Originally posted by Urs
I bet you would do many of us a real favor if you could similarly sketch the content of Vafa's recent work and what its importance is in your opinion.
Hmm... it's almost too much to do so - best thing for you is to
look up his list of papers in spires and check the citations yourself
- this will guide you to some of the important works:
http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+a+vafa%2Cc&FORMAT=WWWCITE&SEQUENCE=
Comparing to others you see from this that he is one of the most
prolific and creative researchers (actually has always been but
currently dominates the field of string theory).
At any rate, here a few highlights:
General theme: topological strings. Not only (often exactly soluble)
toy models for things like large-N dualities, but they also capture
the holomorphic content (BPS sector) of supersymmetric string and
field theories; such as superpotentials and gauge couplings in N=1
SUSY theories, plus an infinite sequence of certain gravitational
couplings in the effective action.
- This framework has allowed to more or less solve the chiral
(holomorphic) sector of N=1 susy gauge theories and in particular
to determine their vauum structure. Most interesting is that all
of this can be phrased in terms of matrix models. This is work done
together with Dijkgraaf and others.
- Non-perturbative physics on Calabi-Yau spaces: mirror symmetry
has been used for a long time to determine effects that are
non-perturbative from the world-sheet point of view, but from the
space time point of view this was mostly tree level - ie, genus
zero in the perturbative expansion. This was useful for eg computing
effective actions of N=2 and N=1 string theories.
The underlying geometry was Calabi-Yau manifolds or "Seiberg-Witten"
Riemann surfaces.
Now the new works (culminating in hep-th/0312085) deal with the
full genus expansion in the string coupling g_s, and this makes
it possible to obtain results that are exact in the string coupling.
All of this has also a close relationship to integrable systems,
and this allows to write down some results explicitly. The underlying
geometry is some sort of quantum deformation of the geometries
indicated above; roughly, if a Calabi-Yau space or Riemann surface
at tree level is given by f(x,y)=0, then the exact quantum geometry
is characterized by a differential operator obtained by letting
x,y become conjugate symplectic variables; ie: f(x,y) -> f(x,d/dx).
The solution of f(x,d/dx) P = 0 is then the exponential of the
all-genus free energy, or exact partition function of the theory!
- In related very recent work hep-th/0312022, it was investigated
what the theory looks like in the strong coupling limit, where g_s
is large. It turns out that the relevant geometry becomes discrete
and has some analogy to melting of crystals. Most interesting in
this is the following: from the exact expressions one can infer
as to what the relevant "geometries" are, one is summing over in
(the relevant topological version of) quantum gravity.
Recall that it is sort of a dogma that one should appropriately
sum over background geometries in quantum gravity. The outcome of
the work under discussion is that the relevant "geometries" are
more general than what one usually calls geometry. [Actually from
the string perspective this is not too much surprising: we know
(esp from the work of Douglas et al) that in stringy D-brane
geometry, more general objects than manifolds or vector bundles
are relevant, rather sheaves and more general objects.]
That is, while a naive quantum gravity person may try to sit down
and sum over some "smooth manifolds", we learn from Vafa-et-al's
work what the right things are to sum over; at least in the
simplified topological context, where one can do exact computations
and obtain explicit results.
These matters are obviously of tremendous conceptional importance,
but since the papers are not easy to read and require quite an
effort to understand even for experts, I am not surprised that many
people are not aware of them. It would have been the duty of Marolf
when writing such a "ressource letter" to point this out to a larger
audience.
