Deforming D‐brane models on orbifolds

Abstract

We review the stabilisation of complex structure moduli in Type IIA orientifolds, especially on T 6 /(Z 2 × Z 6 × ΩR) with discrete torsion, via deformations of Z 2 × Z 2 orbifold singularities. While D6-branes in SO(2N ) and USp(2N ) models always preserve gauge groups ten out of the 15 deformation moduli can be stabilised at the orbifold point.

Massless spectra and gauge couplings at one-loop on non-factorisable toroidal orientifolds

“…A step into this landscape can be provided by combining the resolution techniques studied in e.g. [69,1] with generalisations of the deformation methods initiated in [82][83][84]75].…”

“…0,0 = O(1) of the Möbius strip sector along the second two-torus T 2 (2) , all with the same sign. The gauge couplings of the left-and right-symmetric groups U Sp(2) b ×U Sp(2) c have identical tree-level values (84), identical one-loop beta function coefficients (81) and also identical threshold corrections. The contributions of the first and third two-torus, v 1 and v 3 , are twice as large and with opposite sign compared to those of the 'hidden' gauge group U Sp(4) h .…”

Deformations, moduli stabilisation and gauge couplings at one-loop

“…Solving both conditions allows to extract the unequivocal forms of the various anti-holomorphic involutions [49]. Afterwards, one can determine the fixed loci for each individual anti-holomorphic involution, which will constitute only a subset of all Lag lines on the elliptic curve [49,51,52]. Fortunately for us, the Lag lines defined as fixed loci under σ are in one-to-one correspondence with the torus one-cycles used as building blocks for global intersecting D6-brane models, as can be checked explicitly by virtue of the Weierstrass' elliptic function ℘(z).…”

“…However, in the case of Type IIA model building with fractional D6-branes on orbifolds with discrete torsion, the orbifold singularities have to be deformed rather than blown up, which forces us to consider different tools from algebraic geometry: by viewing two-tori as elliptic curves in the weighted projective space P 2 112 , a factorisable toroidal orbifold with discrete torsion can be described as a hypersurface in a weighted projective space, with its topology being a double cover of P 1 × P 1 × P 1 . Building on this hypersurface formalism first sketched in [48] for the T 6 /(Z 2 × Z 2 ) orbifold with discrete torsion and extended to its T 6 /(Z 2 × Z 2 × ΩR) and T 6 /(Z 2 × Z 6 × ΩR) orientifold versions with underlying isotropic square [49,50] or hexagonal [51,52,50] two-tori, respectively, we focus here on the so far most fertile patch in the Type IIA orientifold landscape with rigid D6-branes [2,1,53,54], the T 6 /(Z 2 × Z 6 × ΩR) orientifold with discrete torsion and one rectangular and two hexagonal underlying two-tori. In this case, the Z (1) 2 -twisted sector conceptually differs from the Z (2) 2and Z (3) 2 -twisted sectors, necessitating separate discussions for the respective deformations and making the deformations of this toroidal orbifold more intricate than the other previously discussed orbifolds with discrete torsion.…”

“…This article is organised as follows: in section 2, we briefly review the hypersurface formulation for describing local deformations of T 6 /(Z 2 × Z 2 × ΩR) singularities as discussed in [48,49,51,52] and then go on to discuss additional constraints imposed by the extra Z 3 symmetry of the T 6 /(Z 2 × Z 6 × ΩR) orientifold. Special attention will be devoted to the sLag cycles used for particle physics model building.…”