Abstract
Experience with Randall–Sundrum models teaches the importance of following how branes back-react onto the bulk geometry, since this can dramatically affect the system's low-energy properties. Yet the practical use of this observation for model building is so far mostly restricted to branes having only one transverse dimension (codimension-1) in the bulk space, since this is where tools for following back-reaction are well developed. This is likely to be a serious limitation since experience also tells us that one dimension is rarely representative of what happens in higher dimensions. Here we summarize recent progress in developing the matching conditions that describe how codimension-2 branes couple to bulk metric, gauge and scalar fields. These matching conditions are then applied to three situations: D7-branes in F-theory compactifications of ten-dimensional (10D) Type IIB string vacua; 3-branes coupled to bulk axions in unwarped and non-supersymmetric six-dimensional (6D) systems; and 3-branes coupled to chiral, gauged 6D supergravity. For each it is shown how the resulting brane–bulk dynamics are reproduced by the scalar potential for the low-energy moduli in the dimensionally reduced, on-brane effective theory. For 6D supergravity, we show that the only 4D-maximally symmetric bulk geometries supported by positive-tension branes are flat.
Highlights
The main obstacle to understanding how properties of higher-codimension branes are related to the bulk geometries they source is the fact that these bulk geometries typically diverge at the position of their sources. (The most familiar example of this for a codimension-3 object is the divergence of the Coulomb potential of a nucleus evaluated at the nuclear position.) It is one of the special features of codimension-1 objects that the bulk fields they source typically do not diverge at their positions
This paper summarizes the bulk-brane matching conditions for codimension-2 objects, and describes several applications to higherdimensional brane systems: F-theory compactifications involving space-filling codimension2 D7-branes situated within 10 dimensions; unwarped 3-brane flux compactifications in 6 dimensional scalar-Maxwell-Einstein theory; and warped and unwarped 3-brane flux compactifications of 6D chiral gauged supergravity
The comparison with the F-theory compactifications provides a sanity check on the junction conditions, since both the brane and bulk actions are explicitly known for Type IIB string vacua [9], as are explicit solutions for the surrounding bulk geometry [21]
Summary
We start by describing the brane-bulk framework within which we work. This starts with a statement of the scalar-metric-Maxwell system whose equations we use, followed by a statement of how the near-brane boundary conditions of the bulk fields are related to the action of the branes which are their source. Γij = gMN ∂ixM ∂jxN is the induced metric, and K is the trace, γijKij, of the extrinsic curvature, of the boundary surface, ∂M This bulk action is chosen to be general enough to include the bosonic part of the supersymmetric theories of interest. Very near a brane, it is useful to further specialize to the most general ansatz consistent with cylindrical symmetry in the two transverse dimensions, {xm, m = n − 2, n − 1} This leads to the following metric: ds2 = dρ2 + e2Bdθ2 + e2W gμν dxμdxν = e2C dr2 + r2dθ2 + e2W gμν dxμdxν (2.9).
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