The Langlands Correspondence in Physics

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In summary, the conversation revolves around the difficulty of understanding quantum field theory in terms of the geometric Langlands program. The participants also discuss the relevance of classical Langlands and the limitations of understanding in terms of continuum geometry. There is also mention of Witten's paper on quantum field theory and the work of Connes, as well as a request for further explanation in physicsy terms.
  • #1
Jim Kata
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Does anybody understand how to put quantum field theory in terms of the geometric Langlands program? I'm kind of reading Witten's paper Quantum Field theory, Grassmanians, and algebraic curves. Also, the more recent work of Connes shows that his group of diffeographisms is motivic galois group. I have not read Witten and Kapustins mammoth paper. Any ideas about Langlands in physics? Teach me what you know. Mind you, my knowledge is very physics based so if you can't explain in physicsy terms what the ring of adele's is you might lose me.
 
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  • #2
Jim Kata said:
Does anybody understand how to put quantum field theory in terms of the geometric Langlands program?

No. Nobody does. Indeed, that's rather like asking "Does anybody have a verifiable theory of quantum gravity?"
 
  • #3
Jim Kata said:
Does anybody understand how to put quantum field theory in terms of the geometric Langlands program?

Why geometric Langlands? Classical Langlands might be more relevant than the kind of continuum geometry that is studied as a result of, say, string theory prejudices.
 

1. What is the Langlands Correspondence in Physics?

The Langlands Correspondence in Physics is a mathematical theory that connects two seemingly unrelated areas of mathematics and physics: number theory and quantum field theory. It was first proposed by mathematician Robert Langlands in the 1960s, and has since been developed and applied by various researchers in both fields.

2. How does the Langlands Correspondence relate to number theory?

The Langlands Correspondence in Physics is based on the idea that there is a deep connection between the properties of prime numbers (which are fundamental in number theory) and the symmetries of quantum field theories. It suggests that certain mathematical objects in number theory, such as modular forms, can be used to describe the behavior of particles in quantum field theories.

3. What is the significance of the Langlands Correspondence in physics?

The Langlands Correspondence in Physics has been described as one of the most profound and mysterious connections between mathematics and physics. It has led to new insights and advancements in both fields, and has the potential to help solve some of the most challenging problems in mathematics and physics.

4. What are some examples of applications of the Langlands Correspondence in physics?

The Langlands Correspondence has been applied to various areas of physics, including string theory, quantum gravity, and condensed matter physics. It has also been used to study topological quantum field theories, which have important applications in areas such as quantum computing and quantum information theory.

5. What are some current developments and open questions in the Langlands Correspondence?

There are ongoing efforts to further develop and understand the Langlands Correspondence in physics. Some open questions include finding a complete and rigorous mathematical proof of the correspondence, extending it to other areas of physics, and exploring its connections to other mathematical theories such as algebraic geometry and representation theory.

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