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Register a new userTwistor Origin of the Superstring
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Nathan Berkovits
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After introducing a d=10 pure spinor $lambda^alpha$, the Virasoro constraint $partial x^m partial x_m =0$ can be replaced by the twistor-like constraint $partial x^m (gamma_m lambda)_alpha=0$. Quantizing this twistor-like constraint leads to the pure spinor formalism for the superstring where the fermionic superspace variables $theta^alpha$ and their conjugate momenta come from the ghosts and antighosts of the twistor-like constraint.
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We describe D=4 twistorial membrane in terms of two twistorial three-dimensional world volume fields. We start with the D-dimensional p-brane generalizations of two phase space string formulations: the one with $p+1$ vectorial fourmomenta, and the second with tensorial momenta of $(p+1)$-th rank. Further we consider tensionful membrane case in D=4. By using the membrane generalization of Cartan-Penrose formula we express the fourmomenta by spinorial fields and obtain the intermediate spinor-space-time formulation. Further by expressing the worldvolume dreibein and the membrane space-time coordinate fields in terms of two twistor fields one obtains the purely twistorial formulation. It appears that the action is generated by a geometric three-form on two-twistor space. Finally we comment on higher-dimensional (D>4) twistorial p-brane models and their superextensions.
We consider a formulation of N=1 D=3,4 and 6 superparticle mechanics, which is manifestly supersymmetric on the worldline and in the target superspace. For the construction of the action we use only geometrical objects that characterize the embedding of the worldline superspace into the target superspace, such as target superspace coordinates of the superparticle and twistor components. The action does not contain the Lagrange multipliers which may cause the problem of infinite reducible symmetries, and, in fact, is a worldline superfield generalization of the supertwistor description of superparticle dynamics.
After a short introduction to Matrix theory, we explain how can one generalize matrix models to describe toroidal compactifications of M-theory and the heterotic vacua with 16 supercharges. This allows us, for the first time in history, to derive the conventional perturbative type IIA string theory known in the 80s within a complete and consistent nonperturbative framework, using the language of orbifold conformal field theory and conformal perturbation methods. A separate chapter is dedicated to the vacua with Horava-Witten domain walls that carry E8 gauge supermultiplets. Those reduce the gauge symmetry of the matrix model from U(N) to O(N). We also explain why these models contain open membranes. The compactification of M-theory on T4 involves the so-called (2,0) superconformal field theory in six dimensions, compactified on T5. A separate chapter describes an interesting topological contribution to the low energy equations of motion on the Coulomb branch of the (2,0) theory that admits a skyrmionic solution that we call ``knitting fivebranes. Then we return to the orbifolds of Matrix theory and construct a formal classical matrix model of the Scherk-Schwarz compactification of M-theory and type IIA string theory as well as type 0 theories. We show some disastrous consequences of the broken supersymmetry. Last two chapters describe a hyperbolic structure of the moduli spaces of one-dimensional M-theory.
We consider possible discretizations for a gauge-fixed Green-Schwarz action of Type IIB superstring. We use them for measuring the action, from which we extract the cusp anomalous dimension of planar $mathcal{N}=4$ SYM as derived from AdS/CFT, as well as the mass of the two $AdS$ excitations transverse to the relevant null cusp classical string solution. We perform lattice simulations employing a Rational Hybrid Monte Carlo (RHMC) algorithm and two Wilson-like fermion discretizations, one of which preserves the global $SO(6)$ symmetry of the model. We compare our results with the expected behavior at various values of $g=frac{sqrt{lambda}}{4pi}$. For both the observables, we find a good agreement for large $g$, which is the perturbative regime of the sigma-model. For smaller values of $g$, the expectation value of the action exhibits a deviation compatible with the presence of quadratic divergences. After their non-perturbative subtraction the continuum limit can be taken, and suggests a qualitative agreement with the non-perturbative expectation from AdS/CFT. Furthermore, we detect a phase in the fermion determinant, whose origin we explain, that for very small $g$ leads to a sign problem not treatable via standard reweigthing. The continuum extrapolations of the observables in the two different discretizations agree within errors, which is strongly suggesting that they lead to the same continuum limit. Part of the results discussed here were presented earlier in arXiv:1601.04670.
The algebra of spacetime supersymmetry generators in the RNS formalism for the superstring closes only up to a picture-changing operation. After adding non-minimal variables and working in the large Hilbert space, the algebra closes without picture-changing and spacetime supersymmetry can be made manifest. The resulting non-minimal version of the RNS formalism is related by a field redefinition to the pure spinor formalism.
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