Real polarization of the moduli space of flat connections on a Riemann surface

We prove that the moduli space of flatSU(2) connections on a Riemann surface has a real polarization, that is, a foliation by lagrangian subvarieties. This polarization may provide an alternative quantization of the Chern-Simons gauge theory in higher genus, in line with the results of [11] for genus one.Supported by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291     

Flat connection contribution to topology changing amplitudes in an ensemble of seifert fibered homology spheres

The Fintushel-Stern pseudofree orbifolds are exploited to construct wave functions of universes created as a result of the interaction of cones on lens spaces. We also study the problem of the definition of topology changing amplitudes for tunneling topology changes, described by cobordisms with Seifert fibered homology sphere boundaries. It is demonstrated that such topology changes are accompanied by creation or annihilation of the lens spaces. The topology-changing amplitude calculations are carried out in the stationary phase approximation for Kodama wave functions. In this approximation the changing amplitudes factorize and they are expressed by means of Chern-Simons invariants of flat connections over Seifert fibered homology spheres and lens spaces.     

On representation spaces in geometric quantization

The structure of the space of wave functions in the representation given by a complete strongly admissible polarization of the phase space is investigated. The conditions that the wave functions should be covariant constant along the real part of the polarization define the Bohr-Sommerfeld set of the representation containing the supports of all wave functions. There is a natural scalar product for the wave functions defined on the Bohr-Sommerfeld set. It is shown, for a real polarization, that the resulting Hilbert space of wave functions is not trivial if and only if the Bohr-Sommerfeld set is not empty.Partially supported by the National Research Council, Grant No. A8091.     

Twistor Geometry of a Pair of Second Order ODEs

We discuss the twistor correspondence between path geometries in three dimensions with vanishing Wilczynski invariants and anti-self-dual conformal structures of signature (2, 2). We show how to reconstruct a system of ODEs with vanishing invariants for a given conformal structure, highlighting the Ricci-flat case in particular. Using this framework, we give a new derivation of the Wilczynski invariants for a system of ODEs whose solution space is endowed with a conformal structure. We explain how to reconstruct the conformal structure directly from the integral curves, and present new examples of systems of ODEs with point symmetry algebra of dimension four and greater which give rise to anti–self–dual structures with conformal symmetry algebra of the same dimension. Some of these examples are (2, 2) analogues of plane wave space–times in General Relativity. Finally we discuss a variational principle for twistor curves arising from the Finsler structures with scalar flag curvature.     

Topological Strings and (Almost) Modular Forms

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H ³(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP ₂ and IP ₁ × IP ₁. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold .     

Consistent canonical quantization of general relativity in the space of vassiliev invariants

We present a quantization of the Hamiltonian and diffeomorphism constraint of canonical quantum gravity in the spin network representation. The novelty consists in considering a space of wave functions based on the Vassiliev invariants. The constraints are finite, well defined, and reproduce at the level of quantum commutators the Poisson algebra of constraints of the classical theory. A similar construction can be carried out in 2+1 dimensions leading to the correct quantum theory.     

Bohr-sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula

We show how the moduli space of flatSU(2) connections on a two-manifold can be quantized in the real polarization of [15], using the methods of [6]. The dimension of the quantization, given by the number of integral fibres of the polarization, matches the Verlinde formula, which is known to give the dimension of the quantization of this space in a Kähler polarization.Supported in part by MSRI under NSF Grant 85-05550Supported in part by an NSF Graduate fellowship, and by a grant-in-aid from the J. Seward Johnson Charitable TrustSupported in part by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291. Address as of January 1, 1993: Department of Mathematics, Columbia University, New York, NY 10027, USA     

Wave functions relative to a real polarization

The structure of the space of wave functions in the representation given by a complete everywhere independent set of commuting observables is analyzed in the framework of geometric quantization. Under the assumptions that the chosen real polarization of the classical phase space is locally trivial and complete, it is shown that the wave functions are generalized sections of an appropriate line bundle with supports determined by generalized Bohr-Sommerfeld conditions. There is a canonical Hilbert subspace of the space of the wave functions with the scalar product defined in terms of the same expressions which appear in the generalized Bohr-Sommerfeld conditions.     

Spinor fields with zero mass in unbounded isotropic media

The Dirac equation for massless fields in unbounded media has solutions similar to the focus wave mode solutions of Maxwell's equations leading to infinite dynamical invariants. We define the splash wave mode solutions as a weighted superposition of the focus wave modes, and discuss the conditions to be fulfilled by the weight functions to make the dynamical invariants bounded. We leave open the physical interpretation of these solutions.     

Invariants of chaotic Hamiltonian systems

The invariants of chaotic bounded Hamiltonian systems and their relation to the solutions of the first variational equations of the equations of motion are studied. We show that these invariants are characterized by the fact that they either lose the property of differentiability as functions on phase space or that a certain formal power series defined in terms of the derivatives of the invariants has zero radius of convergence. For a specific example, we show that the former possibility appears to apply.     

Residue formulas for the largek asymptotics of Witten's invariants of Seifert manifolds. The case ofSU(2)

We derive the largek asymptotics of the surgery formula forSU(2) Witten's invariants of general Seifert manifolds. The contributions of connected components of the moduli space of flat connections are identified. The contributions of irreducible connections are presented in the residue form. This allows us to express them in terms of intersection numbers on their moduli spaces.Address after September 25: L. Rozansky, School of Mathematics, Institute for Advanced Study, Princeton, N.J. 08540, USA.     

Magnetic Phase Transitions to an Incommensurate Structure in LiMn 2 O 4 Compound

Possible second-order magnetic phase transitions to the incommensurate magnetic structure in the orthorhombic phase of LiMn2O4 compound have been investigated. It is shown that “weak Lifshitz condition” holds for this compound (i.e., Lifshits invariants are immaterial in all transitions being considered), and only the incommensurate phase, in which the wavevector near the transition point varies continuously with temperature and pressure, can be formed. A transition is considered both in the exchange approximation occurring in three irreducible representations forming an exchange multiplet as well as in one and two irreducible representations. Expressions have been derived for the mean density of the magnetic moment emerging because of such transitions. It is found that structures of the types of a longitudinal spin wave, a transverse spin wave with polarization along the crystallographic axes perpendicular to the wavevector of the structure, as well as certain superpositions of these waves can be formed in the system.     

利用旋转波片组实现偏振变换器的两种新方法

偏振变换器作为一种重要的光学器件．以其高偏振变换精度、低成本、低插入损耗以及控制算法简单的特性在相干光纤通信和高速光纤通信领域具有十分重要的地位。提出了利用两个λ／4波片和一个λ／2波片的三波片组合实现任意偏振态变换的新方法。用改进的三维庞加莱球方法—偏振态投射平面的二维几何方法分别推导出了λ／4 λ／4 λ／2组合以及λ／2 λ／4 λ／4组合实现任意偏振态变换时每个波片的位置参量计算公式。研究结果表明：两个λ／4波片和一个λ／2波片的任意组合都可以实现任意偏振态的变换;用偏振态投射平面的几何方法处理偏振变换的相关问题比庞加莱球方法和矩阵处理方法简洁明了。研究结果为偏振变换器的设计提供了理论依据。     

Propagator for the time-dependent charged oscillator via linear and quadratic invariants

The problem of a charged particle in the presence of a variable magnetic field is considered. Using the linear and the quadratic invariants as a tool, the wave functions in Fock state as well as in coherent state are obtained. The corresponding propagators which propagate the wave functions in the space-time are derived. Using numerical computations we have managed to draw some plots for the real, imaginary, and absolute values of the propagators. This has been used to analyze the properties of the propagators associated with both of the linear and the quadratic invariants. It has been shown that there is no essential difference between the behavior of the absolute value of the propagators in both of the linear and the quadratic invariants.     

Superconformal Invariance and the Geography¶of Four-Manifolds

The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four-manifold topology. In particular, in the example of gauge group SU(2) with one doublet hypermultiplet, we derive a theorem relating classical topological invariants such as the Euler character and signature to sum rules for Seiberg–Witten invariants. A short account of this paper can be found in [1]. Received: 19 December 1998 / Accepted: 7 March 1999     

Applying supersymmetry to energy dependent potentials

We investigate the supersymmetry properties of energy dependent potentials in the D=1$D=1$ dimensional space. We show the main aspects of supersymmetry to be preserved, namely the factorization of the Hamiltonian, the connections between eigenvalues and wave functions of the partner Hamiltonians. Two methods are proposed. The first one requires the extension of the usual rules via the concept of local equivalent potential. In this case, the superpotential becomes depending on the state. The second method, applicable when the potential depends linearly on the energy, is similar to what has been already achieved by means of the Darboux transform.     

Siebenmann-type cobordisms with borders and topology changes by quantum tunneling

Siebenmann-type cobordisms are constructed to describe topology changes with the Seifert fibered homology spheres in in- and out-states. We study the problem of determining of topology-changing amplitudes for these quantum tunneling processes. The calculations are performed in the stationary phase approximation for Kodama wave functions. In this approximation the amplitudes are expressed in terms of Chern-Simons invariants of flatSU(2)-connections over the cobordism boundary components. The topology-change amplitudes found are factorized into the Kodama wave functions for the lens spaces. The results are compared with those for Fintushel-Stern-type cobordisms which have been previously investigated.