Using a Kaluza-Klein reduction of the Dirac-Born-Infeld and Chern-Simons action we compute the four dimensional N = 1 effective action for the massless modes of a D7brane which is wrapped on a four-cycle of a compact Calabi-Yau orientifold. We do not consider a specific orientifold but instead determine the Kähler potential, the gauge kinetic functions and the scalar potential in terms of geometrical data of a generic orientifold and its wrapped four-cycle. In particular we derive the couplings of the D-brane excitations to the bulk moduli of the orientifold as they are important for the study of soft supersymmetry breaking terms. We relate the resulting Kähler geometry to the N = 1 special geometry of Lerche, Mayr and Warner. Finally we comment on the structure of the D-term which is induced by a Green-Schwarz term in the Chern-Simons action.This analysis begins with a review of compactifying type IIB string theory on a Calabi-Yau orientifold Y in the presence of O3/O7 orientifold planes as has been considered in refs. [44,45,17,20,21,41]. Usually in order to get a low energy effective description of theories originating from type IIB string theory, a natural starting point is the chiral ten dimensional N = 2 type IIB supergravity [46,47]. In our case, however, it is more convenient to start with the democratic type IIB supergravity action [42], because it facilitates the derivation of the whole action containing both the bulk and brane fields. 3 This type IIB supergravity action is comprised of all RR form fields of type IIB string theory at the cost of introducing additional self-duality constraints. Type IIB orientifold spectrum with O3/O7 planesThe massless bosonic spectrum of type IIB string theory in ten dimensions consists of fields from the NS-NS sector and the RR sector. The former sector gives rise to the ten dimensional metric g 10 , the anti-symmetric two tensor B and the dilaton φ. The RR sector contributes all even dimensional anti-symmetric tensors, i.e. the form fields C (0) , C (2) , C (4) , C (6) , and C (8) . The field strengths of these form fields, as they appear 3 The advantage of using the democratic action has also been observed in ref. [48].

In this paper we compute the N = 1 effective low energy action for a stack of N space-time filling D3-branes in generic type IIB Calabi-Yau orientifolds with non-trivial background fluxes by reducing the Dirac-Born-Infeld and Chern-Simons actions. Specifically, we determine the Kähler potential for the excitations of the D-brane including their couplings to all bulk moduli fields. In the effective theory, N = 1 supergravity is spontaneously broken by the presence of fluxes and we compute the induced soft supersymmetry breaking terms. We find an interesting structure in the resulting soft terms with generically universal soft scalar masses.

Many N = (2, 2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N = (2, 2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories -recently computed via localization by Benini et al. and Doroud et al. -yields the exact Kähler potential on the quantum Kähler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kähler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in α ′ . We compute these quantities for the quintic and for Rødland's Pfaffian Calabi-Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P 7 , recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.1 The work of Böhm [24, 25] provides a promising proposal, but we have been unable to implement it well enough to produce a mirror for this example.2 Local special Kähler manifolds are also often called projective special Kähler manifolds and are distinct from special Kähler manifolds -see, e.g., [32].3 This geometric structure gives rise to a Hodge filtration F 3 ⊂ F 2 ⊂ F 1 ⊂ F 0 of weight 3, with F 3 ≃ L, F 2 ≃ V, F 1 ≃ L ⊥ , and F 0 ≃ V C , where L ⊥ is the subspace of V C perpendicular to L with respect to the symplectic pairing ·, · -see, e.g., [33,34].The last expression involves the periods of Ω,B J Ω(ξ) , I, J = 0, . . . , h 2,1 , (2.6) with respect to a canonical symplectic basis (A I , B J ) of H 3 (Y ξ , Z) satisfying A I , B J = δ I J , A I , A J = B I , B J = 0 . (2.7) 2.3 The quantum Kähler moduli space M KählerThe main player of this note is the quantum-corrected Kähler moduli space M Kähler of a Calabi-Yau threefold Y , which is defined as the corresponding space of chiral-antichiral and antichiralchiral moduli of the underlying SCFT. It is also a local special Kähler manifold, parametrizing 15 We thank C. Vafa for pointing this out to us. 16 Note that in spacetime, some of these fields become quaternionic and are subject to further gs corrections.

Using a Kaluza-Klein reduction of the fermionic part of the D-brane action we compute D- and F-terms of the N=1 effective action for generic Calabi-Yau orientifold compactifications in the presence of a space-time filling D7-brane. We include non-trivial background fluxes for the D7-brane U(1) field strength on the internal four-cycle wrapped by the brane. First the four-dimensional fermionic spectrum arising from the D7-brane is derived and then the D- and F-terms are obtained by computing appropriate couplings of these fermionic fields. For specific examples we examine the resulting flux-induced scalar potentials and comment on their relevance in string cosmology.Comment: 1+44 pages, Latex; minor change

For compact Calabi-Yau geometries with D5-branes we study N = 1 effective super- nates. By evaluating these superpotentials at their critical points we reproduce the domain wall tensions that have recently appeared in the literature. Finally we extract orbifold disk invariants from the superpotentials, which, up to overall numerical normalizations, correspond to orbifold disk Gromov-Witten invariants in the mirror geometry.