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Topics in String Phenomenology Topics in String Phenomenology Angel Uranga Abstract Notes taken by Cristina Timirgaziu of lectures by Angel Uranga in June 2009 at the Galileo Galilei Institute School "New Perspectives in String Theory". Topics include in- tersecting D-branes models, magnetized D-branes and an introduction to F-theory phe- nomenology. 1 Introduction String phenomenology deals with building string models of particle physics. The goal is to find a generic scenario or even predictions at TeV scale. Topics in string phenomenology in- clude heterotic strings model building both on smooth (Calabi-Yau) and singular (orbifold) manifolds and model building in Type II strings. The later has several branches: inter- secting D-branes (type IIA strings), magnetized D-branes (type IIB), as well as F-theory models. Common problems in string phenomenology include the issue of supersymmetry break- ing, moduli stabilization, flux compactifications, non perturbative effects and applications to cosmology, in particular string inflation. These lectures are concerned with model building using D-branes. 2 Intersecting D-branes Model building using intersecting D-branes is a very active field in string phenomenology and many reviews are already present in the littereature, including [1]- [5]. For a review of recent progress see [6]. 2.1 Basics of intersecting D-branes In the weak coupling limit D-branes can be well described in the probe approximation as hyperplanes where open strings can end. A number of N overlapping D-branes generates a U(N) gauge theory with 16 supercharges, which corresponds to = 4 supersymmetry in four dimensions. A Dp-brane, extending in the spacial directionsN x0::: xp, breaks half the supercharges and the surving supersymmetry is given by Q = LQL + RQR, where QL 0 1 p 1 and QR are left and right moving spacetime supercharges and L = Γ Γ ::::Γ R. The worldvolume dynamics of a Dp-brane is described by the Born-Infeld and Wess Zumino actions Z Z p+1 −φp 0 S = Tp d ξ e det(G + 2πα F ) + Cp+1; − − where Tp is the tension of the brane, φ is the dilaton, G - the induced metric on the D- brane, F - the field strength of the world volume gauge field and Cp+1 is the p + 1 form that couples to the D-brane. Dp-branes give rise to non-abelian gauge interactions and also to four dimensional chiral fermions provided the = 4 supersymmetry is broken to = 1 at the most. In order to obtain 4d chirality theN six dimensional internal parity mustN be broken, since the 16 supercharges in ten dimensions split as follows under the breaking of the SO(10) symmetry SO(10) SO(6) SO(4); 16 (4; 2L) + (4¯; 2R): ! × ! 1Γi denote de Dirac matrices. 2 Intersecting brane worlds S49 D62 D61 D62 D62 θ2 θ θ 3 M 4 1 D61 D61 R2 R2 R2 Figure 8. A more concrete picture of the configuration of two D6-branes intersecting over a 4D subspace of their volumes. Figure 1: Intersecting D6-branes. The appearanceConsider of two chirality D-branes can intersecting be understood as in from fig. the 1, where fact that a preferred the geometry orientation of the has two been D-branes introducesdefined. The a preferred preferred orientation breaksin the transverse the six dimensional 6D space, parity. namely For by phenomenological considering the relativepurposes rotation we of theneed second to consider D6-brane two stacks with respect of N1 and to theN2 first.type This IIA D6-branes also explains overlapping why one over should choosea 4d subspaceconfigurations and intersecting of D6-branes. at angles For example,θ1, θ2 and twoθ3 in sets three of D5-branes 2-planes. The intersecting spectrum of over 4D doopen not leadstrings to 4Din this chiral configuration fermions, since contains they several do not sectors have enough dimensions to define an orientation in the transverse 6D space. 1-1 strings generate a U(N1) SYM theory in 7 dimensions with 16 supercharges Configurations• of intersecting D6-branes are the topic of coming sections. Indeed, we will describe in section2-2 strings 6.2 that similarly intersecting lead to D-brane a U(N2) modelsSYM theory are dual in 7 dimensions to other configurations with 16 supercharges of D-branes, or• even of other objects in string/M-theory, leading also to chirality. Hence ¯ intersecting D-branes1-2 strings are generate a good massless representative chiral fermions of the in behaviour the (N1; N of2) representation string theory in in 4 di- • mensions2 (since these states are located at the intersection of the two stacks), as configurations with low enough supersymmetry to allow for chirality. well as other states, which could potentially be light scalars Before entering the details, it will be useful to mention the results of the spectrum for this configuration.2-1 strings The open generate string the spectrum antiparticles in a configuration of the states in of sector two stacks 1-2. of n1 and n2 coincident D6-branes• in flat 10D intersecting over a 4D subspace of their volumes consists of 3 three open stringThe sectors: light scalars in the sector 1-2 exhibit the following masses 6161. Strings stretching between D61-branes1 provide U(n1) gauge bosons, three real α0m2 = (θ + θ + θ ) adjoint scalars and fermion superpartners, propagating2 1 2 over3 the seven-dimensional (7D) world volume of the D61-branes. 0 2 1 α m = ( θ1 + θ2 + θ3) 6262. Similarly, strings stretching between D622-branes− provide U(n2) gauge bosons, three 0 2 1 real adjoint scalars and fermion superpartners,α m = propagating(θ1 θ2 + θ3 over) the D62-brane 7D world volume. 2 − 0 2 1 6 6 + 6 6 . Strings stretching betweenα m = both(θ kinds1 + θ2 ofθ D6-brane3) lead to a 4D chiral (1) 1 2 2 1 2 − fermion, transforming in the representation (n , n ) of U(n ) U(n ), and localized at 1 2 1 × 2 the intersection.2These chiral The fermions chirality leave of in the 4d because,fermion due is encodedto the mixed in boundary the orientation conditions defined (Neuman-Dirichlet) by the intersection;of the open this strings will bestretched implicitly between taken the two into stacks account of D-branes, in our discussion. the zero modes of a Ramond fermion, corresponding to the 6d transverse space are not present. 3The oscillator modes in the expansion of the open strings will be shifted by θ as in bα and i −1=2+θi 2.4. Openthis string change spectrum will show for in theintersecting mass of the D6-branes states through contribution to the zero point± energy. In this section, we carry out the computation of the spectrum3 of open strings in the configuration of two stacks of intersecting D6-branes [13]. In particular, we explicitly show the appearance of 4D chiral fermions from the sector of open strings stretching between the different D6-brane stacks. The key point in getting chiral fermions is that the non-trivial angle between the branes removes fermion zero modes in the R sector, and leads to a smaller Clifford algebra. As discussed above, the spectrum of states for open strings stretched between branes in the same stack is exactly as in section 2.2. It yields a U(na) vector multiplet in the 7D world volume of the ath D6-brane stack. We thus centre in the computation of the spectrum of states for open strings stretched between two stacks a and b. The open string boundary conditions for the coordinates along S52 A M Uranga (a) (b) D61 D62 D6 Figure 9. Recombination of two intersecting D6-branes into a single smooth one, corresponding to a vev forFigure a scalar at2: the Recombination intersection. of two branes. recombination is very explicit. It is given by deforming two intersecting 2-planes, described Genericallyby the complex the curvescalaruv spectrum0, to the laid smooth out 2-cycle in (1)uv contains!, with no! corresponding massless states, to the vev in which = = case thereof the is scalar no at supersymmetry the intersection. preserved in the theory. The scalars in the sector 1-2 can also beThe tachyons configuration indicating with tachyonic the instability scalars of corresponds the brane to configuration. situations where In this this case recombination is triggered dynamically. Namely, the recombination process corresponds recombination of the branes will lead to a bound BPS state displaying the phenomenon to condensation of the tachyon at the intersection. It is interesting to point out that in of wallthe crossing degenerate (see case figure where 2). the The intersecting initial configuration brane system becomes is described a brane–antibrane by a scalar system potential, parametrized(e.g., θ byθ a scalar0,θ φ1),with the tachyonscharges 1are and mapped1 under to the the well-studied gauge group tachyon generated of brane– by the 1 2 3 − two D-branes,antibrane=U systems.(1)= U=(1) The situation where all light scalars have positive squared masses × 2 2 corresponds to a non-supersymmetricVD intersection,= ( φ + whichξ) ; is nevertheless dynamically stable against recombination. Namely, the recombinedj j 3-cycle has volume larger than the sum of wherethe volumes Fayet-Illiopoulos of the intersecting term 3-cycles. is related to the intersection angles ξ = θ1 +θ2 +θ3. Three situationsIndeed, can present the different regimes of dynamics of scalars at intersections have a one-to-one mapping with the different relations between the volumes of intersecting and recombined the3-cycles massive [15]. case, Namely,ξ > the0 conditions : the minimum to have or of notVD haveis at tachyons< φ > are= 0 related and theto theU angle(1) U(1) • gaugecriterion group [16] determining is unbroken which particular 3-cycle has smaller volume.
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