The previously promised meditation on the action functional:

1.  Let $\Lambda \subset J^1(X)$ be a Legendrian.  Let $O_\Lambda$ be the space of open strings from $\Lambda$ to itself.  Say I try to compute its cohomology; channeling a competent initiate of Floer theory, I do this by studying the action functional.  (By “the action functional”, I mean whatever thing has Euler-Lagrange equations the Reeb flow.)  The critical points of this are evidently the Reeb chords, and depending on one’s definitions, the length zero paths which just stay on the Legendrian.  (Including or not including these corresponds to having the positive or negative augmentation category: if one wants to include them, one could further perturb by a Morse function on $\Lambda$ and take only chords of strictly positive length.)  In any case, one now tries to write down the differential, product and higher operations in cohomology; following Floer, EGH, etc, one does this by counting certain disks, since these correspond to the Morse flow trees of the action functional.  One encodes these as the differential of the CE dga (or perhaps its corrected version to include the zero paths); if Morse theory worked in this infinite dimensional setting, then the (Koszul) dual of this dga would be the (co?)homology of the path space.

2.  Alas, there is an anomaly.  (I am currently watching this lecture of Eliashberg, where he mentions this observation is originally due to Hofer.)  This anomaly comes about because the differential of the dga has degree zero terms.

3.  An augmentation of the dga is exactly the data needed to remove this anomaly.  I think at this moment we are close enough to physics that Eric or Harold should be able to say what physical theory it is that has moduli space of vacua = Spec(CE-dga)

4.  “Augmentations are sheaves”: i.e., Spec(CE-dga) = sheaves with singular support in $\Lambda$.  (See the upcoming NRSSZ paper for the case $X = \mathbb{R}$, or my previous secret note on this site; I remark here that ultimately I believe there should be a simple argument for this of the form that an augmentation really allows one to treat the action functional just like a finite dimensional generating function, and then use Viterbo’s prescription to get a sheaf from this function.)

…

@Harold & Eric — does this explain any of the connections to various physics that are turning up around the knotty character varieties?  (It’s possible that all of the above is anyway encoded in the sheaves = fukaya category connection, but somehow I am finding the above perspective enlightening lately).