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February 4, 2008

Chern-Simons States from L-infinity Bundles, III: States over the Circle

Posted by Urs Schreiber

I am continuing to look at the structure of (4d)(4-d)-states of Chern-Simons theory over dd-dimensional manifolds by using the L L_\infty-algebraic model of the Chern-Simons 3-bundle (\sim 2-gerbe) with connection over BGB G discussed in section 7.3 of L L_\infty-Connections and Applications (pdf, blog, arXiv).

The program is: transgress this 3-bundle as in section 9.2. to spaces of maps from a dd-dimensional parameter space, and then compute the collection of (4d)(4-d)-sections following the L L_\infty-algebraic description for computing \infty-sections of the general concept of \infty-sections.

Last time I looked at the sections over 2-dimensional surfaces, that computation remaining somewhat inconclusive, as the holomorphic structure on the transgressed 1-bundle does not appear yet.

Today I instead looked at the states over 1-dimensional manifolds: over circles.

I find in section 4.1 of

Sections and covariant derivatives of L L_\infty-algebra connections
(pdf, blog)

that the sections of the 2-bundle obtained from transgression of the Chern-Simons 3-bundle to the configuration space over the circle come from bundles of representations of the Kac-Moody central extension of the Lie algebra of loops over the space of gg-holonomies over the circle.

And I think this is what the result should be, though a couple of details deserve a closer look.

One nice byproduct is this:

the computation explicitly gives a derivation in the present context of the fact that the 2-cocycle on the loop group

tg S 1μ(f,g)= S 1fdg tg_{S^1} \mu(f,g) = \int_{S^1} \langle f \wedge d g \rangle

comes from the very transgression we are talking about of the 3-cocycle

μ(x,y,z)=x,[y,z] \mu(x,y,z) = \langle x, [y,z]\rangle

on the semisiple Lie algebra gg that the Chern-Simons 3-bundle is governed by.

This crucially depends on some gymnastics with that “almost-internal-hom”

(A,B)Ω (maps(A,B)) (A, B) \mapsto \Omega^\bullet(\mathrm{maps}(A,B)) of dg-algebras which we talked about in Transgression of nn-Transport (section 5.1).

Posted at February 4, 2008 5:26 PM UTC

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Read the post Smooth 2-Functors and Differential Forms
Weblog: The n-Category Café
Excerpt: An article on the relation between smooth 2-functors with values in strict 2-groups, and an outline of the big picture that this sits in.
Tracked: February 6, 2008 12:08 PM

Re: Chern-Simons States from L-infinity Bundles, III: States over the Circle

Actually, there is this useful reformulation of the very fact addressed above:

according to section 8 we can regard the Chern-Simons 3-bundle on BGB G as the obstruction to lifting the universal GG bundle EGBGE G \to B G to a String(G)String(G)-2-bundle.

Now as we hit everything in sight with

Ω (maps(,Ω (S 1))) \Omega^\bullet(maps(--, \Omega^\bullet(S^1)))

we transgress everything to loop space:

now we have an ΩG\Omega G-bundle over ΩBG\Omega B G and the CS 3-bundle becomes a 2-bundle aka 1-gerbe which is the lifting gerbe obstructing the lift of the loop group bundle to the corresponding centrally extended loop group bundle.

So the fact that Chern-Simons theory over the circle involves bundles whose fibers are reps of the centrally extended loop group is just another incarnation of the fact that a string-structure on a Spin manifold XX is equivalently

- the obstruction to lifting the Spin-bundle on XX to a String-2-bundle on XX

- the obstruction to lifting the ΩSpin\Omega Spin-bundle on ΩX\Omega X to a Ω^Spin\hat \Omega Spin-bundle.

That’s probably clear to anyone who has ever thought about it, but it’s kind of nice to see how it arises here by just hitting L L_\infty-algebra connection descent objects with the transgression functor

Ω (,Ω (S 1)). \Omega^\bullet(-- ,\Omega^\bullet(S^1)) \,.

Posted by: Urs Schreiber on February 6, 2008 9:06 PM | Permalink | Reply to this

Re: Chern-Simons States from L-infinity Bundles, III: States over the Circle

URS wrote:
a string-structure on a Spin manifold X is equivalently


- the obstruction to lifting the Spin-bundle on X to a String-2-bundle on X

SO IF X HAS A STRING STRUCTURE THEN THE SPIN BUNDLE DOES **NOT** LIFT TO A STRING 2-BUNDLE??


- the obstruction to lifting the \Omega Spin-bundle on\Omega X to a \hat \Omega Spin-bundle.


That’s probably clear to anyone who has ever thought about it,

SO OMEGA DOES NOT EVER KILL THE BOSTRUCTION OR IS THIS SPECIAL TO SPN?

but it’s kind of nice to see how it arises here by just hitting L ∞-algebra connection descent objects with the transgression functor


Ω •(−−,Ω •(S 1))

That notation is a bit inpenetrable

Posted by: jim stasheff on February 7, 2008 12:50 AM | Permalink | Reply to this

Re: Chern-Simons States from L-infinity Bundles, III: States over the Circle

so if XX has a String structure then the Spin bundle does not lift to a String 2-bundle??

It does. The obstructions are the same. And “String structure” means that the obstruction vanishes.

Ω (,Ω (S 1)) \Omega^\bullet( -- , \Omega^\bullet(S^1))

That notation is a bit inpenetrable

In fact it is a typo, too. What I meant to write was

Ω (maps(,Ω (S 1))). \Omega^\bullet(maps( -- , \Omega^\bullet(S^1))) \,.

If it were exactly the internal hom, I could simply write

hom(,Ω (S 1)). hom(--, \Omega^\bullet(S^1)) \,.

But we are still not sure what the best abstract-nonsense name of Ω (maps(,Ω (S 1))) \Omega^\bullet(maps( -- , \Omega^\bullet(S^1))) is, are we?

Posted by: Urs Schreiber on February 7, 2008 1:00 AM | Permalink | Reply to this

the String class

We’ve talked about that privately, but for completeness and in case any of our readers is wondering, let me summarize the situation in more detail, as reviewed and explained nicely in

Katsuhiko Kuribayashi, On the vanishing problem of String classes (pdf)

Originally (I had once collected some list of literature here) at a time when higher bundles hadn’t been much around yet, a String structure on a Spin manifold XX

Spin(n) Q X \array{ Spin(n) &\hookrightarrow& Q \\ && \downarrow \\ && X }

had been conceived as the vanishing of the obstruction of lifting the corresponding loop bundle over loop space ΩX\Omega X

ΩSpin(n) ΩQ ΩX \array{ \Omega Spin(n) &\hookrightarrow& \Omega Q \\ && \downarrow \\ && \Omega X }

through the Kac-Moody central extension

1U(1)Ω^Spin(n)ΩSpin(n)1 1 \to U(1) \to \hat \Omega Spin(n) \to \Omega Spin(n) \to 1

of the loop group.

The obstructing line 2-bundle (“lifting gerbe”) to this lift is classified by a degree 3-class called μ(Q)\mu(Q) in the above text:

μ(Q)H 3(ΩX,). \mu(Q) \in H^3(\Omega X, \mathbb{Z}) \,.

One would expect that this 3-class is the transgression of a 4-class down on XX. And that’s essentially true:

On any Spin bundle QXQ \to X we have a “fractional Pontrjagin” four class denoteded 12p 1(Q)H 4(X,) \frac{1}{2}p_1(Q) \in H^4(X,\mathbb{Z})

and it is true that

(12p 1(Q)=0)(μ(Q)=0). (\frac{1}{2}p_1(Q) = 0) \Rightarrow (\mu(Q) = 0) \,.

Kuribayashi in his article addresses the question whether the converse holds: does μ(Q)\mu(Q) always come from transgressing a four-class down on XX to loop space?

The answer is: yes, if H 4(X,)H^4(X,\mathbb{Z}) is torsion free.

(H 4(X,)torsionfreeandH 2(X,)1)((12p 1(Q)=0)(μ(Q)=0)). (H^4(X,\mathbb{Z}) torsion free and H^2(X,\mathbb{R}) \leq 1) \Rightarrow ( (\frac{1}{2}p_1(Q) = 0) \Leftarrow (\mu(Q) = 0)) \,.

So the vanishing of 12p 1(Q)\frac{1}{2}p_1(Q) is a slightly stronger condition, in general, than the vanishing of μ(Q)\mu(Q). Except when torsion vanishes, then both are equivalent.

It’s an issue familiar throughout the study of nn-bundles: transgressing an nn-bundle to a space of maps with a dd-dimensional source yields an (nd)(n-d)-bundle on the space of maps, but not every (nd)(n-d)-bundle on the space of maps will, in general, arise this way. Only those which have sufficiently high “multiplicative structure” remembering that they came from a higher structure downstairs will.

(If this sounds mysterious: the underlying mechanism is really most simple when you conceive all nn-bundles in terms of their fiber-assigning nn-functors and realize that transgression of these nn-functors is just hitting them with an internal hom. I had once discussed that, with Bruce Bartlett, in a bit more detail for a simple class of cases in the entry Multiplicative structure of transgressed nn-bundles (pdf, blog)

I think what should really be addressed as “the string class” is 12p 1(Q)\frac{1}{2}p_1(Q) instead of just μ(Q)\mu(Q), even though historically it was other way round.

Posted by: Urs Schreiber on February 7, 2008 11:42 AM | Permalink | Reply to this

Re: the String class

Thanks for the historical clarification.
Our readers might also appreciate the earlier resultof McLauglin that 1/2 p_1 and \mu are equivalent when M is 2-connected.
This is a generic result for classes in the stable range, approximately twice the connectivity.

Posted by: jim stasheff on February 7, 2008 12:40 PM | Permalink | Reply to this

Re: Chern-Simons States from L-infinity Bundles, III: States over the Circle

A discussion of the relation:

Pontrjagin class / CS 3-bundle down on XX

\leftrightarrow

String 3-class /Kac-Moody lift obstruction up on LXL X

now appears in the list of example in section 9.3.1.

What is currently equation 488 highlights the source of the derivative terms in the loop group cocycles, i.e. the fact that a Lie algebra cocycle

μ=,[,] \mu = \langle \cdot, [\cdot, \cdot]\rangle

turns, after transgression to loops, into the Kac-Moody 2-cocycle

(f,g) S 1f,g. (f,g) \mapsto \int_{S^1} \langle f, g'\rangle \,.

Posted by: Urs Schreiber on February 14, 2008 6:23 PM | Permalink | Reply to this
Read the post An Exercise in Groupoidification: The Path Integral
Weblog: The n-Category Café
Excerpt: A remark on the path integral in view of groupoidification and Sigma-model quantization.
Tracked: June 13, 2008 6:29 PM

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