The Ginzburg–Landau functional can be formulated in the general setting of a
complex vector bundle over a
compact Riemannian manifold. This is the same functional as given above, transposed to the notation commonly used in Riemannian geometry. In multiple interesting cases, it can be shown to exhibit the same phenomena as the above, including
Abrikosov vortices (see discussion below). For a complex vector bundle E over a Riemannian manifold M with fiber \Complex^n, the order parameter \psi is understood as a
section of the vector bundle E. The Ginzburg–Landau functional is then a
Lagrangian for that section: : \mathcal{L}(\psi, A) = \int_M \sqrt dx^1 \wedge \dotsm \wedge dx^m \left[ \vert F \vert^2 + \vert D \psi\vert^2 + \frac{1}{4} \left(\sigma - \vert\psi\vert^2\right)^2 \right] The notation used here is as follows. The fibers \Complex^n are assumed to be equipped with a
Hermitian inner product \langle\cdot,\cdot\rangle so that the square of the norm is written as \vert\psi\vert^2 = \langle\psi,\psi\rangle. The phenomenological parameters \alpha and \beta have been absorbed so that the potential energy term is a quartic
mexican hat potential; i.e., exhibiting
spontaneous symmetry breaking, with a minimum at some real value \sigma\in\R. The integral is explicitly over the
volume form :*(1) = \sqrt dx^1 \wedge \dotsm \wedge dx^m for an m-dimensional manifold M with determinant |g| of the metric tensor g. The D = d + A is the
connection one-form and F is the corresponding
curvature 2-form (this is not the same as the free energy F given up top; here, F corresponds to the
electromagnetic field strength tensor). The A corresponds to the
vector potential, but is in general
non-Abelian when n> 1, and is normalized differently. In physics, one conventionally writes the connection as d-ieA for the electric charge e and vector potential A; in Riemannian geometry, it is more convenient to drop the e (and all other physical units) and take A = A_\mu dx^\mu to be a
one-form taking values in the
Lie algebra corresponding to the symmetry group of the fiber. Here, the symmetry group is
SU(n), as that leaves the inner product \langle\cdot,\cdot\rangle invariant; so here, A is a form taking values in the algebra \mathfrak{su}(n). The curvature F generalizes the
electromagnetic field strength to the non-Abelian setting, as the
curvature form of an
affine connection on a
vector bundle . It is conventionally written as :\begin{align} F = D \circ D = dA + A \wedge A = \left(\frac{\partial A_\nu}{\partial x^\mu} + A_\mu A_\nu\right) dx^\mu \wedge dx^\nu = \frac{1}{2} \left(\frac{\partial A_\nu}{\partial x^\mu} - \frac{\partial A_\mu}{\partial x^\nu} + [A_\mu, A_\nu]\right) dx^\mu \wedge dx^\nu \\ \end{align} That is, each A_\mu is an n \times n skew-symmetric matrix. (See the article on the
metric connection for additional articulation of this specific notation.) To emphasize this, note that the first term of the Ginzburg–Landau functional, involving the field-strength only, is :\mathcal{L}(A) = YM(A) = \int_M *(1) \vert F \vert^2 which is just the
Yang–Mills action on a compact Riemannian manifold. The
Euler–Lagrange equations for the Ginzburg–Landau functional are the Yang–Mills equations :D^*D\psi = \frac{1}{2}\left(\sigma - \vert\psi\vert^2\right)\psi and :D^*F = -\operatorname{Re}\langle D\psi, \psi\rangle where D^* is the
adjoint of D, analogous to the
codifferential \delta = d^*. Note that these are closely related to the
Yang–Mills–Higgs equations.
Specific results In
string theory, it is conventional to study the Ginzburg–Landau functional for the manifold M being a
Riemann surface, and taking n = 1; i.e., a
line bundle. The phenomenon of
Abrikosov vortices persists in these general cases, including M=\R^2, where one can specify any finite set of points where \psi vanishes, including multiplicity. The proof generalizes to arbitrary Riemann surfaces and to
Kähler manifolds. In the limit of weak coupling, it can be shown that \vert\psi\vert
converges uniformly to 1, while D\psi and dA converge uniformly to zero, and the curvature becomes a sum over delta-function distributions at the vortices. The sum over vortices, with multiplicity, just equals the degree of the line bundle; as a result, one may write a line bundle on a Riemann surface as a flat bundle, with
N singular points and a covariantly constant section. When the manifold is four-dimensional, possessing a
spinc structure, then one may write a very similar functional, the
Seiberg–Witten functional, which may be analyzed in a similar fashion, and which possesses many similar properties, including self-duality. When such systems are
integrable, they are studied as
Hitchin systems. ==Self-duality==