Universality of radiative corrections to gauge couplings for strings with spontaneously broken supersymmetry

I review recent work on computing radiative corrections to non-abelian gauge couplings in four-dimensional heterotic vacua with spontaneously broken supersymmetry. The prototype models can be considered as K3 surfaces with additional Scherk-Schwarz fluxes inducing the spontaneous $\mathcal{N}=2 \to \mathcal{N}=0$ breaking. Remarkably, although the gauge thresholds are no longer BPS protected and receive contributions also from the excitations of the RNS sector, their difference is still exactly computable and universal. Based on a talk presented at the DISCRETE 2014 conference at King's College London.Comment: 18 pages, corrected eq. 4.16 and an overall factor of

A Solution to the Decompactification Problem in Chiral Heterotic Strings

We present a solution to the decompactification problem of gauge thresholds in chiral heterotic string theories with two large extra dimensions, where supersymmetry is spontaneously broken by the Scherk-Schwarz mechanism. Whenever the Kaluza-Klein scale is much lower than the string scale, the infinite towers of heavy states contribute non-trivially to the renormalisation of gauge couplings, which typically grow linearly with the large volume of the internal space and invalidate perturbation theory. We trace the origin of the decompactification problem to properties of the six dimensional theory obtained in the infinite volume limit and show that thresholds may instead exhibit logarithmic volume dependence and we provide the conditions for this to occur. We illustrate this mechanism with explicit string constructions where the decompactification problem does not occur.Comment: 26 pages, 1 figur

On the Rankin-Selberg method for higher genus string amplitudes

Closed string amplitudes at genus $h\leq 3$ are given by integrals of Siegel modular functions on a fundamental domain of the Siegel upper half-plane. When the integrand is of rapid decay near the cusps, the integral can be computed by the Rankin-Selberg method, which consists of inserting an Eisenstein series $E_h(s)$ in the integrand, computing the integral by the orbit method, and finally extracting the residue at a suitable value of $s$. String amplitudes, however, typically involve integrands with polynomial or even exponential growth at the cusps, and a renormalization scheme is required to treat infrared divergences. Generalizing Zagier's extension of the Rankin-Selberg method at genus one, we develop the Rankin-Selberg method for Siegel modular functions of degree 2 and 3 with polynomial growth near the cusps. In particular, we show that the renormalized modular integral of the Siegel-Narain partition function of an even self-dual lattice of signature $(d,d)$ is proportional to a residue of the Langlands-Eisenstein series attached to the $h$-th antisymmetric tensor representation of the T-duality group $O(d,d,Z)$.Comment: 53 pages, 3 figures; v2: various clarifications and cosmetic changes, new appendix B on the Rankin-Selberg transform of the lattice partition function in arbitrary degree, small correction to Figure

N=2⋆\mathcal{N}=2^\star from Topological Amplitudes in String Theory

In this paper, we explicitly construct string theory backgrounds that realise the so-called $\mathcal N=2^\star$ gauge theory. We prove the consistency of our models by calculating their partition function and obtaining the correct gauge theory spectrum. We further provide arguments in favour of the universality of our construction which covers a wide class of models all of which engineer the same gauge theory. We reproduce the corresponding Nekrasov partition function once the $\Omega$-deformation is included and the appropriate field theory limit taken. This is achieved by calculating the topological amplitudes $F_g$ in the string models. In addition to heterotic and type II constructions, we also realise the mass deformation in type I theory, thus leading to a natural way of uplifting the result to the instanton sector.Comment: 27 page

Higher Spins in Hyperspace

We consider the Sp(2n) invariant formulation of higher spin fields on flat and curved backgrounds of constant curvature.In this formulation an infinite number of higher spin fields are packed into single scalar and spinor master fields (hyperfields) propagating on extended spaces, to be called hyperspaces, parametrized by tensorial coordinates.We show that the free field equations on flat and AdS-like hyperspaces are related to each other by a generalized conformal transformation of the scalar and spinor master fields. We compute the four--point functions on a flat hyperspace for both scalar and spinor master fields, thus extending the two-- and three--point function results of arXiv:hep-th/0312244. Then using the generalized conformal transformation we derive two--, three-- and four--point functions on AdS--like hyperspace from the corresponding correlators on the flat hyperspace.Comment: 23 pages, typos corrected, references added. Published versio

Marginal Deformations of Vacua with Massive boson-fermion Degeneracy Symmetry

Two-dimensional string vacua with Massive Spectrum boson-fermion Degeneracy Symmetry (MSDS) are explicitly constructed in Type II and Heterotic superstring theories. The study of their moduli space indicates the existence of large marginal deformations that connect continuously the initial d=2, MSDS vacua to higher-dimensional conventional superstring vacua, where spacetime supersymmetry is spontaneously broken by geometrical fluxes. We find that the maximally symmetric, d=2, Type II MSDS-vacuum, is in correspondence with the maximal, N=8, d=4, gauged supergravity, where the supergravity gauging is induced by the fluxes. This correspondence is extended to less symmetric cases where the initial MSDS symmetry is reduced by orbifolds. We also exhibit and analyse thermal interpretations of some Euclidean versions of the models and identify classes of MSDS vacua that remain tachyon-free under arbitrary marginal deformations about the extended symmetry point. The connection between the two-dimensional MSDS vacua and the resulting four-dimensional effective supergravity theories arises naturally within the context of an adiabatic cosmological evolution, where the very early Universe is conjectured to be described by an MSDS-vacuum, while at late cosmological times it is described by an effective N=1 supergravity theory with spontaneously broken supersymmetry

One-Loop BPS amplitudes as BPS-state sums

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.Comment: 42 pages; v4: a few misprints correcte

Rankin-Selberg methods for closed strings on orbifolds

In recent work we have developed a new unfolding method for computing one-loop modular integrals in string theory involving the Narain partition function and, possibly, a weak almost holomorphic elliptic genus. Unlike the traditional approach, the Narain lattice does not play any role in the unfolding procedure, T-duality is kept manifest at all steps, a choice of Weyl chamber is not required and the analytic structure of the amplitude is transparent. In the present paper, we generalise this procedure to the case of Abelian Z_N orbifolds, where the integrand decomposes into a sum of orbifold blocks that can be organised into orbits of the Hecke congruence subgroup {\Gamma}_0(N). As a result, the original modular integral reduces to an integral over the fundamental domain of {\Gamma}_0(N), which we then evaluate by extending our previous techniques. Our method is applicable, for instance, to the evaluation of one-loop corrections to BPS-saturated couplings in the low energy effective action of closed string models, of quantum corrections to the K\"ahler metric and, in principle, of the free-energy of superstring vacua.Comment: 47 pages, 1 figur

Universality of Gauge Thresholds in Non-Supersymmetric Heterotic Vacua

We compute one-loop threshold corrections to non-abelian gauge couplings in four-dimensional heterotic vacua with spontaneously broken $\cal N = 2 \to \cal N = 0$ supersymmetry, obtained as Scherk-Schwarz reductions of six-dimensional K3 compactifications. As expected, the gauge thresholds are no-longer BPS protected, and receive contributions also from the excitations of the RNS sector. Remarkably, the difference of thresholds for non-abelian gauge couplings is BPS saturated and exhibits a universal behaviour independently of the orbifold realisation of K3. Moreover, the thresholds and their difference develop infra-red logarithmic singularities whenever charged BPS-like states, originating from the twisted RNS sector, become massless at special loci in the classical moduli space.Comment: 12 pages, corrected eqs. 3.19, 3.20 and 3.23 and an overall factor of 2 in all threshold