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Abstract
We analyze the finite temperature behaviour of massless conformally coupled scalar fields in homogeneous lens spaces $S^3/{\mathbb Z}_p$. High and low temperature expansions are explicitly computed and the behavior of thermodynamic quantities under thermal duality is scrutinized. The analysis of the entropy of the different lens spaces in the high-temperature limit points out the appearance of a topological nonextensive entropy, besides the standard Stefan-Boltzmann extensive term. The remaining terms are exponentially suppressed by the temperature. The topological entropy appears as a subleading correction to the free energy that can be obtained from the determinant of the lens space conformal Laplacian operator. In the low-temperature limit the leading term in the free energy is the Casimir energy and there is no trace of any power correction in any lens space. In fact, the remaining corrections are always exponentially suppressed by the inverse of the temperature. The duality between the results of both expansions is further analyzed in the paper.
Highlights
Thermodynamic quantities are highly dependent on the geometric and topological properties of the spatial manifold
Non-trivial topologies of space-time have been suggested as a possible explanation of the presence of anomalies in the spectrum of Cosmic Microwave Background (CMB) observed by WMAP [23, 24]
Using the duality transformation properties of the infinite sums appearing in the evaluation of the logarithm of the determinant, we obtain two equivalent expressions for the effective action on S3 and on any homogeneous lens space, which provides the basis for the low- and high-temperature expansions of the free energy
Summary
We will perform the analytic extension of the zeta function in a way which is adequate to isolate the infinite temperature (β → 0) terms from their corrections. The double infinite sum can be extended analytically as done for similar expressions in the S3 case, to obtain δζSl=3/0Z2q+1(s) = Γ−(s1+6(1μ)(a2)q2s+π132)s2smq=1ml,n∞=1n sin (2q + 1)β n 2al s− Note that this part of the zeta function vanishes at s = 0. As done in the previous cases, we evaluate first the contribution of the l = 0 term which, added to the corresponding term in the effective action of S3/Z2 divided by q, will give rise to the topological subleading entropy. We can determine the contribution to the effective action coming from equations (3.28) and (3.29), which is δSeff, S3/Z2q In this case, the first term is the one remaining in the high-temperature limit, and the others are exponentially decreasing corrections. The first two terms are obtained from the determinant of the spatial operator, and are equivalent to the result in [36]
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