7
$\begingroup$

When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.

One often hears (or at least I often hear) that HH^* corresponds to A-infinity deformations. I am wondering whether there is any reference which works this out precisely. EDIT: This seems to be incorrect (depending on what we mean by "deformation"). See Damien's answer. And see David Ben-Zvi's comment.

$\endgroup$
3
  • 1
    $\begingroup$ Let A be an algebra. If you write down what exactly it means to have an `$A_\infty$ structure on $A\oplus A$ (with the first $A$ in degree zero and the second $A$ in some degree) which extends that of $A$, you get the Hochschild cocycle condition. $\endgroup$ Jul 14, 2010 at 20:59
  • 1
    $\begingroup$ In what sense is this incorrect? the claim is that the moduli stack attached to the shifted Hochschild complex, with its $L_\infty$ structure, is the deformation space of your $A_\infty$-algebra. For the latest words on this general deformation theory see Kontsevich-Soibelman's book and Lurie's ICM. If you interpret this statement correctly it will give the assertion Damien says - points over graded rings correspond to graded points of the Hochschild complex.. $\endgroup$ Jul 15, 2010 at 3:26
  • 1
    $\begingroup$ If you take "deformations" to mean "'derived' deformations", then it is correct. If you take "deformations" to mean "non-'derived' deformations", then it is incorrect. ---- Is this correct? ;-) $\endgroup$ Jul 15, 2010 at 4:11

3 Answers 3

11
$\begingroup$

Well. Even in the case of a DG (or $A_\infty$) algebra $A$, infinitesimal (i.e. 1st order) deformations are classified by $HH^2(A,A)$. Namely, the structure maps (a-k-a Taylor components) of an $A_\infty$-algebra, viewed as elements of the Hochschild cochain complex, do have total degree $2$.

I think that one recovers the full Hochschild cohomolgy $HH^*(A,A)$ by considering "derived" infinitesimal deformations (namely, deformations for which the deformation parameter is allowed to have non zero degree).

In other words, and making use of funny words, $HH^*(A,A)$ is the tangent to the derived stack of associative (better, $A_\infty$) algebras at the point $A$. While $HH^2(A,A)$ can be viewed as the tangent to the coarse moduli space. As an indermediate statement between those two, in his PhD thesis Mathieu Anel computed the tangent complex to the 2-stack of associative algebras (not in the derived context): he found that it is precisely a 2 step complex, obtain as a truncation of the Hochschild complex. See http://arxiv.org/abs/math/0607385 (in french, sorry).

$\endgroup$
6
  • $\begingroup$ How can I view the "Taylor coefficients" or structure maps $m_n$ of an $A_\infty$ algebra as elements of the Hochschild cochain complex? $\endgroup$ Jul 14, 2010 at 22:41
  • 1
    $\begingroup$ Oh, I see. We have $m_n : V^{\otimes n} \to V[2-n]$, which has "Hochschild degree" $n$ but "homological degree" $2-n$, hence total degree $n+2-n = 2$. $\endgroup$ Jul 14, 2010 at 23:10
  • $\begingroup$ Yes, exactly! :-) $\endgroup$
    – DamienC
    Jul 16, 2010 at 20:57
  • $\begingroup$ Do you know any references that talk about "derived" deformations? Is any of this written up anywhere yet? $\endgroup$ Jul 22, 2010 at 7:28
  • 2
    $\begingroup$ Thanks, Damien. By the way, Lurie has a new paper that is probably relevant to this discussion: math.harvard.edu/~lurie/papers/moduli.pdf $\endgroup$ Jul 25, 2010 at 21:29
1
$\begingroup$

You might want to look at 0705.3719.

$\endgroup$
1
  • $\begingroup$ I just read this paper and it does not have what I was looking for. $\endgroup$ Jul 14, 2010 at 18:51
0
$\begingroup$

I'll have to read it more carefully, but this paper of Penkava and Schwarz seems to do it: http://arxiv.org/abs/hep-th/9408064

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.