Is magnetic reconnection reconcilable with magnetic field lines neither starting nor ending? According to Maxwell's equations, magnetic fields are divergence-free: $\nabla \cdot \mathbf{B} = 0$. If I understand this correctly, this means that magnetic field lines do not start or end. How can we reconcile this with magnetic reconnection?
 A: One must be very careful in making the step from $\nabla\cdot\mathbf{B}=0$ to a statement such as "magnetic field lines do not start or end".
Consider the field in the region of an X-point type magnetic null (in two dimensions). Take a 'volume' (i.e. an area) centred on the null point, and look at the field lines through the bounding curve. No matter how small you make the volume, you will see an equal number of field lines of equal strength entering and leaving the volume.
At the point of reconnection (in an idealised case) the field lines 'start' and 'end' at an infinitesimal point. Even in the limit that your volume for the purposes of the calculation tends to zero (which defines the scalar field of divergence), you will still have equal flux 'into' and 'out of' that volume.
Note the sentence in this source, where it is stated that "[f]an field lines and spine field lines are notable exceptions to the general tenet that field lines have no beginning or ending – it seems that certain field lines terminate at null points." There is however, as discussed above, no violation of the condition that the field be divergence-free.
Edit: With the amount of attention this post is getting, I feel I should add a couple of points of clarification.


*

*In no sense am I saying that any such thing as a 'magnetic monopole' exists at a reconnecting X-point. In the resistive MHD picture, at an infinitesimal spatial point and for an infinitesimal time, magnetic field lines essentially lose their identity when they pass through the reconnection region. It makes no sense to talk about 'tracing' a field line across the X-point as we normally do when we plot maps of field lines. All we can say for sure is that the flux into and the flux out of a sufficiently small (formally infinitesimal) volume around the X-point are equal, satisfying $\nabla\cdot\mathbf{B}=0$.

*The sense in which the field lines 'terminate' at the reconnection is a corollary to this; because we can't identify any particular path which carries us smoothly across the X-point along a particular field line, we're forced to admit a discontinuity. This is why MHD equilibrium solvers for example use certain computational tricks to 'skirt round' the X-point in a given configuration rather than modelling the field all the way to the discontinuity.

*The foregoing discussion is valid only as long as the resistive MHD picture is valid; once we get down to scales comparable to the electron gyro-radius, the whole thing requires a self-consistent kinetic approach.

A: $\nabla\cdot\mathbf B=0$ does indicate that there are no magnetic monopoles, so there isn't a "starting" or "ending" point for field lines is mostly correct. So this must mean that magnetic  field  lines  either 


*

*form  closed  loops

*extend  to  infinity

*intersect the domain boundary (wall, stellar surface, etc)


So the "starting & ending points" issue is nuanced beyond what you've stated. With reconnection, we can usually assume the middle option: field lines extend to infinity (though invoking that they intersect the boundary is just as valid). 
For those unawares, magnetic reconnection is the when magnetic field lines pointing in opposite directions pinch together (reconnect) and form new lines:

(source)
To model this, one needs to modify the ideal MHD equations (this is because if we assume $\mathbf B\parallel\delta\mathbf x$ where $\delta\mathbf x$ is some displacement of field lines, it will remain so for all time $t$). For typical plasmas, one uses Faraday's law in conjunction with the Lorentz force to model the magnetic evolution,
$$
\frac{\partial \mathbf B}{\partial t}=-\nabla\times\mathbf E=\nabla\times\mathbf u\times\mathbf B\tag{1}
$$
But when considering magnetic reconnection, the conductivity isn't assumed to be infinite, so we have to use Ohm's law and add the current density, $\mathbf J\sim\nabla\times\mathbf B$:
$$
\frac{\partial \mathbf B}{\partial t}=\nabla\times\left(\mathbf u\times\mathbf B+\eta\nabla\times\mathbf B\right)\tag{2}
$$
where $\eta$ is the magnetic diffusivity. So now the magnetic field can diffuse, rather than simply moving along the flow; this is what allows for reconnection to occur in the plasma.
However, because the divergence of the curl of any vector is identically zero, $\nabla\cdot\nabla\times\mathbf A=0$, both (1) and (2) satisfy the divergence-free condition.
A: Magnetic reconnection comes from a cartoon picture of what magnetic field line motion may portray. This is not based on any physical law. Field lines are not real entity - just a means to display the lines of force when magnetic field is present in space. Field line motion is non-unique also, which is a fundamental flaw for people relying on field line motion to understand physics. That is why the magnetic reconnection is so misleading. It is unphysical to cut and join magnetic field lines. The exact equation for describing the time rate of change of B is not what is given in Eq. (2) by Kyle Kanos. The generalized Ohm's law is far more complicated than representing all the other missing terms in the simplified Ohm's law by a scalar resistivity. One should learn more about how magnetic reconnection is entrenched in space plasma physics - see, e.g., the following publications: note that Hannes Alfvén is a Nobel Laureate and is the one who invented MHD, Alfvén, H., On frozen-in field lines and field-line reconnection, J. Geophys. Res., 81, 4019-4021, 1976; Alfvén, H., Electrical currents in cosmic plasmas, Rev. Geophys., 15, 271-284, 1977; Akasofu, S.-I., Auroral substorms as an electrical discharge phenomenon, Progress in Earth and Planbetary Science, 2:20, doi:10.1186/s40645-015-0050-9, 2015; Lui, A. T. Y., Comparison of current disruption and magnetic reconnection, Geosci. Lett., 2:14, doi 10.1186/s40562-015-0031-2, 2015.
