Local Representations of Loop Braid Groups1
Eric Rowell
Dallas, January 2015
Supported by USA NSF grant DMS1108725
joint work with Zhenghan Wang (Microsoft/UCSB), Paul
Martin (Leeds), Zoltan Kadar (Leeds)
1
http://arxiv.org/abs/1411.3768
Motivation from (2 + 1)TQFT
In 2D anyonic topological quantum computation, braiding gates
give representations of the Braid Group Bn :
Generators σi , i = 1, . . . , n − 1 satisfying:
(R1) σi σi+1 σi = σi+1 σi σi+1
(R2) σi σj = σj σi if |i − j| > 1
1
σi
7→
i
...
i+1
n
...
Motions of n points in a disk.
Computation
Physics
output
measure
apply gates
particle
exchange
initialize
create
particles
vacuum
See Z. Wang’s talk for details...
3D Picture
In principle, (3 + 1)TQFT
representations of the group of
Motions of n component (oriented) unlink in a ball.
…
Two operations:
Loop interchange
si :
↔
and Leapfrogging (read upwards):
σi :
The Loop Braid Group LB n is generated by
s1 , . . . , sn−1 , σ1 , . . . , σn−1 satisfying:
Braid relations:
(R1) σi σi+1 σi = σi+1 σi σi+1
(R2) σi σj = σj σi if |i − j| > 1
Symmetric Group relations:
(S1) si si+1 si = si+1 si si+1
(S2) si sj = sj si if |i − j| > 1
(S3) si2 = 1
Mixed relations:
(M1) σi σi+1 si = si+1 σi σi+1
(M2) si si+1 σi = σi+1 si si+1
(M3) σi sj = sj σi if |i − j| > 1
Remark
Replacing (M1) with (M1’) σi+1 σi si+1 = si σi+1 σi gives isomorphic
group OLB n .
Origins
Let C = C1 ∪ C2 ∪ · · · ∪ Cn denote the n-component unlink.
1. (Dahm 1962 thesis, Goldsmith 1981) Motion groups:
M(R3 , C ) ,→ Aut π1 (R3 − C )
2. (McCool 1986) Basis-Conjugating Automorphisms of Free
Groups (presentation).
3. (Fenn, Rimanyi, Rourke 1997) Welded braid
group=LB n =Braid-permutation group
4. (Kauffman 1999) Virtual braid group has LB n as quotient.
5. (Baez, Crans, Wise 2006) presentation, application to 4d BF
TFT.
Topological Questions
Question
Is there a“trace closure” analogous to the braid picture? A Markov
theorem? Alexander Theorem?
Yes: glue in cylinders. Various authors–Kamada, Kauffman,
Rourke...
Question
What quantum invariants can we expect from (3 + 1)TQFTs?
Question
What is your favorite (3 + 1)TQFT?
Question
Which representations of Bn extend to LB n ?
Example
The (inverse

Ii−1
σi 7→  0
0
of the) Burau
 representation:
0
0
0
1

M
0
where M =
t 1−t
0 In−i−1


Ii−1 0
0
0 1
0  where P =
extends: si 7→  0 P
1 0
0
0 In−i−1
Definition
If R ∈ End(V ⊗ V ) satisfies:
(R ⊗ I )(I ⊗ R)(R ⊗ I ) = (I ⊗ R)(R ⊗ I )(I ⊗ R)
we say (V , R) is a Braided Vector Space.
ρR (σi ) = I ⊗i−1 ⊗ R ⊗ I ⊗n−i−1 gives ρR : Bn → GL(V ⊗n )
Question
For which (V , R) is there an S ∈ End(V ⊗ V ) such that
ρR (si ) = I ⊗i−1 ⊗ S ⊗ I ⊗n−i−1 is an extension to
ρR : LB n → GL(V ⊗n )?
Example

1
0
R=
0
−1

0 0 1
1 −1 0
 cannot be extended.
1 1 0
0 0 1
(1)
Definition
A BVS (V , R) is of (right) group-type if there is an ordered basis
X := [x1 , . . . , xn ] of V and gi ∈ GL(V ) such that
R(z ⊗ xi ) = xi ⊗ gi (z) for all i and z ∈ V .
Fact
I
I
I
(V , R) is of group-type if and only if V ∈ G
G YD for
G = hg1 , · · · , gn i.
If |G | < ∞ then G YD ∼
= Rep(DG )
G
(Galindo-Rowell) If (V , R) is of group-type then ρR (Bn ) is
virtually abelian (i.e. abelian A C ρR (Bn ) with
[ρR (Bn ) : A] < ∞).
Proposition (Kadar,Martin,R,Wang)
If (V , R) is a BVS of right group-type then
ρR (si ) = IV⊗i−1 ⊗ S ⊗ IV⊗n−i−1 with S(xi ⊗ xj ) := xj ⊗ xi defines an
extension of ρR to LB n .
Example
Set hj (xi ) = xti+(1−t)j on Cm with t, t − 1 6≡ 0 (mod m) (indices
modulo m). Image of extension factors over AGLn−1 (Zm ).
Conjecture (Slingerland, modified)
BVS (V , R) extends only if it is of group-type. “Local
representations come from groups.”
Question
What are the images of an extended group-type BVS
LB n -representation?
Question
Do such materials exist in nature?
Maybe...see Owczarek’s talk...
Thank you!