Electromagnetism and Langlands

“Ben-Zvi, Sakellaridis and Venkatesh”

“The paper creates a two-way translation between periods and L-functions by recasting periods and L-functions in terms of a pair of geometric spaces used to study basic questions in physics”

“it makes progress on a long-held dream within a sweeping research initiative in mathematics called the Langlands program”

“Mathematicians working on questions in the program seek to build bridges between disparate areas to show how advanced forms of calculus (where periods originate) can be used to answer fundamental open questions in number theory (the home of L-functions), or how geometry can be brought to bear on basic questions in arithmetic”

“The new paper is one of the first to link the geometric and arithmetic sides of the program, which for decades have advanced largely in isolation from each other”

“It started in 1967 as a 17-page handwritten letter from Langlands, then a young professor at Princeton University, to Andre Weil, who was one of the best-known mathematicians in the world”

“Langlands proposed that there should be a way of pairing important objects from calculus called automorphic forms with objects from algebra called Galois groups”

“Automorphic forms are a generalization of periodic functions like the sine in trigonometry, whose outputs endlessly repeat as inputs grow. Galois groups are mathematical objects that describe how entities called fields (like the real or rational numbers) change when they are extended with new elements”

“Pairings like that between automorphic forms and Galois groups are called dualities. They suggest that different classes of objects mirror each other, which allows mathematicians to study either one in terms of the other.”

“In their 2012 book, Sakellaridis and Venkatesh studied a duality between periods, which are closely related to automorphic forms, and L-functions, which are infinite sums that attach to Galois groups”

“From a mathematical point of view, periods and L-functions are entirely different species of objects with no obvious common traits”

“Periods emerged as objects of mathematical interest in the work of Erich Hecke in the 1930s.”

“L-functions are infinite sums that have been used since the work of Leonhard Euler in the mid-18th century to investigate basic questions about numbers. The most famous L-function, the Riemann zeta function, is at the heart of the Riemann hypothesis, which can be viewed as a prediction about how prime numbers are distributed. The Riemann hypothesis is arguably the most important unsolved problem in math.”

“Because they link objects that, on their face, have nothing in common, dualities are powerful. You could stare at a lineup of mathematical objects forever and not perceive how L-functions and periods match up.”

“To translate between superficially incommensurate things, you need to find common ground. One way to do that for objects like L-functions and periods, which originate in number theory, is to associate them with geometric objects.”

“At one of those early talks, Venkatesh explained the need to find a type of geometric object that could index both periods and L-functions, and he described some of his recent progress in that direction. It involved trying to use geometric spaces from an area of math called symplectic geometry, which Ben-Zvi was familiar with from his work in the geometric Langlands program.”

“Sometimes researchers take inspiration from physics to prove purely mathematical results. For example, in a 2008 paper, the physicists Davide Gaiotto and Edward Witten showed how geometric spaces related to quantum field theories of electromagnetism fit into the geometric Langlands program.”

“These spaces come in pairs, one for each side of the electromagnetic duality: Hamiltonian G-spaces and their dual: Hamiltonian Ğ-spaces (pronounced G-hat spaces).”

“Ben-Zvi had absorbed the Gaiotto-Witten paper when it came out, and he had used the physics framework they provided to think about questions in geometric Langlands. But that work — let alone the physics paper that helped motivate it — had no connection at all to the original Langlands program.”

“That is, until Ben-Zvi found himself in the audience at the IAS listening to Venkatesh. He heard Venkatesh explain that following their 2012 book, he and Sakellaridis had come to believe that the correct geometric way to think about periods was in terms of Hamiltonian G-spaces. But Venkatesh allowed that they didn’t know what kind of geometric object to pair with L-functions.”

“That set off bells for Ben-Zvi. Once Sakellaridis and Venkatesh had connected periods with Hamiltonian G-spaces, it became immediately clear what the dual geometric objects for L-functions should be: those Ğ-spaces that Gaiotto and Witten had said were the dual of G-spaces.”

“For Ben-Zvi, all these dualities, between arithmetic, geometry and physics, seemed to be converging. Even though he didn’t understand all the number theory, he was convinced it was all part of “one big, beautiful picture.””

“the duality between periods and L-functions translates into a geometric duality that makes sense within the geometric Langlands program and originates in the duality between electricity and magnetism”

“Physics and arithmetic become echoes of each other, in a way that echoes across the Langlands program”