String Theory and M-Theory

440 String geometry
SOLUTION
An infinitesimal rotation has the form Rij = δij + aij, where aij is infinitesimal,
and aij = −aji. This acts on the coordinates by y ′i = Rijy j . Now
plug this into the three-form (9.212) and keep only the linear terms in a.
Requiring the three-form to be invariant results in the equations
a14 + a36 + a27 = 0, a15 + a73 + a26 = 0,
a16 + a43 + a52 = 0, a17 + a35 + a42 = 0,
a76 + a54 + a32 = 0, a12 + a74 + a65 = 0,
a13 + a57 + a64 = 0.
These seven constraints leave 21 − 7 = 14 linearly independent rotations
under which the calibration is invariant. This construction ensures that
they generate a group. ✷
Appendix: Some basic geometry and topology
This appendix summarizes some basic geometry and topology needed in this
chapter as well as other chapters of this book. This summary is very limited,
so we refer the reader to GSW as well as some excellent review articles for
a more detailed discussion. The mathematically inclined reader may prefer
to consult the math literature for a more rigorous approach.
Real manifolds
What is a manifold?
A real d-dimensional manifold is a space which locally looks like Euclidean
space ¡ d . More precisely, a real manifold of dimension d is defined by
introducing a covering with open sets on which local coordinate systems are
introduced. Each of these coordinate systems provides a homeomorphism
between the open set and a region in ¡ d . The manifold is constructed by
pasting together the open sets. In regions where two open sets overlap,
the two sets of local coordinates are related by smooth transition functions.
Some simple examples of manifolds are as follows:
• ¡ d and £ d are examples of noncompact manifolds.