2021
DOI: 10.1007/jhep12(2021)211
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Notes on massless scalar field partition functions, modular invariance and Eisenstein series

Abstract: The partition function of a massless scalar field on a Euclidean spacetime manifold ℝdβˆ’1 Γ— 𝕋2 and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL(2, β„€) Eisenstein series with s = (d + 1)/2. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series repre… Show more

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Cited by 7 publications
(4 citation statements)
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“…In Type IIB, SL(2, Z) invariance dictates that the coefficients of higher-derivative terms are written in terms of modular forms. The appearance of the modular function f 0 , the non-holomorphic Eisenstein series of weight 3/2, was first observed in [15,18], while other non-holomorphic modular forms f w first appeared in [16,19], see also [20] for a more recent discussion. For R 4 , leading order D-instanton calculations [21,22] confirmed this result.…”
Section: Jhep08(2022)267 1 Introduction and Summarymentioning
confidence: 99%
“…In Type IIB, SL(2, Z) invariance dictates that the coefficients of higher-derivative terms are written in terms of modular forms. The appearance of the modular function f 0 , the non-holomorphic Eisenstein series of weight 3/2, was first observed in [15,18], while other non-holomorphic modular forms f w first appeared in [16,19], see also [20] for a more recent discussion. For R 4 , leading order D-instanton calculations [21,22] confirmed this result.…”
Section: Jhep08(2022)267 1 Introduction and Summarymentioning
confidence: 99%
“…From a theoretical viewpoint, the exact results derived are closely related to those that appear when generalizing modular invariance from two [22,23] to higher spacetime dimensions [24][25][26][27], combined with the technique of going back to real rather than purely imaginary chemical potential [28][29][30].…”
Section: Discussionmentioning
confidence: 81%
“…With Ξ± ‰ 0, this has been discussed instance in [2]. The detailed derivation by a variety of different methods of the expressions provided below can be found in [17].…”
Section: Partition Functionmentioning
confidence: 99%
“…In order to do so, we follow recent work on the finite temperature electromagnetic Casimir effect and modular invariance in this context [16] to show that this scalar field should satisfy periodic boundary on an interval of double the length of the separation of the plates. This allows us in section 5 to simply infer the exact result for the partition function for gravitons, with a chemical potential for suitably modified spin angular momentum turned on, from the well-studied case of a massless scalar field with linear momentum in the compact direction turned on [2][3][4]17], together with modular properties and consistent high and low temperature expansions.…”
Section: References 1 Introductionmentioning
confidence: 99%

Gravitons in a Casimir box

Alessio,
Barnich,
Bonte
2020
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