Periods of algebraic varieties, and more generally of mixed motives over Q, form an algebra, whose spectrum is conjecturally related to the motivic Galois group. However this relationship is difficult to see.
I will discuss a new way to present homotopy periods of an algebraic variety, which makes this relationship more transparent.
Namely, let X be a regular complex variety. We will introduce a Feynman integral related to X.
Its correlators (defined via a perturbative series expansion) are complex numbers, which we call Hodge correlators.
We show that they are homotopy periods, i.e. the real periods of the rational homotopy type of X;
Moreover, they define a functorial real mixed Hodge structure of the latter.
Notes and references.
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Periods of algebraic varieties, and more generally of mixed motives over Q, form an algebra, whose spectrum is conjecturally related to the motivic Galois group. However this relationship is difficult to see.
I will discuss a new way to present homotopy periods of an algebraic variety, which makes this relationship more transparent.
Namely, let X be a regular complex variety. We will introduce a Feynman integral related to X.
Its correlators (defined via a perturbative series expansion) are complex numbers, which we call Hodge correlators.
We show that they are homotopy periods, i.e. the real periods of the rational homotopy type of X;
Moreover, they define a functorial real mixed Hodge structure of the latter.
Notes and references.
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In the first part of the seminar, P. Etingof will give a short overview of Dunkl-Cherednik operators and flat connections arising from them.
In the second part (to start around 6.10pm),
V. Toledano Laredo will describe the Casimir connection
of a simple Lie algebra $\mathfrak{g}$
and explain how it is related to the Dunkl-Cherednik connection of the
corresponding Weyl group and, when $\mathfrak{g}$ is $sl(n)$,
to the Knizhnik-Zamolodchikov connection
on $n$ points for a dual $sl(k)$. He will also explain how the
monodromy of this connection is described by the quantum Weyl
group operators of the quantum group $U_q(\mathfrak{g})$.
Notes and references.
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Part 1: "Introduction to BPS wall-crossing" - I will discuss the integer invariants that come from quantum field theories. These field theories have continuous moduli, often identified with the moduli of some geometric object, and one wants to know a "wall-crossing formula" describing how the integer invariants jump when one crosses special codimension-1 walls in the moduli space. I will describe two broad classes of example where this problem has been solved. One class was studied by Cecotti-Vafa in the early 1990's and mathematically has to do with singularity theory. The other class was studied by Kontsevich-Soibelman very recently and has to do with the geometry of Calabi-Yau threefolds (or categories).
Part 2: "Hyperkahler geometry and BPS wall-crossing" - Kontsevich-Soibelman's
wall-crossing formula looks rather mysterious when first encountered. I will
explain a way to understand this wall-crossing formula geometrically: it is a
consistency condition for a new construction of hyperkahler spaces. This is
work done in collaboration with Davide Gaiotto and Greg Moore.
Notes and references.
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Part 1: "Introduction to BPS wall-crossing" - I will discuss the integer invariants that come from quantum field theories. These field theories have continuous moduli, often identified with the moduli of some geometric object, and one wants to know a "wall-crossing formula" describing how the integer invariants jump when one crosses special codimension-1 walls in the moduli space. I will describe two broad classes of example where this problem has been solved. One class was studied by Cecotti-Vafa in the early 1990's and mathematically has to do with singularity theory. The other class was studied by Kontsevich-Soibelman very recently and has to do with the geometry of Calabi-Yau threefolds (or categories).
Part 2: "Hyperkahler geometry and BPS wall-crossing" - Kontsevich-Soibelman's
wall-crossing formula looks rather mysterious when first encountered. I will
explain a way to understand this wall-crossing formula geometrically: it is a
consistency condition for a new construction of hyperkahler spaces. This is
work done in collaboration with Davide Gaiotto and Greg Moore.
Notes and references.
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It is well-known that 2d topological sigma-models are related to
geometrically interesting categories, such as the derived category of
coherent sheaves and the Fukaya-Floer category. This relationship
underlies the Homological Mirror Symmetry. In this talk I will discuss a
3d topological sigma-model constructed by Rozansky and Witten and its
geometric interpretation. Going from 2d to 3d is roughly equivalent to
categorifying the objects involved. I will show that the Rozansky-Witten
model attaches a certain 2-category to every complex symplectic manifold.
Its objects are complex Lagrangian submanifolds equipped with complex
fibrations. This 2-category is a categorification of the derived category
of coherent sheaves. It is also related to categories of matrix
factorizations and deformation quantization of the derived category of
coherent sheaves.
Notes and references.
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