Monstrous Moonshine from orbifolds

We consider the Monster Module of Frenkel, Lepowsky, and Meurman as aZ ₂ orbifold of a bosonic string compactified by the Leech lattice. We show that the main Conway and Norton Monstrous Moonshine properties, stating that the Thompson series for each Monster group conjugacy class has a modular invariance group of genus zero, follow from an orbifold construction based on an orbifold group composed of Monster group elements. it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each Thompson series. It is also shown that the Power Map formula of Conway and Norton follows from the same vacuum structure. Finally, we demonstrate the validity of the vacuum conjectures for sectors twisted by Leech lattice automorphisms in many cases.     

Some irrational Generalised Moonshine from orbifolds

We verify the Generalised Moonshine conjectures for some irrational modular functions for the Monster centralisers related to the Harada–Norton, Held, M₁₂ and L₃(3) simple groups based on certain orbifolding constraints. We find explicitly the fixing groups of the hauptmoduls arising in each case.     

Orbifolds,Hopf algebras,and the Moonshine

We describe the modular properties and fusion rules of holomorphic orbifold models by Hopf algebraic techniques, using the representation theory of the orbifold quantum group. We apply this theory to the construction of generalized Thompson series, and discuss its connections with Moonshine.     

On the relationship between Monstrous Moonshine and the uniqueness of the Moonshine Module

We consider the relationship between the conjectured uniqueness of the Moonshine Module,, and Monstrous Moonshine, the genus zero property of the modular invariance group for each Monster group Thompson series. We first discuss a family of possibleZ _n meromorphic orbifold constructions of based on automorphisms of the Leech lattice compactified bosonic string. We reproduce the Thompson series for all 51 non-Fricke classes of the Monster groupM together with a new relationship between the centralisers of these classes and 51 corresponding Conway group centralisers (generalising a well-known relationship for 5 such classes). Assuming that is unique, we consider meromorphic orbifoldings of and show that Monstrous Moonshine holds if and onlyZ _r if the only meromorphic orbifoldings of are itself or the Leech theory. This constraint on the meromorphic orbifoldings of therefore relates Monstrous Moonshine to the uniqueness of in a new way.     

Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of the theory of rational orbifold models in conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise number of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlation functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representation of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway–Norton–Queen and to equivariant elliptic cohomology. Received: 7 January 1999 / Accepted: 14 March 2000     

Boundary Structure and Module Decomposition of the Bosonic Z2 Orbifold Models with R=1/2k

The Z₂ bosonic orbifold models with compactification radius R²=1/2k are examined in the presence of boundaries. Demanding the extended algebra characters to have definite conformal dimension and to consist of an integer sum of Virasoro characters, one arrives at the right splitting of the partition function. This is used to derive a free field representation of a complete, consistent set of boundary states, compatible with the modular transformations of the characters. Finally the modules of the extended symmetry algebra that correspond to the finitely many characters are identified inside the direct sum of Fock modules that constitute the space of states of the theory.     

Remarks on Wilson lines,modular invariance and possible string relics in Calabi-Yau compactifications

When one mods out a (2,2) conformal field theory by the action of a discrete group, it is possible to include Wilson lines to break the gauge symmetry. We simplify and generalize an earlier analysis by Witten of the constraints that modular invariance places on the allowed symmetry breaking patterns. The analysis does not depend on the details of the original conformal field theory. We then consider the fractionally charged states in such theories, first discussed by Wen and Witten. We note that these are rather generic, and consider the possibilities for their detection. We also note that, while in general they are expected to be massive (∼M_Planck), in models based on free fields, such as orbifold compactifications, there are likely to be massless (very light) fractionally charged states.     

Topological Open Strings on Orbifolds

We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the \mathbb C³/\mathbb Z₃{{\mathbb C}^3/{\mathbb Z}_3} orbifold point of local \mathbb P²{{\mathbb P}^2}, where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold Gromov–Witten invariants of \mathbb C³/\mathbb Z₃{{\mathbb C}^3/{\mathbb Z}_3}, or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.     

Affine Orbifolds and Rational Conformal Field Theory Extensions of W 1+∞

Chiral orbifold models are defined as gauge field theories with a finite gauge group Γ. We start with a conformal current algebra associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group Γ of inner automorphisms or (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra of local observables invariant under Γ. A set of positive energy modules is constructed whose characters span, under some assumptions on Γ, a finite dimensional unitary representation of . We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules. As an application we construct a family of rational conformal field theory (RCFT) extensions of W _1+∞ that appear to provide a bridge between two approaches to the quantum Hall effect. Received: 5 December 1996 / Accepted: 1 April 1997     

Non-abelian spin-singlet quantum Hall states: wave functions and quasihole state counting

We investigate a class of non-abelian spin-singlet (NASS) quantum Hall phases, proposed previously. The trial ground and quasihole excited states are exact eigenstates of certain (k+1)-body interaction Hamiltonians. The k=1 cases are the familiar Halperin abelian spin-singlet states. We present closed-form expressions for the many-body wave functions of the ground states, which for k>1 were previously defined only in terms of correlators in specific conformal field theories. The states contain clusters of k electrons, each cluster having either all spins up, or all spins down. The ground states are non-degenerate, while the quasihole excitations over these states show characteristic degeneracies, which give rise to non-abelian braid statistics. Using conformal field theory methods, we derive counting rules that determine the degeneracies in a spherical geometry. The results are checked against explicit numerical diagonalization studies for small numbers of particles on the sphere.     

States and automorphism groups of operator algebras

Suppose that a group of automorphisms of a von Neumann algebraM, fixes the center elementwise. We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state onM, then this state is left invariant by the group of automorphisms. As a result we obtain a “noncommutative” ergodic theorem. The discrete spectrum of an abelian unitary group acting as automorphisms ofM is completely characterized by elements inM. We discuss the KMS condition on the CAR algebra with respect to quasi-free automorphisms and gauge invariant generalized free states. We also obtain a necessary and sufficient condition for the CAR algebra and a quasi-free automorphism group to be η-abelian.     

Three generations in the fermionic construction

We obtain three generation SU(3)_c × SU(2)_L × U(1)_Y string models in all of the exactly solvable (0, 2) constructions sampled by fermionization. None of these examples, including those that are symmetric abelian orbifolds, rely on the ℤ₂ × ℤ₂ orbifold underlying the NAHE basis. We present the first known three generation models for which the hypercharge normalization, k₁, takes values smaller than that obtained from an SU(5) embedding, thus lowering the effective gauge coupling unification scale. All of the models contain fractional electrically charged and vectorlike exotic matter that could survive in the light spectrum.     

Norton's Trace Formulae for the Griess Algebra¶of a Vertex Operator Algebra with Larger Symmetry

Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module V , with the trace formulae obtained in a different way by S. Norton, and the spectrum of some idempotents related to 2A, 2B, 3A and 4A elements of the Monster is determined by the representation theory of the Virasoro algebra at c= 1/2, the W ₃ algebra at c= 4/5 or the W ₄ algebra at c= 1. The generalization to the trace function on the whole space is also given for the composition of two adjoint actions, which can be used to compute the McKay-Thompson series for a 2A involution of the Monster. Received: 24 July 2000 / Accepted: 15 June 2001     

Orbifold models and modular transformation

The orbifold models of the heterotic string are constructed on the quotient spaces of generalized tori by translational and rotational discrete symmetries. In order to obtain the consistent orbifold models, the conditions of the modular invariance are derived from a one-loop vacuum amplitude. Z₃ orbifold models satisfying such conditions are searched systematically. It is shown that there are infinite possible models with N = 2 supersymmetry. Among these models, two examples having E₆ and E₇ gauge groups are discussed. The orbifold models with N = 1 supersymmetry are also discussed in detail. It is shown that there are only five consistent models in the class of these models based on E₈ ⊗ E′₈ heterotic string in which the extra six-dimensional torus and the E₈ ⊗ E′₈ maximal torus are modded out by the rotational and the translational Z₃ symmetries respectively.     

Dirac quantization of open p-brane

We give the scheme of Dirac quantization of open p-brane in the D-brane background. Treating the mixed boundary conditions as primary constraints, we get a set of secondary constraints, then the constraints conditions are shown to be equivalent to orbifold conditions imposed on normal p-brane modes.     

Classification of symmetry breaking by Wilson lines on the Z orbifold

We give a complete classification of gauge symmetry breaking by Wilson lines on the standard Z orbifold by deriving the general formula of the conditions of modular invariance and group invariance in the presence of background gauge fields. All possible E₆×SU(3) breaking in terms of one Wilson line is given. The symmetries of the electroweak and grand unification are obtained by combining two Wilson lines.     

Reconstruction of dynamic contrast enhanced magnetic resonance imaging of the breast with temporal constraints

A number of methods using temporal and spatial constraints have been proposed for reconstruction of undersampled dynamic magnetic resonance imaging (MRI) data. The complex data can be constrained or regularized in a number of different ways, for example, the time derivative of the magnitude and phase image voxels can be constrained separately or jointly. Intuitively, the performance of different regularizations will depend on both the data and the chosen temporal constraints. Here, a complex temporal total variation (TV) constraint was compared to the use of separate real and imaginary constraints, and to a magnitude constraint alone. Projection onto Convex Sets (POCS) with a gradient descent method was used to implement the diverse temporal constraints in reconstructions of DCE MRI data. For breast DCE data, serial POCS with separate real and imaginary TV constraints was found to give relatively poor results while serial/parallel POCS with a complex temporal TV constraint and serial POCS with a magnitude-only temporal TV constraint performed well with an acceleration factor as large as R=6. In the tumor area, the best method was found to be parallel POCS with complex temporal TV constraint. This method resulted in estimates for the pharmacokinetic parameters that were linearly correlated to those estimated from the fully-sampled data, with K^trans,R=6=0.97 K^trans,R=1+0.00 with correlation coefficient r=0.98, k_ep,R=6=0.95 k_ep,R=1+0.00 (r=0.85). These results suggest that it is possible to acquire highly undersampled breast DCE-MRI data with improved spatial and/or temporal resolution with minimal loss of image quality.     

Massless chiral families in superstring theories

The contribution of the U(1) and simple group factors S_k of the background gauge group ifS=Π_kS_k to the number of massless fermion generations in superstrings theories is examined. Unlike the semisimple case when only SU(n) factors contribute to the generation number, if the abelian factors are present, all groups contained in S contribute to it.     

Geometry of the quantum complex projective spaceCP q (N)

We study the constraints from theb→sγ decay in the parameter space of effective supergravities from orbifold string theory and with minimal supesymmetric particle content. Both the general dilaton-dominated universal scenario as well as a non-universal scenario for the soft terms are investigated. It is found that the recently reported CLEO upper and lower bounds constrain the parameter space of the models under scrutiny. In particular we find constraints on the values of the parameter tanβ and the gluino masses. In this class of string scenarios the negative sign of the Higgs mixing parameterμ, is phenomenologically preferred.