32 research outputs found
On Staggered Indecomposable Virasoro Modules
In this article, certain indecomposable Virasoro modules are studied.
Specifically, the Virasoro mode L_0 is assumed to be non-diagonalisable,
possessing Jordan blocks of rank two. Moreover, the module is further assumed
to have a highest weight submodule, the "left module", and that the quotient by
this submodule yields another highest weight module, the "right module". Such
modules, which have been called staggered, have appeared repeatedly in the
logarithmic conformal field theory literature, but their theory has not been
explored in full generality. Here, such a theory is developed for the Virasoro
algebra using rather elementary techniques. The focus centres on two different
but related questions typically encountered in practical studies: How can one
identify a given staggered module, and how can one demonstrate the existence of
a proposed staggered module. The text is liberally peppered throughout with
examples illustrating the general concepts. These have been carefully chosen
for their physical relevance or for the novel features they exhibit.Comment: 54 pages, 6 figures, 16 examples. v2: small changes including new
historical footnote after Eq. (3.6). Not the same as the published version
(we gave up correcting the errors
Hadamard's formula and couplings of SLEs with free field
The relation between level lines of Gaussian free fields (GFF) and
SLE(4)-type curves was discovered by O. Schramm and S. Sheffield. A weak
interpretation of this relation is the existence of a coupling of the GFF and a
random curve, in which the curve behaves like a level line of the field. In the
present paper we study these couplings for the free field with different
boundary conditions. We provide a unified way to determine the law of the curve
(i.e. to compute the driving process of the Loewner chain) given boundary
conditions of the field, and to prove existence of the coupling. The proof is
reduced to the verification of two simple properties of the mean and covariance
of the field, which always relies on Hadamard's formula and properties of
harmonic functions.
Examples include combinations of Dirichlet, Neumann and Riemann-Hilbert
boundary conditions. In doubly connected domains, the standard annulus SLE(4)
is coupled with a compactified GFF obeying Neumann boundary conditions on the
inner boundary. We also consider variants of annulus SLE coupled with free
fields having other natural boundary conditions. These include boundary
conditions leading to curves connecting two points on different boundary
components with prescribed winding as well as those recently proposed by C.
Hagendorf, M. Bauer and D. Bernard.Comment: 26 page