,$$ where $\hat{\bm{e}}_r$ is a unit vector in the $r^{\text{th}}$ spatial direction and the block diagonal form of the supermatrices $\ol T {}^a_C$ is $$\label{eq:qtwist}
\ol T {}^a_C 
= 
\diag 
\left( T^a_C, 0, T^a_C \right)
.$$ Here $T^a_C$ are the elements of the Cartan subalgebra of $U(2)$.[^3] As a consequence of Eq. , the sea quarks are periodic, and hence the twist angles $\bm{\theta}^a$ can be altered without generating new gauge configurations.

Defining new quark fields as $\Qt(x) = V^\dagger(x) Q(x)$, where $V(x) = \exp ( i \bm{\th}^a \cdot  \bm{x} \, \ol T {}^a_C / L )$, we can write the partially quenched Lagrangian as $$\label{eq:Ltwist}
\cL 
= 
\sum_{j,k=1}^6 \ol{\Qt}{}^{\hskip 0.2em j} 
\left(
\Dtslash + m_Q \right)_j^{\hskip 0.3em k} 
\Qt_k
,$$ where all $\Qt$ fields satisfy periodic boundary conditions, and the effect of twisting has the form of a $U(1)$ gauge field: $\Dt_\mu = D_\mu + i B_\mu$, with $B_\mu = (0, \bm{\th}^a \, \ol T {}^a_C / L)$. For convenience we treat the twisting in the flavor basis: $\bm{\th}^a \, \ol T {}^a_C = \diag (\bm{\th}^u, \bm{\th}^d, \bm{0}, \bm{0}, \bm{\th}^u, \bm{\th}^d  )$, and similarly for $B_\mu = \diag (B^u_\mu, B^d_\mu, 0, 0, B^u_\mu, B^d_\mu )$. The constant field $B_\mu$ 