Higher categories and observables for generalised Turaev-Viro models

Transcription

1 Hiher teories nd oservles or enerlised Turev-Viro models Septemer 21, 2010 CLP 7 Cteories, Loi nd Phsis, Birminhm Ctherine Meusurer, Deprtment Mthemtik, Universität Hmur work with John W Brrett (in proress)

2 Motivtion Turev-Viro invrint teories, invrints o 3-mniolds, representtion theor o Hop lers,... stte sum models o (3d) quntum rvit topoloil quntum ield theor onorml ield theor, hiher ue theor losed trinulted mniold omple numer invrint under hne o trinultion trinulted mniold with oundr mp etween Hilert spes ssoited with oundr omponents phsil interprettion (TQT, quntum rvit): trnsition mplitudes etween eometries prolem otinin oservles prtil results or speii ses / ontet o quntum rvit so r: no enerl results or ormlism underlin mthemtil strutures?

3 Contents 1. enerlised Turev Viro models 2. Ide: oservles rom hiher teories 3. Cteor dt or oservles: SphCt 4. Deinition o the model 5. Evlution nd oservles 6. Reltion to Reshetikhin-Turev-Witten invrints 7. Outlook nd onlusions

4 1. enerlised Turev-Viro models [Turev, Viro] [Brrett, Westur] invrints o pieewise liner mniolds rom representtion theor o U q (su(2)) t root o unit enerlistion to non-deenerte semisimple spheril teories non-deenerte semisimple spheril teories [Brrett, Westur] non-deenerte pivotl usion teor in whih tr L tr R monoidl teor with strit duls: monoidl teor (C,, e) α β α α β toether with ontrvrint strit tensor untor : C C op, 1 C pivotl: olletion o morphisms ɛ : e (oevlution) or ll ojets whih stis - (α 1 ) ɛ (1 α ) ɛ omptiilit with morphisms - ɛ z (1 ɛ z 1 ) ɛ omptiilit with tensor produt ɛ (1 ɛ ) (ɛ 1 )1 - α α* z α* α

5 spheril: riht- nd let- tre ree tr L (α) ɛ (1 α) ɛ tr R (α) ɛ (α 1 ) ɛ α α tr L (α) tr R (α) non-deenerte usion teor: - End(e) C - Hom(, ) inite-dimensionl vetor spe over - : C C C, : C C op re C-(i)liner - pirin Hom(, ) Hom(, ) C, (α, β) tr(α β) is non-deenerte - initeness: the set o isomorphism equivlene lsses o simple ojets is inite C J Hom(,) Hom(,) - semisimpliit: there eists set J o non-trivil simple ( End() C ) ojets suh tht Hom(, ) Hom(, ) Hom(, ) is vetor spe isomorphism J enerlise rep. theor o q-deormed universl envelopin lers t roots o unit uildin loks o (enerlised) Turev-Viro invrint

6 enerlised Turev-Viro model [Turev, Viro] [Brrett, Westur] invrints o pieewise liner 3-mniolds rom non-de. semisimple spheril teories inredients trinulted 3-mniold M non-deenerte inite semisimple spheril teories dt (lellin) oriented edes simple ojets (reversed ede dul ojet) oriented trinles morphisms etween simple ojets t oundr (reversed orienttion dul morphism) d δ β tetrhedron mplitude α oriented tetrhedr 6j smol γ { } : Hom(, ) Hom(, d) Hom(, ) Hom( d, ) C d α β γ δ tr (δ (γ 1) (1 β) α) γ α : α δ β mplitude o lelled trinultion lelled trinultion mplitude Z(M, l) oriented tetrhedr t { } t t t d t t t

7 Turev Viro invrint C(M) K v K irreps dim q () 2 Z(M, l) oriented tetrhedr t { } t t t d t t t mplitude or trinultion nd lellin q-dimension dim q () tr(id ) e sum over ssinments o simple ojets to edes l:e e Z(M, l) dim q ( e ) losed trinulted 3-mniold M omple numer C(M) deines invrint o pieewise liner 3-mniolds: trinultions relted inite sequenes o Phner moves C(M) invrint under Phner moves h d j i h,i,j,k k d k h k i d j h i d j 1-4 move orthoonlit o 6j smols 2-3 move Biedenhrn Elliot reltion or mniolds with oundr: C(M) depends onl on dt on oundr

8 2. Ide: Oservles rom hiher teories phsil interprettion topoloil quntum ield theor, quntum rvit Turev-Viro model: quntum theor o Euliden 3d rvit with positive osmoloil onstnt interprettion o C(M) s stte sum (~ sttistil phsis models, disrete pth interl) M 2 M 1 question: oservles? dt t oundr M : phsil sttes (Hilert spe) Turev-Viro invrint: deines trnsition mplitudes, inner produt H M2 Hom(, ) lelled trinles () o M 2 H M1 Hom(, ) lelled trinles () o M 1 M M 1 M 2 nivel: oservles s untions o model dt nd dditionl prmeters epettion vlue insertion into stte sum O K v O(M, l)z(m, l) C(M) l:e e J e dim q ( e ) C(M) :HM 1 H M2 C φ ψ φ, ψ

9 prolem need dditionl dt / prmeters to deine oservles whih dt? prtil results: or U q (su(2)) inlusion o prtiles in spin om models Brrett, ri-isls, ri Mrtins reidel, Loupre, irirn, Livine... enerl pttern? enerl deinition? underlin mthemtil strutures? reltion to Reshetikhin-Turev-Witten invrints (invrints o oloured rion rphs)? ide: oservles rom hiher teories k m l phsil interprettion: oservles s deets in stte sum model α α d enerlised Turev-Viro model model vi luin individul lelled tetrhedr luin ondition: ojets nd morphisms on shred es mth ide: rel luin onditions in ontrolled w vi hiher teor theor

10 reled luin onditions ide: interpret spheril teor s 2-teor with sinle ojet onsider ssoited suteor o 2-Ct luin tetrhedr t es ojets nd morphisms on shred es relted untors : C C (1-morphisms in 2-Ct) k l () θ m (α) () () α β d luin tetrhedr round n ede onsisten ondition: or eh ede: pseudonturl trnsormtion etween ssoited untors ν : H (2-morphisms in 2-Ct) luin tetrhedr round verte onsisten ondition: or eh verte modiition etween ssoited pseudonturl trnsormtions Φ: σ τ µ ν id (3-morphisms in 2-Ct) M τ : L HM () H () H() L ν : H µ : L K K σ : M K

11 3. Cteor dt or oservles: SphCt need: omptiilit with model dt (non-deenerte semisimple spheril teor) untors, pseudonturl trnsormtions, modiitions orm r-teor with dulit opertions: SphCt SphCt otined rom r teor 2-Ct restrition to 2-teories with sinle ojet MonCt imposin omptiilit with struture o spheril teor SphCt ojets: spheril teories C (2-teories with sinle ojet) 1-morphisms: spheril untors : C D strit tensor untors etween spheril teories (α β) (α) (β) (e C )e D omptile with dulit (α ) (α) (ɛ )ɛ () rphil representtion: reions in 2d dirms () () () ()

12 2-morphisms spheril trnsormtions ν : etween spheril untors, : C D pseudonturl trnsormtions: ν () ν ojet ν o D toether with olletion o isomorphisms ν : ν () () ν tht re () () ν spheril ondition: ν () () ν () ν nturl ((α) 1 ν ) ν ν z (1 ν (α)) omptile with unit ν e 1 ν omptile with tensor produt ν z (1 () ν z ) (ν 1 (z) ) ν () () () () () () ν ν ν () ν () () () () ν () () ν () ν 3-morphisms modiitions Φ: ν µ etween spheril trnsormtions ν, µ : ν μ Φ ν () Φ μ () μ ν () Φ () μ μ morphisms Φ: ν µ tht ommute with morphisms ν,µ (1 () Φ) µ ν (Φ 1 () )

13 produt opertions r produt: omposition o spheril untors : C D : C E : D E omposition o spheril untors with spheril trnsormtions nd modiitions ν : H(ν) Hν : H H νk : K K ν H H (Hν) H(ν ) K K, : C D (νk) ν K() H(Φ) H : D E Φ K : B C H(μ) horizontl omposition o spheril trnsormtions nd modiitions µ, ν : π ν π µ : H ρ, π : H H (π µ) (π 1 µ ) (1 π µ ) Φ: ν µ Ψ Φ Ψ: π ρ Ψ Φ: π ν ρ µ ρ μ Ψ Φ (1 π Φ) (Ψ 1 µ ) (Ψ 1 ν ) (1 ρ Φ) μ (vertil) omposition o modiitions µ, ν, ρ : Φ: µ ρ Ψ: ν µ Φ Ψ ν μ Ψ Φ: ν ρ ρ

14 tensortor ρ H(ν) H or spheril trnsormtions ν :, invertile modiition σ ν,µ :(ρ) (Hν) (Kν) (ρ ) ρ : H K K H K K(ν) ρ ommutes with modiitions K ρ H K K(μ) H(μ) H(ν) H(Φ) ρ H ρ H K K(ν) K(Φ) K(μ) K H(ν) H ρ stisies enerlised YBE K(ρ) KN (μ) HM KM ν K(ρ) (μ) KM HM (μ) KN HM KM H(ρ) K(ρ) HN ν HM HN KN KN HM (μ) KN ν (μ) HN KN H(ρ) ν (μ) H(ρ)

15 dulit opertions dt rom SphCt to lel oriented edes nd es need dulit opertions or reversin orienttion strit untor o r teories *: SphCt SphCt vi dulit in spheril teor - trivil on spheril untors: - tion on spheril trnsormtions: ν : ν : (ν ) (ν ) - tion on modiitions: Φ: ν µ Φ : µ ν **1, ontrvrint with respet to horizontl nd vertil omposition spheril ondition morphisms ɛ deine modiition I ν :1 ν ν : e ν ν ν ν identities: restrition to: μ ν non-deenerte spheril teories invertile spheril untors H wek untor o r teories #: SphCt SphCt - tion on spheril untors: # 1 - tion on spheril trnsormtions ν : : ν # # ν # - tion on modiitions Φ: ν µ, ν, µ : : Φ # # Φ # ν ν ν ν ρ Φ Φ pirin o morphisms in spheril teories preserved tr D ( (α) (β)) tr C (α β) ontrvrint with respet to r produt nd horizontl omposition dirms: olds in plnes (invertiilit o untors olds nd usps re trivil!) omptiilit identit: # # 1 ν ρ ν

16 4. Deinition o the model dt or oservles oriented e (trinle) spheril untor : C C (reversed orienttion untor # ) µ H I oriented ede + hoie o djent tetrhedron spheril trnsormtion rom untors o djent es to identit µ : # HIJ # 1 C (reversed orienttion dul spheril trnsormtion µ :1 C # HIJ # ) J verte + hoie o miml tree ( li orderin o inident edes) μ2 μ3 μ4 + hoie o inident tetrhedron ( liner orderin o inident edes) μ1 μ5 modiition etween spheril trnsormtions o inident edes Φ: µ 1 (µ 2 1 )... (µ n n ) 1 1C μ7 μ6

17 prolem: how to inorporte dt or oservles into trinultion? ide reine trinultion: onsider 2nd derived omple (vi rentri sudivisions) rentri sudivision o tetrhedron 2nd derived omple o tetrhedron deomposition o M in polhedr ssoited with verties, edes, trinles nd tetrhedr tetrhedron e ede verte

18 Lellin with dt intrinsi dt oservle dt e j H ede tetrhedron () h α i (i) I J () μ () w n m ρ Ψ (h) j h α Θ μ i Σ z l k z spheril teor C SphCt model without oservles deines oservles e dt simple ojets,, in spheril teor C spheril trnsormtions 1 C 1 C (with simple ojets h,i,j) morphism α in spheril teor C spheril untor : C C ede dt spheril trnsormtion µ : H # I # J 1 C miml tree tetrhedron dt 4 modiitions verte dt modiition Φ: e v ρ Φ µ e id verte

19 5. Evlution nd oservles ormultion in terms o dirms dt ssined to polhedr dirms or SphCt nd spheril teor v w u () μ H() H () Φ t H() h h s z w w r α Θ j i j i h h v u t s w r z z tetrhedron: omposition lon shred edes losed strin dirm lelled tetrhedron t Z(t, l) C stte sum C(M, l Spht )K v lellins with intrinsi dt oriented tetrhedr t Z(t, l) oriented edes e dim q ( e )

20 evlution sum over lellins with intrinsi dt in interior usin identities (dirms) or SphCt nd or spheril teor stte sum or three-ll B 3 iven intrinsi dt on oundr oservle dt in interior nd on oundr evlution projetion o interior dt on oundr strin dirm or SphCt nd spheril teor z u v w h p r k s K H o t d stte sum or S 3 iven oservle dt in interior evlution projetion o interior dt on plne p o strin dirm or SphCt z invrint o rph in pieewise liner 3-mniold invrint under Phner moves whih do not involve edes deorted with oservle dt nd oundr u w r s t q

21 6. Reltion to Reshetikhin-Turev-Witten invrints Chern-Simons theor with ue roup H H (e: Euliden 3d rvit with Λ>0) TV invrints or U q (h) oservles: strin dirms or r teor ssoited with U q (h) quntistion C(M) TV C RT W (M) 2? RTW invrints or D(U q (h)) U q (h) U q (h) oservles: knot invrints or Drinel d doule D(U q (h)) question: reltion o oservles to RTW invrints (knot invrints)? reltion etween representtion teor o Hop ler A nd o Drinel d doule D(A) Hop ler A Drinel d doule onstrution qusitrinulr Hop ler D(A) in.dim.reps in.dim.reps monoidl teor C with duls enter onstrution rided monoidl teor with duls Z(C)

22 ide: onsider model with ll untors trivil Spht redues to enter Z(C) dirm omponents omposition ridin spheril trnsormtions ojets in Z(C) modiitions morphisms in Z(C) identities oservles z z z model with trivil untors on S 3 oservles: losed strin dirms or Z(C) i ll modiitions in dirm ridins link nd knot invrints or Z(C) otin RTW invrints or Z(C) s speil se rom TV model with oservles or C z Yn Bter Eqution z

23 7. Outlook nd onlusions oservles or enerlised Turev Viro models ide: rel luin onditions or tetrhedr deortion o model with dt rom r teor SphCt es spheril untors edes spheril trnsormtions verties modiitions deines invrints o rphs emedded in pieewise liner mniolds onsistent nd enerl notion o oservles or 3d stte sum models reover Witten-Reshetikhin-Turev invrints open questions pplitions to onrete ses ( U q (su(2)), inite roups,...) interprettion in ontet o quntum rvit (rvittionl intertion o point prtiles?)