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Register a new userTwistor-Space Recursive Formulation of Gauge-Theory Amplitudes
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Using twistor space intuition, Cachazo, Svrcek and Witten presented novel diagrammatic rules for gauge-theory amplitudes, expressed in terms of maximally helicity-violating (MHV) vertices. We define non-MHV vertices, and show how to use them to give a recursive construction of these amplitudes. We also use them to illustrate the equivalence of various twistor-space prescriptions, and to determine the associated combinatoric factors.
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We formulate the most general gravitational models with constant negative curvature (hyperbolic gravity) on an arbitrary orientable two-dimensional surface of genus $g$ with $b$ circle boundaries in terms of a $text{PSL}(2,mathbb R)_partial$ gauge theory of flat connections. This includes the usual JT gravity with Dirichlet boundary conditions for the dilaton field as a special case. A key ingredient is to realize that the correct gauge group is not the full $text{PSL}(2,mathbb R)$, but a subgroup $text{PSL}(2,mathbb R)_{partial}$ of gauge transformations that go to $text{U}(1)$ local rotations on the boundary. We find four possible classes of boundary conditions, with associated boundary terms, that can be applied to each boundary component independently. Class I has five inequivalent variants, corresponding to geodesic boundaries of fixed length, cusps, conical defects of fixed angle or large cylinder-shaped asymptotic regions with boundaries of fixed lengths and extrinsic curvatures one or greater than one. Class II precisely reproduces the usual JT gravity. In particular, the crucial extrinsic curvature boundary term of the usual second order formulation is automatically generated by the gauge theory boundary term. Class III is a more exotic possibility for which the integrated extrinsic curvature is fixed on the boundary. Class IV is the Legendre transform of class II; the constraint of fixed length is replaced by a boundary cosmological constant term.
We perform the twistor (half-Fourier) transform of all tree n-particle superamplitudes in N=4 SYM and show that it has a transparent geometric interpretation. We find that the N^kMHV amplitude is supported on a set of (2k+1) intersecting lines in twistor space and demonstrate that the corresponding line moduli form a lightlike (2k+1)-gon in moduli space. This polygon is triangulated into two kinds of lightlike triangles lying in different planes. We formulate simple graphical rules for constructing the triangulated polygons, from which the analytic expressions of the N^kMHV amplitudes follow directly, both in twistor and in momentum space. We also discuss the ordinary and dual conformal properties and the cancellation of spurious singularities in twistor space.
We consider the twistor space ${cal P}^6cong{mathbb R}^4{times}{mathbb C}P^1$ of ${mathbb R}^4$ with a non-integrable almost complex structure ${cal J}$ such that the canonical bundle of the almost complex manifold $({cal P}^6, {cal J})$ is trivial. It is shown that ${cal J}$-holomorphic Chern-Simons theory on a real $(6|2)$-dimensional graded extension ${cal P}^{6|2}$ of the twistor space ${cal P}^6$ is equivalent to self-dual Yang-Mills theory on Euclidean space ${mathbb R}^4$ with Lorentz invariant action. It is also shown that adding a local term to a Chern-Simons-type action on ${cal P}^{6|2}$, one can extend it to a twistor action describing full Yang-Mills theory.
The analog of the Cachazo-Svrvcek-Witten rules for scattering amplitudes with massive quarks is derived following an approach previously employed for amplitudes with massive scalars. A prescription for the external wave-functions is given that leads to a one-to one relation between fields in the action and spin-states of massive quarks. Several examples for the application of the rules are given and the structure of some all-multiplicity amplitudes with a pair of massive quarks is discussed. The rules make supersymmetric relations to amplitudes with massive scalars manifest at the level of the action. The formalism is extended to several quark flavors with different masses.
We construct massless infinite spin irreducible representations of the six-dimensional Poincar{e} group in the space of fields depending on twistor variables. It is shown that the massless infinite spin representation is realized on the two-twistor fields. We present a full set of equations of motion for two-twistor fields represented by the totally symmetric $mathrm{SU}(2)$ rank $2s$ two-twistor spin-tensor and show that they carry massless infinite spin representations. A field twistor transform is constructed and infinite spin fields are found in the space-time formulation with an additional spinor coordinate.
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