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Bobby Acharya: Lectures

April 8, 2019
TITLE: New Local G2 Holonomy Spaces and their Gauge Theory Interpretations

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September 13, 2018
TITLE: Supersymmetry, Special Holonomy, Ricci Flatness and the String Landscape

ABSTRACT: The main reason that physicists have been interested in Ricci flat spaces with special holonomy is supersymmetry. These spaces then lead to supersymmetric physical models defined on Minkowski spaces, primarily because the manifolds admit parallel spinors with respect to the Levi-Cevita connection. The rich relation between physics and mathematics has led to many significant results like mirror symmetry, Gromov-Witten theory, Donaldson-Thomas invariants and Gopakumar-Vafa, which leads one to suspect that special holonomy, Ricci flat manifolds are rather special objects. I will discuss the possibility that, perhaps all compact, simply connected, stable Ricci flat manifolds have special holonomy (i.e., the conjecture that could bear the slogan “supersymmetry equals Ricci flat equals special holonomy”).

I survey some of what is known about this question and discuss obstructions to the existence of Ricci flat metrics with generic holonomy, presenting some classes/constructions of manifolds which cannot admit a Ricci flat metric with generic holonomy. In dimension four, one of the main open questions is: Does S² ✕ S² admit a stable Ricci flat metric? I finish by discussing an instability in string theory that occurs for non-supersymmetric Ricci flat manifolds with finite fundamental group without parallel spinors (e.g. the Enriques surface) and suggest a connection to Witten’s `bubble of nothing.’

If the conjecture were true, it would strongly suggest that the only consistent theories of gravity in Minkowski spacetime are supersymmetric theories.

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January 12, 2018
TITLE: String dualities and calibrated cycles

September 11, 2017
TITLE: M-theory/heterotic/type IIA duality

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June 6, 2017
TITLE: M-theorists’ fantasy shopping list for singular G2 manifolds

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January 9, 2017
TITLE: M theory and Type IIA superstring theory: Six-dimensional limits of G2 Manifolds

ABSTRACT: M theory spacetimes which are “fibered” by a circle have an alternative description in Type IIA
superstring theory in the limit that the circle is small. Supersymmetry in Type IIA theory leads to
spacetimes that are “Calabi-Yau threefolds with anti-holomorphic involutions together with special Lagrangian
D-branes and magnetic flux”. M theory/Type IIA duality requires that these arise as S1 collapsed limits of holonomy
G_2-manifolds. I will try and review this picture and emphasise the many open mathematical problems. Time permitting,
I will also discuss how “open Gromov-Witten theory” could be related to associative sub manifolds in M theory.

September 15, 2016
TITLE: Codimension seven singularities in M-theory compactifications

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September 15, 2016
TITLE: Beyond the standard model

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September 13, 2016
TITLE: The Standard model of particle physics

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September 12, 2016
TITLE: Particle physics for mathematicians

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September 6, 2016
TITLE: G₂-manifolds and Particle Physics

ABSTRACT: G_2-holonomy spaces serve as well motivated models of the seven extra dimensions of space predicted by M-theory. I will review how this leads to a completely geometric picture of all the known elementary forces including gravity. Special kinds of singularity of G_2-spaces play a crucial role in this picture, which one might regard as a generalisation of the McKay correspondence. I will try to describe some mathematical problems which have arisen from this work concerning the existence of G_2-holonomy metrics with singularities and questions about the moduli space of G_2-manifolds.

Slides of lecture