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We present a derivation of the scattering amplitude prescription for the pure spinor superstring from first principles, both in the minimal and non-minimal formulations, and show that they are equivalent. This is achieved by first coupling the worldsheet action to topological gravity and then proceeding to BRST quantize this system. Our analysis includes the introduction of constant ghosts and associated auxiliary fields needed to gauge fix symmetries associated with zero modes. All fields introduced in the process of quantization can be integrated out explicitly, resulting in the prescriptions for computing scattering amplitudes that have appeared previously in the literature. The zero mode insertions in the path integral follow from the integration

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Preprint typeset in JHEP style.- HYPER VERSION

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We study the construction of D–brane boundary states in the pure spinor formalism for the quantisation of the superstring. This is achieved both via a direct analysis of the definition of D–brane boundary states in the pure spinor conformal field theory, as well as via comparison between standard RNS and pure spinor descriptions of the superstring. Regarding the map between RNS and pure spinor formulations of the superstring, we shed new light on the tree level zero mode saturation rule. Within the pure spinor formalism we propose an explicit expression for the D–brane boundary state in a flat spacetime background. While the non–zero mode sector mostly follows from a simple understanding of the pure spinor conformal field theory, the zero mode sector requires a deeper analysis which is one of the main points in this work. With the construction of the boundary states at hand, we give a prescription for calculating scattering amplitudes in the presence of a D–brane. Finally, we also briefly discuss the coupling to the world–volume gauge field and show

We propose a localization formula for the chiral de Rham complex generalizing the well-known localization procedure in topological theories. Our formula takes into account the contribution due to the massive modes. The key to achieve this is to view the non-linear βγ system as a gauge theory. For abelian gauge groups we are in the realm of toric geometry. Including the bc system, the formula reproduces the known results for the elliptic genus of toric varieties. We compute the partition function of several models.

By studying phase transitions in supersymmetric gauge theories with Green-Schwarz anomaly cancellation, a natural relation is found between sigma models on certain non-Kähler manifolds with intrinsic torsion and asymmetric Landau-Ginzburg orbifolds. In these orbifold limits, a quantum anomaly of the orbifold action is cancelled by discrete phases in the partition function. These intrinsic torsion phases are derived by blowing down cycles supporting non-trivial H-flux in the linear model. This correspondence extends the CY-LG correspondence to special non-Kähler manifolds and provides computational Orbifolds provide simple and computationally tractable descriptions of string propagation on non-trivial spacetimes: by concentrating all the curvature at the orbifold fixed points, the bulk of the theory is free, with all non-trivial interactions determined by the structure of the orbifold singularities [1,2]. Finding orbifold limits of smooth manifolds thus allows us to

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