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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 / 32Jones’ PhD Thesis
Jones got his Docteurès Sciences 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
hyperfinite II1 factor R by the 3-cocyles H 3 (G ), inspired by Connes’
classification of cyclic actions.
If you have not heard of subfactors, here are two examples
RG ⊆ R and R ⊆ R o G .
Remark: The two subfactors are Fourier dual to each other.
I will explain the terminology soon.
6 / 32Factors 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 / 32Factors 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 / 32Invariants
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 / 32Invariants
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 / 32Jones Index
A II1 factor N forms a Hilbert space L2 (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.
Example: [R : RG ] = [R o G : R] = |G |.
Example: [R ⊗ M2 (C) : R] = 4.
Theorem (Jones index 1983)
2 π
{[M : N ]} = 4 cos : k ∈ N+ ∪ [4, ∞].
k +2
The Jones index below 4 are “quantum”.
Jones: The most challenging part is constructing these subfactors.
9 / 32Jones Subfactors and Temperley-Lieb Algebras
Subfactor with index λ → Temperley-Lieb algebra TL(λ):
ei2 = ei = ei∗ ;
ei ej = ej ei , |i − j| ≥ 2;
−1
ei ei±1 ei = λ ei .
It has a Markov trace τ ,
τ (xen ) = λ−1 τ (x), ∀x ∈ TLn .
where TLn is the subalgebra generated by {ei : 1 ≤ i ≤ n − 1}.
π
For a Jones index λ = 4 cos2 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 / 32Braid Group πi
Braid Group: λ = (q + q −1 )2 , q = e k+2 ,
q
σ i = −q 2 ei + ,
(q + q −1 )
σi σj = σj σi , |i − j| ≥ 2;
σ i σ i+1 σ i = σ i+1 σ i σ i+1 .
Pictorial representation: σ i braids the i th and i + 1th 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 / 32Jones 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 (xσ n ) = 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 + t 3 − t 4. = t −1 + t −3 − t −4 .
Reflection: t → t −1 .
12 / 32Fields Medal
Jones was awarded the Fields Medal at Kyoto in 1990 for these
breakthroughs.
13 / 32Quantum 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 / 32Representation Theory of Subfactors
Is there an interesting representation theory for subfactors?
15 / 32Representation 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 / 32Free-Group Subfactors
The reconstruction does not work for TL(λ), λ > 4.
ei2 = ei = ei∗ ; ei ej = ej ei , |i − j| ≥ 2; ei ei±1 ei = λ−1 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 / 32Free-Group Subfactors
The reconstruction does not work for TL(λ), λ > 4.
ei2 = ei = ei∗ ; ei ej = ej ei , |i − j| ≥ 2; ei ei±1 ei = λ−1 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 / 32Subfactors, 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 R L(F∞ )
asymptote of matrices random matrices
17 / 32Bimodule 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 M MM ,
generators: N MM and its dual M MN ;
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 / 32Principal 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 ⊗M MN ).
In particular, Jones’ subfactors are of type A, because Γ = Ak+1 ,
π
the type A Dynkin diagram, when [M : N ] = 4 cos2 k+2 .
For its adjacent matrix AΓ ,
kAΓ k2 ≤ [M : N ],
and “ = ” holds for amenable subfactors, e.g. for a finite Γ.
Bipartite graphs Γ, s.t. kAΓ k ≤ 2, are classified by ADE Dynkin
diagrams and extended ones.
Question: How about subfactors with Jones index at most 4?
19 / 32ADE 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 / 32Subfactors in CFT
In 1995, Longo and Rehren established the connection between
subfactors and conformal nets, a chiral CFT on S 1 .
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 S 1 and I c be the complement.
I The local algebra LI SU(2) supported at I is a factor,
I Locality: Jones-Wasserman subfactor LI SU(2) ⊆ LI c SU(2)0 ,
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 / 32ADE classification in CFT
Whether other subfactors below index 4 come from CFT?
22 / 32ADE 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 .
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 S 1
with central charge c < 1 by pairs of A − D2n − E6,8 Dynkin
diagrams whose Coxeter numbers are differed by 1.
22 / 32Big 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 / 32Big 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
Shakespeare: To be or not to be, that’s the question.
We would like to know.
23 / 32Haagerup Subfactor
Haagerup constructed a subfactor N ⊂ M with Jones
In 1994, √
index 5+2 17 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, g ρ, g 2 ρ},
with a non-commutative fusion rule,
g 3 = 1, g ρ = ρg 2 , ρ2 = 1 + ρ + g ρ + g 2 ρ.
One of Jones’ favorite questions:
Is the Haagerup subfactor from CFT?
Can one construct a CFT from the Haagerup subfactor?
24 / 32Small 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 / 32Composition 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
H 3 (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, b 2 = 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 / 32Composition 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
H 3 (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, b 2 = 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 / 32Kazhdan 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,
I RG ⊆ 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 / 32Parameterization
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 / 32Parameterization
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.
Answer: Planar algebras admit parameterizations.
28 / 32Planar 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),
I planar graphs: disjoin union of S 1 ;
1
I partition function Z : δ #S .
I Z has reflection positivity ⇐⇒ λ = δ 2 is a Jones index.
I Relation: S 1 = δ.
Remark: There are other non-trivial relations when λ < 4.
29 / 32Yang-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(δ 1 ) ∗ TJL(δ 2 ); (Z = Z1 Z2 ) [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 / 32Based 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 / 32Thank you for listening!
Thank you, Wendy, for pages 4,5!
Thank you, Vaughan, for everything!
32 / 32You can also read
