We compute Yukawa couplings involving chiral matter fields in toroidal compactifications of higher dimensional super-Yang-Mills theory with magnetic fluxes. Specifically we focus on toroidal compactifications of D=10 super-Yang-Mills theory, which may be obtained as the low-energy limit of Type I, Type II or Heterotic strings. Chirality is obtained by turning on constant magnetic fluxes in each of the 2-tori. Our results are general and may as well be applied to lower D = 6, 8 dimensional field theories. We solve Dirac and Laplace equations to find out the explicit form of wavefunctions in extra dimensions. The Yukawa couplings are computed as overlap integrals of two Weyl fermions and one complex scalar over the compact dimensions. In the case of Type IIB (or Type I) string theories, the models are T-dual to (orientifolded) Type IIA with D6-branes intersecting at angles. These theories may have phenomenological relevance since particular models with SM group and three quark-lepton generations have been recently constructed. We find that the Yukawa couplings so obtained are described by Riemann ϑ-functions, which depend on the complex structure and Wilson line backgrounds. Different patterns of Yukawa textures are possible depending on the values of these backgrounds. We discuss the matching of these results with the analogous computation in models with intersecting D6-branes. Whereas in the latter case a string computation is required, in our case only field theory is needed. JHEP 8. A 3-generation MSSM-like Model 56 9. Final comments and conclusions 59 A. Dimensional Reduction of N=1 Super Yang-Mills 61 B. Fluxes and supersymmetry 64 B.1 Hermitian Yang-Mills Equations 64 B.2 D-brane interpretation 66 B.3 Mirror symmetry and string corrections 67JHEP (i, j, k) dependence of the Yukawa coupling. In a SUSY compactification the (holomorphic) superpotential should be identified with this product of ϑ-functions, whereas the rest of the factors should come from D-term normalization of three-point functions.As we said, the natural setting for studying compactifications of D=10 super Yang-Mills is string theory, since it appears naturally in the effective field theory limit of Type I, Type IIB (with D9-branes) and heterotic strings. In fact, although the mentioned compactification and computations may be performed without any reference to string theory, in the present paper we will have always in mind the string theory point of view. In particular it is well known that Type IIB theory with D9-branes compactified on a (magnetized) T 6 is T-dual to Type IIA theory with D6-branes wrapping 3-cycles on T 6 and intersecting at angles. 3 In recent years string models with intersecting D-branes have been actively pursued in order to obtain realistic models of particle physics [4,6,[18][19][20]. The main motivation for this is that in these schemes chiral fermions naturally appear at the points in compact space in which Dp-branes intersect. Furthermore, since Dp-branes may intersect multiple times, this gives a rationale for the...

We propose a bottom-up approach to the building of particle physics models from string theory. Our building blocks are Type II D-branes which we combine appropriately to reproduce desirable features of a particle theory model: 1) Chirality ; 2) Standard Model group ; 3) N = 1 or N = 0 supersymmetry ; 4) Three quark-lepton generations. We start such a program by studying configurations of D = 10, Type IIB D3-branes located at singularities. We study in detail the case of Z Z N N = 1, 0 orbifold singularities leading to the SM group or some left-right symmetric extension. In general, tadpole cancellation conditions require the presence of additional branes, e.g. D7-branes. For the N = 1 supersymmetric case the unique twist leading to three quark-lepton generations is Z Z 3 , predicting sin 2 θ W = 3/14 = 0.21. The models obtained are the simplest semirealistic string models ever built. In the non-supersymmetric case there is a three-generation model for each Z Z N , N > 4, but the Weinberg angle is in general too small. One can obtain a large class of D = 4 compact models by considering the above structure embedded into a Calabi Yau compactification. We explicitly construct examples of such compact models using Z Z 3 toroidal orbifolds and orientifolds, and discuss their properties. In these examples, global cancellation of RR charge may be achieved by adding anti-branes stuck at the fixed points, leading to models with hidden sector gravity-induced supersymmetry breaking. More general frameworks, like F-theory compactifications, allow completely N = 1 supersymmetric embeddings of our local structures, as we show in an explicit example.

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Intersecting Dp-branes often give rise to chiral fermions living on their intersections.We study the construction of four-dimensional chiral gauge theories by considering configurations of type II D(3 + n)-branes wrapped on non-trivial n-cycles on T 2n × (R 2(3−n) /Z N ), for n = 1, 2, 3. The gauge theories on the four non-compact dimensions of the brane world-volume are generically chiral and non-supersymmetric. We analyze consistency conditions (RR tadpole cancellation) for these models, and their relation to four-dimensional anomaly cancellation. Cancellation of U (1) gauge anomalies involves a Green-Schwarz mechanism mediated by RR partners of untwisted and/or twisted moduli. This class of models is of potential phenomenological interest, and we construct explicit examples of SU (3) × SU (2) × U (1) three-generation models. The models are non-supersymmetric, but the string scale may be lowered close to the weak scale so that the standard hierarchy problem is avoided. We also comment on the presence of scalar tachyons and possible ways to avoid the associated instabilities. We discuss the existence of (meta)stable configurations of D-branes on 3-cycles in (T 2 ) 3 , free of tachyons for certain ranges of the six-torus moduli.

We compute the Yukawa couplings among chiral fields in toroidal Type II compactifications with wrapping D6-branes intersecting at angles. Those models can yield realistic standard model spectrum living at the intersections. The Yukawa couplings depend both on the Kähler and open string moduli but not on the complex structure. They arise from worldsheet instanton corrections and are found to be given by products of complex Jacobi theta functions with characteristics. The Yukawa couplings for a particular intersecting brane configuration yielding the chiral spectrum of the MSSM are computed as an example. We also show how our methods can be extended to compute Yukawa couplings on certain classes of elliptically fibered CY manifolds which are mirror to complex cones over del Pezzo surfaces. We find that the Yukawa couplings in intersecting D6-brane models have a mathematical interpretation in the context of homological mirror symmetry. In particular, the computation of such Yukawa couplings is related to the construction of Fukaya's category in a generic symplectic manifold.