Groupoidification in Physics

Transcription

1 Groupoidificaion in Phyic Jeffrey C. Moron November 11, 2010 Abrac Noe for a eminar in he IST TQFT Club abou he Baez-Dolan groupoidificaion and i exenion, a applied o ome oy model in phyic. 1 Inroducion 1.1 Ouline Moivaion: Caegorify a quanum mechanical decripion of ae and procee. Claical Se S: A e whoe elemen are configuraion of a yem Caegorie X: A groupoid wih: Ob: configuraion Mor: ymmerie of configuraion Quanum L2 (S): Vecor pace of ae (in fac, Hilber pace) Λ(X): 2-vecor pace of ae (in fac, 2-Hilber pace) We propoe ha he configuraion pace of phyical yem hould be repreened a groupoid (or ack), baed on local ymmerie. A proce relaing wo yem hrough ime i decribed uing a groupoid of hiorie in a pan of groupoid, wih map o ar and end configuraion. Thi i doing phyic in he monoidal (2-)caegory Span(Gpd). Degroupoidificaion i a funcor urning hi ino phyic in Vec (or Hilb), a uual in quanum mechanic. 2-Linearizaion give a more complee equivalence-invarian Λ for Span(Gpd). I provide a way o do phyic in 2Hilb. Boh invarian rely on a pull-puh proce, and ome form of adjoinne. Applicaion: Foundaional phyic uch a quanum harmonic ocillaor; Wien-ype ETQFT (help inerpre phyical example). 1

2 2 Groupoid and Span 2.1 Groupoid and Sack Definiion 1 A groupoid G (in Se) i a caegory in which all morphim are inverible. Tha i, a a caegory, coni of wo e G 0 (of objec) and G 1 (of morphim/arrow) ogeher wih rucure map: G 1 G0 G 1, i ( ) G 1 G 0 G 1 1 G 1 (1) which define ource, arge, ideniie, parially-defined compoiion, and invere, ayifying ome properie making a groupoid a muli-objec generalizaion of a group. Morphim (arrow) of a groupoid can be compoed if he ource of one arrow i he arge of he oher. Thi can be defined where G 0 and G 1 are e, opological pace, manifold, ec. (Then he map mu be nice in a uiable ene in each cae.) Definiion 2 There i a 2-caegory Gpd wih: Objec: Groupoid (caegorie whoe morphim are all inverible) Morphim: Funcor beween groupoid 2-Morphim: Naural ranformaion beween funcor Groupoid provide a good way of hinking abou local ymmery. E.g. he ranformaion groupoid S//G come from a e S wih an acion of he group G: objec are elemen of S, morphim correpond o group elemen. Example 1 Some relevan groupoid: Any e S can be een a a groupoid wih only ideniy morphim Any group G i a groupoid wih one objec Given a e S wih a group-acion G S S yield a ranformaion groupoid S /G whoe objec are elemen of S; if g() = hen here i a morphim g : Given a differeniable manifold M, he fundamenal groupoid Π 1(M) which ha objec x M and morphim homoopy clae of pah in M. Given a differeniable manifold M and Lie group G, he groupoid A G(M) of principal G-bundle and bundle map; and he groupoid A G(M) of FLAT G-bundle and map. Phyically, groupoid can decribe configuraion pace for phyical yem. (Many phyically realiic cae will alo be, e.g. ymplecic manifold, whoe poin are he objec of he groupoid). Since groupoid are caegorie, i i uual o hink of hem up o equivalence (he weaker noion of iomorphim), o rea differen bu indiinguihable groupoid a he ame. For opological and mooh groupoid, he be verion of hi i a homoopy -like noion: Definiion 3 Two groupoid G and G are (rongly) Moria equivalen if here i a pair of morphim: X f g G G (2) 2

3 where boh f and g are uiably nice map (oherwie hi i a Moria morphim). A ack i a Moria-equivalence cla of groupoid. Thi implie ha he caegorie of repreenaion are equivalen a caegorie (weak Moria equivalence). Thi definiion coincide wih he more familiar one for C algebra, in he cae of groupoid algebra. Moria equivalen groupoid are phyically indiinguihable. (E.g. full acion groupoid; keleon, wih quoien pace of objec). So our propoal i ha configuraion pace hould be (opological, mooh, ec.) ack. 2.2 Span(Gpd) Definiion 4 A pan in a caegory C i a diagram of he form: X A B A pan map f beween wo pan coni of a compaible map of he cenral objec: X f X A B A copan i a pan in C op (i.e. C wih arrow revered). We ll ue C = Gpd, o and are funcor (i.e. alo map morphim, repreening ymmerie). Definiion 5 The bicaegory Span 2(Gpd) ha: Objec: Groupoid Morphim: Span of groupoid Compoiion defined by weak pullback: S X X S T T α X X (3) A 1 A 2 A 3 2-Morphim : iomorphim clae of pan of pan map monoidal rucure from he produc in Gpd, and dual for morphim and 2-morphim. Noe: Thi weak pullback of groupoid ha objec (x, α, x ), where α : f(x) g(x ), and i morphim are commuing quare. We can look a hi wo way: 3

4 Span C i he univeral 2-caegory conaining C, and for which every morphim ha a (wo-ided) adjoin. The fac ha arrow have adjoin mean ha Span(C) i a -monoidal caegory. Thi i ueful o decribe quanum phyic. (See Abramky and Coecke, Vicary). Phyically, X will repreen an objec of hiorie leading he yem A o he yem B. Map and pick he aring and erminaing configuraion in A and B for a given hiory (in he ene inernal o C). Definiion 6 A ae for an objec A in a monoidal caegory i a morphim from he monoidal uni, ψ : I A. In Hilb, hi deermine a vecor by ψ : C H. In Span(Gpd), he uni i 1, he erminal groupoid, o hi i deermined by: S Ψ A where S i a groupoid, fibred over A. An example i he Baez-Dolan uff ype, where A = FinSe 0. Think of uch a ae a an enemble over he bae groupoid A. Acing on a ae by a pan produce a groupoid whoe objec include a hiory: Thi new ae i an enemble wih hee more complicaed objec, which encode he hiory of he pan (groupoidified operaor) ha have been applied. 3 Repreening Span(Gpd) There i alo a caegory Span 1(Gpd), aking pan only up o iomorphim and neglecing he 2-morphim, bu ill compoing via weak pullback. There are wo inereing funcor for our purpoe. Degroupoidificaidon (Baez-Dolan): and 2-linearizaion (Moron): 3.1 Groupoidificaion D : Span 1(Gpd) Hilb Λ : Span 2(Gpd) 2Hilb Degroupoidificaion work like hi: To linearize a (finie) groupoid, ju ake he free vecor pace on i pace of iomorphim clae of objec, C A. Definiion 7 The cardinaliy of a groupoid G i G = 1 # Au(g) [g] G where G i he e of iomorphim clae of objec of G. groupoid ame if hi um converge. We call a 4

5 Thi ha he nice propery ha i ge along wih quoien : Theorem 1 (Baez, Dolan) If S i a e wih a G-acion G S S, hen S /G = #S #G where # denoe ordinary e-cardinaliy. Then here i a pair of linear map aociaed o map f : A B: f : C B C A, wih f (g) = g f # Au(b) # Au(a) g(a) f : C A C B, wih f (g)(b) = f(a)=b The fir i ju compoiion wih f. The econd i he map ending he vecor δ a o δ f(a). Thee are adjoin wih repec o an inner produc 1 uch ha [g i], [g j] = δi,j. # Au(g i ) Thi give D = a a modified um over hiorie : when he groupoid are e, hi ju coun he number of hiorie from g i o g j. The general cae coun wih groupoid cardinaliy. Definiion 8 The funcor D : Span(Gpd) Vec i defined by wih D(G) = C(G), and D(X)(f)([b]) = [x] 1 (b) # Au(b) # Au(x) [f((x))] In he cae he groupoid are e, hi ju give muliplicaion by a marix couning he number of hiorie from x o y. In general, he marix D(X) ha: D(X) ([a],[b]) = (, ) 1 (a, b) 3.2 The Meaured Groupoid Cae The groupoid cardinaliy i a pecial cae of he volume of a ack, which we need o deal wih phyically inereing example. Definiion 9 A lef Haar yem for a (loc.cp.) groupoid G i a family {λ x } x G0, where λ x i a (poiive, regular, Borel) meaure on G x = 1 (x). Unlike for Haar meaure on a Lie group, a (lef) Haar yem λ x i no uniquely defined. I i only unique up o a (quai-invarian, i.e. equivarian) meaure µ on M. Definiion 10 If G i a groupoid, he pace of objec i a meaure pace (G 0, µ), and λ x i a lef Haar yem, he ack volume of G i: vol(x) = dλ x) 1 dµ X( 1 (x) Thi i a ack invarian. (Baed on Weinein, where meaure come from volume form.) 5

6 3.3 2-Linearizaion Recall ha he 2-morphim of Span 2(Gpd) are (io. clae of) pan of pan map: X σ A Y B τ X Compoiion i by weak pullback aken up o iomorphim. Someime one ju ue pan map: here, we wan 2-morphim a well a morphim o have adjoin. Again: aking pan mean allowing adjoinne! We wan a repreenaion of Span 2(Gpd) ha capure more han D Hilber Space Definiion 11 A finie dimenional Kapranov Voevodky 2-vecor pace i a C-linear finiely emiimple abelian caegory (one wih a direc um, a.k.a. biproduc) generaed by imple objec x, where hom(x, x) = C). A 2-linear map beween 2-vecor pace i a C-linear (hence addiive) funcor. 2Vec i he 2-caegory of KV 2-vecor pace, whoe morphim are 2-linear map and whoe 2-morphim are naural ranformaion. Noe: 2Vec i a monoidal 2-caegory wih he Deligne produc and uni Vec. Lemma 1 If B i an eenially finie groupoid, he funcor caegory Λ(B) = [B, Vec] i a KV 2-vecor pace. Noe: If he auomorphim group of (iomorphim clae of) objec of B are B 1,..., B n, hen we have [B, Vec] = Rep(B j) So he bai elemen (imple objec) in [B, Vec] are labeled by ([b], V ), where [b] B and V an irreducible rep of Au(b). Definiion 12 A 2-Hilber pace i an abelian H -caegory. Unpacking hi definiion, a 2-Hilber pace H i an abelian caegory uch ha: j each hom-e ha he rucure of a Hilber pace, and compoiion of morphim i bilinear. H i equipped wih a ar rucure a conravarian funcor : H H which i he ideniy on objec and 2 = 1 H. The ar rucure on H induce an aninaural iomorphim hom(x, y) = (hom(y, x)) 6

7 In finie dimenion, hi i much like 2Vec, in ha all 2-Hilber pace are equivalen o Hilb n, in which cae 2-linear map are equivalen o marix muliplicaion wih Hilber pace enrie (uing and in place of + and ). Baez, Freidel e. al. conjecure he following for he infinie-dimenional cae (incompleely underood): Conjecure 1 Any 2-Hilber pace i of he following form: Rep(A), he caegory of repreenaion of a von Neumann algebra A on Hilber pace. The ar rucure ake he adjoin of a map. In hi conex: For our phyical inerpreaion A i he algebra of ymmerie of a yem. The algebra of obervable will be i commuan - which depend on he choice of repreenaion! Bai elemen are irreducible repreenaion of he vn algebra - phyically, hee can be inerpreed a uperelecion ecor. Any repreenaion i a direc um/inegral of hee. Then 2-linear map are funcor, bu can alo be repreened a Hilber bimodule beween algebra. The imple componen of hee bimodule are like marix enrie. Special example of hi kind: Rep(X) for a groupoid X, by aking A o be he compleion of he groupoid C -algebra C c(x). Rep(L (X, µ)), for a meaure pace, give he caegory of meaurable field of Hilber pace on (X, µ) Linearizaion Funcor Theorem 2 If X and B are eenially finie groupoid, a funcor f : X B give wo 2-linear map: f : Λ(B) Λ(X) namely compoiion wih f, wih f F = F f and f : Λ(X) Λ(B) called puhforward along f. Furhermore, f i he wo-ided adjoin o f (i.e. boh lef-adjoin and righ-adjoin). In fac, he adjoin map f ac by: f (F )(b) = C[Au(b)] C[Au(x)] F (x) f(x) =b Thi i he lef adjoin. Bu here i alo a righ adjoin: f! (F )(b) = hom C[Au(x)] (C[Au(b)], F (x)) [x] f(x) =b In fac, hi i a wo-ided adjuncion, by uing he Nakayama iomorphim, a canonical iomorphim: N (f,f,b) : f! (F )(b) f (F )(b) 7

8 given by he exerior race map in each facor of he um (which ue a modified group average). 1 N : φ x g φ x(g 1 ) #Au(x) [x] f(x) =b [x] f(x) =b g Au(b) Under hi idenificaion, he lef and righ adjoin are iomorphic. By compoing uni/couni wih N, we ge ha f and f are ambidexrou adjoin. (Noe: In general, Span 2(C) will be he univeral 2-caegory for which morphim in C have ambidexrou adjoin. We wan o preerve hi propery.) Call he adjuncion in which f i lef or righ adjoin o f he lef and righ adjuncion repecively. We wan o ue he couni for he lef adjuncion, which i he evaluaion map: η R(G)(x) :G(x) y f(y) =x hom C[Au(x)] (C[Au(y)], G(x)) v y f(y) =x (g g(v)) and he uni for he righ adjuncion, which ju ue he acion: ɛ L(G)(x) : C[Au(x)] C[Au(y)] f G(x) G(x) [y] f(y) =x g y v f(g y)v [y] f(y) =x [y] f(y) =x Definiion 13 Define he 2-funcor Λ a follow: Objec: Λ(B) = Rep(B) := [B, Vec] Morphim Λ(X,, ) = : Λ(a) Λ(B) 2-Morphim: Λ(Y, σ, τ) = ɛ L,τ N η R,σ : () () ( ) ( ) Picking bai elemen ([a], V ) Λ(A), and ([b], W ) Λ(B), we ge ha Λ(X,, ) i repreened by he marix wih coefficien: Λ(X,, ) ([a],v ),([b],w ) = hom Rep(Au(b)) ( (V ), W ) hom Rep(Au(x)) ( (V ), (W )) [x] (,) 1 ([a],[b]) Thi i an inerwiner pace, given by he analog of he inner produc ψ, φ in a Hilber pace. In he cae where ource and arge are 1, here i only one bai objec in Λ(1) (he rivial repreenaion), o he 2-linear map are repreened by a ingle vecor pace. Then i urn ou: Theorem 3 Rericing o hom Span2 (Gpd)(1, 1): A!! 1 X 1!! B 8

9 where 1 i he (erminal) groupoid wih one objec and one morphim, Λ on 2-morphim i ju he degroupoidificaion funcor D. The groupoid cardinaliy come from he modified group average in N. 4 2-Linearized Phyic 4.1 Harmonic Ocillaor Example 2 In he cae where A = B = FinSe 0 (equivalenly, he ymmeric groupoid n 0 Σn - noe no longer finie), we find D(FinSe 0) = C[[]] where n mark he bai elemen for objec [n]. Thi ge a canonical inner produc and can be reaed a he Hilber pace for he quanum harmonic ocillaor ( Fock Space ). The operaor a = and a = M, generae he Weyl algebra of operaor for he QHO. Thee are given under D by he pan A: FinSe 0 id FinSe 0 FinSe 0 and i dual A. Compoie of hee give a caegorificaion of operaor explicily in erm of Feynman diagram. Such compoie are decribed in erm of groupoid whoe objec look like hi: The ource and arge map for he pan pick he e of ar and end poin. The morphim of he groupoid are graph ymmerie. Degroupoidificaion D calculae operaor which (afer mall modificaion involving U(1)-label) agree wih he uual Feynman rule for calculaing ampliude. An ongoing projec (wih Jamie Vicary) i o udy he 2-caegorical verion of hi picure. There are analog of creaion and annihilaion operaor in oher hom-caegorie han hom(1, 1): FinSe 0 { } id { } FinSe 0 FinSe 0 FinSe 0 {, } { } id {, } FinSe 0 9

10 Thi i a 2-morphim α A : A AAA creae a creaion/annihilaion pair a he 1-morphim level. Compoie of hee ac a rewrie rule on he Feynman diagram like hoe een previouly (now wih boundary map). The image of hi picure under Λ involve repreenaion heory of he ymmeric group a Λ(FinSe 0) = n Rep(Σn), and give rie o paraparicle aiic : C Irreducible repreenaion of FinSe 0 are labelled by Young diagram. Rericion and inducion of repreenaion amoun o couning pah hrough he laice above. The uual boonic Fock pace repreenaion F coni of he ymmeric rep (he exreme lef-hand enrie of each row). Thi applie o ae ψ : C F - wihou he hiory a encoded in he diagram of compoie ae. 4.2 Exended TQFT Example 3 An Exended TQFT (ETQFT) i a (weak) monoidal 2-funcor where ncob 2 ha Z : ncob 2 2Vec Objec: (n 2)-dimenional manifold Morphim: (n 1)-dimenional cobordim (manifold wih boundary, wih M a union of ource and arge objec) 2-Morphim: n-dimenional cobordim wih corner One conrucion ue gauge heory, for gauge group G (here a finie group). Given M, he groupoid A 0(M, G) = hom(π 1(M), G) /G ha: Objec: Fla connecion on M Morphim Gauge ranformaion Then A 0(, G) : ncob 2 Span 2(Gpd), and here i an ETQFT Z G = Λ A 0(, G). Thi relie on he fac ha cobordim in ncob 2 can be ranformed ino produc of copan: Then A 0(, G) map hee ino Span 2 (Gpd). Suppoe S : S 1 + S 1 S 1 i he pair of pan, howing wo paricle fuing ino one. 10

11 ncob 2 Span 2 (T op) S 1 i A (A D) i A i D S 1 S 1 i 1 Y i 2 ι 1 ι 2 S 1 S 1 Y i 2 ι 3 ι 4 M Y i 2 i 1 S 1 i 1 Then we have he diagram: (G G) /G m (G /G) 2 G /G (4) Where he map leave connecion fixed, and ac a he diagonal on gauge ranformaion; and m i he muliplicaion map. View S 1 a he boundary around a yem (e.g. paricle). Irreducible objec of Z G(S 1 ) [G /G, Vec] are labelled by ([g], W ), for [g] a conjugacy cla in G and W an irrep of i abilizer ubgroup For G = SU(2), hi i an angle m [0, 2π], a paricle; and an irrep of U(1) (or SU(2) for m = 0) i labelled by an ineger j Thi heory hen look like 3D quanum graviy coupled o paricle wih ma and pin. wih ma m and pin j Under he opology change of he pair of pan, a pair of uch rep i aken o one wih nonrivial repreenaion (uperelecion ecor) for all [mm ] for any repreenaive of [m], [m ] (each poible oal ma and pin for he combined yem). Phyic in hi Hilber pace arie from he 3D 2-morphim. Reference [1] J. Baez, J. Dolan. From Finie Se o Feynman Diagram. Mahemaic Unlimied And Beyond, Engqui, B., Schmid, W. (Ed.), Springer Verlag, arxiv:mah.qa/

12 [2] John C. Baez and Jame Dolan, Higher-dimenional algebra and opological quanum field heory. J.Mah.Phy, vol 36, pp , arxiv:q-alg/ [3] Dawon, R. J. MacG. and Par, R. and Pronk, D. A., Univeral properie of pan, Theory and Applicaion of Caegorie, Vol 13, no 4, pp61-85, [4] Kapranov, M. and Voevodky, V. 2-caegorie and Zamolodchikov erahedron equaion, Proc. Symp. Pure Mah, vol 56 Par 2, pp , [5] Jeffrey C. Moron, Caegorified Algebra and Quanum Mechanic, Theory and Applicaion of Caegorie, vol 16, pp , arxiv:mah/ [6] Jeffrey C. Moron, 2-Vecor Space and Groupoid, Applied Caegorical Srucure (DOI: / ). arxiv: [7] Jeffrey C. Moron, Exended TQFT, Gauge Theory and 2- Linearizaion. arxiv: