G-flux in F-theory and algebraic cycles

Abstract

We construct explicit G 4 fluxes in F-theory compactifications. Our method relies on identifying algebraic cycles in the Weierstrass equation of elliptic Calabi-Yau fourfolds. We show how to compute the D3-brane tadpole and the induced chirality indices directly in F-theory. Whenever a weak coupling limit is available, we compare and successfully match our findings to the corresponding results in type IIB string theory. Finally, we present some generalizations of our results which hint at a unified description of the elliptic Calabi-Yau fourfold together with the four-form flux G 4 as a coherent sheaf. In this description the close link between G 4 fluxes and algebraic cycles is manifest.

Gauge backgrounds and zero-mode counting in F-theory

“…3 The relation between matter surfaces and gauge backgrounds has first been observed in [26] and further described in [27]. If the homology class of the matter surface is vertical, then the associated flux can alternatively be described as a linear combination of a basis of H 2,2 vert ( Y 4 ), as investigated systematically in [26][27][28][29][30][31][32][33][34][35][36][37][38]. Equipped with this representation of gauge background data, we systematically develop the intersection theoretic operations [22] which allow us to extract the relevant gauge bundles on the matter curves.…”

“…[24,28,33,[87][88][89] and references therein. 15 This property is satisfied for most models discussed so far in the literature, an exception being [90].…”

Towards the Standard Model in F-theory

“…In presence of such fluxes, the chiral index of matter in representation R localised on matter curve C R ⊂ B is given by the topological intersection number [49,50,52,[136][137][138][139][140] …”

“…Indeed, in a more general F-theory model without a smooth Type IIB limit, SU (N ) gauge symmetries are not necessarily accompanied by extra diagonal U (1) factors. 2 The structure of abelian gauge symmetries in F-theory is a particularly rich topic with beautiful connections to algebraic geometry and has been investigated in great detail in the very recent literature [45,[49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66], motivated in part by the need for extra U (1) selection rules in the context of F-theory GUT model building [11,[67][68][69][70][71][72].…”

Tate form and weak coupling limits in F-theory

“…It turns out that this 2-cycle is not manifest in a generic point of the moduli space ofB 3 . But if we constrain the complex structure moduli in a suitable way we are able to write this 2-cycle as a set of three algebraic equations in the ambient fourfold which automatically satisfy (5.10) (see [36,46], where similar techniques are used for elliptic fourfolds). One possible constraint which works is the following (5.12)…”