Twistorial String Field Theory

[Urs Schreiber]

I had a look at

N. Berkovits & L. Motl, Cubic Twistorial String Field Theory,
hep-th/0403187

I have to admit that I am (still) not familiar with strings in
twistor space, so let me ask a stupid question:

What about worldsheet supersymmetry? Is there any? If yes, why does the
BRST operator look the way it does in equation (2.4)? If not, why can we
still have a relation to SYM?

Another question: I have a basic understanding of ordinary cubic
bosonic open string field theory. What is, heuristically, the physical
interpretation of the new cubic interaction term in equation (3.19)?

 

[Lubos Motl]

On Wed, 31 Mar 2004, Urs Schreiber wrote:

> What about worldsheet supersymmetry? Is there any? If yes, why does the
> BRST operator look the way it does in equation (2.4)? If not, why can we
> still have a relation to SYM?

Good point. Unlike the standard topological models - where you start with
worldsheet supercharges that are topologically twisted so that they become
spin 0 BRST-like operators - in Berkovits' model there is no worldsheet
supersymmetry.

(Unless you consider the spin 0 BRST charge Q itself to be a
worldsheet supersymmetry, but this would mean that you would also say that
bosonic string theory has a worldsheet supersymmetry, which is weird.)

Super Yang-Mills has a supersymmetry that is a sort of "target space"
SUSY, and you know that these two things are a bit independent. For
example, type 0 theories have worldsheet SUSY, but no spacetime SUSY. The
full supersymmetry combined with conformal symmetry etc. is the so-called
superconformal symmetry, which in the N=4 Super Yang-Mills case is

SU(4|2,2)

Note that the bosonic subgroup is SU(4) times SU(2,2) - times some U(1)
that I will ignore. Here SU(4) is the R-symmetry spin(6), while SU(2,2) is
isomorphic to spin(4,2), the conformal symmetry in 3+1 dimensions. Then
you have 32 anticommuting generators. SU(4|2,2) is the symmetry in
signature 3+1, but it is useful to go to 2+2 dimensions where it becomes

SL(4|4,R)

It's a different noncompact version of the previous group. SL(4) is now
both the R-symmetry, as well as the conformal symmetry. Note that SL(4,R)
is locally isomorphic to SO(3,3) as well as to SU(3,1) (including the
correct signature etc.), which is in all cases the conformal symmetry in
2+2 dimensions.

The symmetry SL(4|4,R) is manifest in the Berkovits-like models (as well
as other models, where it can become its complexification SL(4|4,C))
because it has a simple geometric action on the superspace RP^{3|4}

(Z^1,Z^2,Z^3,Z^4|psi1,psi2,psi3,psi4).

Note that about 1/2 of generators of SL(4|4,R) are fermionic, but they
anticommute to other bosonic generators of SL(4|4,R); the worldsheet
translation generator does not appear in the anticommutator. Therefore the
superconformal symmetry required from super Yang-Mills is an *internal*
symmetry on the worldsheet, and the worldsheet needs no supersymmetries.

If I understand it well, the topological B-model of Witten has a lot of
extra unnecessary fields - the worldsheet superpartners of the relevant
fields - and these fields play no role in calculating the N=4 amplitudes.

> Another question: I have a basic understanding of ordinary cubic
> bosonic open string field theory. What is, heuristically, the physical
> interpretation of the new cubic interaction term in equation (3.19)?

The heuristic interpretation of a cubic term is that two open strings are
allowed to annihilate the first half of the second string, and the second
half of the first string, and form a single string - which can also split
to two strings by the reverse process.

Your question assumes that (3.12) is "the" standard cubic term, while
(3.19) is something new. Such an understanding is not invariant under many
operations, e.g. parity. The two cubic terms (3.12) and (3.19) are equally
good generalizations of the cubic term from bosonic string field theory.
We have (at least) two such generalizations because compared to the
bosonic string there is an extra local GL(1) symmetry on the worldsheet,
and the interaction midpoints are allowed to change the "picture".

Best wishes
Lubos
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[R0man]

Twistorial String Field Theory (TSFT) is a relatively new approach to string theory that utilizes twistor space, which is a mathematical space that encodes the geometry of spacetime and is closely related to the concept of spin. The paper you mentioned, written by N. Berkovits and L. Motl, presents a cubic formulation of TSFT that allows for a more direct connection to super Yang-Mills theory.

Regarding your first question about worldsheet supersymmetry, it is still present in TSFT, but it is not manifest in the BRST operator. This is because TSFT is formulated in twistor space, which is a different mathematical space than the worldsheet. However, the BRST operator in equation (2.4) is still necessary to ensure gauge invariance and consistency of the theory.

As for your second question, the new cubic interaction term in equation (3.19) has a physical interpretation as a three-string vertex. This means that it describes the interaction between three strings, where each string is represented by a twistor. This interaction term is necessary for the consistency of the theory and allows for a direct connection to super Yang-Mills theory.

Overall, TSFT is a promising approach to string theory that offers new insights and connections to other areas of physics. However, it is still a relatively new and developing field, so there is still much to be explored and understood.