in , and is the eigenvalue of the $H_{ab}$ tensor in and its dependence on $a_i$’s comes from the field strengths, $F^{(i)}$. The $\Lambda$-term in comes from the composite connection in the covariant derivative in and its dependence on $a_i$ is due to the gauge potentials, $A^{(i)}$.

To solve the algebraic equations - we impose projection conditions on the Killing spinors, $$\label{projc}
\gamma^{34}\varepsilon_{(i)}\eql - \omega_{ij}\varepsilon_{(j)}\,,\qquad \gamma^2\varepsilon_{(i)}\eql \omega_{ij}\varepsilon_{(j)}\,,$$ which are unique up to a choice of signs on the right hand side, with different choices leading to equivalent BPS equations. Note that by combining the two projectors in and using that $\gamma^0\gamma^1\gamma^2\gamma^3\gamma^4 = 1$ one finds $$\label{}
\gamma^{01}\varepsilon_{(i)}\eql \varepsilon_{(i)}\,,$$ which shows that the two-dimensional holographic field theory indeed has $(0,2)$ supersymmetry.

Using in we then get $$\label{acond}
a_1+a_2+3 a_3+(a_1+a_2-a_3)\cosh 2\chi \eql {2\over g}\,.$$ Since $a_i$ are constant, supersymmetric flows with a varying field, $\chi$, must satisfy[^10] $$\label{aconstr}
a_3=a_1+a_2\;, \qquad  a_1+a_2 = \dfrac{1}{2g}\;.$$ It is convenient to solve the second constraint in by introducing a single parameter, $\mathfrak a$, $$\label{a1a2a}
a_1 \eql  {1\over g}\Big({1\over 4}+\mathfrak a\Big)\,,  \qquad a_2 \eql  {1\over g}\Big({1\over 4}-\mathfrak a\Big)\,.$$ Note that is tantamount to the topological twist along the Riemann surface. Indeed, it implies a cancellation, in the 