P fluxes and exotic branes

Abstract

Abstract:We consider the N = 1 superpotential generated in type-II orientifold models by non-geometric fluxes. In particular, we focus on the family of P fluxes, that are related by T-duality transformations to the S-dual of the Q flux. We determine the general rule that transforms a given flux in this family under a single T-duality transformation. This rule allows to derive a complete expression for the superpotential for both the IIA and the IIB theory for the particular case of a T 6 /[Z 2 × Z 2 ] orientifold. We then consider how these fluxes modify the generalised Bianchi identities. In particular, we derive a fully consistent set of quadratic constraints coming from the NS-NS Bianchi identities. On the other hand, the P flux Bianchi identities induce tadpoles, and we determine a set of exotic branes that can be consistently included in order to cancel them. This is achieved by determining a universal transformation rule under T-duality satisfied by all the branes in string theory.

Non-geometric fluxes & tadpole conditions for exotic branes

“…It turns out that in IIB one has to include the additional flux P a,b 1 ...b 4 . The expression for the resulting superpotential of each theory is mapped to the one of the other theory under mirror symmetry [17], and in the IIB case it coincides with the one derived in [18] as far as the P fluxes are concerned.…”

“…In particular, T-duality acts according to the rule [17] and the fluxes are such that if a is upstairs it coincides with one of the b indices, while if it is downstairs it differs from all the b indices. Using this rule, one can determine for the T 6 /[Z 2 × Z 2 ] orientifold which fluxes can be included in both the IIB and IIA theory following precisely the same method used for the NS fluxes: one first maps the flux P bc a of IIB to the dual IIA theory by performing three T-dualities, then extends this to include all the allowed IIA fluxes, and finally one maps this back to IIB.…”

“…(1.3) implies that one can map to the IIA theory the tadpole condition for the S-dual of the D7-brane by performing three T-dualities. This results in a tadpole condition for an exotic brane [17], which is an object that in the higher-dimensional theory is a generalised KKmonopole, i.e. an object well-defined only in the presence of isometries [19][20][21][22][23][24].…”

“…The potential E 8 associated to the S-dual of the D7-brane has α = −3, and the other mixed-symmetry potentials with the same value of α are of the form E 8+n,2m,n in IIB and E 8+n,2m+1,n in IIA [26]. In [17] it was shown that on the T 6 /[Z 2 × Z 2 ] orientifold the IIB potential E 8 is mapped by mirror symmetry to specific components of the IIA potential E 9,3,1 , and by adding also all the other components that can be consistently included in IIA, this maps in IIB to components of the potentials E 8,4 , E 9,2,1 and E 10,4,2 . Proceeding this way, one manages to obtain the full set of components of α = −3 mixed-symmetry potentials corresponding to exotic branes of the IIB and IIA theory that can be consistently added in the T 6 /[Z 2 × Z 2 ] orientifold model to cancel the tadpoles generated by the P fluxes.…”

“…Proceeding this way, one manages to obtain the full set of components of α = −3 mixed-symmetry potentials corresponding to exotic branes of the IIB and IIA theory that can be consistently added in the T 6 /[Z 2 × Z 2 ] orientifold model to cancel the tadpoles generated by the P fluxes. 4 The aim of this paper is to complete the analysis carried out in [17] by including all possible non-geometric fluxes and exotic branes in the model. We start by considering the S-dual of P a,b 1 ...b 4 , which following [18] we denote as Q ′a,b 1 ...b 4 .…”