Flux Vacua Statistics for Two-Parameter Calabi-Yau's

We study the number of flux vacua for type IIB string theory on an orientifold of the Calabi-Yau expressed as a hypersurface in WCP^4[1,1,2,2,6] by evaluating a suitable integral over the complex-structure moduli space as per the conjecture of Douglas and Ashok. We show that away from the singular conifold locus, one gets a power law, and that the (neighborhood) of the conifold locus indeed acts as an attractor in the (complex structure) moduli space. We also study (non)supersymmetric solutions near the conifold locus.In the process, we evaluate the periods near the conifold locus. We also study (non)supersymmetric solutions near the conifold locus, and show that supersymmetric solutions near the conifold locus do not support fluxes.Comment: 17 pages, LaTex; Part of AN's BTech end-of-undergraduate-freshman(=first)-year summer project at IIT Roorkee; v3: a reference and clarifying text and equations added, a numerical error correcte

New Attractors and Area Codes

In this note we give multiple examples of the recently proposed New Attractors describing supersymmetric flux vacua and non-supersymmetric extremal black holes in IIB string theory. Examples of non-supersymmetric extremal black hole attractors arise on a hypersurface in $WP^{4}_{1,1,1,1,2}$. For flux vacua on the orientifold of the same hypersurface existence of multiple basins of attraction is established. It is explained that certain fluxes may give rise to multiple supersymmetric flux vacua in a finite region on moduli space, say at the Landau-Ginzburg point and close to conifold point. This suggests the existence of multiple basins for flux vacua and domain walls in the landscape for a fixed flux and at interior points in moduli space.Comment: 16 pages, harvmac. v2: acknowledgement update

On the Existence of Non-Supersymmetric Black Hole Attractors for Two-Parameter Calabi-Yau's and Attractor Equations

We look for possible nonsupersymmetric black hole attractor solutions for type II compactification on (the mirror of) CY_3(2,128) expressed as a degree-12 hypersurface in WCP^4[1,1,2,2,6]. In the process, (a) for points away from the conifold locus, we show that the attractors could be connected to an elliptic curve fibered over C^8 which may also be "arithmetic" (in some cases, it is possible to interpret the extremization conditions as an endomorphism involving complex multiplication of an arithmetic elliptic curve), and (b) for points near the conifold locus, we show that the attractors correspond to a version of A_1-singularity in the space Image(Z^6-->R^2/Z_2(embedded in R^3)) fibered over the complex structure moduli space. The potential can be thought of as a real (integer) projection in a suitable coordinate patch of the Veronese map: CP^5-->CP^{20}, fibered over the complex structure moduli space. We also discuss application of the equivalent Kallosh's attractor equations for nonsupersymmetric attractors and show that (a) for points away from the conifold locus, the attractor equations demand that the attractor solutions be independent of one of the two complex structure moduli, and (b) for points near the conifold locus, the attractor equations imply switching off of one of the six components of the fluxes. Both these features are more obvious using the atractor equations than the extremization of the black hole potential.Comment: 1+29 pages, LaTeX; to appear in Fortsch. der Physi

Enumerating Flux Vacua with Enhanced Symmetries

We study properties of flux vacua in type IIB string theory in several simple but illustrative models. We initiate the study of the relative frequencies of vacua with vanishing superpotential W=0 and with certain discrete symmetries. For the models we investigate we also compute the overall rate of growth of the number of vacua as a function of the D3-brane charge associated to the fluxes, and the distribution of vacua on the moduli space. The latter two questions can also be addressed by the statistical theory developed by Ashok, Denef and Douglas, and our results are in good agreement with their predictions. Analysis of the first two questions requires methods which are more number-theoretic in nature. We develop some elementary techniques of this type, which are based on arithmetic properties of the periods of the compactification geometry at the points in moduli space where the flux vacua are located.Comment: 83 pages, 8 figures, harvmac. v2: references added, typos fixed, brief discussion of complex conjugation in sec. 5 adde

The Non-BPS Black Hole Attractor Equation

We study the attractor mechanism for extremal non-BPS black holes with an infinite throat near horizon geometry, developing, as we do so, a physical argument as to why such a mechanism does not exist in non-extremal cases. We present a detailed derivation of the non-supersymmetric attractor equation. This equation defines the stabilization of moduli near the black hole horizon: the fixed moduli take values specified by electric and magnetic charges corresponding to the fluxes in a Calabi Yau compactification of string theory. They also define the so-called double-extremal solutions. In some examples, studied previously by Tripathy and Trivedi, we solve the equation and show that the moduli are fixed at values which may also be derived from the critical points of the black hole potential.Comment: 32 Pages, 2 Figures, LaTeX; v2: typos corrected, references adde

Warped Vacuum Statistics

We consider the effect of warping on the distribution of type IIB flux vacua constructed with Calabi-Yau orientifolds. We derive an analytical form of the distribution that incorporates warping and find close agreement with the results of a Monte Carlo enumeration of vacua. Compared with calculations that neglect warping, we find that for any finite volume compactification, the density of vacua is highly diluted in close proximity to the conifold point, with a steep drop-off within a critical distance.Comment: 30 pages, 2 figure

Noncommutative Field Theories and (Super)String Field Theories

In this lecture notes we explain and discuss some ideas concerning noncommutative geometry in general, as well as noncommutative field theories and string field theories. We consider noncommutative quantum field theories emphasizing an issue of their renormalizability and the UV/IR mixing. Sen's conjectures on open string tachyon condensation and their application to the D-brane physics have led to wide investigations of the covariant string field theory proposed by Witten about 15 years ago. We review main ingredients of cubic (super)string field theories using various formulations: functional, operator, conformal and the half string formalisms. The main technical tools that are used to study conjectured D-brane decay into closed string vacuum through the tachyon condensation are presented. We describe also methods which are used to study the cubic open string field theory around the tachyon vacuum: construction of the sliver state, ``comma'' and matrix representations of vertices.Comment: 160 pages, LaTeX, 29 EPS figures. Lectures given by I.Ya.Aref'eva at the Swieca Summer School, Brazil, January 2001; Summer School in Modern Mathematical Physics, Sokobanja, Yugoslavia, August 2001; Max Born Symposium, Karpacz, Poland, September, 2001; Workshop "Noncommutative Geometry, Strings and Renormalization", Leipzig, Germany, September 2001. Typos corrected, references adde

Anthropic Distribution for Cosmological Constant and Primordial Density Perturbations

The anthropic principle has been proposed as an explanation for the observed value of the cosmological constant. Here we revisit this proposal by allowing for variation between universes in the amplitude of the scale-invariant primordial cosmological density perturbations. We derive a priori probability distributions for this amplitude from toy inflationary models in which the parameter of the inflaton potential is smoothly distributed over possible universes. We find that for such probability distributions, the likelihood that we live in a typical, anthropically-allowed universe is generally quite small.Comment: 12 pages, 2 tables. v3: Replaced to match published version (minor corrections of form

Moduli Stabilization, Large-Volume dS Minimum Without anti-D3-Branes, (Non-)Supersymmetric Black Hole Attractors and Two-Parameter Swiss Cheese Calabi-Yau's

We consider issues of moduli stabilization and "area codes" for type II flux compactifications, and the "Inverse Problem" and "Fake Superpotentials" for extremal (non)supersymmetric black holes in type II compactifications on (orientifold of) a compact two-parameter Calabi-Yau expressed as a degree-18 hypersurface in WCP^4[1,1,1,6,9] which has multiple singular loci in its moduli space. We argue the existence of extended "area codes" [1] wherein for the same set of large NS-NS and RR fluxes, one can stabilize all the complex structure moduli and the axion-dilaton modulus (to different sets of values) for points in the moduli space away as well as near the different singular conifold loci leading to the existence of domain walls. Using techniques of [3] we explicitly show that given a set of moduli and choice of a gauge(the superpotential) corresponding to an extremal black hole, one can actually work out the corresponding charges (of the extremal black hole) - the so-called "inverse problem". We also show the existence of "fake superpotentials" [4] corresponding to non-BPS extremal black-hole solutions corresponding to the aforementioned Calabi-Yau three-fold. By including non-perturbative alpha' and instanton corrections in the Kaehler potential and superpotential [2], we show the possibility of getting a large-volume non-supersymmetric (A)dS minimum - a dS minimum without the addition of anti-D3 branes a la KKLT. The chosen Calabi-Yau has been of relevance also from the point of other studies of stabilization of the Kaehler moduli via nonperturbative instanton contributions [5] and the possibility of getting non-supersymmetric AdS vacua (and their subsequent dS-uplifts) using (alpha')^3 corrections to the Kaehler potential [6,7,8].Comment: 1+32 pages, LaTeX; published (in NPB) version - title changed and some cosmetic changes made on the reviewer's suggestion