Subfactors-Jones.Pdf

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 DocteurèsSciences in Mathematics from the University of Geneva in 1979, under the supervision of André 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,II∞, 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→∞ 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,II∞, 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→∞ 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

I Scalar: Jones index;

I Graph: principal graph;

I Ring: fusion ring;

I Representation Category: standard invariant.

8 / 32 Jones Index

2 AII1 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 ∈ (0, ∞].

The Jones index of a subfactor N ⊆ M of type II1

[M : N ] := dimN M.

G Example: [R : R ] = [R o G : R] = |G|. Example: [R ⊗ M2(C): R] = 4. Theorem (Jones index 1983)

π {[M : N ]} = 4 cos2 : k ∈ + ∪ [4, ∞]. 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 , |i − j| ≥ 2; −1 ei ei±1ei = λ ei .

It has a Markov trace τ,

−1 τ(xen) = λ τ(x), ∀x ∈ TLn.

where TLn is the subalgebra generated by {ei : 1 ≤ i ≤ n − 1}. 2 π For a Jones index λ = 4 cos k+2 , τ is positive semi-deﬁnite, and {ei : i ≥ n} 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 , |i − j| ≥ 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), ∀x ∈ 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(F∞) 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(F∞), ΨΦPS = I .

Free-Group Subfactors The reconstruction does not work for TL(λ), λ > 4.

2 ∗ −1 ei = ei = ei ; ei ej = ej ei , |i − j| ≥ 2; ei ei±1ei = λ ei .

Markov trace τ is positive, but {ei : i ≥ n} 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 , |i − j| ≥ 2; ei ei±1ei = λ ei .

Markov trace τ is positive, but {ei : i ≥ n} 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(F∞) 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(F∞), ΨΦPS = I . 16 / 32 Subfactors, Free Probability, Random Matrices

The reconstruction maps of Popa, Popa-Shlyakhtenko come from Voiculescu’s free probability theory. Guionnet-Jones-Shlyakhtenko gave another reconstruction map ΦGJS in 2010 using subfactor planar algebras, and investigated further connections with free probability and random matrices.

standard invariant amenable non-amenable subfactors of hyperﬁnite RL(F∞) asymptote of matrices random matrices

17 / 32 Bimodule Category

Hilbert bimodules were studied by Connes and Popa in early 1980s, and used by Ocneanu in subfactor theory in 1994. Bisch 1995: standard invariant ↔ bimodule category. Bimodule category of a ﬁnite-index subfactor N ⊂ M:

I 0-morphism: N , M;

I 1-morphisms: bimodules units: N NN and MMM, generators: N MM and its dual MMN ; I 2-morphism: bimodule map;

I Tensor functor ⊗: Connes’ fusion. Ocneanu 1994: Subfactors −→ bimodule category; (ﬁnite) bimodule category −→ Turaev-Viro 3D TQFT.

I Boltzmann weight of 3-cells: F -symbols.

18 / 32 Principal Graphs

The principal graph Γ of a ﬁnite-index subfactor N ⊆ M is the induction-restriction graph between irreducible N − N bimodules and N − M bimodules. (w.r.t. ⊗N MM and ⊗MMN ).

In particular, Jones’ subfactors are of type A, because Γ = Ak+1, 2 π the type A Dynkin diagram, when [M : N ] = 4 cos k+2 . For its adjacent matrix AΓ,

2 kAΓk ≤ [M : N ],

and “ = ” holds for amenable subfactors, e.g. for a ﬁnite Γ.

Bipartite graphs Γ, s.t. kAΓk ≤ 2, are classiﬁed by ADE Dynkin diagrams and extended ones. Question: How about subfactors with Jones index at most 4?

19 / 32 ADE Classiﬁcation

I All extended ADE Dynkin diagrams are principal graphs. (subgroups of SU(2), McKay correspondence 1980)

I Ocneanu 1988: Only An, D2n, E6 and E8 are principal graphs of subfactors below index 4, no D2n+1 nor E7. (“subgroups” of quantum SU(2), NOT Hopf-subalgebra) Key idea: A subfactor defines a flat connection on its principal graph Γ. There is no flat connection on D2n+1 nor E7. I Bases on a series of work of Jones, Ocneanu, Izumi, Kawahigashi and Popa, amenable subfactors with index λ ≤ 4 are completely classified in 1994.

20 / 32 Subfactors in CFT In 1995, Longo and Rehren established the connection between subfactors and conformal nets, a chiral CFT on S1. Jones’ type A subfactors could be recovered from projective positive energy representations of the loop group LSU(2). Let I be a connect interval of S1 and I c be the complement.

I The local algebra LI SU(2) supported at I is a factor, 0 I Locality: Jones-Wasserman subfactor LI SU(2) ⊆ LI c SU(2) , here ’ indicates the commutant.

I Vacuum representation: trivial subfactor ( ⇐⇒ Haag duality in QFT).

I Standard representation: Jones’ type A subfactors. Remark: The subfactors from CFT are of type III, Jones’ subfactors are of type II1. They are not really isomorphic, but they share the same standard invariant. Remark: The braid group arises naturally in CFT.

21 / 32 Yes.

Z2 orbifold of A4n−3 → D2n. Feng Xu’s α-induction of conformal inclusions in 1998: SU(2)10 ⊂ Spin(5)1 → E6; SU(2)28 ⊂ (G2)1 → E8.

Goddard-Kent-Olive 1986: unitary representations of (super) Virasoro algebras, (minimal models ↔ type A subfactors.) Cappelli-Itzykson-Zuber 1987: ADE classification of modular invariance of SU(2)k WZW models. Böckenhauer-Evans-Kawahigashi fully established the connection between subfactors and modular invariance in several papers around 2000, based on α-induction and Ocneanu’s early observations. Kawahigashi-Longo in 2004 classified local conformal nets on S1 with central charge c < 1 by pairs of A − D2n − E6,8 Dynkin diagrams whose Coxeter numbers are differed by 1.

ADE classiﬁcation in CFT

Whether other subfactors below index 4 come from CFT?

22 / 32 ADE classiﬁcation in CFT

Whether other subfactors below index 4 come from CFT? Yes.

Z2 orbifold of A4n−3 → D2n. Feng Xu’s α-induction of conformal inclusions in 1998: SU(2)10 ⊂ Spin(5)1 → E6; SU(2)28 ⊂ (G2)1 → E8.

22 / 32 Shakespeare: To be or not to be, that’s the question. We would like to know.

Big Question

The ADE classiﬁcation in subfactor theory (1983-2000) has been considered as quantum McKay correspondence. No doubt, the ideas about SU(2)k work for Drinfeld-Jimbo quantum groups Gk of other Lie types. We obtain a large family of subfactors in this way. Big Question: Are all subfactors from CFT? Can one construct a CFT from a subfactor? Jones type A subfactors ←→ minimal models

23 / 32 Big Question

23 / 32 Haagerup Subfactor

In 1994,√ Haagerup constructed a subfactor N ⊂ M with Jones 5+ 17 index 2 and principal graph • • • • • • • • • •

This subfactor has the smallest index above 4 with a ﬁnite principal graph. It has six irreducible N − N bimodules {1, g, g 2, ρ, gρ, g 2ρ}, with a non-commutative fusion rule,

g 3 = 1, gρ = ρg 2, ρ2 = 1 + ρ + gρ + g 2ρ.

One of Jones’ favorite questions: Is the Haagerup subfactor from CFT? Can one construct a CFT from the Haagerup subfactor?

24 / 32 Small Index Subfactors

Subfactors not apparently from CFT, appearing in the small-index classiﬁcation: √ I Haagerup 1994, λ ≤ 3 + 3 , (1) Haagerup 1994, λ ∼ 4.30 (2) Asyeda-Haagerup 1999, λ ∼ 4.56 (3) Extended Haagerup, λ ∼ 4.37, constructed by Biglow-Morrison-Peters-Snyder 2009 √ I Jones-Morrison-Snyder 2014, λ ≤ 5, (= 3 + 4) Izumi’s near groups 1993 which generalized E6 and Haagerup, and related ones (Morita equivalence and orbifolds) √ I Afazly-Morrison-Penneys 2020 or 2021, λ ≤ 5.25, (∼ 3 + 5) Composition subfactors at index: √ π π 3 + 5 = 4 cos2 × 4 cos2 ∼ 2 × 2.618. 4 5

25 / 32 Liu 2015: Subfactors exist ⇐⇒ n = 0, 1, 2, 3. Izumi-Morrison-Penneys proved independently for n ≤ 10. Question: How to implement non-trivial compositions in CFT?

Composition Subfactors Bisch-Haagerup 1996: Classiﬁed RG ⊆ R ⊆ R × H, for a group K generated by ﬁnite subgroups G and H. by certain 3-cocyles in H3(K). First non-group-like composition: N ⊂ P ⊂ M, π [P : N ] = 2, [M : P] = 4 cos2 ∼ 2.618 . 5 Fusion rule of irreducible P − P bimodules:

< a, b : a2 = 1, b2 = b + 1, (ab)n = (ba)n > .

Bisch-Haagerup 1994: Subfactors exist for which n ∈ N? Jones and Bisch-Haagerup bet for 3 boxes of beers on this question.

26 / 32 Composition Subfactors Bisch-Haagerup 1996: Classiﬁed RG ⊆ R ⊆ R × H, for a group K generated by ﬁnite subgroups G and H. by certain 3-cocyles in H3(K). First non-group-like composition: N ⊂ P ⊂ M, π [P : N ] = 2, [M : P] = 4 cos2 ∼ 2.618 . 5 Fusion rule of irreducible P − P bimodules:

< a, b : a2 = 1, b2 = b + 1, (ab)n = (ba)n > .

Bisch-Haagerup 1994: Subfactors exist for which n ∈ N? Jones and Bisch-Haagerup bet for 3 boxes of beers on this question. Liu 2015: Subfactors exist ⇐⇒ n = 0, 1, 2, 3. Izumi-Morrison-Penneys proved independently for n ≤ 10. Question: How to implement non-trivial compositions in CFT?

26 / 32 Kazhdan property T of Subfactors Connes-Jones 1985: An i.c.c. group K has Kazhdan property T ⇐⇒ the factor L(K) has Kazhdan property T . Bisch-Nicoara-Popa 2007: There is a continuous family of non-isomorphic hyperﬁnite subfactors with the same standard invariant. Standard invariants do not classify non-amenable hyperﬁnite subfactors.

I When K has property T , generated by G and H, G I R ⊆ R ⊆ R × H has property T ;(R ⊆ R o K has.) + I A subfactor can be rescaled by a scalar p ∈ R . For the isomorphic ones, the scalars p form a subgroup of R+, called the Murray-von Neumann fundamental group F , which is countable for a property T subfactor. I Example: non-isomorphic subfactors p(RZ2 ⊆ R ⊆ R o Z3), + p ∈ R /F , for a property T quotient K of Z2 ∗ Z3,. Remark: Amenable ⇔ Hyperﬁnite, for factors, not for subfactors. Popa-Vaes 15: non-group-like subfactors with Kazhdan property T . 27 / 32 Answer: Planar algebras admit parameterizations.

Parameterization

Temperley-Lieb algebras TL(λ), as the standard invariants of Jones type A subfactors, can be parameterized by the Jones index λ, leading to the continuous parameter of the Jones polynomial. Question: How to parameterize a family of subfactors, or their standard invariants?

I Beneﬁt: We can study common properties and invariants for a family of subfactors.

I Cost: Positivity will be broken after parametrization. It could be challenging to recover positivity. The Markov trace τ is positive on TL(λ) ⇐⇒ λ is a Jones index.

28 / 32 Parameterization

I Beneﬁt: We can study common properties and invariants for a family of subfactors.

I Cost: Positivity will be broken after parametrization. It could be challenging to recover positivity. The Markov trace τ is positive on TL(λ) ⇐⇒ λ is a Jones index. Answer: Planar algebras admit parameterizations.

28 / 32 Planar Algebras Jones introduced subfactor planar algebras as an axiomatization of the standard invariants in 1999, (Planar Algebras I, 1990-1999,) integrating various ideas in von Neumann algebras, knot theory, representation theory, category theory, TQFT and integrable models. A planar algebra P consists of a set of labels (for vertices, edges and faces) and a multiplicative partition function Z on labelled planar graphs (invariant under planar isotopy). local linear relations=Kernel(Z). Reflection positive Z → Subfactor planar algebras → Subfactors Remark: Reflection positivity is crucial in QFT & statistical physics. Example: Temperley-Lieb-Jones Planar algebras TLJ(δ), 1 I planar graphs: disjoin union of S ; #S1 I partition function Z: δ . 2 I Z has reflection positivity ⇐⇒ λ = δ is a Jones index. 1 I Relation: S = δ. Remark: There are other non-trivial relations when λ < 4. 29 / 32 Yang-Baxter Relation Planar Algebras Skein theory: Evaluate partition function consistently by relations. (Similar to Conway’s linear skein theory and Kuperburg’s spider.) Bisch-Jones initiated in 2000 a classification of subfactor planar algebras generated by single 4-valent label with “simple relations”. Liu in 2015 thesis classified singly generated Yang-Baxter relation planar algebras: (1) TJL(δ1) ∗ TJL(δ2); (Z = Z1Z2) [Bisch-Jones 1997] (2) BMW (q, r); (Z =Kauffman polynomial) [Murakami 1987, Birman-Wenzl 1989; Kauffman 1989.] (3) C (q). (unexpected family NOT from knot theory) Classifying q s.t. Z has reflection positivity was most challenging. Remark: Witten gave a physical interpretation of the parameterized Jones polynomial in 5D TQFT, in his talk on November 13, 2020 in Math-Science Literature Lecture Series. The parameterized family (3) C (q) suggested a parameterization of conformal inclusions SU(n)n+2 ⊂ SU(n(n + 1)/2)1 for n ∈ C. Any physical interpretation? 30 / 32 Based on the type A subfactors, we have reviewed a few connections, certainly not all, between subfactors and several areas:

I von Neumann algebras

I integrable models

I knot theory & lower dimensional topology

I quantum groups & representation theory & category theory

I analytic properties (amenable & Kazhdan property T )

I free probability & random matrices

I TQFT & AQFT & CFT 2015-2020: New connections with

I Fourier analysis (→ Quantum Fourier Analysis)

I Thompson group

I modular forms We will continue the story following Jones’s spirit.

31 / 32 Thank you for listening! Thank you, Wendy, for pages 4,5! Thank you, Vaughan, for everything!

32 / 32