130 research outputs found
Bosonic Ghosts at as a Logarithmic CFT
Motivated by Wakimoto free field realisations, the bosonic ghost system of
central charge is studied using a recently proposed formalism for
logarithmic conformal field theories. This formalism addresses the modular
properties of the theory with the aim being to determine the (Grothendieck)
fusion coefficients from a variant of the Verlinde formula. The key insight, in
the case of bosonic ghosts, is to introduce a family of parabolic Verma modules
which dominate the spectrum of the theory. The results include S-transformation
formulae for characters, non-negative integer Verlinde coefficients, and a
family of modular invariant partition functions. The logarithmic nature of the
corresponding ghost theories is explicitly verified using the
Nahm-Gaberdiel-Kausch fusion algorithm.Comment: 17 pages, one figure; v2: added refs and rewrote a little of the
parabolic subalgebra discussion in Sec. 3 (no change to results); v3: added a
sketch proof of Prop. 1, several clarifications and a few more refs (again,
no change to results
Diagrammatic Kazhdan-Lusztig theory for the (walled) Brauer algebra
We determine the decomposition numbers for the Brauer and walled Brauer
algebra in characteristic zero in terms of certain polynomials associated to
cap and curl diagrams (recovering a result of Martin in the Brauer case). We
consider a second family of polynomials associated to such diagrams, and use
these to determine projective resolutions of the standard modules. We then
relate these two families of polynomials to Kazhdan-Lusztig theory via the work
of Lascoux-Sch\"utzenberger and Boe, inspired by work of Brundan and Stroppel
in the cap diagram case.Comment: 32 pages, 22 figure
Logarithmic Conformal Field Theory: Beyond an Introduction
This article aims to review a selection of central topics and examples in
logarithmic conformal field theory. It begins with a pure Virasoro example,
critical percolation, then continues with a detailed exposition of symplectic
fermions, the fractional level WZW model on SL(2;R) at level -1/2 and the WZW
model on the Lie supergroup GL(1|1). It concludes with a general discussion of
the so-called staggered modules that give these theories their logarithmic
structure, before outlining a proposed strategy to understand more general
logarithmic conformal field theories. Throughout, the emphasis is on continuum
methods and their generalisation from the familiar rational case. In
particular, the modular properties of the characters of the spectrum play a
central role and Verlinde formulae are evaluated with the results compared to
the known fusion rules. Moreover, bulk modular invariants are constructed, the
structures of the corresponding bulk state spaces are elucidated, and a
formalism for computing correlation functions is discussed.Comment: Invited review by J Phys A for a special issue on LCFT; v2 updated
references; v3 fixed a few minor typo
Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories
We study the local properties of a class of codimension-2 defects of the 6d
N=(2,0) theories of type J=A,D,E labeled by nilpotent orbits of a Lie algebra
\mathfrak{g}, where \mathfrak{g} is determined by J and the outer-automorphism
twist around the defect. This class is a natural generalisation of the defects
of the 6d theory of type SU(N) labeled by a Young diagram with N boxes. For any
of these defects, we determine its contribution to the dimension of the Higgs
branch, to the Coulomb branch operators and their scaling dimensions, to the 4d
central charges a and c, and to the flavour central charge k.Comment: 57 pages, LaTeX2
Generating toric noncommutative crepant resolutions
We present an algorithm that finds all toric noncommutative crepant
resolutions of a given toric 3-dimensional Gorenstein singularity. The
algorithm embeds the quivers of these algebras inside a real 3-dimensional
torus such that the relations are homotopy relations. One can project these
embedded quivers down to a 2-dimensional torus to obtain the corresponding
dimer models. We discuss some examples and use the algorithm to show that all
toric noncommutative crepant resolutions of a finite quotient of the conifold
singularity can be obtained by mutating one basic dimer model. We also discuss
how this algorithm might be extended to higher dimensional singularities
Enveloping Algebras and Geometric Representation Theory
The workshop brought together experts investigating algebraic Lie theory from the geometric and categorical viewpoints
Enveloping Algebras and Geometric Representation Theory (hybrid meeting)
The workshop brought together experts investigating algebraic Lie theory from the geometric and categorical viewpoints
-Minkowski Spacetimes and DSR Algebras: Fresh Look and Old Problems
Some classes of Deformed Special Relativity (DSR) theories are reconsidered
within the Hopf algebraic formulation. For this purpose we shall explore a
minimal framework of deformed Weyl-Heisenberg algebras provided by a smash
product construction of DSR algebra. It is proved that this DSR algebra, which
uniquely unifies -Minkowski spacetime coordinates with Poincar\'e
generators, can be obtained by nonlinear change of generators from undeformed
one. Its various realizations in terms of the standard (undeformed)
Weyl-Heisenberg algebra opens the way for quantum mechanical interpretation of
DSR theories in terms of relativistic (St\"uckelberg version) Quantum
Mechanics. On this basis we review some recent results concerning twist
realization of -Minkowski spacetime described as a quantum covariant
algebra determining a deformation quantization of the corresponding linear
Poisson structure. Formal and conceptual issues concerning quantum
-Poincar\'e and -Minkowski algebras as well as DSR theories are
discussed. Particularly, the so-called "-analog" version of DSR algebra is
introduced. Is deformed special relativity quantization of doubly special
relativity remains an open question. Finally, possible physical applications of
DSR algebra to description of some aspects of Planck scale physics are shortly
recalled
Pieri resolutions for classical groups
We generalize the constructions of Eisenbud, Fl{\o}ystad, and Weyman for
equivariant minimal free resolutions over the general linear group, and we
construct equivariant resolutions over the orthogonal and symplectic groups. We
also conjecture and provide some partial results for the existence of an
equivariant analogue of Boij-S\"oderberg decompositions for Betti tables, which
were proven to exist in the non-equivariant setting by Eisenbud and Schreyer.
Many examples are given.Comment: 40 pages, no figures; v2: corrections to sections 2.2, 3.1, 3.3, and
some typos; v3: important corrections to sections 2.2, 2.3 and Prop. 4.9
added, plus other minor corrections; v4: added assumptions to Theorem 3.6 and
updated its proof; v5: Older versions misrepresented Peter Olver's results.
See "New in this version" at the end of the introduction for more detail
Octonionic representations of Clifford algebras and triality
The theory of representations of Clifford algebras is extended to employ the
division algebra of the octonions or Cayley numbers. In particular, questions
that arise from the non-associativity and non-commutativity of this division
algebra are answered. Octonionic representations for Clifford algebras lead to
a notion of octonionic spinors and are used to give octonionic representations
of the respective orthogonal groups. Finally, the triality automorphisms are
shown to exhibit a manifest \perm_3 \times SO(8) structure in this framework.Comment: 33 page
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