The purpose of this website is to provide fully explicit component expressions for kinematic factors in ten-dimensional pure spinor superspace. We provide component expansions for BRST (pseudo-)invariants at various orders of string perturbation theory, ranging from SYM tree amplitudes in 1012.3981 to constituents of the 3-loop amplitude found in 1308.6567. The results are available as plain text files which can be directly loaded with FORM.
Superspace expressions are translated into components using the pure spinor formalism's zero mode integration prescription (see hep-th/0001035)
Polarizations are encoded in ten-dimensional Lorentz vectors (16 component Weyl-spinors): $e^i_m$ (($\chi_i^\alpha$) for the gluon (gluino) polarization vector (spinor) of particle $i$ with momentum $k^i_m$ and dot products are represented by FORM as, say, k2.e3 for $k^2\cdot e^3$. The 16 x 16 gamma matrices are represented by the (antisymmetric) function ga(), e.g., chi1*ga(m,n,p)*chi2 for $(\chi^1\gamma^{mnp} \chi^2)$. In order to simplify and canonicalize the subsequent expressions, we exploit transversality $k^i.e^i=0$ of gluon polarizations and the massless Dirac equation $k^i_m.(\gamma^m\chi^i)_{\alpha}=0$ of their gaugino superpartners. The Mandelstam invariants are $s_{ij} = k^i\cdot k^j$ and they appear as "ssij" in the files. The following shortcuts are also used, $s_i = s_{i,i+1}, t_i=s_{i,i+1,i+2}, u_i = s_{i,i+1,i+2,i+3}$. Note that multiparticle Mandelstams are given by the sum of all possible channels, e.g., $s_{123} = s_{12} + s_{13} + s_{23}$.
All results below were computed using the second version of a program called PSS, whose first version was described in 1007.4999.
Apart from their obvious significance in field theory, there is also a string-theory motivation to study SYM tree amplitudes. As discussed in 1106.2645, 1106.2646 and 1203.6215, they carry the polarization dependence of open superstring N-point tree-level amplitudes and also of the non-anomalous part of one-loop amplitudes. Moreover, as discussed in 1205.1516, 1307.3534 and 1504.02759 bilinears in SYM trees capture the polarization dependence of N-point tree amplitudes in type IIA/IIB superstring theory, of $N< 6$ particle one-loop amplitudes and of $N< 6$ particle two-loop amplitudes in type IIB theory.
The expressions for purely bosonic component amplitudes (in terms of dot products between polarization and momentum vectors) still apply after dimensional reduction to less than 10 dimensions and in non-supersymmetric YM theory, provided that polarizations and momenta are restricted to have nonzero components in spacetime directions only. Also, the expressions for two-fermion components (with appropriate relative chiralities of the chi wavefunctions and appropriate representations for the Pauli matrices) can be thought of as dimension-agnostic, because Clifford algebra identities do not depend on D. Only the components with four and more fermions inherit dimension-sensitivity from Fierz identities.
The scalar BRST invariants were first discussed in 1203.6215.
The vector BRST invariants were defined in 1404.4986.
The tensor BRST pseudo-invariants were defined in 1408.3605.
The scalar BRST pseudo-invariants were defined in 1408.3605.
The scalar and vector building blocks were defined (in minimal pure spinor superspace) in 1505.02746.