From: baez@guitar.ucr.edu (john baez)
Newsgroups: sci.physics,sci.math
Subject: Link invariants and gauge theory
Date: 19 Dec 92 21:14:27 GMT
Followup-To: sci.physics
Organization: University of California, Riverside

What follows is a summary of a paper I have just written, "Link
invariants, holonomy algebras and functional integration."  It's an
attempt to make rigorous some aspects of
the loop representation of quantum gravity.  Please don't ask me for
copies of this paper!  I want to have time to catch typos and receive
criticism from some folks for a while, and then I will post it to hep-th
and gr-qc. So if you subscribe to one of these mailing lists you will
see it turn up sometime in January, and you can get it from there.   If
you don't subscribe to these mailing lists, just wait until mid-January
and then send 'em a message with subject header "find baez" to find the
paper's number.  (If you don't even KNOW about these preprint mailing lists,
read the sci.physics FAQ!)

The following summary is essentially the introduction of my paper, but
with various remarks thrown in a feeble attempt to make it more readable to
the non-expert.

My goal in this paper was to provide a rigorous expression of some ideas
from physics concerning the relation of link invariants to
diffeomorphism-invariant states of gauge field theories.
In quantum field theory it is common to work formally with ``measures''
on the space of field configurations, ignoring the difficulty of
describing these measures in mathematically rigorous terms.  
The simplest example arises in the theory of free boson fields.
Here one treats ``Gaussian measures'' of the form 
                e^{Q(x,x)} dx  
where Q is a complex-valued symmetric bilinear form on the real vector
space V.  When V is finite-dimensional and dx denotes Lebesgue
measure, e^{Q(x,x)} dx is indeed a perfectly nice measure
on V, and it's a finite measure when Q is negative definite, for example
                Q(x,x) = -(x_1^2 + ... + x_n^2)
on R^n.  In the cases interesting to quantum field theory, V is
infinite-dimensional, so there is no Lebesgue measure 
on V, but one may still seek some reasonable way to make sense of
the expression 
           \int_V  f(x)  e^{Q(x,x)} dx ,
at least for some class of complex-valued functions f on V.  The point
is that a nice measure (i.e., a finite regular Borel measure) on a nice
space (i.e., a locally compact Hausdorff space - for example, a finite-
dimensional vector space) allows one to integrate all bounded continuous
functions on the space, but in practice one is willing to settle for
much less.  Typically, one would hope to be able to integrate all
functions that live in some reasonably big algebra.

One approach is as follows.  Let A, the ``exponential algebra,'' be the
algebra of functions on V generated by those of the form 
		 f(x) =  e^{Q(v,x)} 
for v in V.   Note that if V is finite-dimensional and the real part of
Q(x,x) is negative for all nonzero x in V, the normalized integral
	(\int_V  f(x) e^{Q(x,x)} dx) /(\int_V e^{Q(x,x)} dx)
can be explicitly computed, and it's just
			e^{Q(v,v)/4}.
Generalizing from this fact, for an arbitrary vector space
V and quadratic form Q one may simply *define* a
complex-linear functional E on the algebra A, which is often called the
``expectation value,'' by setting
		E(f) =  e^{Q(v,v)/4
where f is as above.  Thus we can ``integrate'' all exponential
functions in a formal manner using the expectation value.  Of course,
one would like to be able to integrate other functions too.  One way to 
do this is by completing the algebra of exponential functions in some
norm, and hoping to extend the expectation value to a continuous linear
functional on the resulting normed algebra.   Two popular kinds
of normed algebras are Banach algebras and C*-algebras, the latter being
a special case of the former.  C*-algebras are especially nice in
mathematical physics because they are equipped with a * operation that
has all the properties of the usual adjoint of matrices.  In fact,
C*-algebras are a very handy sort of "algebra of observables" for both
classical and quantum mechanics.  The Gelfand-Naimark theorem says that
every commutative (classical) C*-algebra is isomorphic to the algebra of
bounded continuous functions on some topological space.  And a
continuous linear functional on a commutative C*-algebra then amounts to
a nice (finite regular Borel) measure on this space.

The case where Q is real-valued and negative definite
has been widely used in constructive quantum field theory, where we take
V to be the configuration space of the *classical* free boson field, and
want to form the Hilbert space L^2(V) to be the space of states for the
quantized free boson field.  In this case the relation of the
expectation value to conventional integration theory is more than a
merely formal one.  In this case
	 E(1) = 1, 
	 f >= 0 implies E(f) >= 0,
and
	 f >= 0 implies  |E(fg)| <= ||g|| E(f) 
where ||g|| is the L^infinity norm of g, that is, the supremum of the
absolute value of g.  Thus a theorem of Segal says that A is isomorphic to an
algebra of functions on some probability measure space (unique up to a
suitable equivalence relation) on which E is represented as integration.
(This can be found in Segal and Kunze's book Integrals and Operators,
where they give a nice treatment of such abstract "integration algebras"
- algebras equipped with nice expectation value functionals.)  
Moreover, one can complete A in the L^infinity norm to obtain a
C*-algebra E_c, and the expectation value extends uniquely to a
continuous linear functional on E_c.

On the other hand, Feynman integrals, in which Q is imaginary, arise
frequently in practical calculations in quantum field theory.  
The question of extending the expectation value E from A to a
larger algebra of functions on V has been studied extensively in this case.
In this case, the expectation value E does not extend
to a continuous linear functional on the completion of A in the 
L^infinity norm, as is easily checked in the case when V is the real
line and Q(v,v) = iv^2, where the L^infinity completion of A is the
space of "almost periodic functions" on the real line.   Since
L^infinity$ norm is the unique C*-norm on A, it follows that A admits no
C*-algebra completion to which E extends to a continuous linear functional.
One can, however, find a Banach algebra completion
of A to which E extends continuously.  For example, one can 
complete E in the norm given by 
           ||\sum_i c_i e^{Q(v_i,x)} || = \sum_i |c_i|   
if the points v_i are distinct.  

In my paper I consider an extension of this idea from 
the linear context of the free boson field to nonabelian
gauge field theories, with particular attention paid to
diffeomorphism-invariant states of gauge field theories, since these are
what arise as states of quantum gravity a la Ashtekar.  An interesting
example of a diffeomorphism-invariant gauge field theory is Chern-Simons
theory.  Given a compact Lie group G (the "gauge group") and a trivial
principal G-bundle P over a 3-dimensional manifold M, the
Chern-Simons action of a connection A on P is given by
   	S(A) = (k/4pi) \int_M tr(A ^ dA + (2/3)A ^ A ^ A) 
where the ``level'' k is a nonnegative integer.   The quantity S(A)
changes by an integer multiple of 2pi under gauge transformations of A, so one
may formally define a ``measure''
             e^{iS(A)} DA
on the space of smooth connections modulo gauge transformations, where
DA is the (purely heuristic) ``Lebesgue measure'' on this space.
Since S(A) is invariant under the action under the group
Diff_0(M) of diffeomorphisms of M that are connected to the
identity, at a formal level we expect the Chern-Simons ``measure'' to be
Diff_0(M)-invariant.  The problem is in making this sort of stuff make
sense, because the space of connections modulo gauge transformations is
a fairly nasty infinite-dimensional space, not even a manifold, so there
really is no Lebesgue measure on it.

The notion of the holonomy around a loop provides an interesting
relation between link invariants and such diffeomorphism-invariant
``measures.''    Given an oriented piecewise smooth loop gamma in 
M, let T(Gamma,A) denote the trace of the holonomy of the
connection A around gamma, relative to some finite-dimensional unitary
representation of G.   T(gamma) = T(gamma, .) is a gauge-invariant
function of A, so it may be regarded as a function on the space of
connections modulo gauge transformations.  It is called a
``Wilson loop'' in physics.  Wilson loops are popular ``gauge-invariant
observables.''  

Ashtekar and Isham, in their paper ``Representations of the holonomy
algebra of gravity and non-abelian gauge theories'' (which will appear
in Jour. Class. Quant. Grav. if it hasn't yet, but is paper 920253 on
hep-th), define the ``holonomy
algebra'' to be the algebra of functions on the space of connections mod
gauge transformation generated by piecewise smooth Wilson loops, and
form the "holonomy C*-algebra" as the completion of the holonomy algebra in
the L^infinity norm.  (In fact, they do this only for the group SU(2),
and their definitions look more complicated, but they boil down to
precisely this.)  One may work with continuous linear functionals on the
holonomy C*-algebra as a rigorous version of ``measures'' on the space
of connections mod gauge transformations.
In particular, suppose that L is an oriented link
in M having components gamma_1, ... , gamma_p, where
the gamma_i are embedded circles.  Then if E is a continuous linear
functional on the holonomy C*-algebra, defining
      F(L) = E(T(gamma_1) ... T(gamma_p)),
it is easy to see that if E is invariant under the action of Diff_0(M),
F(L) is a topological invariant of the link L.  In lowbrow lingo, a
diffeomorphism-invariant state of a gauge field theory defines a link
invariant.  This is why in the loop variables approach to quantum
gravity, one expects states of quantum gravity to be given by link
invariants (satisfying certain constraints that are still not fully
understood).  

However, the calculations of Polyakov (in "Fermi-Bose
transmutations induced by gauge fields") and Witten (in "Quantum field
theory and the Jones polynomial") in Chern-Simons theory show
that making sense of the Chern-Simons ``measure''
	          e^{iS(A)} DA
involves further subtleties.   Polyakov gave a heuristic argument to show
that in the case of the defining (one-dimensional) representation of 
G = U(1), given disjoint embedded oriented circles gamma_1, gamma_2 in S^3, 
one should be able to calculate the integral over the space of connections
mod gauge transformations
         \int T(gamma_1,A) T(gamma_2,A) e^{iS(A)} DA  
in terms of the linking number of gamma_1 and gamma_2.  (This is a
simple topological invariant that counts how many times gamma_1 winds
around gamma_2.)  The argument breaks down when gamma_1 and gamma_2
intersect, since one obtains a divergent integral.   
If gamma is an embedded circle equipped with a framing (that is,
vector field on gamma that's everywhere linearly independent from the
tangent vector of gamma), one could try to rescue the expression
      \int T(\gamma,A)^2 e^{iS(A)}  DA  
by just *defining* it to be the ``self-linking number'' of gamma.  The
self-linking number is an invariant of oriented framed links and one
defines it simply by taking two copies of gamma, pushing one off the
other in the direction of the framing vector, and calculating the
linking number.  However, since the self-linking number involves
the framing of gamma, while the integral above appears not to, 
this may appear rather ad hoc.   It would be nicer to have a formalism
in which this was a *result* rather than a last-minute trick.

A similar problem occurs in SU(2) Chern-Simons theory.
Using techniques from conformal field theory Witten argued that in 
the case of the defining (2-dimensional) representation of G = SU(2), one
should be able to calculate
       \int T(gamma_1,A) T(gamma_2,A) ... T(gamma_p,A) e^{iS(A)} DA 
in terms of another topological invariant, the ``Kauffman bracket'' of
the link in S^3 with components gamma_i.  (The Kauffman bracket is just
a slight variant of the famous Jones polynomial.)
Again, though, the Kauffman polynomial is an invariant of framed links.

In my paper I describe a modified holonomy
C*-algebra, and also a holonomy Banach algebra, such that
diffeomorphism-invariant continuous linear functionals on these algebras
define invariants of framed links.
Our holonomy algebras are completions of the algebra generated by a kind
of regularized Wilson loop, or ``tube.''  This clarifies the general
relation between framing and regularization in diffeomorphism-invariant
gauge theories.

Moreover, I show that diffeomorphism-invariant continuous linear
functionals on these holonomy algebras are *classified* by
the framed link invariants they define.  That is, the
map from such functionals to framed link invariants is one-to-one. 
One reason for working with Banach and C*-algebras 
is to obtain this result, which depends on approximating
arbitrary products of tubes by products of linked tubes.
This result is a step towards finding a useful inverse of the ``loop
transform'' for diffeomorphism-invariant gauge field theories.
But since I don't know *which* framed link invariants come from
continuous linear functionals on these holonomy algebras, there is still a
long way to go.