Moduli in N = 1 heterotic/F-theory duality

Abstract

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 5brane / F-theoretic 3brane impurities. 1 1 60 Λ, and therefore D ′ Z[W/W 0 ] = D ′ 1 60 Λ = 1 60 D ′ Λ = 1 60 60Λ = Λ.) As we promised in the previous section, this provides the justification for the matching of brane impurities.

Matter in transition

“…The net change in matter content is 56) in exact agreement with the expected transition from supergravity. Note that the total number of multiplets is the same before and after the transition.…”

“…As is well-known in the literature (see [20,55,56] for reviews), this limit in parameter space corresponds to the large volume and weak coupling regime in the heterotic theory and is realized geometrically in the F-theory geometry via the following log semi-stable degeneration of the Calabi-Yau manifold, Y :…”

“…Representation Multiplicity (in full multiplets) (56) r To find the number of neutral multiplets, we again count the number of complex degrees of freedom in the Weierstrass model and subtract any redundant degrees of freedom. As with the SU(7) model, the f and g expressions for the SU(8) tuning are invariant under the following transformations in the polynomials:…”

“…where Y (1) and Y (2) are non-CY, dP 9 fibered (n + 1)-folds, glued along a common divisor D [56][57][58]. In the case of heterotic duality, D = X n is a CY variety of one lower dimension than Y n+1 and forms the background of the heterotic geometry.…”

Perspectives on Pfaffians of heterotic world-sheet instantons

“…In case C is ample (positive) one 7 Which are unobstructed [3], however, for C smooth and B rational. 8 As h 0,1 (X) = 0; when applying the same argument to E and c therefore h 0,1 (E ) = 0 will be relevant. 9 Cf.…”