Chiral matter and transitions in heterotic string models

Abstract

In the framework of N = 1 supersymmetric string models given by the heterotic string on an elliptic Calabi-Yau π : Z → B together with a SU(n) bundle we compute the chiral matter content of the massless spectrum. For this purpose the net generation number, i.e. half the third Chern class, is computed from data related to the heterotic vector bundle in the spectral cover description; a non-technical introduction to that method is supplied. This invariant is, in the class of bundles considered, shown to be related to a discrete modulus which is the heterotic analogue of the F -theory four-flux. We consider also the relevant matter which is supported along certain curves in the base B and derive the net generation number again from the independent matter-related computation. We then illustrate these considerations with two applications. First we show that the construction leads to numerous 3 generation models of unbroken gauge group SU(5), SO(10) or E 6 . Secondly we discuss the closely related issue of the heterotic 5-brane/instanton transition resp. the F-theoretic 3-brane/instanton transition. The extra chiral matter in these transitions is related to the Hecke transform of the direct sum of the original bundle and the dissolved 5-brane along the intersection of their spectral covers. Finally we point to the corresponding F -theory interpretation of chiral matter from the intersection of 7-branes where the influence of four-flux on the twisting along the intersection curve plays a crucial role.

Matter in transition

“…There the spectral covers are complex surfaces inside Calabi-Yau 3 folds and once again, codimension two components (here curves) can be separated off one component and the deformed smoothly into the other. One remarkable difference in the 4-dimensional theory however is that in the N = 1 theory, such transitions can also be chirality changing [64,65]. We will return to this point later.…”

“…Examples of other related transitions have been explored in the heterotic literature (see [64,65] for representative examples). As described in those works and in section 4, small instanton transitions on a single E 8 fixed plane can only arise when the vector bundle geometry -an H-principal bundle on X n with H ⊂ E 8 -is reducible.…”

Five-brane BPS states in heterotic M-theory

“…As discussed in [3,18], the number of quark and lepton generations is given by N gen = c 3 (V ) 2 . Using the expression for c 3 (V ), it was shown that the three family condition imposes the further constraint that…”

Perspectives on Pfaffians of heterotic world-sheet instantons

“…first-order deformations. 7 Beginning with the base in (3.8) note that after tensoring the short exact sequence…”

“…As π C * (c 1 (L)) is fixed by the condition c 1 (V ) = 0 the possible L's are parametrized 13 by a discrete part encoded in λ and a continuous part from Jac(C/B) ∼ = P ic 0 (C) ∼ = Jac(C). 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.…”