Skip to main content

Questions tagged [twistor-theory]

For questions related to twistor theory. Twistor theory maps the geometric objects of conventional $3+1$ space-time (Minkowski space) into geometric objects in a $4$ dimensional space with metric signature $(2,2)$. This space is called twistor space, and its complex valued coordinates are called "twistors."

Filter by
Sorted by
Tagged with
0 votes
1 answer
23 views

I am confused about a certain property of spinor transformations that seems to be inconsistent based on my current understanding.

I am considering the transformation of a two dimensional Weyl spinor $\lambda^\alpha$ given by a matrix transformation of the form $p(\lambda)^\beta = \lambda^\alpha m_\alpha^{\ \ \beta}$. Let's say ...
Teddy Baker's user avatar
2 votes
1 answer
125 views

ADHM construction: why two bundles of the monad have the same ranks?

I'm reading The ADHM construction of Yang-Mills instantons by Simon Donaldson. A theorem in the section 4 says: Let $E$ be a rank $r$ holomorphic bundle over $\mathbb{CP}^3$ with $c_2 = k$. Suppose ...
Arith Geo's user avatar
1 vote
0 answers
51 views

Projective curves under twistor fibration

Consider a holomorphic curve $f:\Sigma\to\mathbb{CP}^3$ of degree, say, $d$ from some Riemann surface to projective space. The Penrose twistor fibration $\pi:\mathbb{CP}^3\to\text{S}^4$ then allows us ...
Bob Knighton's user avatar
1 vote
0 answers
163 views

From Monad construction of Instantons to ADHM data

In reference to Atiyah's book - "Geometry of Yang-Mills Fields" (1979). In chapter 5, section 3, he describes how the monad construction for $Sp(n)$ potentials can be interpreted in terms of ...
Siva Tej's user avatar
3 votes
1 answer
164 views

Some questions about Twistor Space of a closed $4$-manifold

Let $(M,g)$ be a closed Riemannian manifold of dimension $4$. We denote its twistor space, the space of almost complex structures on the tangent bundle $TM$ by $Z\xrightarrow{\pi}M$. At any point the ...
Partha's user avatar
  • 1,419
1 vote
0 answers
68 views

Spinor form of the exterior derivative

In the book of Ward and Wells "Twistor Geometry and Field Theory" and also in the paper "Cohomology and Massless Fields" of Eastwood, Penrose and Wells, appears a spinor form of ...
Giovanny Soto's user avatar
0 votes
0 answers
128 views

Transitive action on a flag manifold

Given a 4-dimensional complex space $T$, we consider the flag manifolds $F_{d_1,\ldots,d_m}(T)=\{(S_1,\ldots,S_m)\ |\ \textrm{dim}S_i=d_i, \ \ S_1\subset S_2\subset\ldots\subset S_m\}$. The $S_i$ are ...
Giovanny Soto's user avatar
1 vote
1 answer
179 views

Coordinate charts on a flag manifold

I'm trying to prove that the flag $F_{d_1\ldots d_m}(V)=\{(S_1,\ldots,S_m)|S_1\subset S_2\subset\ldots S_m \ \ \ \ \textrm{with} \ \ \ \ \textrm{dim}S_i=d_i \ \ \ \ \textrm{and}\ \ \ 1<d_1<\...
Giovanny Soto's user avatar
1 vote
0 answers
115 views

Reference for Penrose-Ward transformation

I am currently looking for a reference that covers the Penrose-Ward transformation. For context the Penrose-Ward transformation relates a holomorphic vector bundle $E$ on some twistor space $Z$, with ...
Thomas Kurbach's user avatar
0 votes
1 answer
224 views

Wedge and common notation for "a line between two points"

I'm using a somewhat old presentation from 2011 that covers twistor geometry. It uses the notation "$L = Z_1 \wedge Z_2$" to suggest that the line $L$ is the "join of the twistors $Z_1$ and $Z_2$, ...
Brad's user avatar
  • 65
6 votes
1 answer
177 views

How to see that $\mathbb{C}\mathbb{P}^3\cong\mathrm{SO}(5)/\mathrm{U}(2)$?

I stumbled on this ismorphism in the context of twistor fibrations. See for example 'Twistors in Mathematics and Physics' by Bailey and Baston, p.58. Can anybody provide a construction of this ...
deepfloe's user avatar
  • 456
2 votes
1 answer
182 views

A few general questions on the Penrose transform

Let us consider the Bateman or Whittaker's pioneering examples of a Penrose transform. Starting from a holomorphic function on an open subset of twistor space, they constructed a solution to the ...
Malkoun's user avatar
  • 5,353