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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