Perturbative higher-derivative terms in $$ \mathcal{N} $$ = 6 asymmetric orbifolds

Abstract

We analyse the perturbative four-point amplitudes in the simplest string theory examples of T-fold backgrounds, which enjoy $$ \mathcal{N} $$ N = 6 supersymmetries in four dimensions. There are two theories defined as asymmetric orbifolds of order 2 and 3, respectively. The perturbative spectrum and the one-loop four-point amplitudes are shown to be invariant under an arithmetic symplectic group defined over the Hurwitz (respectively Eisenstein) quaternions. The supersymmetry constraints on the low energy effective couplings are determined and we conjecture the U-duality group of the theory and the non-perturbative completion of the six-derivative coupling as a rank one theta series. We discuss the non-perturbative spectrum of BPS states in the light of our analysis.

Freely acting orbifolds of type IIB string theory on T5

“…Orbifolds whose partition functions give non-integer degeneracies should be excluded. As shown in [34], this only allows the cases Z 2 and Z 3 , which precisely agrees with the T-duality approach that we use to construct our orbifold models, as we saw in the example of 2.5. Now, let us present an example of a Z 2 orbifold, breaking 8 left-moving supersymmetries, with twist vectors ũ " `0, 0, 1 2 , 1 2 ˘and u " p0, 0, 0, 0q, or in terms of the mass parameters m " pπ, 0, 0, 0q.…”

“…For a sensible theory this degeneracy must be an integer number. As shown in [34], this additional integrality constraint can exclude a seemingly consistent model.…”

“…Even in this class, there are surprises. As it was recently shown in [34], asymmetric orbifolds of the type pT 4 ˆS1 q{Z p with N " 6 (24 supersymmetries) only exist for p " 2, 3 but lead to problems with the partition function for p " 4, 6. These partition functions are modular invariant, but have non-integral coefficients in the q q-expansion in the twisted sectors and hence appear to be unphysical.…”

“…This choice of twist vectors also implies that the left-moving momenta are projected out, while there exists an invariant sublattice of right-moving momenta Λ R Ă Γ 4,4 contributing to the partition function Z T 4 rk, ls (for k or l ‰ 0). Following the examples of [26,34], we take the right-moving momenta to lie in the lattice D 4 for p " 2 or A 2 ' A 2 for p " 3. Once we have the twist vectors and the invariant sublattice, we can calculate the degeneracy of the ground state of the twisted sectors, which is given by the number of fixed points (3.27) divided by the volume of the invariant sublattice [21,22].…”

“…It has the same massless spectrum as the p " 2 orbifold and hence the same moduli space, dictated by N " 6 supersymmetry. Asymmetric orbifolds with N " 6, in four dimensions, have also been constructed in [69] and more recently in [34,70,71].…”