What does the eta notation mean in the naming of a transition metal complex? I am reading a paper (Gutel et al., J. Mater. Chem. 2007, 17, 3290-3292) and I have come across the following reaction:


  In organic solvents, the decomposition under mild conditions of
  ($\eta^4$-1,5-cyclooctadiene)($\eta^6$-1,3,5-cyclooctatriene)ruthenium(0),
  Ru(COD)(COT), under a dihydrogen atmosphere gives ruthenium
  nanoparticles (RuNPs), by releasing cyclooctane (COA).
  
  $$\text{Ru(COD)(COT)} + \text{H}_2 \rightarrow \text{RuNPs} + \text{COA}$$


What do the $\eta^4$ and $\eta^6$ notations represent?  Do they somehow specify the coordination "connectivity" of the Ru(0) and the COD and COT coordinating ligands?

Below, I have drawn the (two-dimensional) structures of COD and COT.  How would I draw the connectivity of Ru(COD)(COT), or is it not specified in the given name?
What do the $\eta^4$ and $\eta^6$ notations represent? Do they somehow specify the coordination "connectivity" of the Ru(0) and the COD and COT coordinating ligands?


They do.

The $\eta$-notation denotes the hapticity of the ligand, which is the number of atoms in the ligand bonded to the metal center.



In your image, shown above, cyclooctadiene (COD), has two alkene functional groups. A transition metal can coordinate to just one alkene, which would be $\eta^2$, or to both alkenes, which would be $\eta^4$.

Cyclooctatriene (COT) has three alkenes. A metal center can be coordinated at one ($\eta^2$), two ($\eta^4$), or all three ($\eta^6$) alkenes.