One significant connection to mathematical physics is through <a href="http://en.wikipedia.org/wiki/Vertex_operator_algebra">vertex operator algebras</a>, used in conformal field theory. As algebraic objects, VOA's are fairly complicated, but the operations are conveniently packaged in geometric form, using an operad whose composition is given by sewings of punctured Riemann spheres. See this <a href="http://arxiv.org/PS_cache/hep-th/pdf/9301/9301009v3.pdf">early paper</a> by Lepowsky and Huang for information on this. 

<b>Edit:</b> A more up-to-date account of the operadic story has been given by Huang's student Liang Kong, <a href="http://arxiv.org/PS_cache/math/pdf/0610/0610293v3.pdf">here</a>.