can be traded in for the field $\St$ defined by $\St (x) = V^\dagger(x) \S (x) V(x)$, which is periodic at the boundary [@Sachrajda:2004mi]. In terms of this field, the Lagrangian of  appears as $$\cL = 
\frac{f^2}{8} \str \left( \Dt_\mu \St \Dt_\mu \St^\dagger \right)
- 
\l \, \str \left( m_Q^\dagger \St + \St^\dagger m_Q \right)
.$$ The action of the covariant derivative $\Dt^\mu$ is specified in Eq. .

To include baryons into , one uses rank three flavor tensors [@Labrenz:1996jy]. The spin-$\frac{1}{2}$ baryons are described by the $\bm{70}$-dimensional supermultiplet $\cB^{ijk}$, while the spin-$\frac{3}{2}$ baryons are described by the $\bm{44}$-dimensional supermultiplet $\cT_\mu^{ijk}$ [@Beane:2002vq]. The baryon flavor tensors we use, however, are twisted at the boundary of the lattice. Thus we define new tensors $\cBt^{ijk}$ and $\cTt_\mu^{ijk}$ both having the form [@Tiburzi:2005hg] $$\cBt_{ijk}(x) = V^\dagger_{ii}(x) V^\dagger_{jj}(x) V^\dagger_{kk}(x) \cB_{ijk}(x)
.$$ These baryon fields satisfy periodic boundary conditions and their free and leading-order interaction Lagrangian has been given in [@Tiburzi:2005hg].

In partially quenched QCD, the isovector vector current is defined by $J_\mu^{a}(x) = \ol Q(x)  \, \ol T {}^a  \gamma_\mu  Q(x)$. The choice of supermatrices $\ol T {}^a$ is not unique [@Golterman:2001qj], even when one imposes the condition $\str \, \ol T {}^a = 