)†Dτp[e−S_cˆM(p+1)(τp)O_cˆM(p+1)(τp)](0,…,0). Since the Laplace operator ΔrM on a compact manifold M is an unbounded positive operator with discrete spectrum 0=ν0(ΔrM)<ν1(ΔrM)≤…→∞ and finite dimensional eigenspaces, we will use zeta-function regularization to give the determinants appearing in (2.35) a mathematical meaning. In this formulation the determinant of ΔrM is given by(2.36)detΔrM=exp(−ζ′(0;ΔrM))≡exp(−dds|s=0ζ(s;ΔrM)), where the zeta function of ΔrM is defined by [82,83](2.37)ζ(s;ΔrM)=∑να(ΔrM)≠0να(ΔrM)−s=1Γ(s)∫0∞dtts−1Tr(e−tΔrM−πHr),s∈C. Here the sum runs over the non vanishing eigenvalues of ΔrM only and each eigenvalue appears the same number of times as its multiplicity. The second expression in (2.37) is the Mellin transform of the zeta-function. Notice that by construction ζ(s;ΔrM)=ζ(s;ΔrM|Hr(M)⊥). The zeta function is holomorphic for ℜ(s)>dimM2 and has a meromorphic continuation to C with simple poles at sk=dimM−k2 for k∈N0 and residue Ress=sk[ζ(s;ΔrM)]=ak(ΔrM)Γ(dimM−k2) at s=sk. Here ak(ΔrM) denotes the k-th Seeley coefficient in the asymptotic expansion of the L2-trace of the heat kernel Tr(e−tΔrM) [83], which is given by(2.38)Tr(e−tΔrM)≃∑k=0∞ak(ΔrM)tk−dimM2,fort↓0. The Seeley coefficients ak(ΔrM) are integrals over M of polynomials which depend on the metric and its derivatives. However, ak(ΔrM)=0 whenever k is odd. The zeta function is regular at s=0, yielding(2.39)ζ(0;ΔrM)={−br(M),if dimM=n+1 is odd,an+1(ΔrM)−br(M),if dimM=n+1 is even. In terms of the Seeley coefficients the Euler characteristics χ(M):=∑r=0n+1(−1)rbr(M) of M can be equivalently written as [82,83](2.40)χ(M)=∑r=0n+1(−1)ran+1(ΔrM). From the definition of the determinant (2.36) it follows that for any parameter λ∈R(2.41)det(λΔrM)=λζ(0;ΔrM)detΔrM. In analogy with the finite dimensional case, ζ(0;ΔrM) can thus be interpreted as regularized dimension.From a physical perspective, however, the zeta function regularization seems to have a drawback because the eigenvalues of the Laplace operator have a mass dimension leading to a partition function which is not dimensionless. By including a scale factor μ of mass-dimension [μ]=1 this can be restored. Thus instead of (2.36) we take the dimensionless but scale dependent zeta function ζμ(s;ΔrM)):=∑να(ΔrM))≠0(μ−2να(ΔrM)))−s. Hence ζμ(s;ΔrM))=μ2sζ(s;ΔrM)). Consequently, this leads to the scale dependent but dimensionless determinant(2.42)detμΔrM:=exp[−dds|s=0ζμ(s;ΔrM)], which is related to (2.36) by detμΔrM=μ−2ζ(0;ΔrM)detΔrM. Despite this replacement the partition function is not just yet dimensionless. This traces back to the fact that the gauge fields have dimension and thus the metric hM(r) appearing in (2.35) admits a dimension as well which must be corrected accordingly. Following the usual convention we fix the mass dimension of the p-form gauge field A in n+1 dimensions by [A]=n−12, so that the corresponding field strength FA=dA admits mass dimension [FA]=n+12. Since higher-abelian gauge fields generalize p-form gauge fields, we assign mass dimension [δ1(uˆ)]=n+12 to the field strength δ1(uˆ). The harmonic forms are regarded as specific p-form gauge fields so that they are assigned the same mass dimension as A. Hence [(ρM(p))k]=n−12. Consequently, [(ρM(p+1))l]=n+12 for all k,l. In fact we choose [(ρM(r))k]=n−12+r−p. In order to obtain a dimensionless partition function we have to replace(2.43)detΔrM↦detμΔrMhM(r)↦hM;μ(r)=μ2(p+1−r)hM(r). It follows from the heat kernel expansion (2.38) that the Seeley coefficients of the dimensionless Laplace operator μ−2ΔrM are related to those of ΔrM by (see also [84])(2.44)ak(μ−2ΔrM)=μn+1−kak(ΔrM). Due to the Hodge decomposition and the nilpotency of drM and (drM)† the spectrum of ΔrM|Hr(M)⊥ is the union of the eigenvalues of ΔrM|im(dr+1M)† and ΔrM|im(dr−1M). On the other hand, if ψM(r−1)∈im(drM)† is an eigenform of Δr−1M|im(drM)† with eigenvalue ν, then dr−1MψM(r−1) is an eigenform of ΔrM|im(dr−1M) with the same eigenvalue. This correspondence gives a bijection between the sets of non-vanishing eigenvalues and their related eigenforms of the two operators Δr−1M|im(drM)† and ΔrM|im(dr−1M). Hence the spectrum of ΔrM|Hr(M)⊥ is the union of eigenvalues of ΔrM|im(dr+1M)† and Δr−1M|im(drM)†. This leads to the following factorization(2.45)ζ(s;ΔrM|Hr(M)⊥)=ζ(s;ΔrM|im(dr+1M)†)+ζ(s;Δr−1M|im(drM)†), yielding(2.46)ζ(s;ΔrM|im(dr+1M)†)=∑r′=0r(−1)r−r′ζ(s;Δr′M|Hr′(M)⊥), and finally(2.47)∑r=0p−1(−1)p−rζ(s;ΔrM|im(dr+1M)†)=∑r=0p(−1)p−r(p−r)ζ(s;ΔrM|Hr(M)⊥). Applying the dimensional replacement to the ghost-for-ghost term and using (2.47), we obtain from (2.35) the following general formula for the dimensionless and regularized partition function for a higher-abelian gauge theory governed by the classical action S(uˆp),(2.48)ZM(p)=∏r=0p((detμΔrM|Hr(M)⊥)12(p−r)|Htorr(M;Z)|(det(12πhM;μ(r)))12)(−1)p+1−r×∑cM(p+1)∈Hp+1(M;Z)∫im(dp+1M)†Dτp[e−S_cˆM(p+1)(τp)](0,…,0). According to (2.43) the scale factor which renders the alternating product in (2.48) dimensionless is given by(2.49)μ∑r=0p(−1)p−r(p−r)an+1(ΔrM)+∑r=0p(−1)p−rbr(M). Finally, the formula for the VEV of an observable O can be stated in the following general form(2.50)<O>=∑cM(p+1)∈Hp+1(M;Z)∫im(dp+1M)†Dτp[e−S_cˆM(p+1)(τp)O_cˆM(p+1)(τp)](0,…,0)∑cM(p+1)∈Hp+1(M;Z)∫im(dp+1M)†Dτp[e−S_cˆM(p+1)(τp)](0,…,0). Once the concrete action and the observable are selected, the dimensional replacement (2.43) must be taken into account when performing the τp-integration in (2.48) or (2.50) in order to obtain the correct dimension. This might lead to additional μ dependent terms.2.3The partition function of extended higher-abelian Maxwell theoryIn this section we introduce a concrete field theoretical model by extending the Maxwell theory of electromagnetism to higher degrees. Let us begin with the so-called higher-abelian (generalized) Maxwell theory [54,56,61,66,67] of degree p with action(2.51)S0(uˆp)=qp2‖δ1(uˆp)‖2=qp2∫Mδ1(uˆp)∧⋆δ1(uˆp),uˆp∈Hˆp(M). The coupling constant qp is taken to be dimensionless. A Hamiltonian analysis of this model was given in [66,67] and the functional integral quantization was studied in [61].Now we extend (2.51) by including two arbitrary harmonic forms γp∈Hp+1(M) and θp∈Hn−p(M) representing two de Rham cohomology classes on M and define(2.52)S(γp,θp)(uˆp):=qp2‖δ1(uˆp)−γp‖2+2πi<δ1(uˆp)−γp,⋆θp>. We call this model extended higher-abelian Maxwell theory of degree p. These two harmonic forms - named topological fields - are regarded as non-dynamical (external) fields. Remark that the classical action is invariant under the joint transformation uˆp↦uˆp⋅vˆp and γp↦γp+δ1(vˆp) for all vˆp∈Hˆp(M).88A more detailed discussion of this model, its symmetries respectively further extensions will be given elsewhere.It follows from (2.2) that any tangent vector in TuˆpHˆp(M) has the form uˆp⋅j2([B]) with [B]∈Ωp(M)/dΩp−1(M). Since(2.53)δS0δuˆp[B]=ddt|t=0S0(uˆp⋅j2[tB])=ddt|t=0S(γp,θp)(uˆp⋅j2[tB])=δS(γp,θp)δuˆp[B] both actions S0 and S(γp,θp) lead to the same classical equation of motion, namely(2.54)(dp+1M)†δ1(uˆp)=0. Although classically equivalent, they give rise to different quantum theories. In the next step we will use the general formula (2.48) to determine the corresponding partition function. At first(2.55)S_(γp,θp)cˆM(p+1)(z→p,τp)=qp2<τp,ΔpM|im(dp+1M)†τp>+2πi(−1)(p+1)(n−p)∑i=1bp+1(M)(mp+1i−γpi)θpi+qp2∑i,j=1bp+1(M)(hM(p+1))ij(mp+1i−γpi)(mp+1j−γpj), where the components γpj and θpj are defined with respect to the fixed basis of H•(M) by(2.56)γp=∑i=1bp+1(M)γpi(ρM(p+1))i,θp=∑j=1bn−p(M)θpj(ρM(n−p))j. Let us write γ→p=(γp1,…,γpbp+1(M)) and θ→p=(θp1,…,θpbn−p(M)). The τp integration in (2.48) is Gaussian and using the rule (2.43) and (2.46) one gets(2.57)∫im(dp+1M)†Dτpe−qp2<τp,ΔpM|im(dp+1M)†τp>==(μqp)∑r=0p(−1)p−rζ(0;ΔrM)∏r=0p(detΔrM|Hr(M)⊥)−12(−1)(p−r). The next step is to sum over the topological sectors. The sum over the free part results in a bp+1(M) dimensional Riemann Theta function (A.1) and the sum over the torsion classes yields the order of the (p+1)-th torsion subgroup. Substituting (2.57) into (2.48) finally gives the following dimensionless partition function for the extended higher-abelian Maxwell theory(2.58)ZM(p)(qp;γp,θp)==[2πqp]12∑r=0p(−1)p+1−rbr(M)∏r=0p[(detΔrM|Hr(M)⊥)(p+1−r)2|Htorr(M;Z)|(dethM(r))12](−1)p+1−r×Θbp+1(M)[γ→p0]((−1)(p+1)(n−p)θ→p|−qp2πihM(p+1))|Htorp+1(M;Z)|×μ∑r=0p(−1)p−r(p+1−r)an+1(ΔrM)qp−12∑r=0p(−1)p−ran+1(ΔrM). Due to the modular properties of the Riemann Theta function (A.2), the partition function exhibits the following apparent symmetry properties(2.59)ZM(p)(qp;γp,θp)=ZM(p)(qp;−γp,−θp)ZM(p)(qp;γp+ω,θp)=ZM(p)(qp;γp,θp),for ω∈HZp+1(M)ZM(p)(qp;γp,θp+ω′)=ZM(p)(qp;0,θp),for γp∈HZp+1(M), ω′∈HZn−p(M)ZM(p)(qp;γp,θp)=ZM(p)(qp;0,0)=:ZM(p)(qp),for γp∈HZp+1(M), θp∈HZn−p(M). In odd dimensions one has an+1(ΔrM)=0, so that the partition function becomes independent of the renormalization scale. Since an+1(ΔrM) is an integral over a local polynomial in the metric and its derivatives in even dimensions, the last factor in (2.58) can be absorbed into the action of the theory. Within a local quantum field theory this amounts to add appropriate gravitational counter-terms. So instead of ZM(p)(qp;γp,θp) we regard(2.60)ZˆM(p)(qp;γp,θp):=qp12∑r=0p(−1)p−ran+1(ΔrM)ZM(p)(qp;γp,θp) as the effective partition function for extended higher-abelian Maxwell theory. Due to (2.44) ZˆM(p) is dimensionless. The explicit dependence on μ on the other hand indicates that there is nevertheless an ambiguity which has to be fixed to obtain a physically reasonable result.In the case γp=θp=0, our result for the partition function (2.60) agrees with the dimensionless partition function for higher-abelian Maxwell theory obtained recently in [68].2.4Duality in extended higher-abelian Maxwell theoryDuality between quantized antisymmetric tensor field theories of different degrees and its relation with the Ray-Singer analytic torsion (see below for the definition) were analyzed long time ago by Schwarz in his seminal papers [85,86]. That there exists a duality in higher-abelian Maxwell theory with action (2.51) was already sketched in [56] by means of a specific master equation. The case of acyclic manifolds was discussed in [61].Does there exist a duality between extended higher-abelian Maxwell theories of different degrees as well? Let (qp,γp,θp) be the parameters of the theory of degree p and define the corresponding dual parameters (qn−p−1dual,γn−p−1dual,θn−p−1dual) by1.qn−p−1dual⋅qp=(2π)22.γ→n−p−1dual=(−1)(p+1)(n−p)θ→p3.θ→n−p−1dual=−γ→p. In the following we will compute the quotient of partition functions(2.61)ZˆM(p)(qp;γp,θp)ZˆM(n−p−1)(qn−p−1dual,γn−p−1dual,θn−p−1dual) factor per factor using the expression (2.60). We begin with the alternating product in the partition function (2.58), abbreviated by(2.62)G(p):=∏r=0p[(detΔrM|Hr(M)⊥)(p+1−r)2|Htorr(M;Z)|(dethM(r))12](−1)p+1−r. Since bp+1(M)=bn−p(M) and ⋆(ρM(p))i=∑j=1bp(M)(hM(p))ij(ρM(n+1−p))j, the dual metrics are related by (hM(p+1))−1=hM(n−p). Moreover, we use Htorr+1(M;Z)≅Htorn+1−r(M;Z) which follows from the universal coefficient theorem and Poincare duality (see e.g. [73]). Together with the fact that the Hodge operator commutes with the Laplace operator, one finds after a lengthy calculation(2.63)G(p)G(n−p−1)=∏r=0n+1[(dethM(r))12|Htorr(M;Z)|−1](−1)r+p∏r=0n+1(detΔrM|Hr(M)⊥)r2(−1)r+1+p(dethM(p+1))12|Htorn−p(M;Z)||Htorp+1(M;Z)|. Let us consider the alternating products in (2.63) in more detail: The product in the denominator is nothing but the Ray-Singer analytic torsion [87] of M(2.64)TRS[M]=∏r=0n+1(detΔrM|Hr(M)⊥)r2(−1)r+1=exp(12∑r=0n+1(−1)rrdds|s=0ζ(s;ΔrM|Hr(M)⊥)). Ray and Singer introduced TRS[M] as an analytic analogue of the Reidemeister-Franz torsion (or R-torsion), denoted by TR[M], which is a topological invariant of M. In fact, TR[M] is defined in terms of the combinatorial structure determined by smooth triangulations of the manifold. According to [88], Theorem 8.35, (see e.g. [89] for a review of analytic and Reidemeister torsion) the R-torsion of a compact and closed manifold equals the alternating product of the order of the torsion subgroups of integer cohomology times a so-called regulator. The regulator itself is an alternating product of the volumes of the harmonic tori Hr(M;R)/j⁎(Hr(M;Z)), r=0,…,dimM with respect to the metric induced by identifying real cohomology with harmonic forms. In our notation this volume is given by (2.31) so that the R-torsion admits the form(2.65)TR[M]=∏r=0n+1(dethM(r))12(−1)r∏r=0n+1|Htorr(M;Z)|(−1)r+1. Hence the R-torsion appears in the numerator of the first factor in (2.63). It was conjectured by Ray and Singer and independently proved by Cheeger and Müller [88,90] that the Ray Singer analytic torsion and the R-torsion are equal(2.66)TRS[M]=TR[M]. Thus the first fraction on the right hand side of (2.63) cancels. The quotient of the contributions from the topological sectors yields(2.67)Θbp+1(M)[γ→p0]((−1)(p+1)(n−p)θ→p|−qp2πihM(p+1))|Htorp+1(M;Z)|Θbn−p(M)[γ→n−p−1dual0](θ→n−p−1dual|−qn−p−1dual2πihM(n−p))|Htorn−p(M;Z)|==[2πqp]12bp+1(M)(dethM(p+1))−12|Htorp+1(M;Z)||Htorn−p(M;Z)|e2πi(−1)(p+1)(n−p)γ→pTθ→p, where we have used (A.3) and (A.5). Evidently, (2.61) becomes independent of μ, if the dimension of M is odd. In even dimensions, one has an+1(ΔrM)=an+1(Δn+1−rM) which gives(2.68)∑r=0n+1(−1)p+1−rran+1(ΔrM)=(−1)p+1n+12χ(M). From (2.63), (2.66), (2.67) and (2.68) we finally get the following duality relation(2.69)ZˆM(p)(qp;γp,θp)ZˆM(n−p−1)(qn−p−1dual,γn−p−1dual,θn−p−1dual)=={[qp2π]12(−1)pχ(M)e2πi∫Mθp∧γpμ(−1)p(p+1−n+12)χ(M),if dimM=n+1 is even,e2πi∫Mθp∧γpif dimM=n+1 is odd. The quotient is independent of the metric of M. In odd dimensions, the two extended higher-abelian Maxwell theories are exactly dual whenever the topological fields have integer periods. In the case of vanishing topological fields (i.e. γp=θp=0) we recover once again the result previously derived in [68].3Thermodynamics of higher-abelian gauge theories3.1The free energy - general caseIn the remainder of this paper we want to apply the general setting introduced in the previous sections in order to study the effect of the topology on the vacuum structure and the thermodynamical behavior of higher-abelian gauge theories. Geometrically, a quantum field theory at finite temperature 1/β at equilibrium is usually realized as quantum field theory on the product manifold Xβ:=Tβ1×X. The temperature is included by equipping a 1-torus, denoted by Tβ1, with temperature dependent metric gβ=β2dt⊗dt. Here t is the local coordinate. The n-dimensional compact, connected, oriented and closed Riemannian manifold X with fixed metric gX is the spatial background and Hˆp(Xβ) represents the space of equivalence classes of higher-abelian thermal gauge fields.We begin with the derivation of a general formula for the free energy of a higher-abelian gauge theory with Euclidean action S=S(uˆp), where uˆp∈Hˆp(Xβ). In the second step we will apply this general formula to extended higher-abelian Maxwell theory.According to section 2.1, the corresponding configuration space Ωp(Xβ) of topologically trivial thermal gauge fields is a non-trivializable principal ΩZp(Xβ)-bundle over the gauge orbit space Ωp(Xβ)/ΩZp(Xβ), which itself is a trivializable vector bundle over Tbp(Xβ) with typical fiber im(dp+1Xβ)†. The corresponding family of local trivializations is provided by the diffeomorphisms φa(p)∘φ_a(p) (see (2.20) and (2.25)), but now applied to the “thermal” manifold M=Xβ.The free energy of the higher-abelian gauge theory of degree p is defined by(3.1)FX(p)(β):=−1βlnZXβ(p), where ZXβ(p) is the partition function (2.48) related to the manifold Xβ. In the following we will compute term by term of (2.48) for the thermal setting.Let us equip Xβ with the product metric g=gβ⊕gX. The corresponding volume form volXβ splits into the product(3.2)volXβ=pr1⁎volTβ1∧pr2⁎volX, where pr1:Xβ→Tβ1 and pr2:Xβ→X are the canonical projections and volTβ1, volX are the volume forms on Tβ1 and X, respectively. According to the Künneth theorem there exists the decomposition(3.3)Hr(Xβ;Z)≅Hr(X;Z)⊕Hr−1(X;Z). This implies a split of both the free part Hfreer(Xβ;Z)≅Hfreer(X;Z)⊕Hfreer−1(X;Z) and the torsion part Htorr(Xβ;Z)≅Htorr(X;Z)⊕Htorr−1(X;Z). Hence br(Xβ)=br(X)+br−1(X). A basis for HZr(Xβ), denoted by {(ρXβ(r))I|I=(i,j)}, is generated by(3.4)(ρXβ(r))i:=pr2⁎(ρX(r))i,i=1,…,br(X)(ρXβ(r))j:=pr1⁎ρTβ1(1)∧pr2⁎(ρX(r−1))j,j=1,…,br−1(X), where ρTβ1(1)=1βvolTβ1 is the generator for HZ1(Tβ1) and (ρX(r))k denotes a basis for HZr(X). The dual basis ρXβ(n+1−r) satisfying(3.5)∫Xβ(ρXβ(r))I∧(ρXβ(n+1−r))J=δIJ is then given by(3.6)(ρXβ(n+1−r))j:=pr2⁎(ρX(n+1−r))j,j=1,…,bn+1−r(X)(ρXβ(n+1−r))i:=(−1)rpr1⁎ρTβ1(1)∧pr2⁎(ρX(n−r))i,i=1,…,bn−r(X). Let us denote the Hodge star operator on X associated with metric gX by ⋆_. A direct calculation gives(3.7)⋆pr1⁎ρTβ1(1)=1βpr2⁎volX⋆pr2⁎(ρX(r))i=(−1)rβpr1⁎ρTβ1(1)∧pr2⁎⋆_(ρX(r))i⋆(pr1⁎ρTβ1(1)∧pr2⁎(ρX(r−1))j)=1βpr2⁎⋆_(ρX(r−1))j. Therefore the induced metric hXβ(r) on Hr(Xβ) admits the following matrix representation(3.8)hXβ(r)=(β−1hX(r−1)00βhX(r)), of rank br−1(X)+br(X), where hX(r−1), hX(r) are the induced metrics on Hr−1(X) and Hr(X), respectively. Due to (3.3) every class cXβ(p+1)∈Hp+1(Xβ;Z) has the following (non-canonical) decomposition with integer components(3.9)cXβ(p+1)==(∑i=1bp+1(X)mp+1i(fX(p+1))i+∑j=1rp+1(X)m˜p+1j(tX(p+1))j,∑k=1bp(X)mpk(fX(p))k+∑l=1rp(X)m˜pl(tX(p))l), where the classes (fX(p+1))i and (fX(p))k provide a Betti basis for Hfreep+1(Xβ;Z). The generators for Htorp+1(Xβ;Z) are denoted by (tX(p+1))j and (tX(p))l, respectively. Finally, we choose a family of harmonic background differential characters cˆXβ(p+1)∈Hˆp(Xβ) with characteristic classes δ2(cˆXβ(p+1))=cXβ(p+1) and field strengths(3.10)δ1(cˆXβ(p+1))=∑j=1bp(X)mpjpr1⁎ρTβ1(1)∧pr2⁎(ρX(p))j+∑k=1bp+1(X)mp+1kpr2⁎(ρX(p+1))k. By using (2.42), (3.8) and the Künneth theorem for the torsion subgroup of Xβ, one obtains for the free energy the following formula(3.11)FX(p)(β)=12β∑r=0p(−1)p+1−r(p−r)ζ′(0;ΔrXβ|Hr(Xβ)⊥)−12βlndethX(p)+bp(X)2βln2π−12β(∑r=0p(−1)p−r(p−r)an+1(ΔrXβ)+bp(X))lnμ2+1β(bp(X)2+∑r=0p(−1)(p+1−r)br(X))lnβ+1βln|Htorp(X;Z)|−1βln(∑cXβ(p+1)∈Hp+1(Xβ;Z)∫im(dp+1Xβ)†Dτp[e−S_cˆXβ(p+1)(τp)](0,…,0)). As before the sum over the topological sectors is understood as sum over the integer components of cXβ(p+1). This general formula is the starting point to derive expressions which are adequate for the low- and the high-temperature regimes, respectively.3.1.1Low-temperature regimeIn the next step the Laplace operators appearing in (3.11) are rewritten in terms of the geometry of X. The subspace ⨁r+s=p(pr1⁎Ωr(Tβ1)⊗pr2⁎Ωs(X)) of Ωp(Xβ) is dense. With respect to the product metric g, the Laplace operator ΔpXβ splits into(3.12)ΔpXβ=⨁r+s=p(ΔrTβ1⊗1+1⊗ΔsX). Let να(ΔsX) denote the eigenvalues of ΔsX. The eigenvalues of ΔrTβ1 for r=0,1 are νk(ΔrTβ1)=(2πkβ)2 with k∈Z. Hence the spectrum of ΔpXβ|Hp(Xβ)⊥ is given by the following set of non-vanishing real numbers(3.13)Spec(ΔpXβ|Hp(Xβ)⊥)=={ν(k,α)(r,s):=νk(ΔrTβ1)+να(ΔsX)≠0|r+s=p,(k,α)∈Z×I}. The zeta function of ΔrXβ admits the following representation in terms of these eigenvalues(3.14)ζ(s;ΔrXβ|Hr(Xβ)⊥)={∑(k,α)∈Z×I[ν(k,α)(0,0)]−s,if r=0,∑(k,α)∈Z×I[ν(k,α)(0,r)]−s+∑(k,α)∈Z×I[ν(k,α)(1,r−1)]−s,if r≥1, where each eigenvalue appears as often as its multiplicity. The series converges for ℜ(s)>n+12. Let us introduce the following auxiliary quantity(3.15)I(s;ΔrX):=∑(k,α)∈(Z×I)′[(2πkβ)2+να(ΔrX)]−s, where the prime indicates that the sum runs over all those pairs (k,α) such that the sum (2πkβ)2+να(ΔrX) does not vanish. Thus(3.16)ζ(s;ΔrXβ|Hr(Xβ)⊥)={I(s;Δ0X),if r=0,I(s;ΔrX)+I(s;Δr−1X)if r≥1. Let us now introduce an additional auxiliary quantity to separate the zero-modes(3.17)Iˆ(s;ΔrX):=∑k∈Z∑α∈I[(2πkβ)2+να(ΔrX|Hr(X)⊥)]−s. Hence(3.18)I(s;ΔrX)=Iˆ(s;ΔrX)+2br(X)(β2π)2sζR(2s), where ζR(s) is the Riemann zeta function. Applying the Mellin transform to Iˆ(s;ΔrX), one gets(3.19)Iˆ(s;ΔrX)=1Γ(s)∫0∞dtts−1∑k∈Ze−(2πβ)2k2t∑α∈Ie−να(ΔrX|Hr(X)⊥)t=β2πΓ(s)∫0∞dtts−32∑α∈Ie−να(ΔrX|Hr(X)⊥)t+βπΓ(s)∫0∞dtts−32∑k=1∞e−(k2β24t)∑α∈Ie−να(ΔrX|Hr(X)⊥)t=β2πΓ(s−12)Γ(s)ζ(s−12;ΔrX)+2−s−32βs+12πΓ(s)∑k∈Z∑α∈I[kνα(ΔrX|Hr(X)⊥)]s−12Ks−12(kβνα(ΔrX|Hr(X)⊥)), where Kν(s) is the modified Bessel function of the second kind [91]. Let us expand ζ(s;ΔrX) in a Laurent series at s=−12(3.20)ζ(s−12;ΔrX)=Ress=−12[ζ(s;ΔrX)]s+FPs=−12[ζ(s;ΔrX)]+∑k=1∞σ˜ksk, where the finite part FP of the zeta function at s=−12 is given by(3.21)FPs=−12[ζ(s;ΔrX)]=limϵ→012[ζ(−12−ϵ;ΔrX)+ζ(−12+ϵ;ΔrX)]. To determine the residue of the zeta function, we recall from section 2.2 (but here for the n-dimensional manifold X) that ζ(s;ΔrX) has simple poles at sk=n−k2. Hence the residue at s=−12 is given by(3.22)Ress=−12[ζ(s;ΔrX)]=−an+1(ΔrX)2π. In order to compute the derivative I′(s;ΔrX) at s=0, we make use of the expansions 1Γ(s)=s+γs2+O(s3) and Γ(s−12)=Γ(−12)(1+Ψ(−12)s+O(s2)) for small s, where γ is the Euler-Mascheroni number and Ψ is the Digamma function [91]. Since lims→0dds(h(s)Γ(s))=h(0) holds for any regular function h,99This follows from Γ′(s)Γ(s)=−γ−1s−∑m=1∞[1s+m−1m] and lims→0Γ′(s)Γ(s)2=−1. one obtains(3.23)I′(0;ΔrX)=−2br(X)lnβ−2∑α∈Iln[1−e−βνα(ΔrX|Hr(X)⊥)]−β(FPs=−12[ζ(s;ΔrX)]+2(1−ln2)Ress=−12[ζ(s;ΔrX)]). The sum of the second and third term of (3.23) is a function f of the eigenvalues of ΔrX|Hr(X)⊥. If written schematically as f=f(ΔrX|Hr(X)⊥), this function has the property(3.24)f(ΔrX|Hr(X)⊥)=f(ΔrX|im(dr+1X)†)+f(Δr−1X|im(drX)†). Hence we find for the first term on the right hand side of (3.11)(3.25)∑r=0p(−1)p+1−r(p−r)ζ′(0;ΔrXβ|Hr(Xβ)⊥)=∑r=0p−1(−1)p+1−rI′(0;ΔrX)=−2∑r=0p−1(−1)p+1−rbr(X)lnβ−f(Δp−1X|im(dpX)†). Now it remains to calculate an+1(ΔrXβ) in (3.11). Let us begin by noting that the L2 trace of the heat kernel for ΔrTβ1 has the form(3.26)Tr(e−tΔrTβ1)=Θ1(0|−14πtβ2)=β2πtΘ1(0|−1β24πt)≃β2πt+O(e−1t),fort↓0, where the duality relation (A.4) was used. This implies ak(ΔrTβ1)=β2πδk,0 for r∈{0,1}. From the asymptotic expansions of ΔrXβ and ΔrX, one derives the following relation between the corresponding Seeley coefficients,(3.27)ak(ΔrXβ)={β2πak(Δ0X),for r=0β2π(ak(ΔrX)+ak(Δr−1X)).for r≥1 In addition, the L2 trace of the heat kernels for ΔrX and ΔrX|Hr(X)⊥ are related by(3.28)Tr(e−tΔrX)−dimkerΔrX=Tr(e−tΔrX|Hr(X)⊥). From the asymptotic expansions and the spectral properties of the corresponding operators one gets(3.29)ak(ΔrX)=ak(ΔrX|Hr(X)⊥)+δknbr(X),ak(ΔrX|Hr(X)⊥)=ak(ΔrX|im(dr+1X)†)+ak(Δr−1X|im(drX)†). Together with (3.22) and (3.27) the corresponding alternating sum in (3.11) can be written as(3.30)∑r=0p(−1)p−r(p−r)an+1(ΔrXβ)=−β2πan+1(Δr−1X|im(drX)†)=βRess=−12[ζ(s;Δp−1X|im(dpX)†)]. By substituting (3.23), (3.25) and (3.30) into (3.11), we obtain the following formula for the free energy of an higher-abelian gauge theory with action S(3.31)FX(p)(β)=−12FPs=−12[ζ(s;Δp−1X|im(dpX)†)]−12Ress=−12[ζ(s;Δp−1X|im(dpX)†)]ln(eμ2)2−1β∑α∈Iln[1−e−βνα(Δp−1X|im(dpX)†)]+bp(X)2βln[2πβ]−12βlndethX(p)−bp(X)2βlnμ2+1βln|Htorp(X;Z)|−1βln(∑cXβ(p+1)∈Hp+1(Xβ;Z)∫im(dp+1Xβ)†Dτp[e−S_cˆXβ(p+1)(τp)](0,…,0)). All terms but the last one are of kinematical origin and stem from the ghost-for-ghost contribution, where the alternating product collapses in the thermal case.1010The “collapse” of determinants of the co-exact Laplace operators has been already pointed out long ago for p-form Maxwell theory at finite temperature [51]. This formula - although suited for the low-temperature regime - is valid, however, for all temperatures.3.1.2High-temperature regimeIn the next step we want to derive a formula for the high-temperature limit of the free energy. The aim is to find an analytical continuation of the zeta function which allows for a separation of the leading terms at high temperatures. The starting point is again (3.11). But instead of (3.18), the zero modes are treated differently. Let us introduce the auxiliary quantity(3.32)K(s;ΔrX|Hr(X)⊥):=2∑k=1∞∑α∈I[(2πkβ)2+να(ΔrX|Hr(X)⊥)]−s so that I(s;ΔrX) admits the decomposition(3.33)I(s;ΔrX)=ζ(s;ΔrX|Hr(X)⊥)+2br(X)(β2π)2sζR(2s)+K(s;ΔrX|Hr(X)⊥). For ℜ(s)>n2, the Mellin transformation of K is given by(3.34)K(s;ΔrX|Hr(X)⊥)=2Γ(s)∫0∞dtts−1∑k=1∞e−(2πβ)2k2t∑α∈Ie−να(ΔrX|Hr(X)⊥)t and can be analytically continued elsewhere [92]. Substituting the heat kernel expansion for ΔrX|Hr(X)⊥ and integrating term by term yields the following asymptotic expansion(3.35)K(s;ΔrX|Hr(X)⊥)≃2(2πβ)−2s∑m=0∞am(ΔrX|Hr(X)⊥)(2πβ)n−mΓ(s+m−n2)ζR(2s+m−n)Γ(s). The function Γ(s+m−n2)ζR(2s+m−n) has simple poles at m=n and m=n+1 and can be analytically continued using the well known reflection formula(3.36)Γ(s2)π−s2ζR(s)=Γ(1−s2)π−s−12ζR(1−s) for the Riemann zeta function ζR(s). Hence by taking the Laurent expansion of ζR(s) at the pole s=−1 and using the series expansion for 1Γ(s) once again, a lengthy calculation leads to(3.37)12dds|s=0K(s;ΔrX|Hj(X)⊥)≃∑m=0m≠nm≠n+1∞am(ΔrX|Hr(X)⊥)2n−mπ−12βm−nΓ(n+1−m2)ζR(n+1−m)+an+1(ΔrX|Hr(X)⊥)β2π12(ln(β4π)+γ)−an(ΔrX|Hr(X)⊥)lnβ. In order to extract the leading terms in (3.37), we notice that for m≠n, (3.29) gives(3.38)∑r=0p(−1)p−ram(ΔrX)=am(ΔpX|im(dp+1X)†), which together with a0(ΔrX)=(4π)−n2n!r!(n−r)!vol(X) [83] yields(3.39)a0(ΔpX|im(dp+1X)†)=(4π)−n2(n−1p)vol(X). Substituting (3.33) into (3.11) and using (3.37) and (3.39) give the final formula for the asymptotic expansion of the free energy at high temperatures(3.40)FX(p)(β)≃(n−1p−1)π−n+12Γ(n+12)ζR(n+1)β−(n+1)vol(X)+∑m=1m≠nm≠n+1∞am(Δp−1X|im(dpX)†)2n−mπ−12Γ(n+1−m2)ζR(n+1−m)βm−n−1+12β(ζ′(0;Δp−1X|im(dpX)†)−lndethX(p))−1β(bp(X)2+an(Δp−1X|im(dpX)†))lnβ−Ress=−12[ζ(s;Δp−1X|im(dpX)†)](ln(μβ4π)+γ)+bp(X)2βln(2πμ2)−1βln(∑cXβ(p+1)∈Hp+1(Xβ;Z)∫im(dp+1Xβ)†Dτp[e−S_cˆXβ(p+1)(τp)](0,…,0))+1βln|Htorp(X;Z)|. Actually, we have only provided the asymptotic expansion for the ghost-for-ghost contribution since the dynamical content is not specified yet. Interestingly, we find that the leading term is of Stefan-Boltzmann type in n spatial dimensions, however, with the wrong sign.3.2The free energy - extended higher-abelian Maxwell theoryNow we will focus on the extended higher-abelian Maxwell theory at finite temperature. The starting point is the Euclidean action(3.41)S(γp,θp)(uˆp)=qp2‖δ1(uˆp)−γp‖2+2πi<δ1(uˆp)−γp,⋆θp>,uˆp∈Hˆp(Xβ). In the thermal context we consider the following topological fields(3.42)γp=∑j=1bp+1(X)γpjpr2⁎(ρX(p+1))j∈Hp+1(Xβ),θp=∑k=1bn−p(X)θpkpr2⁎(ρX(n−p))k∈Hn−p(Xβ). In order to determine the corresponding free energy we have to compute just the last term in (3.31). In terms of local coordinates on Tbp−1(X)×Tbp(X)×im(dp+1Xβ)† the action S splits into a topological and a dynamical part,(3.43)S_(γp,θp)cˆXβ(p+1)(y→p−1,z→p,τp)=S(γp,θp)(cˆXβ(p+1))+qp2<τp,ΔpXβ|im(dp+1Xβ)†τp>, where y→p−1∈Tbp−1(X) and z→p∈Tbp(X). The sum over the topological sectors gives(3.44)∑cXβ(p+1)∈Hp+1(Xβ;Z)e−S(γp,θp)(cˆXβ(p+1))=Θbp(X)(θ→p|−qp2πiβ−1hX(p))|Htorp(X;Z)|×Θbp+1(X)[γ→p0](0|−qp2πiβhX(p+1))|Htorp+1(X;Z)|, where (3.8) and (3.42) were used. Due to the modular property (A.2) and (A.6) of the Riemann-Theta function, we can restrict the vectors γ→p and θ→p to {γ→p} and {θ→p}, respectively, which are defined as follows: For any a→=(a1,…,aj,…,ar)∈Rr we introduce {a→}=({a1},…,{aj},…,{ar})∈Rr such that(3.45){aj}={〈aj〉,if 〈aj〉∈[0,12],1−〈aj〉,if 〈aj〉∈[12,1). Here 〈aj〉=aj−max{m∈Z|m≤aj} is the fractional part of aj.We use (2.57), (3.16) and (3.27) to carry out the τp-integration in (3.31). This leads to(3.46)1βln(∫im(dp+1Xβ)†Dτpe−qp2<τp,ΔpXβ|im(dp+1Xβ)†τp>)==12β∑r=0p(−1)p−r[ζ′(0;ΔrXβ|Hr(Xβ)⊥)+ζ(0;ΔrXβ|Hr(Xβ)⊥)ln(μ2qp)]=12βI′(0;ΔpX)+an+1(ΔpX)4πln(μ2qp)−bp(X)2βln(μ2qp). Now we substitute (3.44) and (3.46) together with (3.22), (3.23) into (3.31). In addition we use (A.3) and (A.4) to rewrite the Riemann Theta functions in (3.44) in order to separate the temperature independent part. The resulting expression for the free energy of extended higher-abelian Maxwell theory splits into two parts(3.47)FX(p)(qp;β,γp,θp)=FX;dyn(p)(qp;β,γp,θp)+FX;top(p)(qp;β,γp,θp), where(3.48)FX;dyn(p)(qp;β,γp,θp)=12FPs=−12[ζ(s;ΔpX|im(dp+1X)†)]+12Ress=−12[ζ(s;ΔpX|im(dp+1X)†)]ln(eμ2)2+1β∑α∈Iln[1−e−βνα(ΔpX|im(dp+1X)†)]−12Ress=−12[ζ(s;ΔpX)]lnqp is related to the dynamical (propagating) modes and(3.49)FX;top(p)(qp;β,γp,θp)=2π2qp∑j,k=1bp(X)(hX(p))jk−1{θpj}{θpk}+qp2∑j,k=1bp+1(X)(hX(p+1))jk{γpj}{γpk}−1βlnΘbp(X)(2πiqpβ(hX(p))−1{θ→p}|2πiqpβ(hX(p))−1)−1βlnΘbp+1(X)(qp2πiβhX(p+1){γ→p}|−qp2πiβhX(p+1))−1βln(|Htorp+1(X;Z)|) is related to the topologically inequivalent configurations. In section 2.3 the partition function was redefined in (2.60) in order to absorb the qp dependent local term into the action. In the thermal case this amounts to go over from FX(p) to(3.50)FˆX(p)(qp;β,γp,θp):=FX(p)(qp;β,γp,θp)+12Ress=−12[ζ(s;ΔpX)]lnqp, which affects only the vacuum contribution. Instead of FX(p) we will work in the following with FˆX(p) and refer to it as the effective free energy of extended higher-abelian Maxwell theory. From (3.48) and (3.49) one can read off the corresponding vacuum (Casimir) energy, namely(3.51)FˆX;vac(p)(qp;γp,θp)=12FPs=−12[ζ(s;ΔpX|im(dp+1X)†)]+12Ress=−12[ζ(s;ΔpX|im(dp+1X)†))]ln(eμ2)2+2π2qp∑j,k=1bp(X)(hX(p))jk−1{θpj}{θpk}+qp2∑j,k=1bp+1(X)(hX(p+1))jk{γpj}{γpk}. The topological contribution to the vacuum (Casimir) energy is absent whenever the topological fields vanish or have integer components. On the other hand, the thermal excitations are always affected by a non-trivial topology of X.The free energy has the following symmetry properties(3.52)FˆX(p)(qp;γp,θp)=FˆX(p)(qp;−γp,−θp)FˆX(p)(qp;γp+ω,θp+ω′)=FˆX(p)(qp;γp,θp),ω∈HZp+1(X),ω′∈HZn−p(X). Notice that the expression (3.50) for the free energy is valid for the whole temperature range. However, in order to obtain an expression which is suited to exhibit the high-temperature structure, we substitute (3.33) into (3.46). To extract the relevant contribution from the topological sector, we apply the duality relation (A.5) to the bp+1(X) dimensional Riemann Theta function in (3.44). A lengthy calculation finally yields the following asymptotic expansion for the free energy of extended higher-abelian Maxwell theory in the high-temperature regime(3.53)FˆX(p)(qp;β,γp,θp)≃−(n−1p)π−n+12Γ(n+12)ζR(n+1)β−(n+1)vol(X)−∑m=1m≠nm≠n+1∞am(ΔpX|im(dp+1X)†)2n−mπ−12βm−n−1Γ(n+1−m2)ζR(n+1−m)−12β[ζ′(0;ΔpX|im(dp+1X)†)+ln(det((qp2π)hX(p))det((qp2π)hX(p+1))|Htorp+1(X;Z)|2)]+12β[2an(ΔpX|im(dp+1X)†)+bp(X)+bp+1(X)]lnβ+Ress=−12[ζ(s;ΔpX|im(dp+1X)†)](ln(μβ4π)+γ)−1βlnΘbp(X)(θ→p|−qp2πiβ−1hX(p))−1βlnΘbp+1(X)(γ→p|2πiqpβ−1(hX(p+1))−1). In this limit the contributions from the propagating modes become dominant. The leading term is extensive and exhibits the Stefan-Boltzmann dependency on the temperature in n spatial dimensions. The remaining terms represent modifications due to the topology and geometry of X.In summary, we have shown explicitly how the topology and geometry of X affect both the vacuum (Casimir) energy and the finite temperature part related to the occupied states of the thermal ensemble. The ambiguity in the free energy is expressed by the renormalization scale μ and appears, whenever Ress=−12[ζ(s;ΔpX|im(dp+1X)†)]≠0. In that case renormalization issues have to be taken into account. On the other hand, if ζ(s;ΔpX|im(dp+1X)†) is finite at s=−12, the scale dependency disappears and the free energy is uniquely determined. Finally, let us remark that in the special case p=1 and γ1=0, we recover the expression for the free energy of the photon gas [62], which back then were obtained within the framework of principal U(1)-bundles with connections.Let us recall that in its original meaning the Casimir effect is caused by the change of the vacuum energy or, at finite temperature, of the free energy due to constraints imposed on the quantum fields compared with the free energy of the unconstrained system. The difference between these two free energies is called Casimir free energy (see e.g. [2] for a detailed discussion on this topic). In this respect, the formulae which we derived for the free energy represent always the total free energy of the constrained system. An obvious candidate for a (unconstrained) “reference configuration” would be a gas of higher-abelian gauge fields placed in a large box in flat Euclidean space. In the infinite volume limit all related topological and finite size effects can be neglected. Since we applied zeta function regularization the vacuum energy in the reference configuration is implicitly disregarded. Thus only the thermal contribution of the propagating degrees of freedom remains. In the large volume limit the discrete index α in the third term of (3.48) can be replaced by the continuous n-dimensional wave vector k→∈Rn. Hence the sum can be replaced by an integration with respect to the measure dnk→(2π)n. Using [91] the integration gives for the free energy density(3.54)(n−1p)1β∫Rndnk→(2π)nln(1−e−β|k→|)=−(n−1p)π−n+12Γ(n+12)ζR(n+1)β−(n+1), where the factor (n−1p) is the number of independent propagating degrees of freedom. As expected we get the free energy density of black-body radiation of p-form Maxwell theory in n dimensions.1111Let us remark that there exists another renormalization approach for the Casimir free energy where also the finite temperature contributions are appropriately renormalized [32,36,37,41,42]. The underlining concept is based on the proposal that the renormalized Casimir free energy should satisfy the classical limit at high temperatures.3.2.1The equation of stateIn this section we want to derive the equation of state for the gas of higher-abelian gauge fields. In terms of the (effective) free energy, the internal energy UˆX(p), entropy SˆX(p) and pressure PˆX(p) are given by(3.55)UˆX(p)=∂∂β(βFˆX(p)),SˆX(p)=β2∂∂βFˆX(p),PˆX(p)=−∂∂VFˆX(p). Like the free energy, every thermodynamic function splits into a dynamical and into a topological part. Under a constant scale transformation g↦λ2g, λ∈R, of the background metric of Xβ, one gets β↦λβ andvolX↦λnvolX,νk(i)(ΔrTβ1)↦λ−2νk(i)(ΔrTβ1)hX(p)↦λn−2phX(p),νl(j)(ΔsX)↦λ−2νl(j)(ΔsX). The free energy transforms as(3.56)FˆX(p)↦1λFˆX(p)+lnλλRess=−12[ζ(s;ΔpX|im(dp+1X)†)]+(2π)2qpλ(λ2p+1−n−1)∑j,k=1bp(X)(hX(p))jk−1{θj}{θk}+qp2λ(λn−2p−1−1)×∑j,k=1bp+1(X)(hX(p+1))jk{γj}{γk}+1λβln(Θbp(X)(2πiqpβ(hX(p))−1{θ→}|2πiqpβ(hX(p))−1)Θbp(X)(2πiqpλ2p+1−nβ(hX(p))−1{θ→}|2πiqpλ2p+1−nβ(hX(p))−1))+1λβln(Θbp+1(X)(qp2πiβhX(p+1){γ→}|−qp2πiβhX(p+1))Θbp+1(X)(qp2πiλn−2p−1β(hX(p+1)){γ→}|qp2πiλn−2p−1βhX(p+1))). The main observation is that in general the free energy does not transform homogeneously of degree −1, meaning that the free energy is no longer an extensive quantity. There are two obstructions: The first one is caused by the explicit dependence on the renormalization scale μ. The second one is related to non-trivial Hfreep(X;Z)⊕Hfreep+1(X;Z). The resulting equation of state is obtained from the right hand side of (3.56) by taking the derivative with respect to λ at λ=1. A lengthy calculation yields(3.57)FˆX(p)=nPˆX(p)vol(X)−β−1SˆX(p)+Γ(p), with anomalous term(3.58)Γ(p)=Ress=−12[ζ(s;ΔpX|im(dp+1X)†)]+(2p+1−n)(2π2qp∑j,k=1bp(X)(hX(p))jk−1{θpj}{θpk}−qp2∑j,k=1bp+1(X)(hX(p+1))jk{γpj}{γpk})+2p+1−nβ∂∂λ|λ=1ln(Θbp+1(X)(qp2πiλβhX(p+1){γp→}|−qp2πiλβhX(p+1))Θbp(X)(2πiqpλβ(hX(p))−1{θp→}|2πiqpλβ(hX(p))−1)). Since the thermodynamic functions are interrelated by FˆX(p)=UˆX(p)−β−1SˆX(p), we finally obtain the following equation of state:(3.59)PˆX(p)=1nvol(X)[UˆX(p)−Γ(p)]. The topological contribution to Γ(p) vanishes for n=2p+1. If in addition the zeta function of ΔpX|im(dp+1X)† is finite at s=−12, then the anomalous term Γ(p) vanishes identically.3.2.2Thermal dualityNow we want to discuss the relation between thermodynamic functions of extended higher-abelian Maxwell theories of degree p and n−p−1. Let (qp,γp,θp) be the corresponding set of parameters in degree p. We introduce the dual parameters by1.qn−p−1dual⋅qp=(2π)22.γ→n−p−1dual=θ→p3.θ→n−p−1dual=γ→p. Since Spec(Δn−pX|im(dn−p−1X))=Spec(Δn−p−1X|im(dn−pX)†) and ⋆_ΔpX|im(dp+1X)†=Δn−pX|im(dn−p−1X)⋆_, one has Spec(ΔpX|im(dp+1X)†)=Spec(Δn−p−1X|im(dn−pX)†). Using that Htorp+1(X;Z)≅Htorn−p(X;Z) one can verify directly from the explicit expression (3.47) that(3.60)FˆX(p)(qp;β,γp,θp)=FˆX(n−p−1)((2π)2qp;β,θp,γp). As a consequence, all thermodynamic functions are equal implying that these theories are exactly dual to each other.Alternatively, (3.60) can be obtained directly from the general duality relation (2.69): Since the Euler characteristics of a product splits and χ(Tβ1)=0, one gets χ(Xβ)=χ(Tβ1)χ(X)=0. Due to the specific choice for γp and θp (3.42) the pairing term is absent.3.3Extended higher-abelian Maxwell theory on the n-torusIn this section we want to present an explicit example by studying the extended higher-abelian Maxwell theory on the n-torus X=Tn. The aim is to compute exact expressions for the corresponding thermodynamic functions in the low- and the high-temperature regimes, respectively.Let (t1,…,tn) denote the local coordinates of Tn. We equip the n-torus with the flat metric gTn=L2∑i=1ndti⊗dti, so that vol(Tn)=Ln.The starting point for the computation of the free energy is the general formula (3.47). The eigenvalues of ΔpTn|im(dp+1Tn)† are given by the sequence of positive real numbers νm→(ΔpTn|im(dp+1Tn)†)=(2πL)2∑j=1nmj2, where m→=(m1,…,mn)∈Z0n:=Zn\0. For each fixed m→∈Z0n, let us choose an orthonormal basis εm→r∈m→⊥ with r=1,…,n−1. The components of εm→r are denoted by εm→;jr, with j=1,…,n. Then the co-exact p-forms(3.61)ψr1,…,rp;m→(p):=∑1≤j1<…<jp≤nεm→;j1r1⋯εm→;jprpe2πi∑k=1nmktkdtj1∧…∧dtjp∈im(dp+1Tn)†⊗C are eigenforms of ΔpTn|im(dp+1Tn)† associated to the eigenvalue νm→(ΔpTn|im(dp+1Tn)†). Moreover, for each fixed m→∈Z0n the eigenforms ψr1,…,rp;m→(p) with 1≤r1<…<rp≤n−1 provide a basis for the (n−1p)-dimensional eigenspace, whose dimension gives the number of independent polarization states associated to the propagating degrees of freedom. For the corresponding zeta-function we obtain(3.62)ζ(s;ΔpTn|im(dp+1Tn)†)=(n−1p)∑m→∈Z0n[∑j=1n(2πmjL)2]−s=(2πL)−2s(n−1p)En(s;1,…,1), where En denotes the Epstein zeta function in n dimensions (A.14). Using the reflection formula (A.13), it follows that (3.62) is regular at s=−12, so that its finite part (3.21) reads(3.63)FPs=−12[ζ(s;ΔpTn|im(dp+1Tn)†)]=−1L(n−1p)π−n+12Γ(n+12)En(n+12;1,…,1). As a consequence, the resulting free energy does not depend on the renormalization scale μ. In order to determine the topological contribution, we notice that the p-forms(3.64)(ρTn(p))i1,…,ip=dti1∧…∧dtip,1≤i1<…<ip≤n provide a basis for HZp(Tn) and induce the metric(3.65)(hTn(p))ij=Ln−2pδij,i,j=1,…,(np). Substituting (3.63) and (3.65) into (3.47) leads to the following expression for the free energy(3.66)FˆTn(p)(qp;β,γp,θp)=−12L(n−1p)π−n+12Γ(n+12)En(n+12;1,…,1)+2π2qpL2p−n∑j=1(np){θpj}2+qp2Ln−2p−2∑j=1(np+1){γpj}2+1β(n−1p)∑m→∈Z0nln[1−e−2πβL−1|m→|]−1β∑j=1(np)lnΘ1(2πiqpβL2p−n{θpj}|2πiqpβL2p−n)−1β∑j=1(np+1)lnΘ1(qp2πiβLn−2p−2{γpj}|−qp2πiβLn−2p−2). The contributions (first and fourth term) of the propagating modes depend on the degree p only via the factor giving the number of independent polarization states. When considering the product LFˆTn(p) these contributions become a function of the inverse scaled temperature βL and thus they are scale invariant under the joint transformation β↦λβ and L↦λL for λ∈R. By contrast, the topological part depends intrinsically on degree p and - except in odd dimensions and for degree p=n−12 - spoils this scale invariance. In that exceptional case, the equation of state (3.59) reduces to the “conventional” form(3.67)PˆT2p+1(p)L2p+1=12p+1UˆT2p+1(p), without anomalous term. In the following we will provide low- and high-temperature expansions for the thermodynamic functions.3.3.1Low-temperature regimeLet us denote the sum of the last two Riemann-Theta functions in (3.66) by flow(p). Since {θpj} and {γpj} belong to the interval [0,12], condition (A.9) is satisfied. By applying (A.11) to flow(p), one gets the following series expansion(3.68)flow(p):=1βln2Nθ+Nγ+1β∑m=1∞1m(np)+2(−1)mNθ1−e(2π)2qpβL2p−nm+1β∑m=1∞1m(np+1)+2(−1)mNγ1−eqpβLn−2p−2m+1β∑j=1{θpj}≠12(np)∑m=1∞(−1)m+1me−(2π)2qpβL2p−nm(12+{θpj})+e−(2π)2qpβL2p−nm(12−{θpj})1−e−(2π)2qpβL2p−nm+1β∑j=1{γpj}≠12(np+1)∑m=1∞(−1)m+1me−qpβLn−2p−2m(12−{γpj})+e−qpβLn−2p−2m(12+{γpj})1−e−qpβLn−2p−2m, where Nθ and Nγ are the numbers of components of θ→p and γ→p, such that {θpi}=12 or {γpi}=12. All but the first term decrease exponentially for β→∞.The internal energy can be written in the form(3.69)UˆTn(p)(qp;β,γp,θp)=UˆTn;Cas(p)(qp;γp,θp)+2πL(n−1p)∑k→∈Z0n|k→|e2πβL−1|k→|−1−∂∂β(βflow(p)), where the last two terms decrease exponentially for low temperatures and UˆTn;Cas(p) denotes the regularized vacuum (Casimir) energy, given by(3.70)UˆTn;Cas(p)(qp;γp,θp):=limβ→∞UˆTn(p)(qp;β,γp,θp)=limβ→∞FˆTn(p)(qp;β,γp,θp)==−12L(n−1p)π−n+12Γ(n+12)En(n+12;1,…,1)+2π2qpL2p−n∑j=1(np){θpj}2+qp2Ln−2p−2∑j=1(np+1){γpj}2. Whereas the vacuum energy of the propagating modes is negative, the topological fields give always a positive contribution to the total vacuum energy.The pressure splits into(3.71)PˆTn(p)(qp;β,γp,θp)=PˆTn;Cas(p)(qp;γp,θp)+2πnL−(n+1)(n−1p)∑k→∈Z0n|k→|e2πβL−1|k→|−1+∂∂Lflow(p), where again the last two terms decrease exponentially with increasing temperature. The vacuum (Casimir) pressure is defined as the limit(3.72)PˆTn;Cas(p)(qp;γp,θp):=limβ→∞PˆTn(p)(qp;β,γp,θp)=−12n(n−1p)π−n+12Γ(n+12)En(n+12;1,…,1)L−(n+1)−2π2qpL2(p−n)2p−nn∑j=1(np){θpj}2−qp2L−2(p+1)n−2p−2n∑j=1(np+1){γpj}2. The vacuum pressure exerted by the propagating degrees of freedom is negative for all p. On the other hand, the sign of the pressure induced by the topological fields depends on p. Finally, we find for the entropy(3.73)SˆTn(p)(qp;β,γp,θp)=−(n−1p)∑k→∈Z0nln[1−e−2πβL−1|k→|]+2πβL(n−1p)∑k→∈Z0n|k→|e2πβL−1|k→|−1−β2∂∂β(flow(p)). In the zero temperature limit the entropy related to the propagating modes vanishes. According to the series expansion (3.68), the topological modes (i.e. the last term in (3.73)) contribute, whenever the topological fields have components which are multiples of 12. All other terms decrease exponentially. In summary, the entropy converges to(3.74)SˆTn;Cas(p)(qp;γp,θp):=limβ→∞SˆTn(p)(qp;β,γp,θp)=ln2Nθ+Nγ, indicating that the ground state of the system is degenerate of degree 2Nθ+Nγ. The maximum entropy at zero temperature is obtained for Nθ;max=(np) and Nγ;max=(np+1), resulting in(3.75)SˆTn;Cas(p)(qp;γp,θp)|Nθ;max,Nγ;max=(n+1p+1)ln2.3.3.2High-temperature regimeTo study the high-temperature behavior we will not apply the general formula for the asymptotic expansion (3.53) but derive an exact formula for the free energy and subsequently for the other thermodynamic functions. The starting point is (3.11) together with (3.44) and the second equation in (3.46), but now applied to Tβn:=Tβ1×Tn. The eigenvalues of ΔrTβn|im(dr+1Tβn)† read(3.76)νk0,k→=(2πk0β)2+∑i=1n(2πkiL)2,(k0,k→)=(k0,k1,…,kn)∈Z0n+1, such that the resulting zeta function becomes(3.77)ζ(s;ΔrTβn|im(dr+1Tβn)†)=(np)En+1(s;2πβ,2πL,…,2πL). In the first step we apply the Chowla-Selberg formula (A.20) to (3.77) by setting r=n+1, l=n, c1=…cn=2πL and cn+1=2πβ. In the second step we extract the leading term for the high-temperature regime from the corresponding bp+1(Tn)-dimensional Riemann Theta function in (3.44) by using (A.5). A lengthy calculation using (2.47) gives the following expression for the free energy(3.78)FˆTn(p)(qp;β,γp,θp)=−(n−1p)π−n+12Γ(n+12)ζR(n+1)β−(n+1)Ln−2(n−1p)β−n+22Ln2∑kn+1=1∞∑k→∈Z0nkn+1n2|k→|−n2Kn2(2πkn+1Lβ|k→|)−12β[(n−1p)En′(0;1,…,1)−ln((2π)Λ1LΛ2βΛ3qpΛ4)]−1β∑j=1(np)lnΘ1({θpj}|−qp2πiLn−2pβ)−1β∑j=1(np+1)lnΘ1({γpj}|2πiqpL2p+2−nβ), with the constants(3.79)Λ1:=(np)−(np+1)−2(n−1p)Λ2:=2(n−1p)−(np)(n−2p)+(np+1)(n−2p−2)Λ3:=(np+1)+(np)−2(n−1p)Λ4:=(np+1)−(np). As expected from the general structure, the leading contribution is once again the Stefan-Boltzmann term in n dimensions. Let us denote the last two terms in (3.78) by fhigh(p). Since the condition (A.9) is satisfied, one gets the following series expansion for fhigh(p), using (A.10),(3.80)fhigh(p)=1β[(np)∑m=1∞1m11−eqpLn−2pβm+2∑j=1(np)∑m=1∞(−1)mmeqp2Ln−2pβmcos(2πm{θpj})1−eqpLn−2pβm]+1β[(np+1)∑m=1∞1m11−e(2π)2qpL2p+2−nβm+2∑j=1(np+1)∑m=1∞(−1)mme2π2qpL2p+2−nβmcos(2πm{γpj})1−e(2π)2qpL2p+2−nβm]. All terms of fhigh(p) decrease exponentially for β→0. Since Kν(z)≃π2ze−z(1+O(1z)) for |z|→∞ [91], the summands in the second term of (3.78) show an exponential decrease in the limit β→0 as well. Let us now briefly display the other thermodynamic functions in the high temperature regime. For the internal energy one gets(3.81)UˆTn(p)(qp;β,γp,θp)=n(n−1p)π−n+12Γ(n+12)ζR(n+1)Lnβn+1+Λ32β−∂∂β(βfhigh(p))−4π(n−1p)Ln+22β−n+42∑kn+1=1∞∑k→∈Z0nkn+1n+22|k→|2−n2Kn2−1(2πkn+1Lβ|k→|). Here we have used that ddzKν(z)=−Kν−1(z)−νzKν(z) [91]. In addition to the leading term of Stefan-Boltzmann type and the exponentially decreasing last two terms in (3.81), there appears a topological term linear in the temperature. For the thermal pressure we obtain(3.82)PˆTn(p)(qp;β,γp,θp)=(n−1p)π−n+12Γ(n+12)ζR(n+1)β−(n+1)−Λ22nβLn+1nLn−1∂∂Lfhigh(p)−4πn(n−1p)L2−n2β−n+42∑kn+1=1∞∑k→∈Z0nkn+1n+22|k→|2−n2Kn2−1(2πkn+1Lβ|k→|), where the last two terms show an exponential decrease for β→0. Like before the corresponding topological subleading term is linear in the temperature. Finally, the entropy admits the following form(3.83)SˆTn(p)(qp;β,γp,θp)=(n+1)(n−1p)π−n+12Γ(n+12)ζR(n+1)(Lβ)n−β2∂∂βfhigh(p)+2(n−1p)(Lβ)n2∑kn+1=1∞∑k→∈Z0nkn+1n2|k→|−n2Kn2(2πkn+1Lβ|k→|)−4π(n−1p)(Lβ)n+22∑kn+1=1∞∑k→∈Z0nkn+1n+22|k→|2−n2Kn2−1(2πkn+1Lβ|k→|)+12[(n−1p)En′(0;1,…,1)+Λ3−ln((2π)Λ1LΛ2βΛ3qpΛ4)]. In the high temperature limit all terms except the first and the last one in (3.83) are exponentially suppressed.In summary, we have explicitly shown that in the high temperature limit all thermodynamic functions are dominated by terms of Stefan-Boltzmann type which are related to the propagating degrees of freedom. The topological contributions appear as subleading corrections. Although the formulae for the thermodynamic functions look quite different in the low- and high temperature regimes, each of them is valid for the whole temperature range.Before closing this section we want to address briefly two topics: the thermodynamic limit of the free energy density and the entropy to energy ratio. Concerning the first one the starting point is (3.78). Depending on the degree p one has to use the series expansions either (3.68) or (3.80) to compute the large volume limit for the topological contributions. In summary, it can be shown that the topological effects disappear so that all terms but the Stefan-Boltzmann term vanish and we end up with (see (3.54))(3.84)limL→∞FˆTn(p)/Ln=−(n−1p)π−n+12Γ(n+12)ζR(n+1)β−(n+1). Regarding the second topic we give an estimate for the entropy to internal energy ratio in the high-temperature limit using (3.81) and (3.83), however, neglecting exponentially decreasing terms. This ratio can be written in the form(3.85)SˆTn(p)UˆTn(p)≈n+1nβ+O(βn+1(1+lnβ)),β→0. The leading term is linear in β and derives from the corresponding Stefan-Boltzmann terms. This term is in some sense universal since it depends only on the dimension of the torus and is independent of p. The topological and geometrical properties contribute through further subleading terms in the expansion. For instance, in dimension n=3 and for p=1 the first term agrees exactly with the classical result for the entropy to energy ratio of the black body photon gas enclosed in a large box. For general p, our result compares further to [52], where the same leading term was obtained for the p-form Maxwell theory at finite temperature on hyperbolic spaces. Once again, this result highlights that in the infinite temperature limit the system of higher-abelian gauge fields is controlled by the propagating degrees of freedom.4Polyakov loop operator in higher-abelian gauge theories4.1The static brane antibrane free energySo far we have discussed the impact of the topology of the spatial background on the thermodynamic functions of higher-abelian gauge fields. In the next step we want to probe the topological effect on the two-point correlation function of a higher-abelian generalization of the Polyakov loop operator.In ordinary gauge theory at finite temperature the Polyakov loop operator is a variant of the Wilson loop operator and measures the holonomy of the gauge potential 1-form along the periodic (thermal) time direction Tβ1. The corresponding correlation functions are interpreted as free energy in the presence of static charges relative to the free energy of the pure gauge field background. In particular, the two-point correlation function gives the effective potential between a pair of oppositely charged static particles. Moreover, the Polyakov loop operator defines an order parameter for the confinement-deconfinement transition, even in the abelian case (see e.g. [93–95]).In order to generalize the Polyakov loop operator to higher-abelian gauge fields, we replace the static charge located at a point x∈X (which is a singular 0-cycle) by a static, closed brane, which is represented by a smooth singular (p−1)-cycle Σ∈Zp−1(X;Z). Correspondingly, the world-line Tβ1×{x}∈Z1(Xβ;Z) of the static particle placed at x∈X along periodic (thermal) time is replaced by the world-volume Σβ:=Tβ1×Σ∈Zp(Xβ;Z) of the static brane Σ.For a smooth singular (p−1)-cycle Σ∈Zp−1(X;Z) we define the higher-abelian Polyakov loop operator of degree p by(4.1)Σ↦PΣ(p)(uˆp):=uˆp(Σβ),Σβ:=Tβ1×Σ∈Zp(Xβ;Z) For topologically trivial differential characters which are of the form j2([Ap])∈Hˆp(Xβ), for Ap∈Ωp(Xβ), the higher-abelian Polyakov loop operator reduces to(4.2)PΣ(p)(j2([Ap]))=exp2πi(∫ΣβAp), which is nothing but the minimal coupling of the p-form gauge field Ap to the world-volume Σβ.In the following we will assign charges ±1 to the brane and antibrane, respectively.1212In general we could, however, assign a charge q˜∈Z to the brane by replacing Σ by q˜Σ. Notice that the total brane charge is identically zero on a compact and closed manifold. In analogy to the case of point particles, we introduce the static brane antibrane free energy FΣ(1),−Σ