Heterotic complex structure moduli stabilization for elliptically fibered Calabi-Yau manifolds

Holomorphicity of vector bundles can stabilize complex structure moduli of a Calabi-Yau threefold in N = 1 supersymmetric heterotic compactifications. In principle, the Atiyah class determines the stabilized moduli. In this paper, we study how this mechanism works in the context of elliptically fibered Calabi-Yau manifolds where the complex structure moduli space contains two kinds of moduli, those from the base and those from the fibration. Defining the bundle with spectral data, we find three types of situations when bundles’ holomorphicity depends on algebraic cycles exist only for special loci in the complex structure moduli, which allows us to stabilize both of these two moduli. We present concrete examples for each type and develop practical tools to analyze the stabilized moduli. Finally, by checking the holomorphicity of the four-flux and/or local Higgs bundle data in F-theory, we briefly study the dual complex structure moduli stabilization scenarios.

ATIYAH CLASS AND CHERN CHARACTER FOR GLOBAL MATRIX FACTORISATIONS

We define the Atiyah class for global matrix factorisations and use it to give a formula for the categorical Chern character and the boundary-bulk map for matrix factorisations, generalising the formula in the local case obtained in [12]. Our approach is based on developing the Lie algebra analogies observed by Kapranov [7] and Markarian [9].

Atiyah Classes of Strongly Homotopy Lie Pairs

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an SH Lie algebra A when it is extended to L. In fact, with such an SH Lie pair (L, A) and any A-module E, there is associated a canonical cohomology class, the Atiyah class [αE], which generalizes the earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class [αL/A] induces a graded Lie algebra structure on [Formula: see text], and the Atiyah class [αE] of any A-module E induces a Lie algebra module structure on [Formula: see text]. Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.

Tangent Lie algebra of derived Artin stacks

Since the work of Mikhail Kapranov in [Compos. Math. 115 (1999), no. 1, 71–113], it is known that the shifted tangent complex \mathbb{T}_{X} [-1] of a smooth algebraic variety X is endowed with a weak Lie structure. Moreover, any complex of quasi-coherent sheaves E on X is endowed with a weak Lie action of this tangent Lie algebra. This Lie action is given by the Atiyah class of E. We will generalise this result to (finite enough) derived Artin stacks, without any smoothness assumption. This in particular applies to singular schemes. This work uses tools of both derived algebraic geometry and {\infty} -category theory.