Modularity from monodromy

Abstract

In this note we describe a method to calculate the action of a particular Fourier-Mukai transformation on a basis of brane charges on elliptically fibered Calabi-Yau threefolds with and without a section. The Fourier-Mukai kernel is the ideal sheaf of the relative diagonal and for fibrations that admit a section this is essentially the Poincar sheaf. We find that in this case it induces an action of the modular group on the charges of 2-branes.

Instanton corrections and Emergent Strings

“…The fact that the SL(2, Z) U modular symmetry is unbroken reflects that in type IIA the points z 1 → 0 and z 1 → ∞ are the same and related by a monodromy, the Fourier-Mukai transform. Due to this monodromy the type IIA prepotential is a modular function of z 1 [39] consistent with quantum corrections on the type I side arranging in modular functions of U .…”

Elliptic blowup equations for 6d SCFTs. Part II. Exceptional cases

“…In this section, we provide evidence for the elliptic blowup equations (3.4) by showing that the components of the elliptic blowup equations transforms correctly as weak Jacobi forms. This is established by showing that the weight and the index, in general a quadratic polynomial, of the corresponding components in (3.4) match the predictions for the index and weight made from the 2d and the 6d anomaly polynomial or from the transformation properties of the refined topological string partition function under the S and T monodromies of the Calabi-Yau space X, see [22] and more generally [56]. In general the blowup equations give interesting identities for Jacobi forms, one example is proven in section 3.3.1, see also (3.48).…”

Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities

“…2 A crucial step in generalizing the analysis of [1,2] is to modify and extend these monodromy arguments to genus one fibrations that do not have a section and also to include those parameters that correspond to the volumes of fibral curves and, after circle compactification, to the vaccum expectation values of scalars in vector multiplets. The action of the corresponding monodromies for elliptic fibrations with reducible fibers has been calculated in [13] and we extend the argument to fibrations without sections. Using this generalization, we establish that for genus one fibrations with N -sections and N ≤ 4, the SL(2, Z) part is broken to the finite index subgroup Γ 1 (N ) as well as the fact that the complexified Kähler parameters that correspond to rational fibral curves become JHEP11(2019)170 elliptic parameters of the coefficients Z β (τ, λ, m).…”

“…Instead we work on the stringy Kähler moduli space M ks (M ) of M directly. Extending the method of [2,13,23], our strategy is to identify the Kähler moduli as central charges of 2-branes and to study auto-equivalences of the category of branes that generate an action of Γ 1 (N ). We will then relate those auto-equivalences to the generic monodromies that correspond to the boundary of the geometric cone and to the large volume limiting points and thus show that the Γ 1 (N ) action corresponds indeed to monodromies in the stringy Kähler moduli space.This allows us to use the known automorphic properties of Z top.…”

“…For elliptic fibrations, the most characteristic auto-equivalence is induced by the Fourier-Mukai kernel which is the ideal sheaf I ∆ B of the relative diagonal ∆ B in M × B M [26][27][28][29][30] and it acts as τ → τ N τ +1 on the τ parameter [13]. We will call this the Utransformation or the relative Conifold transformation.…”