STUDY OF BLACK HOLES AND COMPACTIFICATION GEOMETRIES IN STRING THEORY

Abstract

String theory is a theory of interacting relativistic strings. It is the most promising candidate for a consistent quantum theory of gravity. There are many interesting aspects of string theory. In this thesis, I will be interested in discussing issues related to black holes in four and three dimensions arising out of compactifying string theory on suitable manifolds and will also discuss various algebro-geometric properties of suitable geometries relevant for string/NI-theory compactification. I will also discuss some interesting aspects of non-perturbative superpotential arising from wrapping an M2-bran around a supersymmetric non-compact three-cycle embedded in a G2-manifold. We will elaborate on all these as we proceed. Classically, black holes are some solutions to the Einstein's gravitational field equa-tions, which have a horizon. Black holes form an important part of research of the obser-vational astrophysics where a lot of detectors aim at detecting black holes in our universe. There have been strong candidates for black holes which have been observed (for eg: GRS 1915+105) [1], which have mass M 14Ms on and J/GM2 > 0.98, where J is the angular momentum of the black hole. Black holes do not restrict themselves to just astrophysi-cal interest, but also serve as a theoretical laboratory to test many novel ideas of quantum gravity or string theory. Classically, it is well known that nothing can come out of black holes, not even light. But interestingly, after including quantum effects, it can he seen that black holes radiate and hence have temperature associated with them, known as Hawking temperature. There are specific kind of black holes having zero temperature, known as extremal black holes. An interesting feature of extremal black holes is that they exhibit attractor mechanism [2, 3, 4]. This means that the various scalar fields present in the theory, also known as moduli fields, at the horizon are expressed in terms of the mass, charges and angular mo- ii ABSTRACT mentum carried by the black hole and do not depend on their asymptotic values. These moduli arise in the lower dimensional theory as a result of compactifying string theory on some compact manifolds. This feature was initially observed only for supersymmetric extremal black holes but later, in a number of interesting papers [5, 6, 7] on this topic, it was also proved to hold for non-supersymmetric extremal black holes. As observed in [8], the fact that extremal black holes have an infinite throat near the horizon or speak-ing differently that the horizon of an extremal black hole is at an infinite proper distance from any point outside the horizon, is responsible for the attractor feature of an extremal black hole. This feature is present in non-supersymmetric extremal black holes as well and one expects attractor mechanism to hold even for non-supersymmetric extremal black holes which was explicitly observed in [9, 10]. Supersymmetric black holes at the attractor point, correspond to minimizing the central charge and the effective black hole potential, whereas non-supersymmetric attractors [5], at the attractor point, correspond to minimizing only the potential and not the central charge. Attractor equations for (non) supersymmet-ric black holes and flux vacua were given by Kallosh [7] and some examples verifying the same were studied by Giryavets [11], including JIB compactified on one-parameter Calabi-Yau's. Kallosh [7] also observed an interesting resemblance between black holes and flux vacua attractors. However, the flux vacua attractors had some terms which allowed for sta-bilizing the moduli with vanishing central charges which was not possible for BPS black holes. In our work on black hole attractors, we were mainly interested in looking at a par-ticular example of type IIB string theory compactified on a CY3 and looking at various cases in which an attractor solution is possible or not. We consider an example of type DB compactified on (the mirror of) a two-parameter Calabi-Yau CY3 (2, 128) (expressed as a degree-12 hypersurface in a weighted complex projective space WCP4[1, 1,2, 2, 6]) and look for possible non-supersymmetric black hole attractor solutions, both, from the equiv-alent points of view of extremizing an effective black-hole potential and also by using the attractor equations. We get some interesting connections between arithmetic and geometry 111 and non-supersymmetric black-hole attractors. We were mainly interested in examining the complex structure moduli space and seeing if the attractor equations (equations which determine the moduli in terms of the charges carried by the black hole) give attractor so-lutions or not i.e we were interested in the existence of attractor solutions for type IIB compactified on a particular CY3. There are some interesting black holes in three dimensions known as BTZ black holes [12]. Such black holes are asymptotically AdS3, and arise in a three-dimensional theory of (super)gravity with negative cosmological constant, which can be obtained from string theory after compactification on suitable manifold. An interesting property of BTZ black hole is that for theories with extended supergravity, the entropy of this black hole depends just on the coefficients of the gravitational and gauge Chern-Simons terms and do not receive any further higher derivative corrections. This was initially observed by Kraus and Larsen [13] using AdS/CFT correspondence. Such a result was indeed very surprising and needed some better explanation from the point of view of the bulk supergravity action. Later, in [14], this was explained from the bulk point of view for theories having (0,4) supersymmetry or higher. The explanation was that the higher derivative terms that are allowed to be added preserving supersymmetry, can be removed by field redefinition and hence will not affect the entropy (since Wald's formula is invariant under field redefinition). The approach was to construct CFT correlators from gravity and argue their non-renormalization in the presence of higher derivative terms. This, in turn implies the non-renormalization of the central charges in the boundary. Later, a more direct approach was given by Gupta and Sen [15]. They showed that a direct field redefinition can be implemented in the bulk which also shows the non-renormalization directly without going to the boundary conformal field theory. Their result shows the non-renormalization even for lesser extended supergravity theories (i.e (2,0) or (0,2)). The prime motivation for my next work, was to use the correlator approach of [14] for (2,0) theories and gain better insight into some of its structure. In particular, we ob-served that we could not restrict ourselves to just bosonic supergravity as was done in the iv ABSTRACT (4,0) case. We also classified the set of higher derivative terms that preserve supersym-metry with/without modifying the existing supersymmetry transformations that keeps the standard supergravity action, without the higher derivative terms, invariant. We gave an extensive discussion on the first order and second order formulations of the theory with and without Chem-Simons terms and also computed the correlators including the fermions which matched the conformal field theory predictions. String compactification in the presence of fluxes gives a more realistic theory and a theory with minimal supersymmetry in four dimensions. For the requirement of N = 1 the-ories in four dimensions, one needs to compactify M-theory in the presence of G-fluxes, on a seven-fold with either G2-holonomy or SU (3) structure. We have constructed seven-folds with G2-holonomy, using a six-dimensional half-flat manifold (Iwasawa) via Hitchin's flow equations, and manifold with SU(3) structure via generalisation of Hitchin's flow equa-tions. We uplift the Iwasawa manifold to a G2 manifold through "size" deformation (of the Iwasawa metric), via Hitchin's flow equations, showing also the impossibility of the uplift for "shape" and "size" deformations (of the Iwasawa metric). Using results of [16], we also uplift the Iwasawa manifold to a seven-fold with SU(3) structure through "size" and "shape" deformations via generalisation of Hitchin's flow equations. For seven-folds with SU(3)-structure, the result could be interpreted as M5-branes wrapping two-cycles embedded in the seven-fold (as in [16]) -a warped product of either a special hermitian six-fold or a balanced six-fold with the unit interval. We also saw that there can be no uplift to seven-folds of SU(3) structure involving non-trivial "size" and "shape" deformations (of the Iwasawa metric) retaining the "standard complex structure" - the uplift generically makes one move in the space of almost complex structures such that one is neither at the standard complex structure point nor at the "edge". There is another beautiful aspect of string theory which is "Mirror symmetry". It says that the two theories with different descriptions are equivalent. In the context of string propagation on Calabi-Yau manifolds in the presence of RR fluxes, the role of complex deformations on one manifold gets exchanged with the Kahler deformations on the dual V manifold. These dual manifolds are known as "Mirror Pairs". Rigid manifolds can not have Calabi-Yau manifolds as their mirror pairs. To preserve the geometric nature of mirror symmetry, it was proposed in [17] that supermanifolds need to be considered. The mirror of a Calabi-Yau may be a super Calabi-Yau. We then discuss some algebraic geometric aspects of a supermanifold in a super weighted complex projective space. Then using some known techniques [18], we solve the super-Picard Fuchs equation and also obtain the periods for the mirror to the super Calabi-Yau, both in the large and small complex structure limits. Periods are the building blocks for getting the prepotential (and therefore the Miler potential) in N = 2 Type II theories corn-pactified on a Calabi-Yau manifolds and also for the superpotential for few other relevant theories and hence period computation is very relevant in the context of phenomenological model building. We further discuss the monodromies at 0,1 and Co (using again techniques developed in [19, 20]). We also obtain the mirror hyperpsurface involving supermanifolds which are not super Calabi-Yau's. One can identify a non-compact instantonic configuration in M-theory on a 02 mani-fold, with a supersymmetric three-cycle embedded in the same manifold, considering the superpotential of an isolated membrane instanton obtained by wrapping an M2-brane on the supersymmentric three-fold. The heat-kernel-asymptotics method is used for extract-ing the UV-divergent contributions of bosonic and fermionic determinants - the same are encoded in the Seeley-de Witt coefficients. We evaluated the leading- and next-to-leading order contributions in a parameter "zeta" (which is taken to be small in Witten's MQCD) of the Seeley-de Witt coefficients and found a match between fermionic and world-volume-based bosonic contributions up to this order. This led us to the conjecture that the full determinants and not just the UV-divergent pieces, must be equal - and this is a signature of the surviving supersymmetry in the instanton configuration considered. Basically, we did the spectral analysis of an explicit embedding of the non-compact three-cycle in 02 manifold, having turned off the M-theory four-form G-flux