123 research outputs found
Homogeneous components in the moduli space of sheaves and Virasoro characters
The moduli space of framed torsion free sheaves on the
projective plane with rank and second Chern class equal to has the
natural action of the -dimensional torus. In this paper, we look at the
fixed point set of different one-dimensional subtori in this torus. We prove
that in the homogeneous case the generating series of the numbers of the
irreducible components has a beautiful decomposition into an infinite product.
In the case of odd these infinite products coincide with certain Virasoro
characters. We also propose a conjecture in a general quasihomogeneous case.Comment: Published version, 19 page
Free Boson Realization of
We construct a realization of the quantum affine algebra
of an arbitrary level in terms of free boson fields.
In the limit this realization becomes the Wakimoto
realization of . The screening currents and the vertex
operators(primary fields) are also constructed; the former commutes with
modulo total difference, and the latter creates the
highest weight state from the vacuum state of the boson
Fock space.Comment: 24 pages, LaTeX, RIMS-924, YITP/K-101
Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
We introduce a Kazhdan--Lusztig-dual quantum group for (1,p) Virasoro
logarithmic minimal models as the Lusztig limit of the quantum sl(2) at pth
root of unity and show that this limit is a Hopf algebra. We calculate tensor
products of irreducible and projective representations of the quantum group and
show that these tensor products coincide with the fusion of irreducible and
logarithmic modules in the (1,p) Virasoro logarithmic minimal models.Comment: 19 page
Logarithmic extensions of minimal models: characters and modular transformations
We study logarithmic conformal field models that extend the (p,q) Virasoro
minimal models. For coprime positive integers and , the model is defined
as the kernel of the two minimal-model screening operators. We identify the
field content, construct the W-algebra W(p,q) that is the model symmetry (the
maximal local algebra in the kernel), describe its irreducible modules, and
find their characters. We then derive the SL(2,Z) representation on the space
of torus amplitudes and study its properties. From the action of the
screenings, we also identify the quantum group that is Kazhdan--Lusztig-dual to
the logarithmic model.Comment: 43pp., AMSLaTeX++. V3: Some explanatory comments added, notational
inaccuracies corrected, references adde
On the 2D zero modes' algebra of the SU(n) WZNW model
A quantum group covariant extension of the chiral parts of the
Wess-Zumino-Novikov-Witten model on a compact Lie group G gives rise to two
matrix algebras with non-commutative entries. These are generated by "chiral
zero modes" which combine in the 2D model into "Q-operators" which encode
information about the internal symmetry and the fusion ring. We review earlier
results about the SU(n) WZNW Q-algebra and its Fock representation for n=2 and
display the first steps towards their generalization to higher n.Comment: 10 pages, Talk presented by L.H. at the International Workshop LT10
(17-23 June 2013, Varna, Bulgaria
Resolutions and Characters of Irreducible Representations of the N=2 Superconformal Algebra
We evaluate characters of irreducible representations of the N=2
supersymmetric extension of the Virasoro algebra. We do so by deriving the
BGG-resolution of the admissible N=2 representations and also a new
3,5,7...-resolution in terms of twisted massive Verma modules. We analyse how
the characters behave under the automorphisms of the algebra, whose most
significant part is the spectral flow transformations. The possibility to
express the characters in terms of theta functions is determined by their
behaviour under the spectral flow. We also derive the identity expressing every
character as a linear combination of spectral-flow transformed
N=2 characters; this identity involves a finite number of N=2 characters in the
case of unitary representations. Conversely, we find an integral representation
for the admissible N=2 characters as contour integrals of admissible
characters.Comment: LaTeX2e: amsart, 34pp. An overall sign error corrected in (4.33) and
several consequent formulas, and the presentation streamlined in Sec.4.2.3.
References added. To appear in Nucl. Phys.
Analytic Expressions for Singular Vectors of the Superconformal Algebra
Using explicit expressions for a class of singular vectors of the
(untwisted) algebra and following the approach of Malikov-Feigin-Fuchs and
Kent, we show that the analytically extended Verma modules contain two linearly
independent neutral singular vectors at the same grade. We construct this two
dimensional space and we identify the singular vectors of the original Verma
modules. We show that in some Verma modules these expressions lead to two
linearly independent singular vectors which are at the same grade and have the
same charge.Comment: 35 pages, LATE
Generalized twisted modules associated to general automorphisms of a vertex operator algebra
We introduce a notion of strongly C^{\times}-graded, or equivalently,
C/Z-graded generalized g-twisted V-module associated to an automorphism g, not
necessarily of finite order, of a vertex operator algebra. We also introduce a
notion of strongly C-graded generalized g-twisted V-module if V admits an
additional C-grading compatible with g. Let V=\coprod_{n\in \Z}V_{(n)} be a
vertex operator algebra such that V_{(0)}=\C\one and V_{(n)}=0 for n<0 and let
u be an element of V of weight 1 such that L(1)u=0. Then the exponential of
2\pi \sqrt{-1} Res_{x} Y(u, x) is an automorphism g_{u} of V. In this case, a
strongly C-graded generalized g_{u}-twisted V-module is constructed from a
strongly C-graded generalized V-module with a compatible action of g_{u} by
modifying the vertex operator map for the generalized V-module using the
exponential of the negative-power part of the vertex operator Y(u, x). In
particular, we give examples of such generalized twisted modules associated to
the exponentials of some screening operators on certain vertex operator
algebras related to the triplet W-algebras. An important feature is that we
have to work with generalized (twisted) V-modules which are doubly graded by
the group C/Z or C and by generalized eigenspaces (not just eigenspaces) for
L(0), and the twisted vertex operators in general involve the logarithm of the
formal variable.Comment: Final version to appear in Comm. Math. Phys. 38 pages. References on
triplet W-algebras added, misprints corrected, and expositions revise
Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals
The Jack polynomials P_\lambda^{(\alpha)} at \alpha=-(k+1)/(r-1) indexed by
certain (k,r,N)-admissible partitions are known to span an ideal I^{(k,r)}_N of
the space of symmetric functions in N variables. The ideal I^{(k,r)}_N is
invariant under the action of certain differential operators which include half
the Virasoro algebra. Moreover, the Jack polynomials in I^{(k,r)}_N admit
clusters of size at most k: they vanish when k+1 of their variables are
identified, and they do not vanish when only k of them are identified. We
generalize most of these properties to superspace using orthogonal
eigenfunctions of the supersymmetric extension of the trigonometric
Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular,
we show that the Jack superpolynomials P_{\Lambda}^{(\alpha)} at
\alpha=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible superpartitions span
an ideal {\mathcal I}^{(k,r)}_N of the space of symmetric polynomials in N
commuting variables and N anticommuting variables. We prove that the ideal
{\mathcal I}^{(k,r)}_N is stable with respect to the action of the
negative-half of the super-Virasoro algebra. In addition, we show that the Jack
superpolynomials in {\mathcal I}^{(k,r)}_N vanish when k+1 of their commuting
variables are equal, and conjecture that they do not vanish when only k of them
are identified. This allows us to conclude that the standard Jack polynomials
with prescribed symmetry should satisfy similar clustering properties. Finally,
we conjecture that the elements of {\mathcal I}^{(k,2)}_N provide a basis for
the subspace of symmetric superpolynomials in N variables that vanish when k+1
commuting variables are set equal to each other.Comment: 36 pages; the main changes in v2 are : 1) in the introduction, we
present exceptions to an often made statement concerning the clustering
property of the ordinary Jack polynomials for (k,r,N)-admissible partitions
(see Footnote 2); 2) Conjecture 14 is substantiated with the extensive
computational evidence presented in the new appendix C; 3) the various tests
supporting Conjecture 16 are reporte
BRST Analysis of Physical States for 2D (Super) Gravity Coupled to (Super) Conformal Matter
We summarize some recent results on the BRST analysis of physical states of
2D gravity coupled to c<=1 conformal matter and the supersymmetric
generalization.Comment: 11 page
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