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Shifted symplectic structures
This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-shifted symplectic structures (n-symplectic
Langlands duality for Hitchin systems
We show that the Hitchin integrable system for a simple complex Lie group $G$ is dual to the Hitchin system for the Langlands dual group $\lan{G}$. In particular, the general fiber of the connected
Families of K3 surfaces
We use automorphic forms to prove that a compact family of Kaehler K3 surfaces with constant Picard number is isotrivial.
Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models
In this paper we prove the smoothness of the moduli space of Landau-Ginzburg models. We formulate and prove a Tian-Todorov theorem for the deformations of Landau-Ginzburg models, develop the
Non-Birational Twisted Derived Equivalences in Abelian GLSMs
In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions. Examples
Hodge theoretic aspects of mirror symmetry
We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of
Standard Models from Heterotic M-theory
We present a class of N=1 supersymmetric models of particle physics, derived directly from heterotic M-theory, that contain three families of chiral quarks and leptons coupled to the gauge group
Shifted Poisson structures and deformation quantization
This paper is a sequel to ‘Shifted symplectic structures’ [T. Pantev, B. Toën, M. Vaquié and G. Vezzosi, Publ. Math. Inst. Hautes E'tudes Sci. 117 (2013) 271–328]. We develop a general and flexible
Symplectic Lefschetz fibrations with arbitrary fundamental groups
In this paper we give an explicit costruction of a symplectic Lefschetz fibration whose total space is a smooth compact four dimensional manifold with a prescribed fundamental group. We also study