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 $\Sigma$ of the manifold $\mathcal{M}_6$. To leading order in $\alpha'$ (and in the case of vanishing fluxes along the brane) this contributes a source action

$$S_{loc} = -\int\limits_{R^4 \times \Sigma} d^{p+1}\xi T_p \sqrt{-g}\, +\, \mu_p \int\limits_{R^4 \times \Sigma} C_{p+1} \tag{2.16}$$ ... This equation gives a stress tensor $$T_{\mu\nu}^{loc} = -T_p e^{2A} \eta_{\mu\nu}\delta(\Sigma), \qquad T_{mn}^{loc} = -T_p \Pi_{mn}^{\Sigma}\delta(\Sigma),\tag{2.18}$$ where $\delta(\Sigma)$ and $\Pi^\Sigma$ denote the delta function and projector on the cycle $\Sigma$ respectively.

**Question**: What is the expression for the ``projector on the cycle $\Sigma$'' and how does it arise?

For some context, the metric is

$$ds_{10}^2 = e^{2 A(y)} \eta_{\mu\nu}\, dx^\mu dx^\nu + e^{-2A(y)}\tilde{g}_{mn}\, dy^{m}dy^{n}.\tag{2.6}$$

the geometry is a product $M_4 \times \mathcal{M}_6$, where $x^\mu$ are four-dimensional coordinates ($\mu = 0, \ldots, 3$) and $y^m$ are coordinates on the compact manifold $\mathcal{M}_6$. Further, the stress tensor is defined by

$$T_{MN}^{loc} = -\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{MN}},\tag{2.11}$$

where $M, N$ are 10 dimensional indices ($M, N = 0, \ldots, 9$).

This post imported from StackExchange Physics at 2015-06-15 09:17 (UTC), posted by SE-user leastaction