TOPICS IN LARGE VOLUME SWISS-CHEESE COMPACTIFICATION GEOMETRIES

Abstract

Motivated by studies in string cosmology, I started my research focussing on a system-atic study of large volume type JIB string compactifications that addressed issues in string cosmology like obtaining a metastable non-supersymmetric dS minimum with-out adding anti-D3 brane and obtaining slow roll inflation with the required number of 60 e-foldings along with non-trivial non-Gaussianities and gravitational waves. In addition to the complex-structure moduli dependent and the non-perturbative instan-ton generated superpoteritial (as in "KKLT-like scenarios") as well as the inclusion of perturbative a'-corrections (as in "Large Volume Scenarios"), we also included "non-perturbative" Uzi-corrections (world-sheet instanton corrections) in the Kahler potential written out utilizing the (subgroup of) SL(2, Z) symmetry of the underlying type IIB theory. I elaborate below on the abovementioned issues: Although the KKLT-type as well as LVS models could realize dS minimum fix-ing all moduli as well, but with the need of some uplifting mechanism for uplifting AdS minimum to dS minimum to be introduced by hand such as adding anti-D3 brane. In the mid of 2007 we showed the possibility of realizing dS solution in a more natural way in the context of type IIB "Swiss-Cheese" orientifold compactifi-cation in the LVS limit. What makes our setup different from the earlier LVS setups studied is the inclusion of non-perturbative world-sheet instanton corrections to the 11 Kahler potential and the modular completions of the Kahler potential and the su-perpotential. Using these (non-)perturbative a'-corrections to Kahler potential and non-perturbative instanton corrections to superpotential, we discussed the possibility of realizing a non-supersymmetric dS minimum in the LVS limit of the internal man-ifold and there was no need for adding D3-brane la KKLT) by hand. Moving one more step towards realistic stringy cosmological model as a test of string theory, af-ter realizing non-supersymmetric dS minimum, we addressed the issue of embedding slow roll (axionic) inflationary scenarios in our LVS Swiss-Cheese orientifold setup without the "77-problem" . After investigating the eigenvalues and eigenvectors of the hessian of N = 1 scalar potential, we showed that a linear combination of NS-NS axions provides a flat direction for the inflaton field to inflate from a saddle point to some nearest dS minimum. The Gopakumar-Vafa invariants appearing through the non-perturbative world-sheet instanton contributions to the Kahler potential play a very crucial role in solving the /7-problem and we argued that by an appropriate choice of the holomorphic, isometric involution involved in the orientifolding of the Swiss-Cheese Calabi-Yau, one can realize the number of e-foldings Ne ti 60 which is favored by cosmological observations. Motivated by the ongoing PLANCK satellite experiments in cosmology, we turned ourselves towards studies related to some more involved issues in cosmology such as realizing C9(1) non-linearity parameter fNL as a signature of non-Gaussianities and finite/detectable tensor-to-scalar ratio as some signals of gravitational waves. Using 6N-formalism in the context of curvature perturbations also applicable to slow roll violating scenarios as developed by Yokoyama et al, we realized non-Gaussianities parameter fNL ti 0(10') for the slow roll case and fNL ti (9(1) for beyond slow-roll case in our LVS Swiss-Cheese orientifold setup. Further using general (not specific to string theory) considerations of Hamilton-Jacobi formalism and some algebraic geometric inputs, after imposing the freezing out of the curvature perturbations at super horizon scales we showed the possibility of realizing the tensor-to-scalar ratio r ti 0(10-3) along with loss of scale invariance lying within the experimental bounds 111 in R 11 < 0.05. Also we made some interesting observations pertaining to the possibility of inflaton field being a Cold Dark Matter (CDM) candidate in some corner of the moduli space. After working on the aforementioned interesting issues in the context of realis-tic model building in string cosmology for almost two years, in the summer of 2009 we switched gears and started to explore LVS string phenomenology. In studying cosmology and phenomenology within a single string theoretic setup, there has been a long-standing tension between LVS cosmology and LVS phenomenology due to a hierarchy in energy scales involved on both sides, which is equivalent to the hierar-chial requirement for the volume of the Calabi-Yau V -- 106 for LVS cosmology and V 1016 for LVS phenomenology. This tension is reflected equivalently as realizing a 1012 GeV gravitino for cosmology and a TeV gravitino for particle phenomenology. For addressing this issue, we incorporated the effect of a single mobile spacetime-filling D3-brane and stacks of D7-branes wrapping the "big" divisor of the aforementioned Swiss-Cheese Calabi-Yau along with the magnetized fluxes. One also needs to include the "geometric" Kahler potential in the moduli space Kahler potential; using NLSM techniques and the toric data for the Swiss-Cheese, we calculated the same in. LVS limit in terms of derivatives Siegal-theta functions and have concluded that the same are subdominant (in the LVS limit) as compared to the tree level and (non-) per-turbative contributions. The motion of the spacetime-filling D3-brane dictates the gravitino mass through the holomorphic prefactor appearing in the non-perturbative superpotential. We proposed a possible geometric resolution by correlating the 1012 GeV and TeV gravitino masses for Calabi-Yau volume stabilized to V -- 106 with the motion of the spacetime-filling D3-brane from a non-singular elliptic curve to another (non-singular elliptic curve) in the same string theoretic setup. As opposed to the previous LVS models, we considered the wrapping of D7-branes along the big divisor and showed that after constructing local odd involutive one-form, one can realize a competing contribution from the Wilson line modulus implying the possibility of cancelation with the big divisor volume and hence the possibility of iv realizing gym 0(1). This way we proposed an alternate possibility of supporting SM on D7-branes wrapping the big divisor. Supersymmetry breaking is an important issue in realistic model building in string phenomenology. In our abovementioned D3/ D7 LVS type IIB Swiss-Cheese orien-tifold setup, we realized 0(TeV) gravitino and gaugino masses and calculated several other soft terms. We observed a universality in the D3-brane position moduli masses. The physical ft, physical Yukawas and physical Bμ-terms were found to be universal for D3-brane position moduli; the same was not for Wilson line modulus and their mixing terms with D3-brane position moduli. The trilinear A-terms were observed to be universal. The matter fields and other soft-terms corresponding to the D3-brane position moduli were seen to be heavier than gravitino. Further the un-normalized physical /2-parameters for the D3-brane position moduli were found to be O(TeV) which is phenomenologically favored for the electroweak symmetry breaking. Fol-lowed by this, we studied a one-loop RC running to the electroweak scale of the gaugino (as well as slepton and squark) masses in mSugra like models and we found that the one-loop integral related to the running of the square of the product of gauge coupling and the gaugino mass, is 3(TeV)2 which agrees well e.g. with a value of (TeV)2 at an mSUGRA-point on the "SPS la slope". Moreover, we realized very small (negligible) physical Yukawa couplings in our setup and finally we argued that the D3-brane position moduli could be identified with two Higges while Wilson line moduli could be identified with squarks and sleptons of (first two families of) some MSSM like-models. Further we showed the possibility of realizing ferrnions masses of first two generations along with order eV neutrino masses for Calabi Yau volume V ti (105 — 106) and D3-instanton number rts = 2. In the context of proton decay, we argued the absence of SUSY GUT-type dimension-five operators and estimate an upper bound on the proton lifetime to be around 1061 years from a SUSY GUT-type dimension-six operator. Apart from the issues related to string cosmology and string phenomenology, we worked on some other interesting issues on implications of moduli stabilization via inclusion of fluxes in type IIB compactification scenarios. In this context, we discussed the issues of existence of area codes, "inverse-problem" related to non-supersymmetric black hole attractors and existence of fake superpotentials. We argued the existence of "area codes" where complex structure moduli and dilaton modulus can be stabilized at the points near to as well as away from the two singular conifold loci of the Swiss-Cheese Calabi-Yau in WCP4 [1, 1, 1, 6, 9] for the same values of the NS-NS and RR fluxes. We have explicitly solved the inverse problem which is to calculate the electric and magnetic charges of the extremal black hole potential, given the extremum values of the moduli.