We consider deformations of torsion-free G
 2 structures, defined by the G
 2-invariant 3-form φ and compute the expansion of $${\ast \varphi }$$
 to fourth order in the deformations of φ. By considering M-theory compactified on a G
 2 manifold, the G
 2 moduli space is naturally complexified, and we get a Kahler metric on it. Using the expansion of $${\ast \varphi }$$
 , we work out the full curvature of this metric and relate it to the Yukawa coupling.

Local Geometry of the G 2 Moduli Space

The complicated nature of calculations in general relativity was one of the driving forces in the early development of computer algebra (CA). CA has become widely used in gravity research (GR) and its use can be expected to grow further. Here the general nature of computer algebra is discussed, along with some aspects of CA system design; features particular to GR’s requirements are considered; information on packages for CA in GR is provided, both for those packages currently available and for their predecessors; and applications of CA in GR are outlined.

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Computer algebra in gravity research

We consider deformations of torsion-free G2 structures, defined by the G2-invariant 3-form $\phi$ and compute the expansion of the Hodge star of $\phi$ to fourth order in the deformations of $\phi$. By considering M-theory compactified on a G2 manifold, the G2 moduli space is naturally complexified, and we get a Kahler metric on it. Using the expansion of the Hodge star of $\phi$ we work out the full curvature of this metric and relate it to the Yukawa coupling.

Local geometry of the G2 moduli space

The d=6 trace anomaly from quantum field theory four-loop graphs in one dimension

In nD differential geometry, basic geometric structures and properties are described locally by differentiable functions and equations with indices that obey Einstein summation convention. Although symbolic manipulation of such indexed functions is one of the oldest research topics in computer algebra, so far there exists no normal form reduction algorithm to judge whether two indexed polynomials involving indices of different coordinate systems are equal or not. It is a challenging task in computer algebra.

R. Portugal

Papers

The Riegeom package: abstract tensor calculation