2-Monoid of Observables on String G

Transcription

1 2-Monoid of Obsevables on Sting G Scheibe Novembe 28, 2006 Abstact Given any 2-goupoid, we can associate to it a monoidal categoy which can be thought of as the 2-monoid of obsevables of the 2-paticle popagating on that 2-goupoid Hee we show that fo the 2-goupoid Σ(Sting G ) this monoidal categoy is the categoy ΛRep(ΛSting G ) of loops in epesentations of the loop goupoid of Sting G We ague that epesentation of ΛSting G ae twisted equivaiant bundles on G Intoduction Fo vaious easons, I find the following geneal concept useful, which hee I want to apply to a special case elated to loop goups and epesentations Let pa be a 1-categoy, called the paamete space Let P be a smooth 2-categoy, called the taget space Let ta : P Bim(Vect ) be a smooth 2-functo to the 2-categoy whose mophisms ae bimodules, called a 2-vecto bundle with connection on taget space Let 1 : P Bim(Vect ) be the tenso unit in the monoidal 2-categoy of 2-vecto bundles with connection, ie the 2-functo that sends eveything to the identity on Let conf [pa, P] be the 2-categoy of 2-functos fom paamete space to taget space, called the configuation space Let ta : conf [pa, Bim] be the 2-functo on configuation space obtained by postcomposing with ta This can oughly be thought of as the tansgessed 2-vecto bundle Let sect = [1, ta ] be the categoy of mophisms fom the tivial 2-vecto bundle on configuation space to the tansgessed 2-vecto bundle This I call the space of sections on configuation space Let = End(1 ) be the monoidal categoy of endomophisms of 1, called the monoid of obsevables lealy, sect is a module categoy fo usscheibe at mathuni-hambugde 1

2 Hee I would like to undestand the categoy fo the following setup Let G be a simple, simply connected and compact Lie goup, and let k H 3 (G, Z) be a level Fom the centally extended loop goup, ˆΩ k G, we can fom the goupoid Sting G P G ˆΩ k G P G ove based paths in G This goupoid can be egaded fom two points of view As a centally extended goupoid, it is the canonical bundle gebe with class k ove G The goupoid has a stict monoidal stuctue, with stict monoidal inveses Theefoe it can also be egaded as a stict 2-goup Being monoidal, we can fom the suspension Σ(Sting G ), which is a 2- categoy with a single object We want to egad this as ou taget space, in the above sense, and study the monoid of obsevables on the configuation space of 2-paticles popagating on this taget space Notice that in as fa as BG ΣG, we can think of Sting G as a twisted vesion of BG This means we set P = Σ(Sting G ) and pa = Σ(Z) In this case, I seem to find the following esult: Poposition 1 The goupoid ΛSting G [Σ(Z), Σ(Sting G )] / obtained by identifying isomophic 1-mophisms in configuation space is a cental extension of the the loop goupoid of G ΛG [Σ(Z), Σ(G)] Poposition 2 The monoidal categoy is = [Σ(Z), Rep(ΛSting G )] Poposition 3 The categoy Rep(Sting G ) is the categoy of equivaiant gebe modules on G Note that fo G finite, Simon Willeton agued that ΛG, which is nothing but the action goupoid of the adjoint action of G on itself, plays the ole of the loop goup of G, by noticing that BΛG LBG Given a goup 3-cocycle κ on G, hence a goupoid 2-cocycle on ΛG, one can theefoe addess the twisted epesentations Rep κ (ΛG) both as twisted epesentations of the loop goup of G - in the above sense - as well as twisted equivaiant vecto bundles on G What I descibe hee looks like a Lie goup analog of this pespective on the Feed-Hopkins-Teleman theoem 2

3 Definition 1 Fo G 2 any stict 2-goup, the loop goupoid of G 2 is the 1- goupoid obtained by identifying isomophic 1-mophisms in conf = [Σ(Z), Σ(G 2 )] Poposition 4 The categoy of endomophisms of 1 is, as a monoidal categoy, equivalent to the categoy of loops in the categoy of epesentations of the loop goupoid of G 2 : End(1 ) [Σ(Z), Rep(ΛG 2 )] Poof An object in End(1 ) is, being a pseudonatual tansfomation, a functoial assignment of 1-mophisms in conf to squaes in [Σ(Z), Bim] V : ( g 1 h g 2 ) V g1, V (h) V g2 which is compatible with 2-mophisms This compatibility hee just says that V is invaiant on 1-mophisms that ae connected by a 2-mophisms But this means that V is a epesentation of ΛG 2 Moeove, the mee existence of the squae on the ight says that V g1 V (h) V g1 V g2 V g1 ( ) = V (h) V g1 V g2, V g2 V g2 ( ) which means that g 1 V g1 ( ) is a natual automophism of this epesentation of ΛG 2 Notice that the V g hee ae -bimodules, hence vecto spaces, while V (h) is a -bimodule homomophism, hence a linea map Next, a mophism in End(1 ) is a modification, hence an assignment V g g k g V g The tin can equation fo this says that k is a natual isomophism fom the epesentation V to the epesentation V Moeove, the mee existence of 3

4 k g above says that this natual isomophism is compatible with the natual automophism V ( ) and V ( ) But this means nothing but that k encodes a mophism in [Σ(Z), Rep(ΛG 2 )] Now let G 2 = Sting G be the stict 2-goup coesponding to the cossed module ˆΩ k G P G, whee G is any simple, simply connected and compact Lie goup Poposition 5 ΛSting G is isomophic to a cental extension of ΛG Poof It suffices to check this fo the vetex goups of connected components So fix any element P G This amounts to choosing any g G and a path connecting it to the neutal element All choices of paths with fixed endpoint coespond to the same connected component An automophism in ΛG 2 [,] is epesented by a 2-cell in Sting G Fo this to exist, the endpoints of and have to commute But with the endpoint fixed, the path is abitay, because the above 2-cell is to be identitfied with a 1 a fo any loop a Hence all paths with the same endpoint ae to be identified hoosing a fixed epesentative of the family of all paths with the same endpoint, 4

5 the 2-mophism hee is a uniquely detemined loop, togethe with an element in the U(1)-toso ove that loop Theefoe any vetex goup of ΛSting G is a cental extension of the coesponding vetex goup of ΛG Fo any element g G, fix an element in P G In othe wods, choose a section s P G G This will only locally be smooth, of couse We can identify two goupoid mophisms into ΛSting G, namely Sting G ΛSting G with and with (P G/G) s ΛSting G Ad g Poposition 6 The loop goupoid ΛSting G is geneated by the images of Sting G and (P G/G) s unde these two functos Poof Fo any mophism in ΛSting G, we can choose a epesentative whose vetical mophisms lie in the image of the chosen section s: 5

6 We can then wite = 1 Ad ( ) 1, and the ight hand side is manifestly a composite of the two types of geneatos Now, notice that, as a centally extended goupoid, Sting G is nothing but the canonical bundle gebe on G Accodingly, a goupoid epesentation of Sting G is the same as a module fo that gebe, altenatively known as a twisted bundle on G It is known that the decategoification of gebe modules on a space, Rep(Sting G ), is the same as the twisted K-theoy of that space But poposition 6 says that a epesentation of ΛSting G is a epesentation of Sting G, which at the same time caies the stuctue of a epesentation of the adjoint action of G on P G It hence looks as if Rep(ΛSting G ) would give ise to the twisted and Ad G - equivaiant K-theoy of G 6