What kind of branes are there in Berkovitz' Twistor Strings?

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What are the dimensions of the branes that to which it attaches? Can a string attach to other string as it were a brane? I mean, the string is 2d space time object, the twistor space is 4D, so the complement should be another string.

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The twistor string was originally a topological string in a twistor space. A topological string is a simplified version of the (super)string which is effectively decoupled from the metric of the space through which it moves. Usually, the superstring is studied on a manifold M x X, where M is 4D Minkowski space and X is a 6D Calabi-Yau space, and the topological string is studied just on X, the compactified extra dimensions.

The D-branes of superstring theory are hypersurfaces on which the open strings end. For the open topological string, the branes are special submanifolds of the space X. The topological string comes in two forms, A model and B model; the A-branes are "Lagrangian", the B-branes are "holomorphic" (these labels are shorthand for the detailed properties).

Witten's twistor string is a topological B model on CP(3|4), which is a supertwistor space containing extra fermionic degrees of freedom. Berkovits's construction works differently and I don't understand it. But they are both discussed in arXiv:0708.2276, where it says that Witten's twistor string involves both D1-branes and D5-branes.

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Yes, these days I've made some research and I figured a tiny bit of those things out. I emailed 3 people in the research of twistor strings. One told me that the space CP(3|4) could be thought of a kind of complex 3 brane, because that's the space where the strings are attached. The other person told me that there couldn't possibly be analogues of branes in twistor space because there isn't quadratic terms in the worldsheet. The third told me my question would depend how I would see the problem.

Further, I think both of them are right, since CP(3|N) is not really a brane, but more a necessary boundary condition. It also seems the choice of supersymmetry for the world sheet is extremely flexible and there is some that conjecture that N=8 version yields supergravity in 4D.

As far as I could try to understand, the ontological difference between Witten's and Berkovit's it is that the former is made from complicated wrapping of superstring branes which are in the end indentified with a topological string embed in twistor space whereas in latter is a sigma model living in twistor space.

Berkovits's model must involve space-filling branes. That way, the open strings can go anywhere in the twistor space. It also has a closed string sector like a normal string theory.

There's also a modification due to Abou-Zeid et al which a few people are studying.

There must be a deep connection between twistor strings and the usual string theory. Twistor strings and Maldacena's construction both describe N=4 SYM, just at different couplings. So either twistor strings are a flawed form of string theory (evidence for this is that they contain conformal supergravity which is not unitary) or they are part of the big unified string theory, maybe from an unusual perspective. The ++-- twistor signature (two times) reminds me of F-theory and Itzhak Bars.