The Viterbo Transfer as a Map of Spectra

Let $L$ and $N$ be two smooth manifolds of the same dimension. Let $j\colon L\to T^*N$ be an exact Lagrange embedding. We denote the free loop space of $X$ by $\Lambda X$. C. Viterbo constructed a transfer map $(\Lambda j)^! \colon H^*(\Lambda L) \to H^*(\Lambda N)$. This transfer was constructed using finite dimensional approximation of Floer homology. In this paper we define a family of finite dimensional approximations and realize this transfer as a map of Thom spectra: $(\Lambda j)_! \colon (\Lambda N)^{-TN} \to (\Lambda L)^{-TL+\eta}$, where $\eta$ is a virtual vector bundle classified by the tangential information of $j$.