Projector and delta function on a cycle $\Sigma$ of a manifold $\mathcal{M}_6$

Question

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

Answer 1

The projector onto the cycle is the covariant splitting of the ambient tangent space into the normal and tangent bundles of $\Sigma$. You may be able to give a formula for it in a patch where you have some coordinates with the first few lying along $\Sigma$ and the rest normal. Then the projector just projects onto the tangent directions in the first coordinates.

(Note that such an object is not automatically holomorphic, which is important for these applications.)

Comments

Thanks Ryan. Do you have a reference for this?