This query stems from Witten's paper, ''Solutions of Four-Dimensional Field Theories via M Theory'' (hep-th/9703166). Specifically, consider Type IIA string theory, and suppose one has a stack of NS5 branes (''fivebranes'') of Type IIA with D4 branes (''fourbranes'') suspended between them.

The fourbrane worldvolume is parametrized by the coordinates $(x^0, x^1, x^2, x^3, x^6)$. The fivebrane worldvolume is parametrized by $(x^0, x^1, x^2, x^3, x^4, x^5)$. Let $v \equiv x^4 + i x^5$.

Quoting the reference (cf. page 4 of the above paper)

...note that $x^6$ is determined as a function of $v$ by minimizing the total fivebrane worldvolume. For large $v$ the equation for $x^6$ reduces to a Laplace equation,

$$\nabla^2 x^6 = 0$$

Here $\nabla^2$ is the Laplacian on the fivebrane worldvolume. $x^6$ is a function only of the directions normal to the fourbrane ends, that is only of $v$ and $\bar{v}$.

Is it obvious that the coordinate $x^6$ should be harmonic?

For branes, one usually writes a metric and the fact that the supersymmetry variation of the gravitino is zero for this background metric gives rise to an integrability condition, which in turn yields a differential equation which tells us that some function of the coefficients in the metric is harmonic. But that function is usually a function of the coordinates transverse to the brane (which of course $v$ and $\bar{v}$ are in this case). But why is it $x^6$ which must satisfy such an equation?

Does this have to do with the fact that when one compactifies on a circle, $x^6$ is like a scalar field?


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