Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence Alexander B. Atanasov 1 1 Dept. of Mathematics Yale University May 13, 2018 Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Overview 1 Overview of the Langlands Program 2 S -duality in the twisted 4D N = 4 theory 3 Instantons and Monopoles in Gauge Theory 4 ‘Hooft Lines Revisited Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Goal: Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Goal: To understand the Langlands correspondence in terms of topologically twisted N = 4 super Yang-Mills gauge theory Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Conjecture (Langlands) To each n-dimensional representation of the absolute Galois group, there is a corresponding automorphic representation of GL n ( Q ) so that the Frobenius eigenvalues of the Galois representation agree with the Hecke eigenvalues of the automorphic representation. Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Q: What are Galois representations? Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Q: What are Galois representations? A: They are n-dimensional representations of Gal ( Q / Q ) . Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Q: What are automorphic representations? Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Adeles Definition (Ring of adeles) The ring of adeles of Q is defined as res � A Q := R × Q p , p prime where Q p denotes the p -adic completion of the rationals. Here R can be viewed as the completion at p = ∞ and the above product is restricted in the sense that: res � � Q p := ( x p ) ∈ Q p | x p ∈ Z p for all but finitely many p . p prime p prime Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Automorphic Representations GL n ( Q ) � GL n ( A Q ) � GL n ( Q ) . Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Automorphic Representations GL n ( Q ) � GL n ( A Q ) � GL n ( Q ) . So we have GL n ( Q ) � Fun ( GL n ( Q ) \ GL n ( A Q )) Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Automorphic Representations GL n ( Q ) � GL n ( A Q ) � GL n ( Q ) . So we have GL n ( Q ) � Fun ( GL n ( Q ) \ GL n ( A Q )) This can be decomposed into irreducible representations, which are known as the automorphic representations of GL n ( Q ). Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Automorphic Representations GL n ( Q ) � GL n ( A Q ) � GL n ( Q ) . So we have GL n ( Q ) � Fun ( GL n ( Q ) \ GL n ( A Q )) This can be decomposed into irreducible representations, which are known as the automorphic representations of GL n ( Q ). Though not absolutely precise, this is a good first-order description of what an automorphic representation is. Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Langlands over Finite Fields Definition (Adele Ring for F q ( t )) The ring of adeles of F q ( t ) is defined as res � F q (( t − x )) A F q ( t ) := x ∈ P 1 ( F q ) and the above product is restricted as before in the sense that all but finitely many terms in this product over x lie in F q [[ t − x ]]. Here the completion at the point at infinity corresponds to F q ((1 / t )). Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Langlands over Finite Fields We naturally have that � O F q ( t ) := F q [[ t − x ]] x ∈ P 1 ( F q ) sits inside A F q ( z ) . Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Langlands over Finite Fields Automorphic representations → GL n ( O F )-invariant functions on GL n ( F ) \ GL n ( A F ) Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft Langlands over Finite Fields Automorphic representations → GL n ( O F )-invariant functions on GL n ( F ) \ GL n ( A F ) Galois representations → representations of ´ etale fundamental group (in the unramified case) Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft How does this translate into geometry? Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft How does this translate into geometry? Guiding principle 1: Weil’s Uniformization Theorem Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft How does this translate into geometry? Guiding principle 1: Weil’s Uniformization Theorem Theorem (Weil Uniformization) Take F the function field for a curve C over F q . There is a canonical bijection as sets between G ( F ) \ G ( A F ) / G ( O F ) and the set of G-bundles over C. Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft How does this translate into geometry? Guiding principle 1: Weil’s Uniformization Theorem Theorem (Weil Uniformization) Take F the function field for a curve C over F q . There is a canonical bijection as sets between G ( F ) \ G ( A F ) / G ( O F ) and the set of G-bundles over C. Moreover, there exists an algebraic stack denoted by Bun G ( C ) whose set of F q points are in canonical bijective correspondence with this set. Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft How does this translate into geometry? Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Overview of the Langlands Program S -duality in the twisted 4D N = 4 theory Instantons and Monopoles in Gauge Theory ‘Hooft How does this translate into geometry? Guiding principle 2: ´ Etale Fundamental Group Alexander B. Atanasov VFU Magnetic Monopoles, ‘t Hooft Lines, and the Geometric Langlands Correspondence
Recommend
More recommend
