Theorem: Let $X$ be a complete, non-singular algebraic curve of genus
$2$. Let $U(2, \Theta)$ be the space of $S$-equivalence classes of semi-stable vector
bundles of rank $2$ and degree $\Theta$. The group $\Gamma$ of elements of order $2$ in $J$
acts on $PH^0(J^1, L\_\Theta^2)$ in a natural way; let $A$ be the associated projective bundle on $J/\Gamma \cong J$. Then $U(2, 0)$ is canonically isomorphic to $A$. In other words, $U(2, 0)$ is canonically isomorphic to the space of positive divisors on $J^1$ algebraically equivalent to $2\Theta$.


Moduli of Vector Bundles on a Compact Riemann Surface 


Author(s): M. S. Narasimhan and S. Ramanan


Source: Annals of Mathematics, Second Series, Vol. 89, No. 1 (Jan., 1969), pp. 14-51


Question: Is the projective bundle in theorem comes from some vector bundle on $J$. Means is it $P(E)$ of some bundle $E$ over $J$.



Confusion in known result about moduli space of vector bundle of rank 2 degree 0 vector bundles over smooth curve of genus 2
EDIT: After I posted my answer, I realized that you were very close to answering your own question---the vector bundle you want is the one whose fiber over $\alpha \in {\rm Jac}(C)$ is $H^{0}(\mathcal{O}(2\Theta) \otimes \alpha)^{\vee}.$ However, this is equivalent via Strange Duality to what I have written below.


In general, if $C$ is a smooth projective curve of genus $g \geq 2$ $U\_{C}(r,0)$ is the moduli space of S-equivalence classes of vector bundles of rank $r$ and degree $0$ on $C,$ and we fix a line bundle $L$ on $C$ of degree $g-1,$ the locus
$$\Theta\_{L} = \{E \in U\_{C}(r,0) : h^{0}(E \otimes L) > 0 \}$$
is an ample divisor on $U\_{C}(r,0).$ Recall that the fibers of the determinant map ${\rm det} : U\_{C}(r,0) \to {\rm Jac}(C)$ are each isomorphic to ${\rm SU}\_{C}(r)$, which parametrizes rank-$r$ vector bundles with trivial determinant. For any two line bundles $L,L'$ of degree $g-1$ on $C,$ the restrictions of $\Theta\_L$ and $\Theta\_{L'}$ to any fiber of ${\rm det}$ are linearly equivalent; the resulting ample line bundle on ${\rm SU}\_{C}(r)$, which is denoted by $\mathcal{L},$ is called the $\textit{determinant bundle},$ and it generates ${\rm Pic}(SU\_{C}(r)).$ 


For $k \geq 1$, the $\textit{Verlinde bundle}$ on the Jacobian ${\rm Jac}(C)$ corresponding to the rank-level pair $(r,k)$ (and our choice of $L$, typically a theta-characteristic) is the vector bundle $$\mathbf{E}\_{r,k} := {\rm det}\_{\ast}\mathcal{O}(k\Theta\_{L})$$


For each $\alpha \in {\rm Jac}(C)$ the fiber of $\mathbf{E}\_{r,k}$ over $\alpha$ is isomorphic to 
$$H^{0}({\rm SU}\_{C}(r,\alpha),\Theta\_{L}|\_{{\rm SU}\_{C}(r,\alpha)}) \cong H^{0}({\rm SU}\_{C}(r),\mathcal{L}).$$ 
In the case where $g=r=2,$ the paper of Narasimhan-Ramanan that you are reading proves ${\rm SU}\_{C}(2) \cong \mathbb{P}^{3}.$ In particular, the determinant bundle $\mathcal{L}$ on ${\rm SU}\_{C}(2)$ is exactly the hyperplane bundle $\mathcal{O}(1)$ on $\mathbb{P}^{3}.$ The map ${\rm det}$ then realizes $U\_{C}(2,0)$ as a $\mathbb{P}^3$-bundle over ${\rm Jac}(C)$; this is the projectivization of the rank-4 Verlinde bundle $\mathbf{E}\_{2,1}.$


