I have a negative definite 4-manifold $X$ with a rational homology sphere boundary $Y$. For simplicity, I'm assuming also that $H_1(X)=0$. Given a spin^c-structure $\mathfrak s$ on $X$ that restricts to $\mathfrak t$ on $Y$, there is an upper bound for the Heegaard Floer homology correction term $$ 4 d(Y, {\frak t}) \geq c_1({\frak s})^2 + b_2(X).$$
My question is how should I interpret $c_1({\frak s})^2$ here?
I've seen in practice that by thinking of $H^2(X) \subset Hom(H_2(X);\mathbb{Z})$ the intersection form of $X$ induces a rational valued pairing on $H^2(X)$. Then $c_1(\frak s)^2$ is computed by squaring $c_1({\frak s})$ using this pairing. However, I'm not sure what the topological interpretation on $c_1({\frak s})^2$ should be to make this the correct thing to do.
I suspect there must be something simple that I'm missing here. Thanks in advance.