Geometric Langlands From Six Dimensions (0905.2720v1)

Abstract: Geometric Langlands duality is usually formulated as a statement about Riemann surfaces, but it can be naturally understood as a consequence of electric-magnetic duality of four-dimensional gauge theory. This duality in turn is naturally understood as a consequence of the existence of a certain exotic supersymmetric conformal field theory in six dimensions. The same six-dimensional theory also gives a useful framework for understanding some recent mathematical results involving a counterpart of geometric Langlands duality for complex surfaces. (This article is based on a lecture at the Raoul Bott celebration, Montreal, June 2008.)

• The paper presents a novel embedding of the geometric Langlands duality within a six-dimensional supersymmetric conformal field theory.
• It demonstrates how lower-dimensional categorical structures emerge from higher-dimensional quantum field theories, connecting string theory and Yang-Mills symmetries.
• Witten’s analysis highlights electric-magnetic duality and SL(2,Z) symmetry, offering fresh insights into the duality framework in gauge theories.

Overview of "Geometric Langlands From Six Dimensions" by Edward Witten

Edward Witten's paper "Geometric Langlands From Six Dimensions" provides an intricate exploration into the relationship between the geometric Langlands duality and theories within the context of gauge and conformal field theories. The primary narrative of this paper unfolds by embedding the geometric Langlands correspondence, traditionally construed on the framework of Riemann surfaces, within a six-dimensional exotic supersymmetric conformal field theory setting.

Witten's theoretical investigation begins by acknowledging that while geometric Langlands duality has connections to four-dimensional gauge theory through electric-magnetic duality, its broader implications and understanding might be more comprehensively constructed from a six-dimensional supersymmetric theory. This theory not only stretches its interpretative power over Riemann surfaces but also potentially extends to the field of complex surfaces, thereby contributing to a richer mathematical framework that accommodates recent advancements involving the Langlands program.

The paper provides a robust foray into the formulation of quantum field theory (QFT), detailing how a quantum field theory defined on a d-dimensional space assigns structures to manifolds of lower dimensions. Witten elaborates that a category can be associated to a two-dimensional manifold in a quantum field theoretical context and posits that this constitutes an essential component of understanding geometric Langlands as an equivalence between categories.

In describing non-standard dimensions and their associated vector spaces or categories, Witten examines physical examples such as string theory, where 2-dimensional QFT delineates string propagation, or the richer structures seen in three-dimensional Chern-Simons gauge theories.

A significant move in Witten's analysis is reinterpreting the statement of the geometric Langlands correspondence in specific contexts -- notably, how gauge theories with G and its Langlands dual group $G^\vee$ possess equivalent categorical structures in four-dimensional spaces. Witten leans heavily on N = 4 supersymmetry and the twisting associated with it to formulating and elucidating these equivalencies.

The connection to four-dimensional Yang-Mills theories introduces a classical example, yet Witten's exposition bears more straightforward results via the lens of six-dimensional theories. Specifically, he identifies a certain six-dimensional conformal field theory, suspected to encode these properties inherently and representing quantum "selfdual gerbes," as a foundational framework for these dualities.

Witten’s approach utilizes this six-dimensional backdrop to expound on the electric-magnetic duality, which, when considered under the maximal symmetry conditions (e.g., when G is simply-laced), emerges naturally as SL(2, Z) actions in the coupling parameters, resulting in the parallel symmetry of a duality in supersymmetric Yang-Mills theories systematically engendering the Langlands duality.

In the thorough examination of subdimensions within four and six-dimensional contexts, the paper analyzes manifold operators of differing codimensions. This work sets the foundation to explore defects and their role in higher-dimensional theories by understanding them from varied manifold perspectives. The resulting cascade of realizations from ten-dimensional string theory directly supports these six-dimensional constructs, giving a coherent narrative to higher-dimensional duality in seamless collaboration with known string theory frameworks.

In conclusion, Witten’s paper is a cornerstone contribution to the theoretical physics and mathematics community’s comprehension of duality symmetries in quantum field theories. It provides both a sophisticated synthesis of geometric Langlands ideas using higher-dimensional field theories and broad implications for future explorations in quantum field theory, mathematical physics, and beyond. Potential advancements will likely involve deeper examinations of the six-dimensional theories or extensions into the theoretic domains posited by string theory or ten-dimensional analogs.