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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!