In the paper ``Hierarchies from Fluxes in String Compactifications'' by Giddings, Kachru and Polchinski, the following example is considered for a localized source that may have negative tension (my question has more to do with math than string theory):
..consider a p-brane wrapped on a (p−3) cycle Σ of the manifold M6. To leading order in α′ (and in the case of vanishing fluxes along the brane) this contributes a source action
Sloc=−∫R4×Σdp+1ξTp√−g+μp∫R4×ΣCp+1
... This equation gives a stress tensor Tlocμν=−Tpe2Aημνδ(Σ),Tlocmn=−TpΠΣmnδ(Σ),
where δ(Σ) and ΠΣ denote the delta function and projector on the cycle Σ respectively.
Question: What is the expression for the ``projector on the cycle Σ'' and how does it arise?
For some context, the metric is
ds210=e2A(y)ημνdxμdxν+e−2A(y)˜gmndymdyn.
the geometry is a product M4×M6, where xμ are four-dimensional coordinates (μ=0,…,3) and ym are coordinates on the compact manifold M6. Further, the stress tensor is defined by
TlocMN=−2√−gδSδgMN,
where M,N are 10 dimensional indices (M,N=0,…,9).
This post imported from StackExchange Physics at 2015-06-15 09:17 (UTC), posted by SE-user leastaction