Geometric Langlands, Part 5: Affine Lie Algebras, a.k.a. 
In Defense of Central Extensions.
--------------------------------

(an informal diatribe)

The representation theory of finite dimensional (semisimple)
Lie algebras is a well developed and rich topic, as you all fully
know. What is less universally familiar is the incredibly rich 
theory of affine Lie algebras, the simplest class of infinite
dimensional (Kac-Moody) Lie algebras, and their extraordinarily
deep relations with various areas of classical mathematics, 
especially the study of Riemann surfaces, modular forms, integrable
systems, special functions, and the Monster (not to mention, or rather
to mention their
central role in relation to quantum groups, string theory/conformal
field theory, elliptic cohomology/K-theory etc. through which
some of the above are best understood "in vivo"). 

Well enough of the introductory BS. What are affine Lie algebras?
These are essentially just the loop algebras Lg of maps from the circle
into g, our favorite simple Lie algebra, which form a Lie algebra
by  pointwise bracket. Thus loop algebras are the one-dimensional
case of current algebras, namely the Lie algebras Map(X,g) where
X is any manifold say. (One can get fussy about the kind of maps
one allows, but we won't - take them to be polynomial maps.)

Before we lose the algebraic geometers among you, note that the loop
group is simply the group of K-points of the algebraic group G, where
K is the (local) field C(t) of rational (complex-valued) functions 
in t (corresponding to finite Fourier series on the circle.
(One often considers instead the group corresponding to the 
complete local field C((t)), with closely related representation
theory.) Thus one may expect that many of the phenomena encountered in
the representation theory of groups over p-adic fields (like the
Langlands picture) will be paralleled for loop/affine algebras -
and indeed there are also analogs (namely going to quantum affine
algebras) to reduction modulo p and rep theory over finite fields -
except that the geometry of primes is replaced with the geometry of
Riemann surfaces (for which I won't try to hide a strong bias :-)
In any case everything said and thought about double coset spaces
etc are local-field phenomena and appear in this context as well.

Well I won't try to motivate loop algebras much more.. what is more
crucial is the appearance of the affine algebras as central extensions
of loop algebras, which I know might seem initially less natural
and perhaps less immediately appealing - so let me try to convince you
of the importance, naturality and beauty of passing to the affine
algebras (which I have yet to define..)

First for some technical motivations. By the theorems of Weyl etc, a simple
Lie algebra has vanishing H1 and H2 (Lie algebra cohomology.) It does on
the other hand have a one dimensional H3 (with a canonical generator).
I don't quite understand the appearance of this H3 (one can write down
a nice cocycle in terms of Killing form/Casimirs/something that
generates it..) but geometrically it corresponds to our good friend the
three-sphere: Lie algebra cohomology agrees with the (topological)
cohomology of the corrseponding complex (simply connected) Lie group.
Now these Lie groups always contain SL2's, which as you know are the key
to their representation theory.. => thus they also contain SU2's, the
maximal compact of SL2  - indeed once can find a canonical one after
choosing a set of positive roots (=a notion of highest weight)..
This three sphere then contributes a Z to H3, and it turns out that's
all there is.

Now when looping a topological space, its cohomology groups slide down
one spot. Thus we see that the loop group LG (and its Lie algebra Lg)
has vanishing H1 but a nonvanishing (one-dimensional) H2. Lie algebra H2
classifies central extensions: an element of H2 can be represented by a
2-cocycle, so we define a new Lie algebra consisting (as vector space)
of the direct sum of our algebra and a one-dimensional center, while the
commutator [l,m] in the old algebra is changed to [l,m]+
cocycle(l,m) times the central element (generator).
It follows that Lg has a UNIVERSAL (one-dimensional) central extension,
which as a vector space consists of LG direct sum CK, the
one-dimensional vector space generated by the central element K.. this
new algebra is called the affine algebra g^.

Now why would one want to centrally extend your algebra anyway? Because
you can? not very convincing. However you have already encountered
central extensions in a natural context, albeit with Lie groups not Lie
algebras.. The Lie algebra so3 can be integrated to the Lie group SO3,
however something is wrong with this construction since not all
(fin-dim) reps of so3 integerate to SO3 - there is the famous spin
representation, which becomes doubly-valued when extended to SO3. That
is, we get not a well-defined representation of SO3 but rather a
projective representation, one defined up to +-1.. We all know how to
fix this - pass to the (Z/2) central extensions of SO3, namely Spin1
(=SU2), and spinors become legal again. Rather than think of this as
passing to the universal cover, we might want to consider this as a
first motivating example for central extensions - namely projective
representations  (defined up to a constant multiplier / cocycle....)
correspond to representations of a central extension of our group or
algebra. 

Thus the first reason we want to consider g^ rather than Lg is that many
naturally defined constructs will only be PROJCETIVE reps of LG, and
thus correspond to representations of g^ - we'd have a big unseemly gap
in the edifice of reps if we only considered Lg. 
An irreducible rep of g^ will have the central element K acting by a
scalar (Schur's lemma), so we haven't veered all that far from Lg..
except that 0 is usually not the most interesting value of this scalar -
rather there is a "critical level" - reps on which K acts by -h^, minus
the dual Coxeter number, where everything gets REALLY interesting,
completely new geometry appears, exciting connections with other
"critical" phenomena, etc etc..

Now projective representations are extremely ubiquitous in
infinite-dimensional geometry. The most famous manifestation of this is
the Hilbert-space formulation of quantum mechanics, where states are
defined not as vectors in our Hilbert space H, but rather as rays -
one-dimensional subspaces. In other words the ntaural representations of
quantum mechanics are often naturally projective representations, since
the space of states is the projective space of H and this is where the
physics occurs.

Another manifestation of projectivity in quantum mechanics is the
Heisenberg uncertainty principle, as encoded in the Heisenberg algebra.
This is a "two-step nilpotent" Lie algebra - in other words the
commutator of any two elements (like x and d/dx) lies in the center =>
i.e. the Heisenberg algebra is a central extension of an abelian Lie
algebra. In infinite dimensional geometry the Heisenberg algebra is used
as the substitute for abelian Lie algebras as a "base case", and thus
naturally one is led to consider central extensions of our algebra of
intersest in order to embed it in Heisenberg. Classically this
corresponds to the Borel-Weil construction: G acts on the flag variety
G/B and all interesting reps are realized as line bundles etc on this
space.. now G/B contains an open cell (the "big cell") B id B, the orbit
of the identity coset under B -- which is an affine space. G of course
doesn't preserve this open cell but g, the Lie algebra, does - so
we get g acting as vector fields on the big cell (and hence realized on
sections of our line bundle, restricted to this affine space).
Thus we seek to embed g into the algebra of differential operators on
the affine space - which is precisely the Heisenberg (aka Weyl)
algebra.. The analog of this embedding (lifting vector fields to
diffops) does NOT work in infinite dimensions unless we in fact
centrally extend our Lie algebra in a fashion parallel to that of the
Heisenberg algebra   ==> g^..

The appearance of central extensions in infinite dimensional geometry is
also deeply linked with the phenomenon of renormalization: certain
operations that make perfect sense in finite dimensions are "suddenly"
ill-defined in infinite dimensions and require a clever
"renormalization", that is the substitution in some canonical way of a
finite quantity that will satisfy the same role. One common example of
renormalization is the "zeta-function renormalization", which uses the
time-honored principle of analytic continuation: if we have an infinite
series/product/expression which diverges in the range we're interested
in, perhaps it is because we are simply using a Taylor series expansion
(i.e. concrete realization) of the function we're really interested in,
but outside its domain of convergence.. so to get the right answer we
should take this expression in the radius of convergence and
analytically continue it to the point of relevance, getting a nice
finite answer.. This is the technique used to define for example the
Quillen metric mentioned last time.

Now in the case of Lie algebras, the reason H2(g) is zero is because we
can construct a cochain whose coboundary gives any 2-cocycle -i.e.
two-cocycles are all trivial. Now when we try to write the same standard
cocycle as a coboundary in infinite dimensions, we have a problem: the
cochain whose coboundary this would be has blown up, poof!, exploded,
diverged - it no longer exists. We thus get a nontrivial class in H2
which gives us the desired central extension - there is an "anomaly", or
cohomology obstruction, in the renormalized theory (or something to that
effect...don't quote me here!) This can be expressed algebraically by
the failure of traces to be defined in general, and thus for traces of
commutators to be trivial. (This can be expressed in many different
ways, in terms of Fredholm operators, or Tate's linear algebra, etc. -
again this is quite a universal phenomenon and characteristic of
infinite dimensions.) If we look at the Lie algebra gl of \infty by
\infty matrices, and let p be the projection on the "upper left
quadrant" i,j<0, this cocycle can be written as
Tr(p([f,g])-[p(f),p(g)]), which is a coboundary when Tr makes sense on
all matrices (i.e. fin dim case)..

The renormalization/central extension phenomenon 
can also be expressed in the language of gerbes, if you are
friendly with them.. Recall a "torsor" or "principal space" for a group
G is a space on which G acts simply transitively, but which does not
necessarily have a canonical identification with G itself - these make
up H1(G).. alas I have no space/time to go into these now. A G-gerbe is
then the next step of separation away from G - it is a category, all of
whose Hom sets arre in fact G-torsors (one can think of a torsor as
being a one-element category, whose Hom set is identified with G..) 
G-gerbes classify geometrically the SECOND cohomology group of G (i.e.
the second nonabelian cohomology..)  What does this have to do with
anything? One finds often in renormalization, e.g. in trying to define
path integrals etc., that the objects we're trying to define make no
sense in themselves, but their RATIOS do - we have a natural class of 
objects, these infinite-dimensional integrals, which form a C* torsor -
in other words it makes sense to say one is 3+2i times another, but not
what the value of one is. Following this reasoning along it follows that
much of quantum field theory-type mathematics can/should be expressed in
the language of torsors/gerbes/2-gerbes etc corresponding to a rigorous
formulation of "ill-defined objects".. and that the H2 class
corresponding to our favorite central extensions fit nicely in this
framework - note the analogy C*-torsor <=> projective representation
(defined up to constant).. I don't know how to formalize this, so I'll
leave it as a (literature-search) exercise for the reader.

This same renormalized cocycle problem is concretely apparent in
representation theory in the form of "normal ordering", aka "Wick
ordering". This arises in representation theory, where one wishes to
consider highest weight representations (more generally "category O") as
a natural habitat for representation theorists, where one can work like
in finite dimensions. As for finite dimensional Lie algebras, this is
expressed by having raising and lowering operators, and our
representations are then such that we can kill any vector by raising it
enough times. (Physicists reverse the arrows and have lowest weights,
corresponding to "positive energy" requirements, I believe.) So we can
most usefully pass to a completion of our (universal envelopping)
algebra w.r.t. the ideal of raisings, 
since any vector in our rep will be killed by sufficiently high powers
of this ideal and so these infinite sums become finite and well defined
on any given vector. However we must of course not allow infinitely many
lowerings, since then we won't get well defined operators on our
representation. This leads to the operation of normal ordering, which
takes some "word" in the universal envelopping algebra and switches the
order of operators so raisings come first and then lowerings, thus
making it well - defined... This renormalization procedure, which is
obligatory if one wishes to define such key objects as Casimir operators
etc., once again corresponds to the cocycle defining the central
extension of our Lie algebra - but again I can elaborate no further.

The desire to repreoduce highest-weight theory in infinite dimensions
led to the discovery of the class of Kac-Moody Lie algebras, which have
a structure theory completely analogous to that of simple
finite-dimensional Lie algebras - they too are given by finite (nxn)
Cartan matricces and the corresponding Dynkin diagrams/Coxeter groups,
the basis in terms of hi,ei, and fi, with the expanded sl2 relations
(aka Serre relations)..nondegenerate Cartan matrices correspond to
simple Lie algebras, while nxn matrices with rank n-1 give a natural
next class to study, and have Weyl groups which are semidirect products
of standard Weyl groups by affine lattices..these are known as affine
algebras and are discovered to agree with the centrally extended loop
algebras g^  --  a natural algebraic reason to  pass to affine algebras.

The central extension 2-cocycle can
also be traced back to the Cauchy integral theorem, namely to the
phenomenon of residues.. this is most concretely apparent in the theory
of vertex operator algebras(VOAs)/operator product expansions (OPEs) -
here we study the infinite words considered above in rep theory as an
algebraic structure in themselves, namely formal power series with
values in operators on our representation; normal ordering forces us to take
commutators (this is what happens when you switch the order of things),
and the commutators of the formal  power series are determined formally
by Cauchy integrals, i.e. by the residue (1/z) terms in the operator
product expansion.. This can be said very formally and in great
generality, as was done by Tate (yes this comes up in number theory
too..) - theory of "Tate's residues", "Tate central extensions" etc..
this is not an "analytic" phenomenon (these power series are formal) but
rather a basic property of function fields, be they over Fp or complex or
whatever.

Finally we conclude this diatribe with some vague geometric motivation
for the appearance of central extensions. The geometry behind affine
algebras is that of Riemann surfaces, namely CONFORMAL geometry (aka
Conformal Field Theory, CFT). A conformal structure is a metric, but
defined only up to a constant multiplier (at each point - namely up to
multiplication by a function.) This might again indicate that
representations that appear naturally in conformal geometry will be
projective representations of the (infinite-dimensional) infinitesimal
symmetry groups/algebras of Riemann surfaces or e.g. vector bundles over them
(this is precisely the origin of CFT) - one defines a field theory (or
any object) concretely in terms of the metric, but cares about
preserving it only up to a constant locally..  

In classical geometry this appears in the form of the Schwarzian
derivative. This differential operator measures how far a given
conformal map is from being a Mobius transformation. Since we can
normalize a Mobius transformation to have arbitrary first and second
order terms, the Schwarzian should be a quadratic expression -
essentially the second-order jet of our conformal map. But Schwarzian
derivatives of functions are NOT quadratic differentials, but rather
"projective connections" - they have a more complicated, nonlinear
transformation property which corresponds to a "up to projective
transformation" in the definition... Thus the algebra of local
symmetries of a punctured disc, the basic infinite-dimensional algebra
which is the building block for all conformal geometry, is given not by
vector fields on C* as one might expect 
(PS vector fields on C* are the dual to quadratic differentials - contract 
your quadratic differential with a vector field and you get a one-form,
which you can integrate around the circle, namely take a residue...)

---- but rather the dual space to the space of PROJECTIVE CONNECTIONS,
which is a (universal one-dimensional) central extension of the Lie algebra
of vector fields, and is known as the Virasoro algebra - the undisputed
star of conformal field theory, and an intimate friend of the affine Lie
algebras, which have their central extension for the same reason (as
symmetries of bundles over the above picture.)

And that's all for now, folks.