Subfactors in Memory of Vaughan Jones Zhengwei Liu Tsinghua University Math-Science Literature Lecture Series November 23, 2020, Harvard CMSA and Tsinghua YMSC 1 / 32 Sir Vaughan Frederick Randal Jones, a great New Zealand mathematician, suddenly passed away at Nashville, Tennessee, in the US on September 6, 2020, due to an ear infection and complications. This talk is dedicated to my advisor Vaughan Jones. Jones initiated the modern theory of subfactors in early 1980s and investigated this area for his whole academic life. Subfactor theory has both deep and broad connections with various areas in mathematics and physics. I will review some highlights in the developments of subfactors, based on insightful examples{Jones style. I am sorry for not mentioning many experts who have made substantial contributions in this area. 2 / 32 2020 ! 1952 (Time Machine in the Japanese Cartoon \Doraemon") 3 / 32 Childhood ! B.S. & M.S. I Jones was born in Gisborne, New Zealand on December 31, 1952 to parents Jim Jones and Joan Jones (ne Collins). I He developed his lifelong interest in math and science at St. Peters School and Auckland Grammar School. I After graduating from the University of Auckland with a B.Sc. in 1972 and an M.Sc. with First Class Honours in 1973, he was awarded a Swiss Government Scholarship. 4 / 32 Marriage While pursuing his PhD at University of Geneva in Switzerland, Jones met his future wife, Martha, whom he forever called Wendy. They married in Wendy's hometown of Westfield, New Jersey in 1979. 5 / 32 Jones' PhD Thesis Jones got his Docteur`esSciences in Mathematics from the University of Geneva in 1979, under the supervision of Andr´e Haefliger and Alain Connes. In his thesis he classified outer actions of a finite group G on the 3 hyperfinite II1 factor R by the 3-cocyles H (G), inspired by Connes' classification of cyclic actions. If you have not heard of subfactors, here are two examples G R ⊆ R and R ⊆ R o G: Remark: The two subfactors are Fourier dual to each other. I will explain the terminology soon. 6 / 32 Answer: Subfactors, e.g. R ⊆ R o G, for a countable group G. I A subfactor is an inclusion of factors N ⊆ M, which can be regarded as an action of a \quantum group" G on N , even though we do not see G and its action directly. Factors and Subfactors A factor is a von Neumann algebra with trivial center. Example: B(H), bounded operators on a Hilbert space. Murray-von Neumann classified factors by types In, II1, II1, III. A factor of type II1 is infinite dimensional with a (unique) trace. Examples: L(K) (acting on L2(K)), for an i.c.c. group K. L( lim Sn) L(F2): n!1 Connes 1973: Hyperfinite () Amenablity () · · · Hyperfinite II1 factor R: unique, smallest and universal for amenable groups. Question: How to recover group symmetries from R? 7 / 32 Factors and Subfactors A factor is a von Neumann algebra with trivial center. Example: B(H), bounded operators on a Hilbert space. Murray-von Neumann classified factors by types In, II1, II1, III. A factor of type II1 is infinite dimensional with a (unique) trace. Examples: L(K) (acting on L2(K)), for an i.c.c. group K. L( lim Sn) L(F2): n!1 Connes 1973: Hyperfinite () Amenablity () · · · Hyperfinite II1 factor R: unique, smallest and universal for amenable groups. Question: How to recover group symmetries from R? Answer: Subfactors, e.g. R ⊆ R o G, for a countable group G. I A subfactor is an inclusion of factors N ⊆ M, which can be regarded as an action of a \quantum group" G on N , even though we do not see G and its action directly. 7 / 32 Answer: Invariants. I Scalar: Jones index; I Graph: principal graph; I Ring: fusion ring; I Representation Category: standard invariant. Invariants Constructing a subfactor N ⊆ M by generators may look simple from its definition. (v.s. constructing a subgroup of a group.) However, one would get the whole factor by a random choice of generators. Question: How to detect different subfactors? 8 / 32 Invariants Constructing a subfactor N ⊆ M by generators may look simple from its definition. (v.s. constructing a subgroup of a group.) However, one would get the whole factor by a random choice of generators. Question: How to detect different subfactors? Answer: Invariants. I Scalar: Jones index; I Graph: principal graph; I Ring: fusion ring; I Representation Category: standard invariant. 8 / 32 Jones Index 2 A II1 factor N forms a Hilbert space L (N ) measured by its trace. It is the 1-dim standard N −module, denoted by N N . Hilbert N -modules are classified by the dimension d 2 (0; 1]. The Jones index of a subfactor N ⊆ M of type II1 [M : N ] := dimN M: G Example: [R : R ] = [R o G : R] = jGj. Example: [R ⊗ M2(C): R] = 4. Theorem (Jones index 1983) π f[M : N ]g = 4 cos2 : k 2 + [ [4; 1]: k + 2 N The Jones index below 4 are \quantum". Jones: The most challenging part is constructing these subfactors. 9 / 32 Jones Subfactors and Temperley-Lieb Algebras Subfactor with index λ ! Temperley-Lieb algebra TL(λ): 2 ∗ ei = ei = ei ; ei ej = ej ei ; ji − jj ≥ 2; −1 ei ei±1ei = λ ei : It has a Markov trace τ, −1 τ(xen) = λ τ(x); 8x 2 TLn: where TLn is the subalgebra generated by fei : 1 ≤ i ≤ n − 1g. 2 π For a Jones index λ = 4 cos k+2 , τ is positive semi-definite, and fei : i ≥ ng generate a factor Rn, (of hyperfinite type II1), by GNS construction. The subfactor R2 ⊆ R1 has Jones index λ. 10 / 32 Braid Group πi Braid Group: λ = (q + q−1)2, q = e k+2 , q σ = −q2e + ; i i (q + q−1) σi σj = σj σi ; ji − jj ≥ 2; σi σi+1σi = σi+1σi σi+1: th th Pictorial representation: σi braids the i and i + 1 strings. σ = ; σ−1 = ; Reidemeister move II: = ; Reidemeister move III: = . Remark: Jones constructed braid group representations and developed deep connections with integrable models and Drinfeld-Jimbo quantum groups. 11 / 32 Jones Polynomial Jones met Birman in 1984: Markov trace on the braid group ) knot invariant It was surprising that the Markov trace naturally comes from the trace of the II1 factor, τ(xσn) = τ(x); 8x 2 TLn; −! = ; (Reidemeister move I) therefore leading to a knot invariant, well-known as the Jones polynomial, by which Jones answered a series of old questions in knot theory in 1985. = t + t3 − t4: = t−1 + t−3 − t−4: Reflection: t ! t−1. 12 / 32 Fields Medal Jones was awarded the Fields Medal at Kyoto in 1990 for these breakthroughs. 13 / 32 Quantum Symmetries The early work of Jones in 1980's led surprising connections not only between von Neumann algebras and knot theory, but also quantum groups, lower dimensional topology, integrable systems, algebraic/conformal/topological quantum field theory. (AQFT, CFT, TQFT) The most surprising connection was Witten's interpretation of Jones polynomial (at roots of unity) as a path integral in Chern-Simon theory. These connections have deep influence in development of various modern areas of mathematics, such as quantum topology. Subfactors capture quantum symmetries, beyond group symmetries. 14 / 32 Yes, it is very rich for finite-index subfactors (of type II1). It includes the unitary representation theory of finite groups and quantum groups. Representation Ψ (Amenable) (Amenable) Subfactor Standard Invariant Reconstruction Φ Example: Jones subfactors with index λ ≤ 4 ! TL(λ). Popa 1994: ΨΦ = I and ΦΨ = I . An amenable subfactor can be reconstructed from its standard invariant. (Quantum Tannaka-Krein duality) Representation Theory of Subfactors Is there an interesting representation theory for subfactors? 15 / 32 Representation Theory of Subfactors Is there an interesting representation theory for subfactors? Yes, it is very rich for finite-index subfactors (of type II1). It includes the unitary representation theory of finite groups and quantum groups. Representation Ψ (Amenable) (Amenable) Subfactor Standard Invariant Reconstruction Φ Example: Jones subfactors with index λ ≤ 4 ! TL(λ). Popa 1994: ΨΦ = I and ΦΨ = I . An amenable subfactor can be reconstructed from its standard invariant. (Quantum Tannaka-Krein duality) 15 / 32 Yes, non-amenable subfactors, precisely free-group subfactors. Representation Ψ L(F1) Subfactor Standard Invariant Reconstruction ΦPS Popa introduced standard λ-lattices as an axiomatization of the standard invariant in 1995. Popa-Shlyakhtenko 2003: All standard invariants come from subfactors of L(F1), ΨΦPS = I . Free-Group Subfactors The reconstruction does not work for TL(λ), λ > 4. 2 ∗ −1 ei = ei = ei ; ei ej = ej ei ; ji − jj ≥ 2; ei ei±1ei = λ ei : Markov trace τ is positive, but fei : i ≥ ng do not generate a factor. Question: Is TL(λ) a standard invariant of a subfactor with index λ? 16 / 32 Free-Group Subfactors The reconstruction does not work for TL(λ), λ > 4. 2 ∗ −1 ei = ei = ei ; ei ej = ej ei ; ji − jj ≥ 2; ei ei±1ei = λ ei : Markov trace τ is positive, but fei : i ≥ ng do not generate a factor. Question: Is TL(λ) a standard invariant of a subfactor with index λ? Yes, non-amenable subfactors, precisely free-group subfactors. Representation Ψ L(F1) Subfactor Standard Invariant Reconstruction ΦPS Popa introduced standard λ-lattices as an axiomatization of the standard invariant in 1995. Popa-Shlyakhtenko 2003: All standard invariants come from subfactors of L(F1), ΨΦPS = I . 16 / 32 Subfactors, Free Probability, Random Matrices The reconstruction maps of Popa, Popa-Shlyakhtenko come from Voiculescu's free probability theory.