On Lie N-tegration and Rational Homotopy Theory

October 20, 2007

On Lie N-tegration and Rational Homotopy Theory

Posted by Urs Schreiber

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In rational homotopy theory one studies spaces “up to finite ambiguity” as Dennis Sullivan put it, namely by considering all forms of homotopy and (co)homology over the rationals (i.e discarding all torsion information).

For an overview see for instance

Kathryn Hess
Rational Homotopy Theory: A Brief Introduction
(2000)
(pdf).

The crucial insight of Dennis Sullivan described in

Dennis Sullivan
Infinitesimal computations in topology
Publications mathématiqeu de l’ I.H.É.S., tome 47 (1977), p. 269-331
(NUMDAM)

was that all rational spaces are obtained from integrating Lie $n$ -algebras.

Of course Sullivan didn’t put it that way, nor do many people in rational homotopy theory. Instead they are talking about differential graded commutive algebras, which are freely generated in positive degree, as graded commutative algebras.

Here at the $n$ -Café we call (following Jim Stasheff’s suggestion) such dg-algebras “quasi free differential algebras” (qDGCAs) and are fond of the fact that they are dual to codifferential coalgebras, which are the same as $L_\infty$ -algebras, which are the same as Lie $n$ -algebras, which are $n$ -fold categorifications of Lie algebras. For a quick reminder on how this works, see Lie $n$ -algebra cohomology. For a bestiary of examples, most of them described in both languages, try Zoo of Lie $n$ -algebras. For more try section Plan, subsection The bridge as well as the section Lie $n$ -algebra cohomology here.

It was Ezra Getzler who explained that what Sullivan did with qDGCAs was essentially the integration of the corresponding Lie $n$ -algebras:

Ezra Getzler
Lie theory for nilpotent $L_\infty$ -algebras
arXiv:math/0404003

The basic idea of this integration process, vividly but ultra-tersely sketched on the first two pages of

Pavol Ševera
Some title containing the words “homotopy” and “symplectic”, e.g. this one
arXiv:math/0105080

and less vividly, but in more detail, described in

André Henriques
Integrating $L_\infty$ algebras
arXiv:math/0603563

is that from any Lie $n$ -algebra $g_{(n)}$ we form the simplicial space whose set of $k$ -simplices is the set of Lie $n$ -algebroid morphisms from the tangent algebroid of the standard $k$ -simplex to the given Lie $n$ -algebroid

$[k] \mapsto \mathrm{Hom}_{n\mathrm{Lie}}(T \Delta_k , g_{(n)})$

which in terms of the dual qDGCAs reads

$[k] \mapsto \mathrm{Hom}_{qDGCA}(g_{(n)}^* , \Omega^\bullet(\Delta_k) ) \,,$

where $\Omega^\bullet(X)$ is simply the deRham differential algebra of forms on $X$ .

Notice that the Lie $n$ -algebroid morphisms appearing here are, morally, the differential version of smooth pseudofunctors

$\Pi_1(\Delta_k) \to \Sigma G_{(n)}$

from the fundamental groupoid of the $k$ -simplex to the one-object $n$ -groupoid of the Lie $n$ -group integrating our Lie $n$ -algebra: hence nothing but a flat $G_{(n)}$ -valued parallel $n$ -transport on $\Delta_k$ .

For too long to comfortably admit, I didn’t get the intuitive and conceptual gist underlying this construction. While a good pedagogical account still needs to be written, as far as I can see, I personally profited a lot from realizing how the above procedure reproduces ordinary integration of Lie algebras when we keep the nonabelian Stokes theorem in mind (I talked about that here. For a description of the nonabelian Stokes theorem see section Parallel $n$ -transport, subsection 2-Functors and differential 2-forms here) and that we might profitably think of the space built by the above procedure as the fundamental $n$ -groupoid

$\Pi_n(\mathbf{BG}_{(n)})$

of the “generalized smooth classifying space” given by the sheaf on manifolds

$\mathbf{BG}_{(n)} : U \mapsto \mathrm{Hom}_{n\mathrm{Lie}}(T U , g_{(n)}) \,.$

This is not (yet) supposed to be a precise statement, but if you are like me in that you need to have the feeling to know why we are doing something in order to enjoy doing it, I suggest this as a good working assumption (originally mentioned here).

The central issue of Ezra Getzler’s above article is to reduce the size of $\Pi_n(\mathbf{GB}_{(n)})$ by strictifying it a lot, such that, in particular, it becomes finite dimensional at each stage.

André Henriques instead works with the unrestricted space. His main application was the integration of the String Lie 2-algebra (section Lie $n$ -algebra cohomology, subsection String, Chern-Simons and Chern Lie $n$ -algebras here) and the demonstration that it is the 3-connected cover of $\mathrm{Spin}$ .

Notice that, using Sullivan’s age-old theorem (8.1),v) from the above article, this becomes essentially a triviality:

the qDGCA defining the String Lie 2-algebra is simply that of the underlying Lie algebra together with a single additional degree 2-generator $b$ whose differential is required $d b = \mu$ to be the canonical 3-cocycle $\mu = \langle \cdot , [\cdot, \cdot] \rangle$ on the underlying semisimple Lie algebra. But that says nothing but that the generator of the third cohomology becomes cohomologically trivial!

(For more on the Lie 2-algebra cohomology of the String Lie 2-algebra see Cohomology of the String Lie 2-algebra. Danny Stevenson has meanwhile started to check (some terms still need to be done) that the qDGCA computation discussed there is indeed reproduced by a computation of the (rational) cohomology of the total space of the String group as constructed by Henriques and in From Loop groups to 2-groups).

Here I start with having a closer look at parts of this literature by going through Sullivan’s old paper and highlighting the Lie $n$ -algebra theory he discusses, without saying so at that time.

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p. 277 of Sullivan’s article: action Lie $n$ -algebroids

The first comment concerns the “twisted cohomology of differential algebras” that Sullivan discussed in section 1, p. 277, and which also plays a major role as a starting point for Ezra Getzler (see the first couple of pages of his introduction). The same phenomenon also appears in various texts on BV-quantization.

I am claiming that we might profitably think of Lie action algebroids here, namely Lie algebroids obtainable from differentiating action Lie groupoids obtained from a group acting on something.

Let $V$ be a vector space. Then the Lie algebra

$gl(V)$

is dually described by the qDGCA on $\wedge^\bullet (V \otimes V^*)$ with the differential on the generator with respect to a given basis $\{e_i\}$ ov $V$

$\tensor{a}{_i_^j} := e_i \otimes e^j$

given by

$d_{gl(V)} \tensor{a}{_i_^j} = \sum_k \tensor{a}{_i_^j} \wedge \tensor{a}{_j_^k} \,.$

Now, a representation of any other Lie algebra $g$ on $V$ is nothing but a morphism $\rho : g \to gl(V)$ of Lie algebras. Hence a morphism $\rho^\dual : (\wedge^\bullet V \otimes V^*, d_{gl(V)}) \to (\wedge^\bullet g^*, d_g)$ of the qDGCA described above to the Chevalley-Eilenberg algebra of $g$ .

Being a morphism of free graded-commutative algebras means that this is given by a set of elements of $g^*$ ,

$\tensor{\Theta}{_i_^j} \,,$

one for each generator of $gl(V)^*$ . But, by the above definition of $d_{gl(V)}$ , respect for the differentials then requires that these generators satisfy

$d_g \tensor{\Theta}{_i_^j} = \sum_k \tensor{\Theta}{_i_^k} \wedge \tensor{\Theta}{_k_^j} \,.$

This is the Maurer-Cartan-like equation which pervades Getzler’s discussion. One notices that a collection of such “Maurer-Cartan elements” $\Theta$ can be canonically used to construct a new algebroid, namely that dually given by the Chevalley-Eilenberg algebra of $g$ for Lie algebra cohomology with values in the module $V$ . See p. 277 of Sullivan’s article. His “twisted cohomology” is, I claim, to be regarded as the qDGCA dual to the action Lie algebroid of $g$ acting on $V$ .

This is the same kind of reasoning underlying the interpretation of the BV formalism which I mentioned in BV-Formalism, Part IV.

p. 278 of Sullivan’s article: quasi-free graded commutative algebras

Here in the very last paragraph on that page the condition is first mentioned that we want to restrict attention to differential graded-commutative algebras which are free as graded commutative algebras (and generated in non-negative degree).

These are known to be those related dually to “semistrict” Lie $n$ -algebras. These are Lie $n$ -algebras for which the Jacobi identity is allowed to hold up to coherently weak equivalence, but where the skew symmetry of the bracket functor is required to hold strictly.

One might ask:

a) to what kind of Lie $n$ -algebras would, dually, dg-algebras non-free as graded commutative algebras correspond?

b) to what kind of dg-algebras would Lie $n$ -algebras with coherently weak skew symmetry correspond.

Neither answer is known. But I am speculating that the answer to a) is b) and the answer to b) is a), essentially.

See the discussion at Categorified Clifford Algebra and weak Lie $n$ -Algebras and the table in this comment. This has grown out of thinking about

Dmitry Roytenberg
On weak Lie 2-algebras
(pdf)

p. 279 of Sullivan’s article strict Lie $n$ -algebras

On the top of p. 279 Sullivan restricts attention to those dg-algebras $A$ whose differential satisfies $d A \subset A \wedge A \,.$

This says that only binary operations exist in the dual $L_\infty$ -algebra. And this means that the Jacobiator and all its higher coherences vanish: we are dealing with a strict Lie $n$ -algebra. Equivalently, with a dg-Lie algebra.

Sullivan calls qDGCAs satisfying this condition minimal.

Later on he proves that all qDGCAs come from tensoring minimal ones with free ones. In my language this should mean: all Lie $n$ -algebras come from strict ones tensored with those obtained from weak cokernels for an identity morphism on a Lie $n$ -algebra. (Because these are precisely those corresponding to free dg-algebras, as described in Structure of Lie $n$ -algebras. (Remind me to update this file with my latest polished version.))

p . 281 the first hint on Lie $n$ -tegration

After having discussed qDGCAs, Sullivan notices a couple of striking analogies with homotopy theory of spaces. In particular he notices that the fundamental group of a space plays the role of the Lie algebra dual to a qDGCA. This is the first hint for $G_{(n)} = \Pi_{(n)}(U \mapsto \mathrm{Hom}_{qDGCA}(g_{(n)}^*, \Omega^\bullet(U))) \,.$

p. 283: obstruction theory

Now we are already discussing obstruction theory. At that time diagrams were not yet fashionable, I guess, but no amount of prose can substutute them:

given qDGCAs $A$ , $B$ and $C$ , and given morphisms

$\array{ && A \\ && \uparrow \\ C &\leftarrow& B }$

we are asking for the obtruction to lifting to a morpism

$\array{ && A \\ &\swarrow& \uparrow \\ C &\leftarrow& B } \,.$

Notice that, dually, we have a sequence of Lie $n$ -algebras

$\array{ k \\ \downarrow \\ g \\ \downarrow \\ b }$ together with a connection taking values in one of them $\array{ &&k \\ &&\downarrow \\ &&g \\ &&\downarrow \\ T X &\to&b }$

and are asking for the obstruction to lifting this $b$ -connection to a $g$ -connection through our extension

$\array{ &&k \\ &&\downarrow \\ &&g \\ &\nearrow&\downarrow \\ T X &\to&b } \,.$

I need to think about how Sullivan’s component-based answer relates to the one I talk about in section String and Chern-Simons $n$ -transport, subsection obstruction theory here, based on the idea of weak cokernels which has grown out of the discussion of Obstructions for $n$ -bundle lifts, Obstructions, Tangent Categories and Lie $n$ -tegration , Obstructions to $n$ -Bundle Lifts Part II: The BIG Diagram.

section 8: integration and cohomology

In section 8 (p. 300) finally the integration process is discussed. Sullivan does not think of his qDGCAs as Lie $n$ -algebras, but does emphasize (that’s the whole point) how their cohomology can be used to study the rational cohomology of spaces (“integrating them”, as we would now say).

The striking statement is that at the beginning of p. 301, which later becomes item v) of theorem (8.1) on p. 304:

the cohomology of the space “integrating” a qDGCA is precisely that of the qDGCA.

The familiar example for that is Lie algebra cohomology: the cohomology of the Chevalley-Eilenberg algebra, which is the qDGCA dual to a Lie algebra and noting but what is known as Lie algebra cohomology, is the same as the cohomology of the simply connected compact Lie group integrating it.

The generalization of this fact means that computations as in Cohomology of the String Lie 2-algebra indeed compute the cohomology of the group integrating the String Lie 2-algebra.

And indeed, Danny started to check (looks good so far but some more terms need to be checked), for String, that this is true.

But at some point here I am slightly confused about dimensions, re cohomology of $G$ versus that of $B G$ . Need to think about that.

Posted at October 20, 2007 1:56 PM UTC

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Re: On Lie N-tegration and Rational Homotopy Theory

I’m not an expert on this, and hopefully Jim Stasheff can clarify, but from what I’ve gathered the equivalence of the homotopy theories of rational homotopy types, comm dgas and dg Lie algebras goes back to Quillen’s 1968 (?) paper on rational homotopy theory, where he is pretty explicit that what we’re doing is a direct generalization of the classical integration of Lie algebras. Certainly the whole modern approach to deformation theory, as explained by Kontsevich (and others, Deligne, Drinfeld, Feigin — don’t know the history unfortunately) is built on this part of Quillen. To use modern language, Lie and Comm are Koszul dual operads, with the Koszul duality realized geometrically by the operations “differentiate” (aka take tangent complex) and “integrate” (aka solve Maurer-Cartan equations). I believe all the other general statements follow from interpreting this correctly (eg resolving the operads themselves to get homotopically correct versions etc).

The work of Schechtman, Hinich, Getzler and others makes this into a nice geometric statement (at least morally, I don’t know if we need some boundedness etc). As Hinich explains it is just “Lie’s theorem”: rationally Lie algebras and formal groups are the same thing, and likewise L_∞ algebras can be integrated to geometric objects (formal derived stacks - functions on which give commutative dgas), geometric objects can be differentiated to L_∞-algebras (tangent complexes), and this gives an equivalence between the two homotopy theories.

(The subtle questions treated by Getzler etc, have to do as far as I understand with the kind of models we can build for the integration, and with the subtleties of positively graded vs arbitrary complexes - Quillen was dealing with spaces, ie connective things, while Getzler isn’t. Henriques is interested in getting more than formal integration, requiring a different set of ideas.)

Posted by: David Ben-Zvi on October 21, 2007 7:05 PM | Permalink | Reply to this

Re: On Lie N-tegration and Rational Homotopy Theory

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David,

thanks for the reply. I’ll get back to some of the points later, after I checked some things in the literature.

But here are some quick questions:

a) you mention

the whole modern approach to deformation theory

Can you suggest some good introduction to “deformation theory” that emphasizes the Lie $n$ -algebraic/groupoid point of view?

The Homotopy Hypothesis says that

$n$ -groupoids are the same as “homotopy $n$ -types” — nice spaces whose homotopy groups above the $n$ th vanish for every basepoint

What’s known about the relation of this to rational homotopy theory (RHT)? Sounds like in some sense RHT might be the restriction of the homotopy hypothesis to Lie $n$ -groupoids and rational homotopy types. Could that be roughly right?

$L_\infty$ algebras can be integrated to geometric objects

So who noticed first that they should actually integrate to Kan complexes, i.e. to (realizations of) $n$ -groupoids? It seems there are plenty of texts on rational homotopy theory that do not ever mention the word “groupoid” or “Kan complex”.

Posted by: Urs Schreiber on October 21, 2007 8:51 PM | Permalink | Reply to this

Re: On Lie N-tegration and Rational Homotopy Theory

Urs -
First thing to say is I was probably fairly glib (my excuse is lack of caffeine) in the previous post, and should not be taken too literally. In particular I think Ezra’s paper is a lot more central than it might appear from my post (hence its appearance in Annals..) Also I should say I’m very much a novice in this area, and might have a very skewed (and wrong) POV. But let me try to explain what I’m trying to say and one of the experts can correct me as needed.

First the modern approach to deformation theory as Lie theory is explained in a course on Deformation Theory by Kontsevich, available at the Chicago Geometric Langlands webpage. I think it developed in the early ’90s from communications among Kontsevich, Drinfeld, Deligne, Feigin and others. There is a series of papers of Hinich-Schechtman and then of Hinich working this out in detail - the key term being “Deligne groupoid”. I’d recommend some of Hinich’s recent arxiv papers, in particular “Formal stacks as dg coalgebras”. The idea is that to a nonnegatively graded L_infty algebra you can define a formal groupoid of solutions to the Maurer-Cartan equations – a generalization of integrating a Lie algebra to the dg setting (with a shift – ie the formal group of the Lie algebra appears as pi_1, not pi_0). To get an equivalence of categories you have to remember the higher cohomologies of the Lie algebra – the groupoid itself only knows about H^0 and H^1 (with obstructions from H^2) — hence the notion of dg (or derived) groupoid. This encapsulates deformation theory, since the formal completion of any moduli problem is a formal groupoid. The derived structure accounts for things like the “virtual fundamental class”, higher obstruction theory for moduli.

Anyway if you have negative homotopy groups, you can generalize this as follows. a formal groupoid is a functor from Artin rings to groupoids (while its derived version changes the source from ARtin rings to dg Artin rings). If you have negative components then you won’t
get a groupoid but a higher groupoid - there’s morphisms between the morphisms between Maurer-Cartan solutions coming from degree -1 and so on. This can be captured by saying we have a functor to higher groupoids, or to Kan complexes, or to all simplicial sets — all are the same on the level of homotopy theories.
(by which I mean quasicategories or some
equivalent notion — if you’d like I’m
taking the “homotopy hypothesis” as the definition of higher groupoids - which maybe is anathema on this cafe, sorry..)
There are various papers with Deligne 2-groupoids and so on, I think Ezra’s is the first methodical version that goes “all the way down”.

An account of this is also in Toen’s wonderful survey on higher and derived stacks (section 4.4 part 2), where he leaves it as an open question whether the kind of equivalences I mentioned are true - so certainly I have no idea what the technical issues to be resolved are.

But the idea that higher groupoids or Kan complexes etc are the global versions of L_infty algebras is behind all of derived algebraic geometry, appearing in many of Simpson’s papers and in the work of Toen-Vezzosi and Lurie. Derived stacks are
the kind of objects whose tangent spaces are naturally L_infty algebras, and conversely L_infty algebras give formal
derived stacks by solving MC equations. What’s maybe not fully resolved is the
extent to which these are equivalences in
some settings - and one can only imagine that will appear in Lurie’s DAG-n for higher n.

As to why groupoids don’t appear classically in rational homotopy theory I think that’s an issue of language. Certainly Quillen’s point was that the theory of rational spaces behaves just like usual Lie theory – to a space X we assign a group, ie based loops in X. This group has a Lie algebra, namely the Whithead Lie algebra (rational homotopy groups of X – shifted by one since we’ve looped – form a Lie algebra). To go back you take Maurer-Cartan solutions– ie you formally exponentiate, with a shift.
More precisely MC gives the CLASSIFYING
SPACE of the formal group of your Lie algebra, so in other words we exponentiate to get based loops on X and then deloop.
If you accept the homotopy hypothesis as a definition of higher groupoids, which I do, then this is already in ‘68 a Lie theory for higher groupoids…

Again sorry for glibness (now with coffee
so less of an excuse) and the many false statements.

Posted by: David Ben-Zvi on October 21, 2007 9:35 PM | Permalink | Reply to this

Re: On Lie N-tegration and Rational Homotopy Theory

I hardly know where to begin! Rational homotopy theory provides a particularly fertile ground for deformation theory, viz.

“The Lie Algebra Structure of Tangent Cohomology and Deformation Theory,”

Journal of Pure and Applied Algebra 38 (1985) 313-322. (with Michael Schlessinger)

There’s a cahier secret by the same two authors which has been held up for 2 decades because I didn’t speak stacks fluently.

The stucture of the moduli space as a varierty modulo a unipotent group action was clear - only the language of Deligne groupoid and/or stacks was causing expository problems! A still incomplete version is available on request.

Quillen worked in the dg lie category; Sullivan’s models as quasi-free comm dgas
had L_\infty algebras `inside’.

It’s the transition from dg algebra to rational homotopy types OF SPACES
where the fuss about non-neg or non-pos grading comes in.

Posted by: jim stasheff on October 22, 2007 3:17 AM | Permalink | Reply to this

Read the post On BV Quantization, Part VIII
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Excerpt: Towards understading BV by computing the charged n-particle internal to Z-categories, secretly following AKSZ.
Tracked: November 29, 2007 10:29 PM

plain ordinary integration

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Over in other threads, we are wrecking our brains trying to understand what “integration” as in “path integration” might really mean. All that BV-effort is heading towards that big question.

In the light of the discussion in the above thread on integration of $L_\infty$ -algebras, let’s see if maybe we are all just missing the obvious.

We generally assume that the integration of an $n$ -form over an $n$ -dimensional manifold yields a number.

Sure. But recall we also assumed that the integration of a Lie algebra valued 1-form yields an elements of Lie group.

But then the above procedure was invented and we found: not so.

As I emphasized somewhere, the cool aspect of the above way of integrating Lie-algebra valued 1-forms is that you actually never do.

Instead of sending a Lie-algebra-valued 1-form on the interval $I$ to an element of the group, Lie $n$ -tegration teaches us to instead send it to the equivalence class of Lie algebra-valued forms on the interval that would integrate to the same Lie group element.

Now take the Lie group to be $U(1)$ , and obtain a notion of integration without integration.

Let me spell that out for a simple case, just to clarify what I am talking about.

Here is the interval: $I = [0,1] \,.$

Suppose we have a $\mathrm{Lie}(U(1))$ -valued 1-form, i.e. just a plain 1-form $\omega \in \Omega^1(I) \,.$

We want to find its integral $\int_I \omega$ over $I$ . But instead of saying that this integral is a number we’ll say: this integral is an equivalence class: the equivalence class of all 1-forms that would integrate to the same number, would we actually compute that.

The point is that we can say when two 1-forms on $I$ would yield the same number if we integrated them:

$\omega$ and $\omega'$ in $\Omega^1(I)$ would integrate to the same number if and only if there is a closed 1-forms $\hat \omega \in \Omega^1(I^2/~)$ on the standard bigon (unit square with vertical boundaries shrunk to a point) which restricts on the upper boundary to $\omega$ and on the lower to $\omega'$ . By Stokes theorem.

I am not sure if I can convey the thrill of this here. It’s all about that point made in the pdf provided at Transgression, I pointed out that the usual notion of transgression of forms, including, noticably, the step where we do the fiber integration are implicit in forming an internal hom from the corresponding transport functors.

It’s that same thing here, and for the analogous reason I think.

So if I’d integrate a 1-form $\omega$ over the interval $I$ , I would pick this 1-form in the quasi inner hom $\Omega^\bullet(CE(u(1)), \Omega^\bullet(I)\otimes \Omega^\bullet(-))$ which, by the argument provided here is the Lie algebroid that integrates to the Lie groupoid whose objects are 1-forms on the interval and whose morphisms given by the kind of equivalence relation discussed here.

Okay, now replace $\Omega^\bullet(I)$ here with any dgca. The discussion still goes through. Take the dgca of forms on the space of paths of something and obtain…

Posted by: Urs Schreiber on January 17, 2008 11:24 PM | Permalink | Reply to this

Re: plain ordinary integration

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Above I had expressed the idea that it might be very useful to apply the idea of “integration without integration” of $L_\infty$ -algebras also to ordinary integration of $n$ -forms over $n$ -dimensional manifolds:

instead of really computing the integral, we instead pass to equivalences classes of all those forms that would yield the same integral, had we computed it.

To my delight, Jim Stasheff just writes in to tell me that this is, apparently, precisely the point of view advocated and developed by Leonid Dickey et al.

Apparently there is a textbook around, which describes this.

I’ll try to search for more information on this now. Would be very grateful for any hints anyone may have to offer, though.

In particular: has this ever been used in attempts to define integrals over things that are not manifolds? Has this ever been used to study path integrals?

Posted by: Urs Schreiber on January 24, 2008 1:01 PM | Permalink | Reply to this

Re: plain ordinary integration

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In particular: has this ever been used in attempts to define integrals over things that are not manifolds? Has this ever been used to stude path integrals?

Ah, found something:

L. Kauffmann, VASSILIEV INVARIANTS AND FUNCTIONAL INTEGRATION WITHOUT INTEGRATION

Abstract: This paper is an exposition of the relationship between Witten’s functional integral and the theory of Vassiliev Invariants of knots and links in three dimensional space. We show how to conceptualize the functional integral in terms of equivalence classes of functionals of gauge fields. This approach makes it possible to discuss heuristics for functional integration in a mathematical framework.

There is also a website about a talk on this

Conference on Knot Theory and its Ramifications, which has a more detailed abstract:

Abstract: There is a deep inteconnection between invariants of knots and links and the use of functional integrals as a heuristic (they do not exist). We replace the non-existent functional integral by a class of functions of gauge fields: defining F equivalent to G if F-G = DH where F,G,H are functions of a gauge field A that are “rapidly vanishing at infinity” in the sense that they go zero rapidly when an appropriate norm of A goes to infinity. DH denotes a functional derivative of F with respect to one of the gauge coordinates.We then define INT(F) to be the equivalence class of F. The talk will discuss how link invariants and Vassiliev invariants are intertwined with these INTegrals.

That sounds like it is exactly what I am looking for.

Posted by: Urs Schreiber on January 24, 2008 3:53 PM | Permalink | Reply to this

Read the post Integration without Integration
Weblog: The n-Category Café
Excerpt: On how integration and transgression of differential forms is realized in terms of inner homs applied to transport n-functors and their corresponding Lie oo-algebraic connection data.
Tracked: January 24, 2008 9:23 PM

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Weblog: The n-Category Café
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Tracked: January 30, 2008 10:16 AM

Read the post Smooth 2-Functors and Differential Forms
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Excerpt: On how to interpret the geometric construction by Brylinksi and McLaughlin of Cech cocycles classified by Pontrjagin classes as obstructions to lifts of G-bundles to String(G)-2-bundles.
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October 20, 2007