What are the mechanisms for capture of As(III) and As(V) by magnetite from water?
Unfortunately there are no (or only very hard to find) studies to the mechanism of the actual sorption of arsenic (acids) to magnetite. Most literature does describe it as a simple Ligand exchange, although I find that hard to believe. From what I have read[1-4] I will try to compose some hints on what may happen.

First of all lets have a look at arsenic acid. It is a medium strong acid and a quite well oxidising agent.
\begin{align}\ce{
H3AsO4 &amp;\equiv AsO(OH)3\\
AsO(OH)3 + H2O &amp;&lt;=&gt; H3+O + AsO(O^- )(OH)2\\
AsO(OH)3 + 2H3+O + 2e- &amp;&lt;=&gt; As(OH)3 + 3H2O
}\end{align}

In presence of any electron rich material and acidic environment it will therefore not be unusual that arsenous acid will be formed.[5] We may therefore also focus on this, amphoteric molecule:
\begin{align}\ce{
H3AsO3 &amp;\equiv As(OH)3\\
As(OH)3 + H3+O &amp;&lt;=&gt; As+(OH)2 + 2H2O\\
As(OH)3 + {}^{-}OH &amp;&lt;=&gt; AsO^{-}(OH)2 + H2O
}\end{align}

Depending on the pH of the solution one would expect to find many of different of these  species. Arsenic is also somewhat oxophil, meaning that the $\ce{As-O}$-bond is quite strong.

The magnetite in aqueous solution reacts basic, also forming complex like structures or hydroxides, for example this fully hydrated cluster (ref. [2], stoichiometry omitted):
\begin{aligned}\ce{
 Fe3O4 + $x$\, H2O &amp;&lt;=&gt;  Fe2(OH)2(H2O)8^4+
}\end{aligned}

Of course one has to pay attention, that the used ferric nano particles for sorpting arsen(o)ic acid(s) are far larger than the used model complex. However, we can see, that there are functional groups like $\ce{-OH}$ or $\ce{-OH2+}$ (or maybe even $\ce{-O-}$) covering the surface. 

Using this as prerequisites we can think about what might happen next. I generally would suggest to start with physisorption, i.e. the cluster and the acid molecules come in close range. Since we are dealing with contaminated water it is maybe safe to assume, that we can deal with soluble salts of arsenoic acid, hence focusing on $\ce{AsO3^{3-}}$.

When the complexes come into close range hydrogen bonds will form, with 
$
\newcommand{\nanoFe}{\bbox[lightgrey,3pt,border:1px solid black]{\ce{Fe}}}
\nanoFe$ being the nano particle:
$$\ce{\nanoFe-OH\cdots{}O-AsO2^{3-}}$$

Since the whole surface is covered with hydroxide groups, several of these hydrogen bonds may form. As we can see, with this we may now enter the realm of chemisorption. Once these complexes have formed, it is from an entropic point of view favoured to release smaller molecules, water in this case. Also the oxygen atoms bound to arsenic are now in close range to iron atoms. Since we are dealing with nano particle, there will be non-coordinated irons in proximity. This allows for a more permanent complet to form:
$$\ce{\nanoFe-O-AsO^3- -O\cdots{}HO-\nanoFe}$$

With this monodentate complex we have an already stable complex of a bound arsenic to an iron. It is furthermore possible, and probably favoured, to form a bidentate complex with a somewhat similar mechanism.

Let us also consider a more basic scenario. In this case the surface of the magnetite will not be entirely covered with hydroxide atoms, but more freely accessible oxygen atoms will be present. Direct bond formation between $\ce{\nanoFe-O\cdots{}AsO3^{3-}}$ may occur. 



After writing all this I also found a credible source that somehow proves my theory. In &ldquo;Understanding Arsenate Reaction Kinetics with Ferric Hydroxides&rdquo;[6] they used a model system of $$\ce{HAsO4^2- + Fe2O10H15+ -&gt; Fe2O10H15-HAsO4-}$$ to explain the mechanism. 

In this they describe that physisorption is barrierless, leading fast to the adduct that is $\Delta G^\circ_\mathrm{r}=-58~\mathrm{kJ/mol}$ more stable in energy. However, chemisorption and developing the equilibrium is a more lengthy process.

When the formed complexes become neutral, the activation barrier to release arsenate is very high, compared to adsorption. I will stop right here, please read the publication for further details.



Notes, references and further reading:


Dimitri A. Sverjensky, Keisuke Fukushi, Geochimica et Cosmochimica Acta 2006, 70 (15), 3778-3802.
Nianliu Zhang , Paul Blowers , and James Farrell, Environ. Sci. Technol. 2005, 39 (13), 4816–4822.
Arnold F Holleman; Egon Wiberg; Nils Wiberg: Lehrbuch der Anorganischen Chemie. 102. Auflage. Walter de Gruyter &amp; Co., Berlin, New York, 2007. (german)
Egon Wiberg, Nils Wiberg: Inorganic Chemistry. Academic Press, 2001. (english)
Thompson, A. &amp; Goyne, K. W. (2012) Introduction to the Sorption of Chemical Constituents in Soils. Nature Education Knowledge 3(6):15
Heavier halogens ($\ce{X=I, Br, Cl}$) should do the job and they will most certainly be present in usual ground water (more or less acidic solution) :
$$\ce{AsO(OH)3 + 2H3+O + 2KX &lt;=&gt; As(OH)3 + 3H2O + X2 ^ + 2K+}$$
James Farrell and Binod K. Chaudhary, Environ. Sci. Technol. 2013, 47 (15), 8342–8347.
Also interesting: (a) Suvasis Dixit and Janet G. Hering, Environ. Sci. Technol. 2003, 37 (18), 4182–4189 (b) Dimitri A. Sverjensky and Keisuke Fukushi, Environ. Sci. Technol. 2006, 40 (1), 263–271.