TITLE: What does it mean to extend a 2d (topological) conformal field theory to Deligne-Mumford space?
QUESTION [5 upvotes]: For 2D (topological) conformal field theory the corresponding moduli space is the space of Riemann surface with boundary, right? What does it mean by extending the theory to Deligne-Mumford space? How can we do that in terms of the methods we have by now? Thanks for any expository answers!

REPLY [3 votes]: Cohomological Conformal Field theory (CCFT) introduced by M. Kontsevich, Yu. Manin as follows which is the  extension of a 2D (topological) conformal field theory to Deligne-Mumford space
Let $(M,\omega)$ be a symplectic manifold with choice of complex structure $J$.
Define $ev:\overline{\mathcal  M}_{g,k}(M,J,\beta)\to M^k$ and $d:\overline{\mathcal  M}_{g,k}(M,J,\beta)\to\overline{\mathcal  M}_{g,k}$
Let $\Lambda_0$ be Novikov ring, now define the maps 
$$I_{g,k}:H^*(\overline{\mathcal  M}_{g,k})\otimes H^*(M)\to \Lambda_0$$
as follows
$$<\alpha_1,\cdots\alpha_k>_{g,\beta}^\psi :=\sum_{\beta\in H_2(M)}\int_{\overline{\mathcal  M}_{g,k}}d^*\psi\wedge ev^*(\alpha_1\otimes\cdots\otimes\alpha_k)T^{\omega(\beta)}\in\Lambda_0$$
Moreover, we add the following axioms to introduce cohomological Conformal Field theory
1)  $\overline{\mathcal  M}_{g_1,k_1+1}\times \overline{\mathcal  M}_{g_2,k_2+1}\to \overline{\mathcal  M}_{g_1+g_2,k_1+k_2}$
these induces maps,
2) $ H^*(\overline{\mathcal  M}_{g_1+g_2,k_1+k_2})\to H^*(\overline{\mathcal  M}_{g_1,k_1+1})\otimes H^*(\overline{\mathcal  M}_{g_2,k_2+1})$
and 
3) $\overline{\mathcal  M}_{g,k+2}\to \overline{\mathcal  M}_{g+1,k}$
As Kontsevich-Manin interpreted , the algebraic package of a genus zero Cohomological Field theory as follows
We can define a $\Lambda_0$-linear maps, 
$$\tilde{I_k}:H_*(\overline {\mathcal M_{0,k+1}})\to Hom(H^*(M,\Lambda_0)^{\otimes k},H^*(M,\Lambda_0))$$
via
$$<\tilde{I_k}(S)(\alpha_1,\cdots,\alpha_k),\alpha_{k+1}>:=<\alpha_1,\cdots,\alpha_{k+1}>_{g=0}^{P.D.(S)}$$
where P.D. , here means Poincare dual.