Current Research Projects in Prof. Nahm's Group
compact overview
Conformal Field Theory
Andreas Recknagel, Daniel Roggenkamp, Volker Schomerus
On relevant boundary perturbations of unitary
minimal models
Nucl.Phys. B588 (2000) 552-564
We consider unitary Virasoro minimal models on the disk with Cardy boundary
conditions and discuss deformations by certain relevant boundary operators,
analogous to tachyon condensation in string theory. Concentrating on the
least relevant boundary field, we can perform a perturbative analysis of
renormalization group fixed points. We find that the systems always flow
towards stable fixed points which admit no further (non-trivial) relevant
perturbations. The new conformal boundary conditions are in general given
by superpositions of 'pure' Cardy boundary conditions.
Sayipjamal Dulat
The Orbifolds of N=2 Superconformal Theories
with c=3
J.Phys. A33 (2000) 5345
We construct Z_M, M=2,3,4,6 orbifold models of the N=2 superconformal
field theories with central charge c=3. Then we check the description of
the Z_3, Z_4 and Z_6 orbifolds by the N=2 superconformal Landau-Ginzburg
models with c=3, by comparing the spectrum of chiral fields, the Witten
index Tr(-1)^F and the chiral ring with the chiral operator algebra.
Sayipjamal Dulat, Katrin Wendland
Crystallographic Orbifolds: Towards a Classification
of Unitary Conformal Field Theories with Central Charge c=2
JHEP 0006 (2000) 012
We study the moduli space C^2 of unitary two-dimensional conformal field
theories with central charge c=2. We construct all the 28 nonexceptional
nonisolated irreducible components of C^2 that may be obtained by an orbifold
procedure from toroidal theories. The parameter spaces and partition functions
are calculated explicitly, and all multicritical points and lines are
determined. We show that all but four of the 28 irreducible components of C^2
corresponding to nonexceptional orbifolds are directly or indirectly connected
to the moduli space of toroidal theories in C^2. We relate our results to those
by Dixon, Ginsparg, Harvey on the classification of c=3/2 superconformal field
theories and thereby give geometric interpretations to all nonisolated
orbifolds discussed there.
David Brungs, Werner Nahm
The associative algebras of conformal field
theory
Lett.Math.Phys. 47 (1999) 379-383
Modulo the ideal generated by the derivative fields, the normal
ordered product of holomorphic fields in two-dimensional conformal
field theory yields a commutative and associative algebra. The zero
mode algebra can be regarded as a deformation of the latter.
Alternatively, it can be described as an associative quotient of
the algebra given by a modified normal ordered product. We clarify
the relation of these structures to Zhu's product and Zhu's algebra
of the mathematical literature.
String Theory
Werner Nahm, Katrin Wendland
Mirror Symmetry on Kummer Type K3 Surfaces
hep-th/0106104
We investigate both geometric and conformal field theoretic aspects of
mirror symmetry on N=(4,4) superconformal field theories with central
charge c=6. Our approach enables us to determine the action of mirror
symmetry on (non-stable) singular fibers in elliptic fibrations of Z_N
orbifold limits of K3. The resulting map gives an automorphism of order
4,8, or 12, respectively, on the universal cover of the moduli space.
We explicitly derive the geometric counterparts of the twist fields
in our orbifold conformal field theories. The classical McKay
correspondence allows for a natural interpretation of our results.
Holger Eberle
Twistfield Perturbations of Vertex Operators in the Z_2
Orbifold Model
JHEP 06 (2002) 022
We apply Kadanoff's theory of marginal deformations of conformal field
theories to twistfield deformations of Z_2 orbifold models in K3 moduli
space. These deformations lead away from the Z_2 orbifold sub-moduli-space
and hence help to explore conformal field theories which have not yet been
understood. In particular, we calculate the deformation of the conformal
dimensions of vertex operators for p^2<1 in second order perturbation theory.
Werner Nahm, Katrin Wendland
A Hiker's Guide to K3 - Aspects of N=(4,4)
Superconformal Field Theory with central charge c=6
Comm.Math.Phys. 216 (2001) 85-138
We study the moduli space M of N=(4,4) superconformal field
theories with central charge c=6. After a slight emendation of its
global description we find the locations of various known models in the
component of M associated to K3 surfaces. Among them are the Z_2 and
Z_4 orbifold theories obtained from the torus component of M. Here,
SO(4,4) triality is found to play a dominant role. We obtain the B-field
values in direction of the exceptional divisors which arise from orbifolding.
For Z_4 orbifolds this yields an unexpected result. We prove T-duality
for the Z_2 orbifolds and use it to derive the form of M purely
within conformal field theory. For the Gepner model (2)^4 and some of its
orbifolds we find the locations in M and prove isomorphisms to nonlinear
σ models. In particular we prove that the Gepner model (2)^4
has a geometric interpretation with Fermat quartic target space.
Monika Lynker, Rolf Schimmrigk, Andreas Wißkirchen
Landau-Ginzburg Vacua of String, M- and F-Theory at
c=12
Nucl.Phys. B550 (1999) 123-150
Poster
Theories in more than ten dimensions play an important role in
understanding nonperturbative aspects of string theory. Consistent
compactifications of such theories can be constructed via
Calabi-Yau fourfolds. These models can be analyzed particularly
efficiently in the Landau-Ginzburg phase of the linear sigma model,
when available. In the present paper we focus on those sigma models
which have both a Landau-Ginzburg phase and a geometric phase
described by hypersurfaces in weighted projective five-space. We
describe some of the pertinent properties of these models, such as
the cohomology, the connectivity of the resulting moduli space, and
mirror symmetry among the 1,100,055 configurations which we have
constructed.
Ralph Blumenhagen, Andreas Wißkirchen
Spectra of 4D, N=1 Type I String Vacua on
Non-Toroidal CY Threefolds
Phys.Lett. B438 (1998) 52-60
We compute the massless spectra of some four dimensional, N=1
supersymmetric compactifications of the type I string. The
backgrounds are non-toroidal Calabi-Yau manifolds described at
special points in moduli space by Gepner models. Surprisingly, the
abstract conformal field theory computation reveals Chan-Paton
gauge groups as big as SO(12) x SO(20) or SO(8)^4 x SO(4)^3.
Mathematics
Werner Müller, Katrin Wendland
Extremal Kähler metrics and Ray-Singer
analytic torsion
Contemporary Mathematics, American Mathematical Society,
Providence R.I. 1999, p.129-154
Let (X,[ω]) be a compact Kähler manifold with a
fixed Kähler class [ω].
Let K_ω be the set of all Kähler metrics on X whose
Kähler class equals [ω]. In this paper we investigate
the critical points of the functional Q(g)= ||v||_g √(T_0(X,g))
for g∈K_ω, where v is a fixed nonzero vector of the
determinant line λ(X) associated to H*(X) and T_0(X,g) is
the Ray-Singer analytic torsion. For a polarized algebraic manifold
(X,L) we consider a twisted version Q_L(g) of this functional and
assume that c_1(L)=[ω]. Then the critical points of Q_L are
exactly the metrics g∈K_ω of constant scalar curvature.
In particular, if c_1(X)=0 or if c_1(X)<0 and [ω]=-2πc_1(X),
then K_ω contains a unique Kähler-Einstein metric
g_{KE} and Q_L attains its absolut maximum at g_{KE}.
F. Laytimi, W. Nahm
Vanishing theorems for products of exterior and
symmetric powers
For ample vector bundles E over compact complex varieties X and
a Schur functor S_I corresponding to an arbitrary partition I
of the integer |I|, one would like to know the optimal vanishing
theorem for the cohomology groups H^{p,q}(X, S_I(E)), depending
on the rank of E and the dimension n of X. Three years ago (Nov. 1995),
in an unpublished paper one of us (W.N.) proved a vanishing theorem for
the situation where the partition I is a hook.
Here we give a simpler proof of this theorem. We also treat the
same problem under weaker positivity assumptions, in particular
under the hypothesis of ample Λ^{m}(E) with m∈N*. In
this case we also need some bound on the weight |I| of the
partition. Moreover, we prove that the same vanishing condition
applies for H^{q,p}(X, S_I(E)), with p,q interchanged.
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June 19th, 2002
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