Black hole attractor varieties and complex multiplication (original) (raw)
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Complex multiplication symmetry of black hole attractors
Nuclear Physics B, 2003
We show how Moore's observation, in the context of toroidal compactifications in type IIB string theory, concerning the complex multiplication structure of black hole attractor varieties, can be generalized to Calabi-Yau compactifications with finite fundamental groups. This generalization leads to an alternative general framework in terms of motives associated to a Calabi-Yau variety in which it is possible to address the arithmetic nature of the attractor varieties in a universal way via Deligne's period conjecture.
From wrapped M-branes to Calabi-Yau black holes and strings
Journal of High Energy Physics, 2003
We study a class of D = 11 BPS spacetimes that describe M-branes wrapping supersymmetric 2 and 4-cycles of Calabi-Yau 3-folds. We analyze the geometrical significance of the supersymmetry constraints and gauge field equations of motion for these spacetimes. We show that the dimensional reduction to D = 5 yields known BPS black hole and black string solutions of D = 5, N = 2 supergravity. The usual ansatz for the dimensional reduction is valid only in the linearized regime of slowly varying moduli and small gauge field strengths. Our identification of the massless D = 5 modes with D = 11 quantities extends beyond this regime and should prove useful in constructing non-linear ansatze for Calabi-Yau dimensional reductions of supergravity theories.
Protein Science, 2006
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.
Dilatonic black holes in theories with moduli fields
Phys Rev D, 1993
We discuss the low-energy effective string theory when moduli of the compactified manifold are present. Assuming a nontrivial coupling of the moduli to the Maxwell tensor, we find a class of regular black-hole solutions. Both the thermodynamical and the geometrical structure of these solutions are discussed.
New compactifications on Calabi-Yau manifolds
Physics Letters B, 1985
We find nontrivial configurations for antisymmetric tensor gauge fields on Calabi-Yau manifolds, which have vanishing stress-energy tensor. Our construction is a complex version of the Freund-Rubin ansatz. For certain ten-dimensional models, these configurations provide preferential compactification to anti-de Sitter × Calabi-Yau. It is possible that strings [1-4] provide a consistent and calculable description of quantum gravity. The known consistent string models are formulated in dimensions higher than four, and therefore require compactification on some compact space K. The requirement that the compactified string model be free of ghosts restricts the choice of K [3-6]. The special case where K is a Calabi-Yau manifold [7] was studied by Candelas et al. [6]. Calabi-Yau spaces are Ricci-flat K~tler manifolds having three complex dimensions, which are further characterized by the property of admitting a covafiantly constant holomorphic three-form. On the other hand, the low energy limit of superstring models includes a rank-two antisymmetric tensor gauge field B, whose (modified) field strength H is a three-form. It is quite natural to consider the field configuration in which H is set equal to the Calabi-Yau three-form. This is a complex version of the Freund-Rubin [8] ansatz. It is this construction and its generalization which we investigate here. We find that the stress-energy tensor for such configurations vanishes; hence, these field configurations are in fact solutions of the Einstein-Kalb-Ramond equations. Moreover, configurations of this type can provide preferential compactification for certain ten-dimensional models to anti-de Sitter × Calabi-Yau. Unfortunately, the (known) limiting field theory of the SO(32) or E8 × E8 superstring
HKT and OKT geometries on soliton black hole moduli spaces
Nuclear Physics B, 1997
We consider Shiraishi's metrics on the moduli space of extreme black holes. We interpret the simplification in the pattern of N-body interactions that he observed in terms of the recent picture of black holes in four and five dimensions as composites, made up of intersecting branes. We then show that the geometry of the moduli space of a class of black holes in five and nine dimensions is hyper-Kähler with torsion, and octonionic-Kähler with torsion, respectively. For this, we examine the geometry of point particle models with extended world-line supersymmetry and show that both of the above geometries arise naturally in this context. In addition, we construct a large class of hyper-Kähler with torsion and octonionic-Kähler with torsion geometries in various dimensions. We also present a brane interpretation of our results.
On invariant structures of black hole charges
Journal of High Energy Physics, 2012
We study "minimal degree" complete bases of duality-and "horizontal"-invariant homogeneous polynomials in the flux representation of two-centered black hole solutions in two classes of D = 4 Einstein supergravity models with symmetric vector multiplets' scalar manifolds. Both classes exhibit an SL (2, R) "horizontal" symmetry.
Lessons on black holes from the elliptic genus
Journal of High Energy Physics, 2014
We further study the elliptic genus of the cigar SL(2, R) k /U (1) coset superconformal field theory. We find that, even in the small curvature, infinite level limit, there are holomorphic and non-holomorphic parts that are due to the discrete states and a mismatch in the spectral densities of the continuum, respectively. The mismatch in the continuum is universal, in the sense that it is fully determined by the asymptotic cylindrical topology of the cigar's throat. Since modularity of the elliptic genus requires both the holomorphic and non-holomorphic parts, the holomorphic term is universal as well. The contribution of the discrete states is thus present even for perturbative strings propagating in the background of large Schwarzschild black holes. We argue that the discrete states live at a stringy distance from the tip of the cigar both from the conformal field theory wave-function analysis and from a holonomy space perspective. Thus, the way string theory takes care of its self-consistency seems to have important consequences for the physics near horizons, even for parametrically large black holes.
Nuclear Physics B, 2008
We consider two sets of issues in this paper. The first has to do with moduli stabilization, existence of "area codes" [1] and the possibility of getting a non-supersymmetric dS minimum without the addition of D3-branes as in KKLT for type II flux compactifications. The second has to do with the "Inverse Problem" [2] and "Fake Superpotentials" [3] for extremal (non)supersymmetric black holes in type II compactifications. We use (orientifold of) a "Swiss Cheese" Calabi-Yau [4] expressed as a degree-18 hypersurface in WCP 4 [1, 1, 1, 6, 9] in the "large-volume-scenario" limit . The main result of our paper is that we show that by including non-perturbative α ′ and instanton corrections in the Kähler potential and superpotential [6], it may be possible to obtain a large-volume non-supersymmetric 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 Kähler moduli stabilization via nonperturbative instanton contributions [7] and non-supersymmetric AdS vacua (and their subsequent dS-uplifts) using (α ′ ) 3 corrections to the Kähler potential .