Holomorphic Coordinates Research Articles

Coherent states for PT-/non-PT-symmetric and non-Hermitian Morse potentials via the path integral method

We discuss the coherent states for PT-/non-PT-symmetric and non-Hermitian generalized Morse potentials obtained by using path integral formalism over the holomorphic coordinates. We transform the action of generalized Morse potentials into two harmonic oscillators with a new parametric time to establish the parametric time coherent states. We calculate the energy eigenvalues and the corresponding wave functions in parabolic coordinates.

Fiber structure and local coordinates for the Teichmüller space of a bordered Riemann surface

We show that the infinite-dimensional Teichmüller space of a Riemann surface whose boundary consists of n n closed curves is a holomorphic fiber space over the Teichmüller space of an n n -punctured surface. Each fiber is a complex Banach manifold modeled on a two-dimensional extension of the universal Teichmüller space. The local model of the fiber, together with the coordinates from internal Schiffer variation, provides new holomorphic local coordinates for the infinite-dimensional Teichmüller space.

Coherent states for the Hartmann potential

We obtain the coherent states for a particle in the noncentral Hartmann potential by transforming the problem into four isotropic oscillators evolving in a parametric time. We use path integration over the holomorphic coordinates to find the quantum states for these oscillators. The decomposition of the transition amplitudes gives the coherent states and their parametric-time evolution for the particle in the Hartmann potential. We also derive the coherent states in the parabolic coordinates by considering the transition amplitudes between the coherent states and eigenstates in the configuration space.

Nonanticommutative deformation of Script N = 4 SYM theory: the Myers effect and vacuum states

We propose a deformation of ${\cal N}=4$ SYM theoery induced by nonanticommutative star product. The deformation introduces new bosonic terms which we identify with the corresponding Myers terms of a stack of D3-branes in the presence of a five-form RR flux. We take this as an indication that the deformed lagrangian describes D3-branes in such a background. The vacuum states of the theory are also examined. In a specific case where the U(1) part of the gauge field is nonvanishing the (anti)holomorphic transverse coordinates of the brane sit on a fuzzy two sphere. For a supersymmetric vacuum the antiholomorphic coordinates must necessarily commute. However, we also encounter non-supersymmetric vacua for which the antiholomorphic coordinates do not commute.

Integrability of rough almost complex structures

We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.

Parametric-time coherent states for Morse potential

We transform the Lagrangian of the Morse-potential problem into two harmonic oscillators in a new parametric time and quantize this system by using path integrals over holomorphic coordinates of oscillators and derive coherent states.PACS Nos.: 31.15-p, 03.65Ca, 03.65Ge

Normal Coordinates in Kahler Manifolds and the Background Field Method

Riemann normal coordinates (RNC) are unsuitable for \kahler manifolds since they are not holomorphic. Instead, \kahler normal coordinates (KNC) can be defined as holomorphic coordinates. We prove that KNC transform as a holomorphic tangent vector under holomorphic coordinate transformations, and therefore that they are natural extensions of RNC to the case of \kahler manifolds. The KNC expansion provides a manifestly covariant background field method preserving the complex structure in supersymmetric nonlinear sigma models.

Parametric time-coherent states for the hydrogen atom

We obtained coherent states for the hydrogen atom by transforming the problem into four oscillators in the parametric time at a classical level, and quantizing these oscillators by using path integration over their holomorphic coordinates. We showed that for the negative- (positive-) energy coherent states the mean values of the physical position of the electron satisfy the Keplerian ellipses (hyperbolas) and that their dispersions oscillate (increase) in parametric time.

Microlocal complex foliation of R-Lagrangian CR submanifolds

Let $X$ be a complex manifold, $M$ a real analytic submanifold of $X^{R}$, $T^{*}X$ the cotangent bundle to $X$, $T_{M}^{*}X$ the conormal bundle to $M$ in $X$. Assume that $T_{M}^{*}X$ is regular and $CR$ in $T^{*}X$. We then show that $T_{M^{*}}X$ is locally defined as the zero-set of the real and/or imaginary part of holomorphic symplectic coordinates of $T^{*}X$. It is well known that the similar description of $M$ in local complex coordinates of $X$ is true only if $M$ is Levi flat. As an application we obtain a generalization of the celebrated edge of the wedge Theorem.

On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties

We construct a family of Kählerian quasi-Einstein metrics with an isometry group U( n) acting linearly on the holomorphic coordinates. Suitable restrictions on the parameters give rise to complete non-compact as well as compact metrics whose geometrical structure is studied in detail. The two-loop renormalizability properties are discussed according to whether one considers the bosonic σ-models or their (2,2) supersymmetric extension.

TWISTING OF N=1 SUSY GAUGE THEORIES AND HETEROTIC TOPOLOGICAL THEORIES

It is shown that D=4N=1 SUSY Yang-Mills theory with an appropriate supermultiplet of matter can be twisted on a compact Kähler manifold. The conditions for cancellation of anomalies of BRST charge are found. The twisted theory has an appropriate BRST charge. We find a nontrivial set of physical operators defined as classes of the cohomology of this BRST operator. We prove that the physical correlators are independent of the external Kähler metric up to a power of a ratio of two Ray-Singer torsions for the Dolbeault cohomology complex on a Kähler manifold. The correlators of local physical operators turn out to be independent of antiholomorphic coordinates defined with a complex structure on the Kähler manifold. However, a dependence of the correlators on holomorphic coordinates can still remain. For a hyper-Kähler metric the physical correlators turn out to be independent of all coordinates of insertions of local physical operators.

The bosonic string measure at two and three loops and symplectic transformations of the volume form

Symplectic modular invariance of the string integral has been verified at genus 2 and 3 using the period matrix coordinatization of moduli space. A calculation of the transformation of the product of holomorphic coordinates Π i⩽j dτ ij shows that an extra phase arises together with the factor associated with a specific modular weight; the phase is cancelled in the transformation of the entire volume element including the complex conjugate. An argument is given for modular invariance of the reggeon measure at genus 12.

Local heterotic geometry in holomorphic coordinates

In the same spirit as for N=2 and N=4 supersymmetric nonlinear models in two spacetime dimensions by Zumino and by Alvarez-Gaumé and Freedman, we analyse the (2,0) and (4,0) heterotic geometry in holomorphic coordinates. We study the properties of the torsion tensor and give the conditions under which (2,0) geometry is conformally equivalent to a (2,2) one. Using additional isometries, we show that it is difficult to equip a manifold with a closed torsion tensor, but for the real four-dimensional case where we exhibit new examples. We show that, contrarily to Callan et al's claim for real four-dimensional manifolds, (4,0) heterotic geometry is not necessarily conformally equivalent to a (4,4) Kähler--Ricci flat geometry. We rather prove that, whatever the real dimension is, they are special quasi-Ricci flat spaces, and we exemplify our results on Eguchi--Hanson and Taub-NUT metrics with torsion.

Coherent states for quantum compact groups

Coherent states are introduced and their properties are discussed for all simple quantum compact groups. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit and interpret the coherent state as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compact R--matrix formulation (generalizing this way the $q$--deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel--Weil construction) are described using the concept of coherent state. The relation between representation theory and non--commutative differential geometry is suggested.}

Analytical anomaly and heterotic string in the formalism of continual integration

P-loop statistical sums for the superstring and for the heterotic string may be represented as integrals over the space of moduli of Riemann surfaces: ∫Mp|dμss(y)|2/(det Im T)5(6) and ∫Mpdμss(y)dμbos(y) FΓ(y)/(det Im T)5(6), dμss(y), dμbos(y), FΓ(y) being analytical functions of the holomorphic coordinates y on the Teichmüller space. At the same time, the statistical sum of the 26-dimensional bosonic string, compactified on the 16-dimensional torus T16 = R16/Γ in the case of even self-dual lattices Γ8×Γ8 and Γ16, is proportional to ʃMp[|dηbos(y)|2(detImT )5(6)∑i|ƒΛ, i(y)|2. The function FΓ(y) entering the formula for the heterotic string is equal to FΓ(y) = ∑iƒΓ, i(y). There is a similar relation between dμss(y and the statistical sum, arising from the continual integral, based on the supersymmetrized action of the bosonic string. This relation between the sum of the analytical functions and the sum of their square moduli may provide an argument against compactification on six-dimensional tori.

Mass degeneracy and the super-Higgs mechanism

We study the conditions for the existence of a sector with mass-degenerate supersymmetric partners in the presence of the spontaneous breaking of local supersymmetry, as the first step to characterize theories with residual global supersymmetry. The mass difference between fermions and scalars in each chiral multiplet is related to the Riemann curvatures tensor of the scalar field manifold, providing a necessary condition of geometrical nature on theories and chiral multiplets with degeneracy properties. From the study of the corresponding Riemann tensors, we determine the scalar curvature and the chiral multiplets consistent with mass degeneracy or all irreducible symmetric spaces. The solutions always correspond to a specific group coset space and are compared to the conditions for the existence of massless particles (Goldstone bosons, chiral fermions). We also discuss, on more general grounds, the possible choices of the superpotential and the appropriate holomorphic coordinates leading both to zero cosmological constant and mass degeneracy. Starting from their realization in models with flat potentials - in particular the SU( n,1)/U( n) ones - a general characterization is attempted. In this context, non-compact global symmetries of the whole Kähler potential emerge as a useful framework.

Some Integration Problems in Almost-Complex and Complex Manifolds

The theory of almost-complex manifolds leads to a number of problems which properly belong to the field of differential equations. This paper deals with three such problems. The first concerns differential equations in the exterior operator d (frequently also denoted 8) on differential forms; the other two deal with holomorphic coordinates and holomorphic curves. The latter are first stated in differential-geometric terms, and then re-stated as problems in differential equations. The equivalence of these formulations is established in a separate section. The differential equations problems (Theorems I, II', III') and their proofs can be read independently. Two appendices contain underlying facts that are either not available in the form required, or can be obtained only from widely scattered sources.