Re: Question for Thomas Larsson re Virasoro algebras
- From: thomas_larsson_01@xxxxxxxxxxx
- Date: Sun, 11 Dec 2005 23:02:59 +0000 (UTC)
DRLunsford skrev:
> Recently it came to my attention that you regard your work to be a sort
> of quantum analog of tensor analysis, a very interesting idea.
>
> Have you thought about the idea of a tensor density in this formalism
> and the idea of weight in general - that is can your work be
> generalized in that direction?
>
That tensor calculus, and more generally differential geometry, can
be regarded as the representation theory of the diffeomorphism group
is a correct, although not very original, observation. This is
because tensor calculus is the theory of well-defined objects, which
by definition transform as representatations under diffeomorphisms.
Thus, a tensor field is a module, the exterior derivative is the
unique morphism between tensor modules intertwining with the group
action, a connection (+ the unit) carries a reducible but
indecomposable rep, etc.
More generally, special kinds of geometry (symplectic, contact, ...)
can be identified with the representation theory of the appropriate
subgroup (symplectomorphisms, contactomorphisms, ...). This is
well-known among algebraists. E.g., Victor Kac writes in
http://www.arxiv.org/abs/math.QA/9912235 , p 20:
"Each of the four type W, S, H, K of simple primitive Lie algebras
correspond to the four most important types of geometrics of
manifolds: all manifolds, oriented manifolds, symplectic and contact
manifolds."
The insight from conformal field theory is that the representations
of the Virasoro algebra, which is the diffeomorphism algebra in 1D,
are of two qualitatively different types: classical and quantum. The
classical reps are called primary and secondary fields, which are
nothing but scalar densities and their derivatives, i.e. the tensor
fields in 1D. However, there are also quantum reps, which are of
lowest-weight type (the L_0 eigenvalue is bounded from below), and
which have a non-zero central charge. There are different types of
quantum reps (Verma modules, Fock modules, minimal models, coset
models, ...), and it is those which are of direct relevance to
quantum theory.
The lowest-energy reps of the multi-dimensional Virasoro algebra thus
generalize the quantum reps to higher dimensions, in the same sense
as tensor calculus generalize the theory of scalar densities. Hence
it appropriate to think of it as quantum tensor calculus.
This is the basic motivation for its application to quantum gravity.
If classical tensor calculus is useful in classical gravity, which I
think it is (otherwise, why is tensor calculus taught in GR
classes?), then it is not too daring to expect that quantum tensor
calculus is important to quantum gravity.
It should be emphasized that the theory is much harder, and hence
much less developed, in multi-dimensions. In particular, the
irreducible reps, i.e. the analog of the minimal models, are not
known. There seems to be three qualitatively different levels of
complication:
1. Finite-dimensional algebras, like Poincare. The irreps act on
particle states.
2. Infinite-dimensional chiral algebras, like Virasoro and affine
Kac-Moody. The irreps act on free fields.
3. Infinite-dimensional algebras in multi-dimensions. The irreps
act on interacting fields.
Fortunately, the third class of algebras also act (reducibly) on free
jets, which approximate fields in the sense that finite Taylor series
approximate functions. This is how one can get a hold on the
representation theory.
.
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- From: DRLunsford
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