String theory and loop space index theorems

We study index theorems for the Dirac-Ramond operator on a compact Riemannian manifold. The existence of a group action on the loop space makes possible the definition of a character valued index which we calculate by using a two-dimensional sigma model withN=1/2 supersymmetry. We compute the Euler characteristic, the Hirzebruch signature and the Dirac-Ramond genus of loop space. We compare our results to the calculations made by using the Atiyah-Singer character-valued index theorem.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY85-15857. Fermilab is operated by the Universities Research Association, Inc., under contract with the United States Department of Energy     

Higher-Degree Analogs of the Determinant Line Bundle

 In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families index theorem. In the second part of the paper, given a smooth family of Dirac-type operators whose index lies in the subspace of the reduced K-theory of the parametrizing space, we construct a set of Deligne cohomology classes of degree i whose curvatures are the i-form component of the Atiyah-Singer families index theorem. Received: 27 September 2001 / Accepted: 5 April 2002 Published online: 22 August 2002     

S 1 actions and elliptic genera

A proof is given of Witten's conjectures for the rigidity of the index of the Dirac-Ramond operator on the loop space of a spin manifold which admits anS ¹ symmetry.Research supported in part by the National Science Foundation     

The Pfaffian line bundle

We analyze the holomorphic Pfaffian line bundle defined over an infinite dimensional isotropic Grassmannian manifold. Using the infinite dimensional relative Pfaffian, we produce a Fock space structure on the space of holomorphic sections of the dual of this bundle. On this Fock space, an explicit and rigorous construction of the spin representations of the loop groupsLO _n is given. We also discuss and prove some facts about the connection between the Pfaffian line bundle over the Grassmannian and the Pfaffian line bundle of a Dirac operator.Supported by a National Science Foundation Graduate Fellowship     

Anticommuting variables, fermionic path integrals and supersymmetry

Fermionic Brownian paths are defined as paths in a space parametrised by anticommuting variables. Stochastic calculus for these paths, in conjunction with classical Brownian paths, is described; Brownian paths on supermanifolds are developed and applied to establish a Feynman-Kac formula for the twisted Laplace-Beltrami operator on diffential forms taking values in a vector bundle. This formula is used to give a proof of the Atiyah-Singer index theorem which is rigorous while being closely modelled on the supersymmetric proofs in the physics literature.     

Criterion for the existence of a continuous embedding of a weighted Sobolev class on a closed interval and on a semiaxis

A criterion for the existence of a continuous embedding of a weighted Sobolev class in a weighted L _p space is obtained, i.e., the existence of an index n for which the Kolmogorov n-diameter is finite. For the case in which a continuous embedding exists, the reduced Sobolev class is constructed together with a continuous operator of a natural embedding of the class in a weighted L _p -space.     

Electric-magnetic duality in non-abelian gauge theories

The duality transformation of the vacuum expectation value of the operator which creates magnetic vortices (the 't Hooft loop operator in the Higgs phase), is performed in the radial gauge (x_uA_u^a(x) = 0). It is found that in the weak coupling region (small g) of a pure Yang-Mills theory the dual operator creates electric vortices whose strength is $\text{1}\text{g}$. The theory is self-dual in this region, and the effective coupling of the dual Lagrangian is $\text{1}\text{g}$. (It is self-dual also in the extreme strong coupling region.) Thus the above duality transformation reduces to electric-magnetic duality where the electric field in the 't Hooft loop operators transforms into a magnetic field in the dual operator. In a spontaneously broken gauge theory these results are valid only within the region where the vortices (or the monopoles) are concentrated, or in directions of the algebra space of unbroken symmetry, as self-duality holds only for this subset of fields. Noting that the 't Hooft loop operator project into the subspace of these field configurations we find that it is an electric-magnetic duality for the spontaneously broken theory as well. In the strong coupling region a strong coupling expansion in powers $\text{1}\text{g}$ is suggested.     

Variational Operators,Symplectic Operators,and the Cohomology of Scalar Evolution Equations

For a scalar evolution equation u_t = K(t, x, u, u_x, . . . , u_2m+1) with m ≥ 1, the cohomology space H^1,2() is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for u_t = K for which the equation is Hamiltonian is also shown to be isomorphic to the space H^1,2() and subsequently can be naturally identified with the space of variational operators. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The variational operator (or symplectic) nature of the potential form of a bi-Hamiltonian evolution equation is also presented in order to generate examples of interest.     

Equivariant determinant bundles and supersymmetric closed strings

This work is to point out the relationship between the geometric approach to the string theories of Bowick and Rajeev, and Pilch and Warner and the recent research on the index of the Dirac operator in loop space. We consider character-valued Dirac index bundles instead of the vacuum bundle over some parameter space such as Diff S¹/S¹. Thus, some necessary conditions for the absence of the Virasoro anomaly can be derived from some formulas presented here.     

Canonical conjugate momentum of discrete label operators in quantum mechanics I: formalism

We discuss the inadequacy of the standard definition of canonical conjugation for a quantum operator having adiscrete spectrum. A different definition is proposed, based on the analogy betweencontinuous anddiscrete translations (or rotations). This definition can be applied to special operators which we calllabel operators. The general form of the conjugate momentum of a label operator is found and the resulting canonical commutation rules are discussed. It is shown that the canonical commutator acts like ac number in itsdomain _I , but the domain does not coincide with the whole Hilbert space. The properties of the subspace _I are also discussed.     

几种典型背景中Green-Schwarz超弦的κ-对称性

Metsaev和Tseytlin(MT)给出的AdS₅S⁵背景中Green-Schwarz(GS) IIB超弦的Polyakov作用量可以写成等价的Nambu-Goto形式.对于这种形式，给出了新的与靶空间的流有关的投影算子，并用其构造了使作用量不变的局域κ-变换.κ-对称性的这种新方案是由Schwarz对于GS模型提出的.由于MT模型与GS模型有所不同，文中所构造的局域κ-变换有一些新的特点，且适用于其他类似于MT模型的系统.文中分别以AdS₅S¹背景中IIB弦及Polyakov新提出的模型为例，构造了κ-对称性的靶空间形式. 关键词： Green-Schwarz超弦 κ-对称性')" href="#">κ-对称性 AdS₅S⁵')" href="#">AdS₅S⁵ AdS₅S¹')" href="#">AdS₅S¹     

Fermions and parity violation in dimensional reduction schemes

Dimensional reduction, previously applied to Yang-Mills theories, is extended to gauge theories with spinor fields. It is shown that a fairly realistic model in Minkowski space can be obtained from the simplest initial lagrangian, defined in a space-time with extra, compact dimensions. Left-right asymmetry in the fermion sector in four dimensions is possible, and its occurrence is related to a non-vanishing Atiyah-Singer index.     

Quantum Theory of Path-Dependent Field Operator Based on Characteristics of Displacement Operators

The path-dependent operator formalism of quantum electrodynamics proposed by Mandelstam is reformulated through quantum field theory based on characteristics of displacement operators in Minkowski space. It is shown that total energy- and total angular-momentum operators can generate inhomogeneous Lorentz transformations on any local operator including path-independent bilinear forms constructed of path-dependent electron operator Ψ(x, P), but that generators for Ψ(x, P) itself are only their Ψ(x, P)-dependent parts. Such an unfamiliar feature is characteristic of the path-dependent operator formalism. The present approach possesses unique merits in making the logic of the formalism transparent as described in the following: i) Quantum electrodynamics can be formulated but for the help of potential operator even as a tool for calculation up to a final step. ii) Some restriction, which can be used to discuss propriety of gauge conditions, can be figured out. iii) By introducing a path-rearrangement operator, we can keep infinite variety of space-like pathes with the same end point throughout our formulation as they stand. iv) Several points which must be modified in the presence of magnetic monopole are closed up.     

Discrete Magnetic Laplacian

We consider a 2-dimensional discrete operator which we call the Discrete Magnetic Laplacian (DML); it is an analogue of the magnetic Schrödinger operator. It follows from well known arguments that DML has the same spectrum (as a subset inR) as the Almost Mathieu operator (AM). They also have the same Integrated Density of States (IDS) which is known to be continuous. We show that DML is an element in a II₁-factor and its IDS can be expressed through the trace in the II₁-factor. It follows that DML never has anyL ²-eigenfunctions (i.e. has no point spectrum). Then we formulate a natural algebraic conjecture which implies that the spectrum of DML (hence the spectrum of AM) is a Cantor set.Supported by NSF grant DMS-9222491     

The index of the scattering operator on the positive spectral subspace

We construct the scattering operator for a spinor field in a time dependent background by the Dyson expansion. Then we show that the restriction of the scattering operator to the positive spectral subspace (with respect to a reference Hamiltonian) is Fredholm. The computation of the index of this restriction is reduced to the index computation for an elliptic pseudodifferential operator of order zero. We obtain the index in terms of a cohomological formula by means of the Atiyah-Singer index theorem.     

度量算符对Gauss编织态的本征作用及自旋几何

对圈量子引力中标架度量矩阵算符对Gauss编织态的作用为本征作用，提供了完整的证明.求得了全部标架度量矩阵算符的表示矩阵，及其期望值.利用自旋几何定理，在内腿颜色k=0和k=2两种情况下，算得了Gauss编织态顶角毗邻的4条腿(P=1)的相位位形切方向间的全部夹角，以及切矢量的长度. 关键词： 度量算符的表示矩阵 度量期望值 切方向间夹角 切矢量长度     

Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem

Fermionic quantization, or Clifford algebra, is combined with pseudodifferential operators to simplify the proof of the Atiyah-Singer index theorem for the Dirac operator on a spin manifold.     

The Dirac Quantisation Condition for Fluxes on Four-Manifolds

A systematic treatment is given of the Dirac quantisation condition for electromagnetic fluxes through two-cycles on a four-manifold space-time which can be very complicated topologically, provided only that it is connected, compact, oriented and smooth. This is sufficient for the quantised Maxwell theory on it to satisfy electromagnetic duality properties. The results depend upon whether the complex wave function needed for the argument is scalar or spinorial in nature. An essential step is the derivation of a "quantum Stokes' theorem" for the integral of the gauge potential around a closed loop on the manifold. This can only be done for an exponentiated version of the line integral (the "Wilson loop") and the result again depends on the nature of the complex wave functions, through the appearance of what is known as a Stiefel-Whitney cohomology class in the spinor case. A nice picture emerges providing a physical interpretation, in terms of quantised fluxes and wave-functions, of mathematical concepts such as spin structures, spin_C structures, the Stiefel-Whitney class and Wu's formula. Relations appear between these, electromagnetic duality and the Atiyah-Singer index theorem. Possible generalisation to higher dimensions of space-time in the presence of branes are mentioned.     

Operation and physics potential of Tevatron Run 2

We consider a lattice scalar field model with higher derivative terms in the action whose phase diagram contains a tricritical point which is also a triple point between the paramagnetic, ferromagnetic and antiferromagnetic phases. The continuum limit is defined by approching the tricritical point from the paramagnetic side. Contrary to the lattice tricritical g₆ϕ⁶ model we can do a perturbative computation in dimension four. The non-perturbative aspect of the theory relies on the dispersion relation which has the particular feature of having several minima similar to the propagator of lattice fermions. It is shown that this new model is perturbatively renormalizable and provides a non trivial mass spectrum. The positive norm Hilbert space and the unitarity of the time evolution operator in Minkowski space is established by means of the reflection positivity property.     

The Atiyah-Singer index theorem as passage to the classical limit in quantum mechanics

A new approach to the Atiyah-Singer index theorem is described, using the technique of continuous fields ofC ^*-algebras. The proof is given in the case of elliptic pseudodifferential operators on ℝ ⁿ .