Compactifications of F-theory on Calabi-Yau threefolds (II)

We continue our study of compactifications of F-theory on Calabi-Yau threefolds. We gain more insight into F-theory duals of heterotic strings and provide a recipe for building F-theory duals for arbitrary heterotic compactifications on elliptically fibered manifolds. As a byproduct we find that string/string duality in six dimensions gets mapped to fiber/base exchange in F-theory. We also construct a number of new N = 1, d = 6 examples of F-theory vacua and study transitions among them. We find that some of these transition points correspond upon further compactification to 4 dimensions to transitions through analogues of Argyres-Douglas points of N = 2 moduli. A key idea in these transitions is the notion of classifying (0,4) fivebranes of heterotic strings.     

Five-brane superpotentials and heterotic/F-theory duality

Under heterotic/F-theory duality it was argued that a wide class of heterotic five-branes is mapped into the geometry of an F-theory compactification manifold. In four-dimensional compactifications this identifies a five-brane wrapped on a curve in the base of an elliptically fibered Calabi–Yau threefold with a specific F-theory Calabi–Yau fourfold containing the blow-up of the five-brane curve. We argue that this duality can be reformulated by first constructing a non-Calabi–Yau heterotic threefold by blowing up the curve of the five-brane into a divisor with five-brane flux. Employing heterotic/F-theory duality this leads us to the construction of a Calabi–Yau fourfold and four-form flux. Moreover, we obtain an explicit map between the five-brane superpotential and an F-theory flux superpotential. The map of the open–closed deformation problem of a five-brane in a compact Calabi–Yau threefold into a deformation problem of complex structures on a dual Calabi–Yau fourfold with four-form flux provides a powerful tool to explicitly compute the five-brane superpotential.     

Vector Bundle Moduli

We give the general presciption for calculating the number of moduli of irreducible, stable U(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi–Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B = F _r. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory.     

Heterotic non-Kähler geometries via polystable bundles on Calabi-Yau threefolds

In arXiv:1008.1018 it is shown that a given stable vector bundle V on a Calabi-Yau threefold X which satisfies c₂(X)=c₂(V) can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explicit examples of polystable bundles on elliptically fibered Calabi-Yau threefolds where it applies. The polystable bundle is given by a spectral cover bundle, for the visible sector, and a suitably chosen bundle, for the hidden sector. This provides a new class of heterotic flux compactifications via non-Kähler deformation of Calabi-Yau geometries with polystable bundles. As an application, we obtain examples of non-Kähler deformations of some three generation GUT models.     

Constraints on heterotic M-theory from s-cobordism

We interpret heterotic M-theory in terms of h-cobordism, that is the eleven-manifold is a product of the ten-manifold times an interval is translated into a statement that the former is a cobordism of the latter which is a homotopy equivalence. In the non-simply connected case, which is important for model building, the interpretation is then in terms of s-cobordism, so that the cobordism is a simple-homotopy equivalence. This gives constraints on the possible cobordisms depending on the fundamental groups and hence provides a characterization of possible compactification manifolds using the Whitehead group - a quotient of algebraic K-theory of the integral group ring of the fundamental group - and a distinguished element, the Whitehead torsion. We also consider the effect on the dynamics via diffeomorphisms and general dimensional reduction, and comment on the effect on F-theory compactifications.     

Compactifications of the heterotic string with unitary bundles

We describe a large new class of four‐dimensional supersymmetric string vacua defined as compactifications of the E₈ × E₈ and the SO(32) heterotic string on smooth Calabi‐Yau threefolds with unitary gauge bundles and heterotic five‐branes. The conventional gauge symmetry breaking via Wilson lines is replaced by the embedding of non‐flat line bundles into the ten‐dimensional gauge group, thus opening up the way for phenomenologically interesting string compactifications on simply connected manifolds. After a detailed analysis of the four‐dimensional effective theory we exemplify the general framework by means of a couple of explicit examples involving the spectral cover construction of stable holomorphic bundles. As for the SO(32) heterotic string, the resulting vacua can be viewed, in the S‐dual Type I picture, as a generalisation of magnetized D9/D5‐brane models. In the case of the E₈ × E₈ string, we find a natural way to construct realistic MSSM‐like models, either directly or via a flipped SU(5) GUT scenario.     

Vector Bundles and F Theory

To understand in detail duality between heterotic string and F theory compactifications, it is important to understand the construction of holomorphic G bundles over elliptic Calabi-Yau manifolds, for various groups G. In this paper, we develop techniques to describe these bundles, and make several detailed comparisons between the heterotic string and F theory. Received: 6 February 1997 / Accepted: 29 May 1997     

Moduli in N = 1 heterotic/F-theory duality

The moduli in a 4D N = 1 heterotic compactification on an elliptic CY, as well as in the dual F-theoretic compactification, break into “base” parameters which are even (under the natural involution of the elliptic curves), and “fiber” or twisting parameters; the latter include a continuous part which is odd, as well as a discrete part. We interpret all the heterotic moduli in terms of cohomology groups of the spectral covers, and identify them with the corresponding F-theoretic moduli in a certain stable degeneration. The argument is based on the comparison of three geometric objects: the spectral and cameral covers and the ADE del Pezzo fibrations. For the continuous part of the twisting moduli, this amounts to an isomorphism between certain abelian varieties: the connected component of the heterotic Prym variety (a modified Jacobian) and the F-theoretic intermediate Jacobian. The comparison of the discrete part generalizes the matching of heterotic 5-brane/F-theoretic 3-brane impurities.     

Noncommutative Circle Bundles and New Dirac Operators

We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. We analyze in details the example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus and find all connections compatible with an admissible Dirac operator. Conversely, we find a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection.     

Heterotic modular invariants and level-rank duality

New heterotic modular invariants are found using the level-rank duality of affine Kac-Moody algebras. They provide strong evidence for the consistency of an infinite list of heterotic Wess-Zumino-Witten (WZW) conformal field theories. We call the basic construction the dual-flip, since it flips chirality (exchanges left and right movers) and takes the level-rank dual. We compare the dual-flip to the method of conformal subalgebras, another way of constructing heterotic invariants. To do so, new level-one heterotic invariants are first bound; the complete list of a specified subclass of these is obtained. We also prove (under a mild hypothesis) an old conjecture concerning exceptional A_r,k invariants and level-rank duality.     

World-volume and target-space anomalies in the D = 10 super-fivebrane sigma model

The fields of the conjectured “heterotic” super-fivebrane sigma model in ten dimensions are made out of a well-known gravitational sector, the X and the ϑ, and of a still unknown heterotic sector which should be coupled to the Yang-Mills fields. We compute the one-loop d = 6 world-volume and D = 10 target-space Lorentz anomalies which arise from the gravitational sector of the heterotic super-fivebrane sigma model, using a method which we developed previously for the Green-Schwarz heterotic superstring. These anomalies turn out to carry an overall coefficient which is half of that required by the string-fivebrane duality conjecture. As a consequence, the world-volume anomaly vanishes if the heterotic fields consist of 16 (rather than 32) complex Weyl fermions on the world-volume. This implies that the string-fivebrane duality conjecture can not be based on a “heterotic” super-fivebrane sigma model with only fermions in the heterotic sector. Possible implications of this result are discussed.