One-loop BPS amplitudes as BPS-state sums

Journal of High Energy Physics (Impact Factor: 6.11). 03/2012; 2012(6). DOI: 10.1007/JHEP06(2012)070
Source: arXiv


Recently, we introduced a new procedure for computing a class of one-loop
BPS-saturated amplitudes in String Theory, which expresses them as a sum of
one-loop contributions of all perturbative BPS states in a manifestly T-duality
invariant fashion. In this paper, we extend this procedure to all BPS-saturated
amplitudes of the form \int_F \Gamma_{d+k,d} {\Phi}, with {\Phi} being a weak
(almost) holomorphic modular form of weight -k/2. We use the fact that any such
{\Phi} can be expressed as a linear combination of certain absolutely
convergent Poincar\'e series, against which the fundamental domain F can be
unfolded. The resulting BPS-state sum neatly exhibits the singularities of the
amplitude at points of gauge symmetry enhancement, in a chamber-independent
fashion. We illustrate our method with concrete examples of interest in
heterotic string compactifications.

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    • "Threshold corrections are important observables in string compactifications and there has been a recent revival in studying properties of these observables mainly due to the work of [21] [22] [23] [24]. Let us examine the threshold corrections evaluated in K3 × T 2 compactifications which we will generalize in this work to CHL orbifolds of K3. "
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    ABSTRACT: We study ${\cal N}=2$ compactifications of heterotic string theory on the CHL orbifold $(K3\times T^2)/\mathbb{Z}_N$ with $N= 2, 3, 5, 7$. $\mathbb{Z}_N$ acts as an involution on $K3$ together with a shift of $1/N$ along one of the circles of $T^2$. These compactifications generalize the example of the heterotic string on $K3\times T^2$ studied in the context of dualities in ${\cal N}=2$ string theories. We evaluate the new supersymmetric index for these theories and show that their expansion can written in terms of the McKay-Thompson series associated with the $\mathbb{Z}_N$ involution embedded in the Mathieu group $M_{24}$. We then evaluate the difference in one-loop threshold corrections to the non-Abelian gauge couplings with Wilson lines and show that their moduli dependence is captured by Siegel modular forms related to dyon partition functions of ${\cal N}=4$ string theories.
    Preview · Article · Oct 2015
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    • "The modular integral with measure dμ = τ −2 2 dτ 1 dτ 2 is to be performed over the SL(2; Z) fundamental domain F , and we invoke the modular-invariant regularisation prescription of [71] [72] [61] to treat the infra-red divergences ascribed to the massless string states. Henceforth, we shall not explicitly display the R.N. symbol in front of modular integrals, but we shall tacitly assume that all integrals be regularised according to [71] [72] [61]. "
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    ABSTRACT: We study one-loop quantum corrections to gauge couplings in heterotic vacua with spontaneous supersymmetry breaking. Although in non-supersymmetric constructions these corrections are not protected and are typically model dependent, we show how a universal behaviour of threshold differences, typical of supersymmetric vacua, may still persist. We formulate specific conditions on the way supersymmetry should be broken for this to occur. Our analysis implies a generalised notion of threshold universality even in the case of unbroken supersymmetry, whenever extra charged massless states appear at enhancement points in the bulk of moduli space. Several examples with universality, including non-supersymmetric chiral models in four dimensions, are presented.
    Full-text · Article · Aug 2015 · Nuclear Physics B
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    • "+ n relevant for the expansion (3.15), the summand in (3.20) can be rewritten in terms of elementary functions [19]. "
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    ABSTRACT: After integrating over supermoduli and vertex operator positions, scattering amplitudes in superstring theory at genus $h\leq 3$ are reduced to an integral of a Siegel modular function of degree $h$ on a fundamental domain of the Siegel upper half plane. A direct computation is in general unwieldy, but becomes feasible if the integrand can be expressed as a sum over images under a suitable subgroup of the Siegel modular group: if so, the integration domain can be extended to a simpler domain at the expense of keeping a single term in each orbit -- a technique known as the Rankin-Selberg method. Motivated by applications to BPS-saturated amplitudes, Angelantonj, Florakis and I have applied this technique to one-loop modular integrals where the integrand is the product of a Siegel-Narain theta function times a weakly, almost holomorphic modular form. I survey our main results, and take some steps in extending this method to genus greater than one.
    Preview · Article · Jan 2014
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