Intrinsic Torsion Research Articles

Mirror Symmetric SU(3)-Structure Manifolds with NS Fluxes

When string theory is compactified on a six-dimensional manifold with a nontrivial NS flux turned on, mirror symmetry exchanges the flux with a purely geometrical composite NS form associated with lack of integrability of the complex structure on the mirror side. Considering a general class of T3-fibered geometries admitting SU(3) structure, we find an exchange of pure spinors (e iJ and Ω) in dual geometries under fiberwise T–duality, and study the transformations of the NS flux and the components of intrinsic torsion. A complementary study of action of twisted covariant derivatives on invariant spinors allows to extend our results to generic geometries and formulate a proposal for mirror symmetry in compactifications with NS flux.

Effects of small neutral osmolytes on the supercoiling free energy and intrinsic twist of p30delta DNA.

Both theory and experiments are employed to investigate the effects of small neutral osmolytes on the average intrinsic twist (l0), the torsion and bending elastic constants, and the twist energy parameter (ET) that governs the supercoiling free energy. The experimental data for ethylene glycol and acetamide at 37 degrees C suggest, and are interpreted in terms of, a model wherein the DNA exhibits an equilibrium between two distinct conformational states that possess different numbers of bound water molecules and exhibit different intrinsic twists and torsion and bending elastic constants. Expressions are derived to relate the effective ET and l0 to the equilibrium constant, water activity (aw), and number (n) of bound water molecules released per cooperative domain undergoing the two-state transition. The variations of l0 and ET with -ln(aw) are similar for acetamide and ethylene glycol at 37 degrees C. Fitting the theory to those data yields the range n = 103-125 for ethylene glycol and n = 71-113 for acetamide, depending on the assumed value of ET for the dehydrated state. The cooperative domain size of the two-state transition is estimated to exceed about 25-30 base pairs (bp). Between 0 and 19.4 w/v % ethylene glycol, the torsion elastic constant, measured by time-resolved fluorescence polarization anisotropy (FPA), increases by 1.37-fold, whereas the measured ET decreases by 1.15-fold over that same range. The implied decrease in bending rigidity over that range is by a factor of about 0.7. The variations of l0 and ET with increasing -ln(aw) due to added ethylene glycol at 37 degrees C are far smaller than the corresponding variations observed previously at 14 and 15 degrees C. However, at 21 degrees C, upon adding either ethylene glycol or acetamide, l0 and ET initially decline steeply with increasing -ln(aw), with slopes possibly comparable to those seen at 14 and 15 degrees C, but then flatten out and follow curves similar to those at 37 degrees C. Possible origins of such mixed behavior are discussed. The effects of betaine at both 37 and 21 degrees C differ qualitatively and quantitatively in various respects from those of ethylene glycol and acetamide. Upon adding sucrose, l0 initially jumps to higher plateaus at both 37 and 21 degrees C, but its effects on ET cannot be reliably assessed, due to the limited range of -ln(aw).

The general form of supersymmetric solutions of N = (1, 0) U(1) and SU(2) gauged supergravities in six dimensions

We obtain necessary and sufficient conditions for a supersymmetric field configuration in the N = (1, 0) U(1) or SU(2) gauged supergravities in six dimensions, and impose the field equations on this general ansatz. It is found that any supersymmetric solution is associated with an structure. The structure is characterized by a null Killing vector which induces a natural 2 + 4 split of the six-dimensional spacetime. A suitable combination of the field equations implies that the scalar curvature of the four-dimensional Riemannian part, referred to as the base, obeys a second-order differential equation; surprisingly, for a large class of solutions the equation in the SU(2) theory requires the vanishing of the Weyl anomaly of N = 4 SYM on the base. Bosonic fluxes introduce torsion terms that deform the structure away from a covariantly constant one. The most general structure can be classified into terms of its intrinsic torsion. For a large class of solutions the gauge field strengths admit a simple geometrical interpretation: in the U(1) theory the base is Kähler, and the gauge field strength is the Ricci form; in the SU(2) theory, the gauge field strengths are identified with the curvatures of the left-hand spin bundle of the base. We employ our general ansatz to construct new solutions; we show that the U(1) theory admits a symmetric Cahen–Wallach4 × S2 solution together with a compactifying pp-wave. The SU(2) theory admits a black string, whose near horizon limit is AdS3 × S3, which is supported by a self-dual 3-form flux and a meron on the S3. In the limit of the zero 3-form flux we obtain the Yang–Mills analogue of the Salam–Sezgin solution of the U(1) theory, namely R1,2 × S3. Finally we obtain the additional constraints implied by enhanced supersymmetry, and discuss Penrose limits in the theories.

On mirror symmetry with fluxes

AbstractWe review a proposal for mirror symmetry on general six‐dimensional backgrounds involving manifolds admitting SU(3) structure and NS three‐form flux. The action of twisted covariant derivatives on invariant spinors leads to definition of complexified intrinsic torsion and most of the information about the geometry and the flux is encoded in terms of (real part of) a certain six‐by‐six matrix. Mirror symmetry exchanges the (1,1) and (2,0) components of this matrix.

Almost Quaternion-Hermitian Manifolds

Following the point ofview of Gray and Hervella, we derive detailed conditions whichcharacterize each one of the classes of almostquaternion-Hermitian 4n-manifolds, n > 1. Previously, bycompleting a basic result of Swann, we give explicitdescriptions of the tensors contained in the space of covariantderivatives of the fundamental form Ω and split thecoderivative of Ω into its Sp(n)Sp(1)-components. For 4n > 8, Swann also proved that all theinformation about the intrinsic torsion ∇Ω iscontained in the exterior derivative dΩ. Thus, wegive alternative conditions, expressed in terms of dΩ, to characterize the different classes of almostquaternion-Hermitian manifolds.

A Bochner formula for almost-quaternionic-Hermitian structures

An integral formula is derived, relating the six irreducible components of the intrinsic torsion of an Sp n Sp 1 structure on a compact 4 n-dimensional manifold with the Riemann curvature tensor. Some consequences of the formula are studied.

Superstrings with intrinsic torsion

We analyze the necessary and sufficient conditions for the preservation of supersymmetry for bosonic geometries of the form ${R}^{1,9\ensuremath{-}d}\ifmmode\times\else\texttimes\fi{}{M}_{d},$ in the common Neveu-Schwarz--Neveu-Schwarz (NS-NS) sector of type II string theory and also type I or heterotic string theory. The results are phrased in terms of the intrinsic torsion of G structures and provide a comprehensive classification of static supersymmetric backgrounds in these theories. Generalized calibrations naturally appear since the geometries can always arise as solutions describing NS or type I or heterotic fivebranes wrapping calibrated cycles. Some new solutions are presented. In particular we find $d=6$ examples with a fibered structure which preserve $\mathcal{N}=1,2,3$ supersymmetry in type II and include compact type I or heterotic geometries.

N=1geometries for M theory and type IIA strings with fluxes

We derive a set of necessary and sufficient conditions for obtaining $\mathcal{N}=1$ backgrounds of M theory and type IIA strings in the presence of fluxes. Our metrics are warped products of four-dimensional Minkowski space-time with a curved internal manifold. We classify the different solutions for irreducible internal manifolds as well as for manifolds with ${S}^{1}$ isometries by employing the formalism of group structures and intrinsic torsion. We provide examples within these various classes along with general techniques for their construction. In particular, we generalize the Hitchin flow equations so that one can explicitly build irreducible 7-manifolds with a 4-form flux. We also show how several of the examples found in the literature fit in our framework and suggest possible generalizations.

Supersymmetric intersecting D6-branes and fluxes in massive Type IIA string theory

We study N=1 supersymmetric four-dimensional solutions of massive Type IIA supergravity with intersecting D6-branes in the presence NS–NS three-form fluxes. We derive N=1 supersymmetry conditions for the D6-brane and flux configurations in an internal manifold X 6 and derive the intrinsic torsion (or SU(3)-structure) related to the fluxes. In the absence of fluxes, N=1 supersymmetry implies that D6-branes wrap supersymmetric three-cycles of X 6 that intersect at angles of SU(3) rotations and the geometry is deformed by SU(3)-structures. The presence of fluxes breaks the SU(3) structures to SU(2) and the D6-branes intersect at angles of SU(2) rotations; non-zero mass parameter corresponds to D8-branes which are orthogonal to the common cycle of all D6-branes. The anomaly inflow indicates that the gauge theory on intersecting (massive) D6-branes is not chiral.

Gstructures, fluxes, and calibrations in M theory

We study the most general supersymmetric warped M-theory backgrounds with a non-trivial G flux of the type ${R}^{1,2}\ifmmode\times\else\texttimes\fi{}{\mathrm{M}}_{8}$ and ${\mathrm{AdS}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathrm{M}}_{8}.$ We give a set of necessary and sufficient conditions for preservation of supersymmetry which are phrased in terms of G structures and their intrinsic torsion. These equations may be interpreted as calibration conditions for a static ``dyonic'' M-brane, that is, an M5-brane with a self-dual three-form turned on. When the electric flux is turned off we obtain the supersymmetry conditions and non-linear PDEs describing M5-branes wrapped on associative and special Lagrangian three-cycles in manifolds with ${G}_{2}$ and $\mathrm{SU}(3)$ structures, respectively. As an illustration of our formalism, we recover the 1/2 BPS dyonic M-brane, and also construct some new examples.

Supersymmetric M-theory compactifications with fluxes on seven-manifolds andG-structures

We consider Minkowski compactifications of M-theory on generic seven-dimensional manifolds. After analyzing the conditions on the four-form flux, we establish a set of relations between the components of the intrinsic torsion of the internal manifold and the components of the four-form flux needed for preserving supersymmetry. The existence of two nowhere vanishing vectors on any seven-manifold with G_2 structure plays a crucial role in our analysis, leading to the possibility of four-dimensional compactifications with N=1 and N=2 supersymmetry.

Nontrivial RR two‐form field strength and SU(3)‐structure

AbstractWe discuss how in the presence of a nontrivial RR two‐form field strength and nontrivial dilaton the conditions of preserving supersymmetry on six‐dimensional manifolds lead to generalized monopole and Killing spinor equations. We construct an SU(3)‐structure on these manifolds and give the explicit expression of its intrinsic torsion in terms of the RR two‐form field strength and dilaton. We show that the manifold is Kähler if F(1,1) = 0.

Non-Kähler string backgrounds and their five torsion classes

We discuss the mathematical properties of six-dimensional non-Kähler manifolds which occur in the context of N=1 supersymmetric heterotic and type IIA string compactifications with non-vanishing background H-field. The intrinsic torsion of the associated SU(3) structures falls into five different classes. For heterotic compactifications we present an explicit dictionary between the supersymmetry conditions and these five torsion classes. We show that the non-Ricci-flat Iwasawa manifold solves the supersymmetry conditions with non-zero H-field, so that it is a consistent heterotic supersymmetric groundstate.

Bochner formulae for orthogonal G-structures on compact manifolds

A general technique is introduced for deriving Bochner type formulae on a compact riemannian manifold, relating its curvature tensor with the intrinsic torsion of a compatible (orthogonal) G-structure. The technique is illustrated for the groups G=U n ,SU n ,G 2 and Spin 7, with various applications of the derived formulae in these cases.

Dynamics of helical strips

The dynamics of inertial elastic helical thin rods with noncircular cross sections and arbitrary intrinsic curvature, torsion, and twist is studied. The classical Kirchhoff equations are used together with a perturbation scheme at the level of the director basis, and the dispersion relation for helical strips is derived and analyzed. It is shown that all naturally straight helical strips are unstable whereas free-standing helices are always stable. There exists a one-parameter family of stationary helical solutions depending on the ratio of curvature to torsion. A bifurcation analysis with respect to this parameter is performed, and bifurcation curves in the space of elastic parameters are identified. The different modes of instabilities are analyzed.

Development of a New Force Field for Polynorbornene

A new force field has been customized for the variation of polynorbornene that contains a bicycloheptane group in the backbone structure. The force field was developed from ab initio density functional theory (DFT), and semiempirical electronic structure calculations for both stereochemical dimers of the 2,3 exo−exo isomer of polynorbornene. The bond length and bond angle parameters were determined using the HF/6-311G** ab initio SCF method. The intrinsic torsion potential was determined using the AM1 semiempirical method and the van der Waals parameters are kept same as in the Dreiding 2.21 force field. Both the bonded and torsional energy functions compared well to DFT calculations. The equilibrium geometry and the torsional energetics of the customized force field differ significantly from generic force fields such as Dreiding. Comparisons to experimentally determined geometry and infrared and Raman spectra were used to determine the optimum ab initio and semiempirical method to use for force field par...

On the origin of the temperature dependence of the supercoiling free energy

Monte Carlo simulations using temperature-invariant torsional and bending rigidities fail to predict the rather steep decline of the experimental supercoiling free energy with increasing temperature, and consequently fail to predict the correct sign and magnitude of the supercoiling entropy. To illustrate this problem, values of the twist energy parameter (E(T)), which governs the supercoiling free energy, were simulated using temperature-invariant torsion and bending potentials and compared to experimental data on pBR322 over a range of temperatures. The slope, -dE(T)/dT, of the simulated values is also compared to the slope derived from previous calorimetric data. The possibility that the discrepancies arise from some hitherto undetected temperature dependence of the torsional rigidity was investigated. The torsion elastic constant of an 1876-bp restriction fragment of pBR322 was measured by time-resolved fluorescence polarization anisotropy of intercalated ethidium over the range 278-323 K, and found to decline substantially over that interval. Simulations of a 4349-bp model DNA were performed using these measured temperature-dependent torsional rigidities. The slope, -dE(T)/dT, of the simulated data agrees satisfactorily with the slope derived from previous calorimetric measurements, but still lies substantially below that of Duguet's data. Models that involve an equilibrium between different secondary structure states with different intrinsic twists and torsion constants provide the most likely explanation for the variation of the torsion constant with T and other pertinent observations.

A class of exact dynamical solutions in the elastic rod model of DNA with implications for the theory of fluctuations in the torsional motion of plasmids

New explicit solutions are obtained for the nonlinear equations of Kirchhoff’s theory of the dynamics of inextensible elastic rods without neglect of rotatory inertia. These exact solutions describe a class of motions possible in closed circular rings possessing a uniform distribution of intrinsic curvature ku and intrinsic torsion. When ku≠0, the motions in this class are such that the axial curve of the ring remains stationary while the cross sections rotate about their centers in such a way that the angle ψ of rotation is independent of axial location and is governed by the nonlinear pendulum equation. When ku=0, such uniform rotation of cross sections can occur at an arbitrary steady rate. The methods of classical equilibrium statistical mechanics yield the following conclusions for canonical ensembles of rings for which the motion is this type of pure homogeneous torsion. When 1/ku=11.85 nm (i.e., when the intrinsic curvature ku is among the highest observed in naturally occurring, approximately uniformly curved, stress-free DNA segments), if the flexural rigidity is assigned a value usually accepted for duplex DNA, at T=298 K the root-mean-square value, 〈ψ2〉1/2, of the angle ψ is 11.2°. For motions in this class, the heat capacity per ring, as a function of T/ku, shows a maximum which, when T=298 K, occurs where 1/ku=127 nm and corresponds to an ensemble of rings of which approximately 1% have sufficient energy for escape over the barrier associated with the separatrix between periodic and monotone solutions of the nonlinear pendulum equation; for that ensemble of rings, 〈ψ2〉1/2=43.3°.

Evaluation of spinal deformity by means of intrinsic curvature and torsion

Evaluation of spinal deformity by means of intrinsic curvature and torsion