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August 25, 2007

The G and the B

Posted by Urs Schreiber

We have a pretty good understanding of what the “BB-field” in string theory really is, in terms of arrow-theory.

This nicely explains a bunch of things. I used to be very annoyed with myself, though, for not understanding, on this nice abstract level, one of the more intriguing aspects of the BB-field:

beyond its mere nature as being the 2-categorical version of a line bundle with connection, it turns out that the connection 2-form BB here unifies in an intriguing way with a Riemannian metric (and with a dilaton field, in fact).

Both these rank-2 tensors BB and gg sum up to an object g+B g + B which is known as the “open string metric” to string theorists, and which happens to have a surprisingly nice and natural geometrical interpretation in the context of what is called generalized complex geometry.

In this approach, due to Nigel Hitchin, one studies the geometry of manifolds XX all in terms of the sum of their tangent and cotangent bundle T *XTX, T^* X \oplus T X \,, making use of various kinds of useful natural structures present on this bundle, like its canonical bilinear pairing as well as the Courant bracket.

As is indicated to some extent in section 3.8 of

Marco Gualtieri
Generalized Complex Geometry

this bundle TXT *XT X \oplus T^* X is to be thought of as the Atiyah Lie 2-algebroid of an abelian gerbe on XX, hence as the infinitesimal version of something like the 2-groupoid of automorphisms of the corresponding ΣU(1)\Sigma U(1) 2-bundle.

The more or less obvious nn-algebroid structures on TX n1T *X T X \oplus \wedge^{n-1} T^* X should corespond to the Atiyah Lie nn-algebroids of Σ n1\Sigma^{n-1}-nn-bundles.

While I do follow all the algebra underlying generalized complex geometry, I had always said to myself that at some point I should better understand what all this really means. Given the available nice categorical picture of line 2-bundles, there should be a nice arrow-theoretic integrated version of understanding all this, showing how a line 2-bundle with connection knows something about Riemannian structures.

Unfortunately, I had never really taken the time to think this through.

Now, today Marco Gualtieri approached me with a question. Since he is already happy with having unified the 2-form connection with the Riemannian metric in his formalism, he was trying to get hints for how all the other string background fields would fit into this mathematical picture, like the RR-forms and in particular the dilaton.

This made me really angry with myself. I wished I had figured out at least some of the answers to these obviously open and crucial questions before he asked me! I knew these needed to be answered at some point.

As it goes, this kind of frustration sometimes is the best motivatin to get going. While S. Merkulov was teaching us this morning about operads, Poisson structures and graph complexes, I believe I made some progress with understanding what’s going on.

The key is, it seems, to first figure out what the Atiyah nn-groupoid of an nn-bundle really is. I’ll discuss this using tangent nn-categories and flows on nn-categories, as described in Arrow-theoretic differential geometry.

I shall try to indicate what I mean by that, and how it gives rise to the appearance of the structures we see in generalized complex geometry.

So let XX be a smooth manifold. Let X×XX \times X be its pair Lie groupoid.

Notice that we obtain the corresponding Lie algebroid as the one whose sections are the \mathbb{R}-flows on X×XX \times X TXv:=INN(X×X), T X \ni v : \mathbb{R} = \to \mathrm{INN}(X \times X) \,, using the reasoning described here, and using the notion of inner automorphisms (n+1)(n+1)-groups as described in The inner automorphism 3-group of a strict 2-group.

Now let PXP \to X a principal GG-bundle on XX and consider its integrated Atiyah sequence AdPP× GPX×X. \mathrm{Ad} P \to P \times_G P \to X \times X \,.

Here P× GPP \times_G P is the Atiyah groupoid of PP: its objects are the fibers of PP, its morphism are all fiber morphisms (necessarily isomorphisms).

The sections of the corresponding Lie algebroid are again the \mathbb{R}-flows on this groupoid, hence the smooth group homomorphisms INN(P× GP). \mathbb{R} \to \mathrm{INN}(P \times_G P) \,.

Locally that’s clearly sections of TXLie(G)T X \oplus \mathrm{Lie}(G). I think globally it correctly TP/GT P / G, as it should be.

When passing to 2-bundles now, I will simply my and your lifes by only working locally, for the time being.

So for G 2:=ΣU(1) G_2 :=\Sigma U(1) the Lie 2-group shifted U(1)U(1), consider a trvial principal G 2G_2-2bundle X×G 2X X \times G_{2} \to X with the corresponding Atiyah Lie 2-groupoid, which is simply the cartesian product (X×X)×(Σ 2U(1)). (X \times X) \times (\Sigma^2 U(1) ) \,.

A smooth \mathbb{R}-flow on this 2-groupoid INN((X×X)×(Σ 2U(1))) \mathbb{R} \to \mathrm{INN}( (X \times X) \times (\Sigma^2 U(1) )) is the following:

a vector field vΓ(TX) v \in \Gamma(T X) as before, encoding the flow on X×XX \times X itself, together with a functorial smooth assignment of rectangles Id Id exp(i γA v,t) Id Id \array{ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet \\ \downarrow^{\mathrm{Id}} &\Downarrow^{\exp(i \int_\gamma A_{v,t})}& \downarrow^{\mathrm{Id}} \\ \bullet &\stackrel{\mathrm{Id}}{\to}& \bullet } in Σ 2U(1)\Sigma^2 U(1) to paths (xγy), (x \stackrel{\gamma}{\to} y) \,, coming from a 1-form AΩ 1(X) A \in \Omega^1(X) smoothly parameterized by tt, by the usual logic.

So indeed, a smooth \mathbb{R}-flow on the Atiyah 2-groupoid of a trivial ΣU(1)\Sigma U(1)-2-bundle is a section of TXT *X. T X \oplus T^* X \,.

By precisely the same kind of argument we’d get TX (n1)T *XT X \oplus \wedge^{(n-1)} T^* X for nn-bundles with structure nn-group Σ n1U(1)\Sigma^{n-1} U(1). (To see that in detail, you need to either work with pseudofunctors or pull back the Atiyah groupoid to nn-paths along P n(X)X×XP_n(X) \to X \times X first.)

That sets the scene. Next I should try to reformulate all the stuff done in generalized complex geometry using arrow-theoretic differential theory.

Posted at August 25, 2007 12:42 PM UTC

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7 Comments & 2 Trackbacks

Re: The G and the B

That method of flows on categories is really useful, it seems. For instance consider the strict version of the String 2-group. Smooth morphisms INN(String k(G)) \mathbb{R} \to \mathrm{INN}(\mathrm{String}_k(G)) (notice: not INN 0(String k(G))\mathrm{INN}_0(\mathrm{String}_k(G))!, we are really mapping into all automorphisms which are connected by a pseudonatural transformation to the identity) come, for each element pp of the path group of GG, from a smooth family of squares p p g(t) ω(t) p g(t) \array{ \bullet &\stackrel{p}{\to}& \bullet \\ \downarrow^{p_g(t)} &\Downarrow^{\omega(t)}& \downarrow^{p_g(t)} \\ \bullet &\to& \bullet } with tp g(t)t \mapsto p_g(t) denoting a smooth 1-parameter family of based paths in GG, starting, for t=0t=0 at the constant path on the identity, with the endpoint evolving as tg(t)t \mapsto g(t).

Then the element ω(t)\omega(t) of the centrally extended loop group is already essentially fixed up to the central part. In the limit t0t \to 0 only the endpoint of the path contributes (all wiggling of the path is of higher than linear order in tt), hence we get at once that \mathbb{R}-flows on the String 2-group live in the vector space Lie(G). \mathrm{Lie}(G) \oplus \mathbb{R} \,. And that’s just the vector space on which, indeed the skeletal weak version of the Lie 2-algebra of the string group is built.

(And the grading is clearly that induced by the the degree kk of kk-morphisms: the Lie(G)\mathrm{Lie}(G)-part comes from 1-morphisms and is degree 1, while the \mathbb{R}-part comes from the 2-morphisms.)

Posted by: Urs Schreiber on August 25, 2007 9:44 PM | Permalink | Reply to this

Re: The G and the B

Make sure you ask Marco to whistle for you!

Posted by: Allen Knutson on August 25, 2007 10:40 PM | Permalink | Reply to this

Re: The G and the B

Make sure you ask Marco to whistle for you!

Too late, my train leaves in a few hours. Next time I see him I’ll ask him if he has meanwhile found the tune of the dilaton in GCG.

Posted by: Urs Schreiber on August 25, 2007 11:03 PM | Permalink | Reply to this

Re: The G and the B

Maybe I am beginning to see how the Riemannian metric comes in from just the arrows.

Check section 6.1 in Gualtieri’s math.DG/0401221 and in particular proposition 6.6, the upshot of which is that with b b any 2-form on XX and with g g any Riemannian metric on XX, we may regard g±b:TXT *X g \pm b : T X \to T^* X as a metric (that “open string metric”) in the context of generalized complex geometry.

Consider first the case that the Riemannian metric is absent g=0. g = 0 \,. And notice that the 2-form bb has to be interpreted as a connection on our trivial ΣU(1)\Sigma U(1)-bundle, defining a 2-functor tra b:P 2(X)(P 2(X)×Σ 2U(1)). \mathrm{tra}_b : P_2(X) \to (P_2(X)\times \Sigma^2 U(1)) \,. Let’s abbreviate our Atiyah 2-groupoid of the trivial ΣU(1)\Sigma U(1)-2-bundle on XX as At:=(P 2(X)×Σ 2U(1)). \mathrm{At} := (P_2(X)\times \Sigma^2 U(1)) \,. Then notice that, with our interpretation of TXT *XT X \oplus T^* X as the space of smooth 1-parameter families of inner automorphisms of the Atiyah 2-groupoid as above (now I do work in the picture where I have pulled back along P 2(X)X×XP_2(X) \to X \times X) we can interpret the map b:TXT *X b : T X \to T^* X in terms of arrows simply as mapping flows on P 2(X)P_2(X) to flows on the Atiyah 2-groupoid according to Id P 2(X) P 2(X) Ad exp(v) tra b At=P 2(X) tra b Id At At . \array{ & {}^{\;\;\;}\nearrow \searrow^{\mathrm{Id}} \\ P_2(X) &\Downarrow& P_2(X) \\ & {}_{\;\;\;}\searrow \nearrow_{\mathrm{Ad}_{\exp(v)}} \\ && \downarrow^{\mathrm{tra}_b} \\ && At } \;\;\;\;\;\; = \;\;\;\;\;\; \array{ P_2(X) \\ \downarrow^{\mathrm{tra}_b} \\ & {}^{\;\;\;}\nearrow \searrow^{\mathrm{Id}} \\ At &\Downarrow& At \\ & {}_{\;\;\;}\searrow \nearrow_{} } \,.

So this does map flows on the base to flows on the total Atiyah 2-groupoid, and by unravelling what all these arrows mean one finds that this does indeed precisely correspond to the map b:TXT *X. b : T X \to T^* X \,.

(Compare this to the diagram in What is a Lie derivative really? to get a better idea for what’s going on.)

So among all maps from flows on the base to flows on the Atiyah 2-groupoid, those induced by a transport 2-functor as above do consistently correspond to maps induced from the corresponding connection 2-form.

But there are more morphisms from base flows to flows on the Atiyah 2-groupoid than that. Indeed, each gS 2T *Xg \in S^2 T^* X does procide one. The underlying formula for the components is pretty much as above, only that now there is no 2-functor P 2(X)AtP_2(X) \to At defined by this anymore.

So, I am thinking that a generalized metric in the sense of generalized complex geometry, hence an “open string metric” is – while in general no longer a transport 2-functor with values in the line 2-bundle, instead a functor from flows on the base to flows on the Atiyah Lie 2-groupoid of the line 2-bundle.

I’ll provide you with the precise diagrammatics of this next week.

Posted by: Urs Schreiber on August 25, 2007 10:58 PM | Permalink | Reply to this

Re: The G and the B

I feel like a layman staring at awe at an abstract piece of art and commenting on how beautiful it is only to be told the painting is upside down.

I have no idea what you just said Urs, but it is beautiful to look at :)

Posted by: Eric on August 26, 2007 6:39 AM | Permalink | Reply to this

Open String Metric

Both these rank-2 tensors BB and gg sum up to an object g+Bg+B which is known as the “open string metric” to string theorists…

Umh… not quite.

The open string metric is G=gBg 1B G = g - B\cdot g^{-1} \cdot B Its inverse is closer to the thing you’re considering G 1=[(g+B) 1] sym G^{-1} = {\left[(g+B)^{-1}\right]}_{\text{sym}} The bivector, [(g+B) 1] anti-sym {\left[(g+B)^{-1}\right]}_{\text{anti-sym}} is also significant.

Posted by: Jacques Distler on September 3, 2007 6:19 AM | Permalink | PGP Sig | Reply to this

Re: Open String Metric

known as the “open string metric” to string theorists…

not quite

Ah, thanks for the correction!

Posted by: Urs Schreiber on September 3, 2007 7:46 AM | Permalink | Reply to this
Read the post Arrow-Theoretic Differential Theory IV: Cotangents
Weblog: The n-Category Café
Excerpt: Cotangents and morphisms of Lie n-algebroids from arrow-theoretic differential theory.
Tracked: September 3, 2007 4:33 PM
Read the post BV for Dummies (Part V)
Weblog: The n-Category Café
Excerpt: Some elements of BV formalism, or rather of the Koszul-Tate-Chevalley-Eilenberg resolution, in a simple setup with ideosyncratic remarks on higher vector spaces.
Tracked: October 30, 2007 10:26 PM

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