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November 3, 2006

A 3-Category of Twisted Bimodules

Posted by Urs Schreiber

Those readers not yet bored to death by my posts might recall the following:

I was arguing that the 3-group controlling Chern-Simons theory (and maybe also the gauge structure of the Green-Schwarz mechanism #) is a sub-3-group of the inner automorphism 3-group #

(1)INN(String G)AUT(String G), \mathrm{INN}(\mathrm{String}_G) \subset \mathrm{AUT}(\mathrm{String}_G) \,,

of the String 2-group - for GG an ordinary Lie group (here assumed to be compact, simple and simply connected).

Part of the evidence (I, II) I presented was the observation that the canonical 2-representation # of String G\mathrm{String}_G on

(2)Bim(Hilb) HilbMod \mathrm{Bim}(\mathrm{Hilb}) \stackrel{\subset}{\to} {}_\mathrm{Hilb}\mathrm{Mod}

apparently extends to a representation of INN(String G)\mathrm{INN}(\mathrm{String}_G) on “twisted” bimodules, and that this representation seems to exhibit the expected structures #.

Like Bim(C)\mathrm{Bim}(C) can be thought of as coming from lax functors into Σ(C)\Sigma(C), for CC a 2-monoid (an abelian monoidal category), twisted bimodules

(3)TwBim(C) \mathrm{TwBim}(C)

can be thought of as coming from lax functors into the endormorphism 3-monoid

(4)END(C) \mathrm{END}(C)

of CC - in a way that is analogous to the step from the 2-group String G\mathrm{String}_G to its automorphism 3-group AUT(String G)\mathrm{AUT}(\mathrm{String}_G).

3-morphisms in TwBim(C)\mathrm{TwBim}(C) look a little like the fundamental disk correlator with one bulk insertion in rational CFT #: a disk, bounded by bimodules, with a ribbon colored in CC running perpendicular through the disk’s center. (And this is not supposed to be a coincidence #.)

This picture suggests an obvious 3-category structure. That however is slightly oversimplified. On the other hand, the description in terms of lax functors into END(C)\mathrm{END}(C) is a little too unwieldy.

Hence my goal here is to write down precisely and explicitly what TwBim(C)\mathrm{TwBim}(C) looks like and how compositions are defined. Diagrams can be found in these notes:

\;\;\;a 3-category of twisted bimodules

The hard part is to check coherent weak properties, like the exchange law. I have checked what looked nontrivial - and am hoping that I haven’t overlooked anything. But if anyone has seen before anything like the 3-category TwBim(C)\mathrm{TwBim}(C) that I am trying to describe here, please drop me a note.

A crucial point in all these constructions is the restriction to inner automorphisms and inner endomorphisms.

Analogous to the restriction to inner automorphisms of the structure 2-group

(1)INN(G 2)AUT(G 2), \mathrm{INN}(G_2) \subset \mathrm{AUT}(G_2) \,,

the endomorphisms of CC that are used to obtain TwBim(C)\mathrm{TwBim}(C) are all “inner”.

Recall how this can be motivated # by applying Schreier theory to the Atiyah groupoid sequence.

An ordinary principal G-bundle

(2)P X \array{ P \\ \downarrow \\ X }

gives rise to the Atiyah sequence of groupoids (the “exponentiated Atiyah sequence” #)

(3)AdP Trans(P) P 1(X) = = = P× GG P× GP X×X. \array{ \mathrm{Ad} P &\to& \mathrm{Trans}(P) &\to& P_1(X) \\ = & & = & & = \\ P \times_G G &\to& P \times_G P &\to& X \times X } \,.

Crucially, the GG action on GG itself used in forming P× GGP \times_G G is the adjoint action by GG on itself. This is where the inner automorphisms enter the game.

Schreier theory then suggests (as emphasized by Danny Stevenson # ) that a connection on PP is a pseudofunctor

(4)(tra,curv tra):X×XAUT(P× GG), (\mathrm{tra},\mathrm{curv}_\mathrm{tra}) : X \times X \to \mathrm{AUT}(P \times_G G) \,,

where the target is the 2-groupoid whose objects are points in XX, whose morphisms are fiber isomorphisms

(5)(P× GG) x(P× GG) y (P \times_G G)_x \to (P \times_G G)_y

and whose 2-morphisms are natural isomorphisms of these (where we regard the fibers - which are groups - as categories with a single object).

But this implies that an ordinary connection

(6)tra:P 1(X)Trans(X), \mathrm{tra} : P_1(X) \to \mathrm{Trans}(X) \,,

acting by isomorphisms of fibers of PP, will act by inner automorphisms on the × GG\times_G G factor.

More precisely, if we locally trivialize the pseudofunctor (tra,curv tra)(\mathrm{tra},\mathrm{curv}_\mathrm{tra}), it indeed takes values in

(7)Σ(INN(G)) \Sigma(\mathrm{INN}(G))

instead of in all of

(8)Σ(AUT(G)). \Sigma(\mathrm{AUT}(G)) \,.

Similar comments apply to the categorified setup, where PP is replaced by a principal 2-bundle and GG by a 2-group.

The 3-category of twisted bimodules that I am talking about # is supposed to be relevant for the associated 2-vector description of this principal setup.

Posted at November 3, 2006 12:41 PM UTC

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