a compact surface or 
. We need areas, so we consider a volume form on
. 
INI Seminar/Talk 20181003 T. Levy
2d Yang–Mills holonomy process
a compact surface or . We need areas, so we consider a volume form on
.
a compact Lie group, connected. 
a Lie algebra with scalar product 
.
Example 
and 
.
connections on the principal bundle 
.
: continuous loops on with finite length, up to re-param. (finite lenght is
a diffeo invariant notion).
We want to describe a collection of –valued
random variables 
such that their distribution is
heuristically given by
This is a probability measure on (–valued)
-forms on the manifold which is invariant under certain transformations.
Multiplicativity
If 
are two loops based on the same point then we
must have
where 
is the concatenation of the two loops and
is the loop run in opposite sense.
The curvature 
of the random connection is
distributed like white noise. Since curvature controls the infinitesimal
holonomies we should expect this to show up in 
.
We imagine: 
is Brownian motion/bridge on indexed by with the area
playing the role of time.
Lattice Yang–Mills theory
(Finite dimensional marginals of the holonomy process)
We start from our surface and discretize it by
drawing a graph 
over it (e.g. a triangulation).
It is a real embedded graph. 
where 
are points of the surface and 
which are edges embedded in the surface. Therefore it has faces 
. (we assume orientation for each
edge in )
Configuration space 
. Let
be a path in ,
that is 
: 
and 
an orientation. Then we consider the
discrete holonomy
This is a discrete version of a connection. There is a natural
probability measure on 
by taking the product
measure of Haar measure on .
To make it more interesting we take the heat kernel on :
fundamental solution of the heat equation
It satisfies
Sengupta's formula (Migdal, Witten, Driver)
This expression has the advantage that it gives a consistent family of probability measures, namely it is invariant under subdivisions.
Consider two graphs, 
finer than 
. We have a natural map 
by decimation which preserves the measure: 
.
Consider a face 
of which
is split in two in as in this figure:
Here 
. Now 
and 
, 
and 
. The convolution of the
heat–kernel allows for the following computation
This is the reason to use the heat kernel in the definition of 
.
Theorem of 
-valued
r.v.s, unique in distribution, which is consistent with the lattice
theory and stochastically continuous.
Ilya Chevyrev has a recent preprint where he defines a space of
distributional connections on which he is able
to define a probability measures and holonomies whose law coincide with
the holonomy process.
Example 
. Two loops with disjoint
areas of size 
and 
,
then
so 
are independent and distributed like the
Brownian motion on at time
and . If the two loops have
non disjoint areas then we can write them as three loops 
with disjoint areas of size 
such
that 
, 
and now
Example 
with area 
. Two loops with disjoint areas of size
and . Then outside there is
third face of area 
. In this
case Sengupta's formula gives:
with 
. So now 
where 
is a Brownian bridge on conditioned to return to 
at
time .
The large N limit.
We consider expectations of Wilson's loops:
We are going to do some simple computations of numbers of this kind which reveal some combinatorical phenomena which will “resurface” also in MM equations.
Take 
. . 
a loop of area 
on
where 
is the Brownian motion on 
. We use the notation 
(normalized trace) so that 
.
The BM on is given by an SDE. Take a linear
Brownian motion 
on 
such
that
Given this we can compute
so
Let us try to compute
In order to compute 
we take an ONB 
of and we say that 
. We therefore conclude that
We use now the following identity
which is proven by proving first that it does not depend on the basis and then using your preferred basis to do the computation.
 |
(1) |
We are now going to evaluate with similar method 
and now use that
which gives a second equation:
 |
(2) |
and (1) (2) form a system of ODEs which can be solved to give:
These relations express connections between unitary Brownian motions and the combinatorics of permutations and random walks in the symmetric group.
Take 
and a permutation 
cycle lenghts 
Then I claim that we have
where 
indicate when 
are
in the same cycle of 
.
Therefore, in principle, one can solve this huge family of equations and
taking the 
limit one can prove the following
theorem:
Theorem
Another theorem says:
Theorem with group . For any loop 
Makeenko–Migdal equation
This last theorem gives a sort of law of large numbers in which it plays a role the function
The MM equations give informations about 
.
Take a loop , we want to
compute 
It depends on the combinatorial
structure of the loop and the area of the faces it identifies
The MM equations tells us how this number changes when we distort the loop. Look at an intersection
Where 
are the loops obtained by removing the
intersection as shown in the picture. The MM equations are known to be
true on the plane and on the sphere and the finite 
version are also known on certain surfaces (which?).
The value of on a simple loop is 
. Loops with one intersection point and
disjoint areas give
A loop with on intersection point and one area inside the other (large
area 
and small area ). Call the value of
Then
When we change the area of the unbounded face we get
so we can solve and obtain
which give the limiting value we found before.