Theta line-bundle variation conjecture for moduli spaces on abelian surfaces

Let $v$ be a Mukai vector, and let $F_{1}$ and $F_{2}$ be sheaves with the same Mukai vector orthogonal to $v$. Let $\Theta_{F_i}$ denote the corresponding theta line bundles on the moduli spaces $\mathsf M^{+}_{v}$, $\mathsf M^{-}_{v}$ and $\mathfrak M_{v}$, and let $\alpha^{+}$ and $\alpha^{-}$ be the morphisms defined in the paper. Let $\mathbf R\mathcal S$ denote the Fourier–Mukai transform, and let $(-1)$ denote inversion on the relevant dual abelian variety. Theta line-bundle variation conjecture. The following identities should hold:

1. On $\mathsf M^{+}_{v}$,

$$\Theta_{F_{1}}=\Theta_{F_{2}}\otimes (\alpha^{-})^{\star}(\det F_{1}\otimes \det F_{2}^{-1}).$$

2. On $\mathsf M^{-}_{v}$,

$$\Theta_{F_{1}}=\Theta_{F_{2}}\otimes ((-1)\circ\alpha^{+})^{\star}(\det \mathbf R\mathcal S(F_{1})\otimes \det \mathbf R\mathcal S(F_{2})^{-1}).$$

3. If $c_{1}(v)=0$, then on $\mathfrak M_{v}$,

$$\Theta_{F_{1}}=\Theta_{F_{2}}\otimes ((-1)\circ\alpha^{+})^{\star}(\det \mathbf R\mathcal S(F_{1})\otimes \det \mathbf R\mathcal S(F_{2})^{-1})\otimes (\alpha^{-})^{\star}(\det F_{1}\otimes \det F_{2}^{-1}).$$

The formulas are proposed to describe how theta line bundles vary with the auxiliary sheaf $F$; the supplied text gives supporting computations and explicitly presents them as a speculation, but does not prove the full statement.