Twistor Theory Research Articles

Holomorphic currents and duality inN= 1 supersymmetric theories

Twisted supersymmetric theories on a product of two Riemann surfaces possess non-local holomorphic currents in a BRST cohomology. The holomorphic currents act as vector fields on the chiral ring. The OPE's of these currents are invariant under the renormalization group flow up to BRST-exact terms. In the context of electric-magnetic duality, the algebra generated by the holomorphic currents in the electric theory is isomorphic to the one on the magnetic side. For the currents corresponding to global symmetries this isomorphism follows from 't Hooft anomaly matching conditions. The isomorphism between OPE's of the currents corresponding to non-linear transformations of fields of matter imposes non-trivial conditions on the duality map of chiral ring. We consider in detail the $SU(N_c)$ SQCD with matter in fundamental and adjoint representations, and find agreement with the duality map proposed by Kutasov, Schwimmer and Seiberg.

Twistor theory of hyper-Kähler metrics with hidden symmetries

We review the hierarchy for the hyper-Kähler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-Kähler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-Kähler metric admits a hidden symmetry if it admits a certain Killing spinor. We show that if the hidden symmetry is tri-holomorphic, then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a “twistor group” action as introduced by Bielawski [Twistor Quotients of Hyper-Kähler Manifolds (World Scientific, River Edge, NJ, 2001)]. This leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalized Legendre transform for the hyper-Kähler metric itself given by Ivanov and Rocek [Commun. Math. Phys. 182, 291 (1996)]. We show that the ALE spaces are examples of hyper-Kähler metrics admitting three tri-holomorphic Killing spinors. These metrics are in this sense analogous to the “finite gap” solutions in soliton theory. Finally we extend the concept of a hierarchy from that of our earlier work [Commun. Math. Phys. 213, 641 (2000)] for the four-dimensional hyper-Kähler equations to a generalization of the conformal anti-self-duality equations and briefly discuss hidden symmetries for these equations.

Harmonic functions, central quadrics and twistor theory

Solutions to the n-dimensional Laplace equation which are constant on a central quadric are found. The associated twistor description of the case n = 3 is used to characterize Gibbons–Hawking metrics with tri-holomorphic SL(2, ℂ) symmetry.

Darboux transformation for MZM-I, II equations

The Darboux transformations for MZM (Manakov, Zakharov and Mikhailov) equations will be constructed in this Letter, and U( N), SU( N) solitons are obtained. Through a trick from twistor theory, the Darboux transformation will correspond to a holomorphic map from C to Grassmannian manifold.

On a geometrically exact curved/twisted beam theory under rigid cross-section assumption

A geometrically exact curved/ twisted beam theory, that assumes that the beam cross-section remains rigid, is re-examined and extended using orthonormal frames of reference starting from a 3-D beam theory. The relevant engineering strain measures with an initial curvature correction term at any material point on the current beam cross-section, that are conjugate to the first Piola-Kirchhoff stresses, are obtained through the deformation gradient tensor of the current beam configuration relative to the initially curved beam configuration. The stress resultant and couple are defined in the classical sense and the reduced strains are obtained from the three-dimensional beam model, which are the same as obtained from the reduced differential equations of motion. The reduced differential equations of motion are also re-examined for the initially curved/twisted beams. The corresponding equations of motion include additional inertia terms as compared to previous studies. The linear and linearized nonlinear constitutive relations with couplings are considered for the engineering strain and stress conjugate pair at the three-dimensional beam level. The cross-section elasticity constants corresponding to the reduced constitutive relations are obtained with the initial curvature correction term. Along with the beam theory, some basic concepts associated with finite rotations are also summarized in a manner that is easy to understand.

Complex Analysis: A Brief Tour into Higher Dimensions

1. INTRODUCTION. Why is it that most graduate students of mathematics (and many undergraduates as well) are exposed to complex analysis in one variable, yet only a small minority of students or, for that matter, professional mathematicians ever learn even the most basic corresponding theory in several variables? Think about it another way: Could anyone seriously argue that it might be sufficient to train a mathematics major in calculus of functions of one real variable without expecting him or her to learn at least something about partial derivatives, multiple integrals, and some higher dimensional version of the Fundamental Theorem of Calculus? Of course not, the real world is not one-dimensional! But neither is the complex world: witness classical applications of complex analysis to quantum field theory, more recent uses in twistor and gravitation theory, and the latest developments in string theory! Multidimensional complex analysis is an indispensable tool in modern theoretical physics. (See, for example, Green, Schwarz, and Witten [6], Manin [12], Henkin and Novikov [9].) Aside from questions of applicability, shouldn’t the pure mathematician’s mind wonder about the restriction to functions of only one complex variable? It should not surprise anyone that there is a natural extension of complex analysis to the multivariable setting. What is surprising is the many new and intriguing phenomena that appear when one considers more than one variable. Indeed, these phenomena presented major challenges to any straightforward generalization of familiar theorems. Amazing progress was made in the 1940s and 1950s, when the area was enriched by new and powerful tools such as coherent analytic sheaves and their cohomology theory. Yet for decades these apparently quite abstract techniques constituted a formidable barrier that deterred analysts from exploring this territory unless they were committed to a research career in multidimensional complex analysis. Fortunately, many of the technical hardships of the pioneering days can now be overcome with much less effort by approaching the subject along different routes. The purpose of this article is to lead you, as painlessly as possible, on a tour through some of the foundations of complex analysis in several variables, and to take you to some vantage points from which we can enjoy some of the fascinating sights that remain hidden from us as long as we restrict ourselves to the complex plane. Along the way, we shall encounter a few of the unexpected higher dimensional phenomena and explore fundamental new concepts, always trying to gain an understanding of the underlying difficulties. Good walking shoes (i.e., an elementary introduction to basic complex analysis in one variable, and standard multivariable real calculus) are all the equipment that is needed, no technical gear or skills will be required. I will be rewarded if, at the end of our hike, many of you will recommend this tour to friends and colleagues. Perhaps a few of you will be sufficiently intrigued and challenged that you will pick up one of the excellent books that are available to guide you through more advanced terrain in order to reach the “high peaks” of the subject.

On rack cohomology

We prove that the lower bounds for Betti numbers of the rack, quandle and degeneracy cohomology given in Carter et al. (J. Pure Appl. Algebra, 157 (2001) 135) are in fact equalities. We compute as well the Betti numbers of the twisted cohomology introduced in Carter et al. (Twisted quandle cohomology theory and cocycle knot invariants, math. GT/0108051). We also give a group-theoretical interpretation of the second cohomology group for racks.

Twisted homotopy theory and the geometric equivariant 1-stem

We develop methods for computing the equivariant homotopy set [M,SV]G, where M is a manifold on which the group G acts freely, and V is a real linear representation of G. Our approach is based on the idea that an equivariant invariant of M should correspond to a twisted invariant of the orbit space M/G. We use this method to make certain explicit calculations in the case dimM=dimV+dimG+1.

`Twist theory' for the roles of estrogen and testosterone in the process of spermatogenesis

Estrogen and testosterone have different effects on different steps of spermatogenesis and for each step optimal levels of estrogen and testosterone are different. The effects of these hormones behave as if `twisted' from one end to the other end of the spectrum of steps of spermatogenesis. A graphic model is presented to explain this `Twist Theory'.

Contact metric 5-manifolds, CR twistor spaces and integrability

The CR twistor space is defined over a contact metric 5-manifold M. Like the 4-dim twistor theory, the integrability of the almost CR twistor structure is discussed in terms of the Weyl conformal curvature and also the scalar curvature of M.

Métriques autoduales sur la boule

A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball. When the data on the boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishes), one may hope to find in the conformal class of the filling metric an Einstein metric, with a pole of order 2 on the boundary. We determine which conformal metrics on the 3-sphere are boundaries of such selfdual Einstein metrics on the 4-ball. In particular, this implies the Positive Frequency Conjecture of LeBrun. The proof uses twistor theory, which enables to translate the problem in terms of complex analysis; this leads us to prove a criterion for certain integrable CR structures of signature (1,1) to be fillable by a complex domain. Finally, we solve an analogous, higher dimensional problem: selfdual Einstein metrics are replaced by quaternionic-Kahler metrics, and conformal structures on the boundary by quaternionic contact structures (previously introduced by the author); in contrast with the 4-dimensional case, we prove that any small deformation of the standard quaternionic contact structure on the (4m−1)-sphere is the boundary of a quaternionic-Kahler metric on the (4m)-ball.

Twists and generalized Zolotarev polynomials

A generalization of the so-called Zolotarev polynomial is investigated though the theory of twists and that of generalized Jacobian varieties. As an application, extremal properties, rationality, and a relation with multigrades are revealed.

M-fivebranes wrapped on supersymmetric cycles. II

We construct D=11 supergravity solutions dual to the twisted field theories arising when M-theory fivebranes wrap supersymmetric cycles. The cases considered are M-fivebranes wrapped on (i) a complex Lagrangian four-cycle in a D=8 hyper-Kaehler manifold corresponding to a D=2 field theory with (2,1) supersymmetry (ii) a product of two holomorphic two-cycles in a product of two Calabi-Yau two-folds corresponding to a D=2 field theory with (2,2) supersymmetry and (iii) a product of a holomorphic two-cycle and a SLAG three-cycle in a product of a Calabi-Yau two-fold and a Calabi-Yau three-fold corresponding to a quantum mechanics with two supercharges. In each case we construct BPS equations and find IR superconformal fixed points corresponding to new examples of AdS/CFT duality arising from the twisted field theories.

Twisted quandle homology theory and cocycle knot invariants

The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the definition of state-sums. The invariants are used to derive information on twisted cohomology groups.

Membranes wrapped on holomorphic curves

We construct supergravity solutions dual to the twisted field theories arising when M-theory membranes wrap holomorphic curves in Calabi-Yau n-folds. The solutions are constructed in an Abelian truncation of maximal D=4 gauged supergravity and then uplifted to D=11. For four-folds and five-folds we find new smooth AdS/CFT examples and for all cases we analyse the nature of the singularities that arise. Our results provide an interpretation of certain charged topological AdS black holes. We also present the generalised calibration two-forms for the solutions.

M-fivebranes wrapped on supersymmetric cycles

We construct supergravity solutions dual to the twisted field theories arising when M-theory fivebranes wrap general supersymmetric cycles. The solutions are constructed in maximal D=7 gauged supergravity and then uplifted to D=11. Our analysis covers Kahler, special Lagrangian and exceptional calibrated cycles. The metric on the cycles are Einstein, but do not necessarily have constant curvature. We find many new examples of AdS/CFT duality, corresponding to the IR superconformal fixed points of the twisted field theories.

Supergravity duals of gauge theories from F(4) gauged supergravity in six dimensions

We construct supergravity solutions dual to twisted field theories that are the worldvolume theories of D4-branes wrapped on 2, 3-cycles, and NS-fivebranes on 2-cycles. We first obtain the solutions for the Romans' six-dimensional gauged supergravity theories and then up-lift them to ten dimensions. In particular, we find solutions for field configurations with either non-abelian fields orB-fields being excited. One of these solutions, in the massless case, is up-lifted to the massless type IIA string theory. This is the first example of such a kind. The cases studied provide new examples of the AdS/CFT duality involving twisted field theories.

Remarks on twisted theories with matter

We investigate some aspects of N=2 twisted theories with matter hypermultiplets in the fundamental representation of the gauge group. A consistent formulation of these theories on a general four-manifold requires turning on a particular magnetic flux, which we write down explicitly in the case of SU(2k). We obtain the blowup formula and show that the blowup function is given by a hyperelliptic sigma-function with singular characteristic. We compute the contact terms and find, as a corollary, interesting identities between hyperelliptic Theta functions.

On Twistors in Cantorian [formula omitted] space

This short note considers Penrose Twistor theory from the E (∞) Cantorian space-time view point. It is found that the expectation value of the Hausdorff dimension of E (∞) could be written in terms of Twistors, quantities as D= R ( c) αβ αβ+ R ( c)αβ αβ= 1 Z α Z α = 1 2(s) =4+φ 3.

Einstein–Weyl geometry, the dKP equation and twistor theory

It is shown that Einstein–Weyl (EW) equations in 2+1 dimensions contain the dispersionless Kadomtsev–Petviashvili (dKP) equation as a special case: if an EW structure admits a constant-weighted vector then it is locally given by h= dy 2−4 dx dt−4u dt 2 , ν=−4u x dt , where u= u( x, y, t) satisfies the dKP equation ( u t − uu x ) x = u yy . Linearised solutions to the dKP equation are shown to give rise to four-dimensional anti-self-dual conformal structures with symmetries. All four-dimensional hyper-Kähler metrics in signature (++−−) for which the self-dual part of the derivative of a Killing vector is null arise by this construction. Two new classes of examples of EW metrics which depend on one arbitrary function of one variable are given, and characterised. A Lax representation of the EW condition is found and used to show that all EW spaces arise as symmetry reductions of hyper-Hermitian metrics in four dimensions. The EW equations are reformulated in terms of a simple and closed two-form on the CP 1 -bundle over a Weyl space. It is proved that complex solutions to the dKP equations, modulo a certain coordinate freedom, are in a one-to-one correspondence with mini-twistor spaces (two-dimensional complex manifolds Z containing a rational curve with normal bundle O(2) ) that admit a section of κ −1/4, where κ is the canonical bundle of Z . Real solutions are obtained if the mini-twistor space also admits an anti-holomorphic involution with fixed points together with a rational curve and section of κ −1/4 that are invariant under the involution.