Thermodynamics of abelian forms in real compact hyperbolic spaces

Thermodynamics of abelian forms in real compact hyperbolic spaces

A. A. Bytsenko^I; V. S. Mendes^I; A. C. Tort^II

^IDepartamento de Física, Universidade Estadual de Londrina, Londrina, Brazil

^IIDepartamento de Física Teórica, Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

ABSTRACT

We analyze gauge theories based on abelian p–forms in real compact hyperbolic manifolds. The explicit thermodynamic functions associated with skew–symmetric tensor fields are obtained via zeta–function regularization and the trace tensor kernel formula. Thermodynamic quantities in the high–temperature expansions are calculated and the entropy/energy ratios are established.

1 Introduction

It is known that the thermodynamics of quantum fields in an Einstein universe for some radius is equivalent to that of an instantaneously static closed Friedmann-Robertson-Walker universe. The field thermodynamics of positive curvature Einstein spaces was discussed by several authors before. In particular, the so-called entropy bounds or entropy to thermal energy ratios were calculated and compared with known bounds such as the Bekenstein bound or the Cardy-Verlinde bound. For example, for a massless scalar field in S³ space this was done in [1] and for a massive scalar field in [2]. Here we wish to extend the evaluation of those type of bounds to the case of skew symmetric tensor fields in real hyperbolic spaces.

We shall work with a D–dimensional compact hyperbolic space X with universal covering M and fundamental group G. We can represent M as the symmetric space G/K, where G = SO₁(D,1) and K = SO(D) is a maximal compact subgroup of G. Then we regard G as a discrete subgroup of G acting isometrically on M, and we take X to be the quotient space by that action: X = G\M = G\G/K. Let t be an irreducible representation of K on a complex vector space V_t, and form the induced homogeneous vector bundle G × _K V_t (the fiber product of G with V_t over K) ® M over M . Restricting the G action to G we obtain the quotient bundle E_t = G\(G × _K V_t) ® X = G\M over X. The natural Riemannian structure on M (therefore on X) induced by the Killing form ( , ) of G gives rise to a connection Laplacian on E_t. If W_K denotes the Casimir operator of K – that is W_K = –å, for a basis {y_j} of the Lie algebra k₀ of K, where (y_j ,) = –, then t(W_K) = l_t for a suitable scalar l_t. Moreover for the Casimir operator W of G, with W operating on smooth sections G^¥ E_t of E_t one has = W – l_t1 ; see Lemma 3.1 of [3]. For l > 0 let

be the space of eigensections of corresponding to l. Here we note that since X is compact we can order the spectrum of – by taking 0 = l₀ < l₁ < l₂ < ...; lim_j®¥l_j = ¥. It will be convenient moreover to work with the normalized Laplacian _p = –c(D) where c(D) = 2(D – 1) = 2(2N – 1). _p has spectrum where the multiplicity m_j of the eigenvalue c(D)l_j is given by m_j = dim G^¥ .

It is easy to prove the following properties for operators and forms: dd = dd = 0, d = (–1)^Dp+D+1*d*, **w_p = (–1)^p(D–p)w_p. Let a_p, b_p be p–forms; then the invariant inner product is defined by (a_p, b_p): = ò_M a_p Ù *b_p. The operators d and d are adjoint to each other with respect to this inner product for p–forms: (da_p, b_p) = (a_p, db_p). In quantum field theory the Lagrangian associated with w_p takes the form: L = dw_p Ù *dw_p (gauge field); L = dw_p Ù *dw_p (co-gauge field). The Euler-Lagrange equations supplied with the gauge give _pw_p = 0 , dw_p = 0 (Lorentz gauge); _pw_p = 0 , dw_p = 0 (co-Lorentz gauge). These Lagrangians give possible representation of tensor fields or generalized Abelian gauge fields. The two representations of tensor fields are not completely independent. Indeed, there is a duality property in the exterior calculus which gives a connection between star-conjugated gauge tensor fields and co-gauge fields.

2 The trace formula applied to the tensor Kernel

We can apply the version of the trace formula developed by Fried in [4]. First we define additional notation. For s_p the natural representation of SO(2N – 1) on L^pC^2N–1, one has the corresponding Harish-Chandra-Plancherel density, given for a suitable normalization of Haar measure dx on G by

for 0 < p < N – 1, where

is an even polynomial of degree 2N – 2. One has that and for N < p < 2N – 1. Now define the Miatello coefficients (see the ref. [5]) for G = SO₁(2N + 1, 1) by , 0 < p < 2N – 1 . Let Vol(G\G) denote the integral of the constant function 1 on G\G with respect to the G - invariant measure on G\G, induced by dx. For 0 < p < D – 1, the Fried trace formula [4] applied to the tensor kernel associated to the Laplace operator on co-exact forms is [6,7]:

where b_p are the Betti numbers. In the above formula and are the identity and hyperbolic orbital integrals, respectively.

3 The spectral functions of exterior forms

The spectral zeta function related to the Laplace operator

_j can be represented by the inverse Mellin transform of the heat kernel _t = Tr exp( –t

_j). Using the Fried formula, we can write the zeta function as a sum of two contributions:

For the identity component we have

where V_G = c(1)Vol(G\G)/4p, and we define = b^(j) + (r₀ – j)², r₀ = (D – 1)/2 and b^(j) are constants. Replacing the Harish-Chandra-Plancherel measure, we obtain two representations for (s,j) which hold for odd and even dimension

where B_n is the n-th Bernoulli number. Moreover, we define

In fact, we do not need the hyperbolic component (s,j) since in the high temperature limit (see next section), only the function (s,j) will present contribution.

4 The high temperature limit

Using the Mellin representation for the zeta function one can obtain useful formulae for the non-trivial temperature dependent part of the identity and hyperbolic orbital components of the free energy (for details see the ref. [8,9,10])

A tedious calculation gives the following result:

The contribution associated to the hyperbolic orbital component is negligible small.

4.1 The thermodynamic functions and the entropy bound

In the context of the Hodge theory, the physical degrees of freedom are represented by the co-exact forms. Thus the free energy becomes

However, if we perform the sum explicitly, we see that

In the high temperature limit (b ® 0) we have

where, for the even dimensional case

and, for the odd dimensional case

In fact, in the sums (12) - (15) only terms containing the Miatello coefficients survive and define the coefficients A₁ and A₂. The entropy and the total energy can be obtained with the help of the following thermodynamic relations: S^(D)(b) = b²¶ ^(D)(b) /¶b, E^(D)(b) = ¶(b ^(D)(b)) /¶b. Therefore,

The entropy/energy ratio becomes

5 Concluding remarks

We have obtained the high-temperature expansion for the entropy/energy ratios of abelian gauge fields in real compact hyperbolic spaces. The dependence on the Miatello coefficients related to the structure of the Harish-Chandra-Plancherel measure starts from the second term of the expansion. In the case of scalar fields (p = 0) we have eq. (18) with

where = + m² ( = for the massless case). For three-dimensional hyperbolic manifolds the Miatello coefficients reads [11]: = = 1 and therefore S⁽³⁾(b)/E⁽³⁾(b) = (4/3)b+(10/3p²) (2–) b³+ (b⁵). This formula is in agreement with result obtained in [2] where entropy bounds have been calculated for spherical geometry and where the dependence on the geometry of the background also starts from the second term of the expansion.

Acknowledgements

A.A.B. would like to thank Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP/Brazil) and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq/Brazil) for partial financial support, and the Instituto de Física Teórica (IFT/UNESP) for kind hospitality.

Received on 4 December, 2003

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Publication Dates

• Publication in this collection
  04 Nov 2004
• Date of issue
  Sept 2004

History

• Received
  04 Dec 2003