from Langlands Functoriality as a Boundary Condition: Why the Geometric Langlands Program Requires a Finite Base-2 Cutoff

OVERVIEW: Wikipedia https://en.wikipedia.org/wiki/S-duality: Kapustin and Witten showed that one can construct a pair of quantum field theories in two-dimensional spacetime.

THE PAPER: Electric-Magnetic Duality And The Geometric Langlands (2007) by Anton Kapustin and Edward Witten. Abstract. In this paper, we recall quantum gauge theory with discrete electric and magnetic fluxes and review the main results of Vafa-Witten theory when the gauge group

Inspire^HEP renders S-duality in Vafa-Witten theory for non-simply laced gauge

[hep-th/9505186] On S-Duality in Abelian Gauge Theory

EARLIER:

• arXiv https://arxiv.org › hep-th by E Witten · 1995 · Cited by 439 — On S–Duality in Abelian Gauge Theory, by Edward Witten is known to possess an electric-magnetic duality symmetry.

• EM Duality: Yu Leon Liu, https://leon2k2k2k.github.io The current plan is to understand S duality in 4D N=4 Yang Mills. From there, we follow Kapustin–Witten to see how S–duality connects to geometric Langlands.

• Witten Geometric Langlands Talk and Paper
• Columbia Math Department: https://www.math.columbia.edu/~woit/wordpress. Apr 22, 2006 — The electric-magnetic duality coming from S–duality in the gauge theory relates electric and magnetic eigenbranes.
• On S-duality in Abelian gauge theory. Princeton University https://collaborate.princeton.edu. by E Witten · 1995 · In this paper, the duality is studied on a general four-manifold and it is shown that the partition function is not a modular-invariant function but transforms…
• What evidence do we have for S-duality: Physics Stack Exchange, https://physics.stackexchange.com/questions/ Oct 26, 2011 — Do we have anything resembling a proof*? Or is it just a collection of “coincidences”? Also, do we have evidence from lattice gauge theory computations? 1 answer There is a “proof” of S-duality in free abelian gauge theory. See, for example, Witten’s On S-Duality in Abelian Gauge Theory or his IAS lecture.
• S-duality in nLab: nLabhttps://ncatlab.org › nlab › show › S-duality
• Sep 4, 2025 — The relation of S–duality to geometric Langlands duality was understood in. Anton Kapustin, Edward Witten, Electric-Magnetic Duality And The …Read moreIdeaIn (super) Yang-Mills theoryIn string theory. Overview
• Testing S-duality with non-orientable surfaces, Harvard University, https://ui.adsabs.harvard.edu › abs › arXiv:1811
• Kapustin and Witten showed that a twisted version of N=4 gauge theory in four dimensions compactifies to a two-dimensional sigma-model. Scholarly articles for Kapustin-Witten S-Duality:
  • S-dual of Hamiltonian G spaces and relative Langlands … – ‎Nakajima 
  • The Kapustin–Witten equations and nonabelian Hodge (2022)  – ‎Liu 
  • Phases of N= 4 SYM, S–duality and nilpotent cones – ‎Balasubramanian

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