# Instantons and Fivebranes

What is the general relationship between instantons and fivebranes?

In the paper Magnetic Monopoles in String Theory'' by Gauntlett, Harvey and Liu, the authors state the fivebrane ansatz of heterotic supergravity as

$$F_{\mu\nu} = \pm \frac{1}{2}{\epsilon_{\mu\nu}}^{\lambda\sigma}F_{\lambda\sigma}$$ $$H_{\mu\nu\lambda} = \mp {\epsilon_{\mu\nu\lambda}}^{\sigma}\partial_\sigma \phi$$ $$g_{\mu\nu} = e^{2\phi}\delta_{\mu\nu}, \qquad g_{ab} = \eta_{ab}$$

where $\mu, \nu, \ldots$ denote "transverse space", and $a, b, \ldots$ denote "orthogonal space".

They go on to say that the first equation can be solved by an instanton configuration in an SU(2) subgroup of the gauge group.

Instantons are solutions of the self/anti-self duality equations in Euclidean space.

(a) Is it possible to define instanton solutions because $g_{\mu\nu}$ is conformally Euclidean in this case?

(b) The one-instanton solution with unit winding is given by

$$F_{\mu\nu} = \frac{2\bar{\sigma}_{\mu\nu}\rho^2}{(x^2 + \rho^2)^2}$$

Now, in Euclidean space $\sigma_{\mu\nu} = (\sigma_1, \sigma_2, \sigma_3, i\mathbb{I})$ where the $\sigma_i$ denote the three (usual) 2x2 Pauli matrices and $\mathbb{I}$ is the 2x2 identity. Likewise, $\bar{\sigma}_{\mu\nu} = (\sigma_1, \sigma_2, \sigma_3, -i\mathbb{I})$.

I am assuming this is what was used here.

The authors find

$$\nabla_\rho \nabla^\rho \phi = \mp \frac{\alpha'}{120}\epsilon^{\mu\nu\lambda\sigma}Tr(F_{\mu\nu}F_{\lambda\sigma})$$

and apparently solve this for a charge one self-dual instanton to get

$$e^{2\phi} = e^{2\phi_0} + 8\alpha'\frac{x^2 + 2\rho^2}{(x^2+\rho^2)^2}$$

(Note that $\nabla$ is the Laplace-Beltrami operator, or the Laplacian defined with the respect to the metric $g_{\mu\nu}$).

The steps between the last two equations are what I want to fill in for myself, and its a bit unclear how instantons are being introduced.