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\title{DennisFest Notes}
\date{May 28, 2011}
\author{Gabriel C. Drummond-Cole}
\begin{document}
\maketitle
\section{Kevin Costello III}
So I know I've been barraging you guys with definitions and I can't expect you to have absorbed everything.  What I've done so far is introduced language to describe moduli spaces of solutions to elliptic equations at a formal level.  

So $M$ is the derived moduli space of solutions to an elliptic equation.  We were thinking we should define this in terms of a Lie algebra, $\Ll_M$ is a sheaf of $L_\infty$ algebras on $X$.  Oh, sorry, Dennis.  Can you see?

[You should write a bit larger.] [Dennis moves.] [No, that's good.] [Laughter.]

$\Ll_M(X),d$ is an elliptic complex and there are some conditions, \wh, so maybe an example, if $X$ is a four-manifold, I can look at the instanton equations, solutions, self-dual connections, and the Lie algebra $\Ll_{M_{\text{inst}}}$, in degree $0$ is the gauge group, in degree one is the ways of making connections, and in degree $2$ I want the anti-self dual part to vanish.

\[\Omega^0(M,\ggg)\to \Omega^1(M,\ggg)\to \Omega^2_-(X,\ggg)\]

The reason this is a good space is because this is an elliptic complex.  $H_1$ of the complex is finite dimensional if $M$ is compact.

We were also interested in geometric structures on these guys, like symplectic forms.  So a symplectic form of degree $k$ on $M$ corresponds to an $\Ll$-invariant pairing with a shift of degrees (degree $k-2$).  We're interested in symplectic forms because classical field theories are critical points of a function, which correspond in this derived world to symplectic forms of degree $-1$.  One example to bear in mind, if I take any elliptic equation, I can take the cotangent bundle, that has a symplectic form, so I can take the shifted cotangent bundle.  If $M$ is any moduli problem then $T^*[-1]M$ is a classical field theory. 

Suppose I take the instanton equations and take its cotangent bundle.  The Lie algebra $\Ll_{T^*[-1]M_{\text{inst}}}$ has two pieces, the original one and then the duals of these ones:
\[
\xymatrix{&\Omega^2_-\ar[r]&\Omega^3\ar[r]&\Omega^4\\
\Omega^0\ar[r]&\Omega^1\ar[r]&\Omega^2_-}
\]
We shifted it because we wanted a pairing of degree $-3$.  In physics this is the self-dual Yang Mills, actual Yang Mills is a deformation of this. 

[Does actual Yang Mills fit in your framework?]

You can do it for $\rR^4$, I don't know how to do it in an arbitrary $4$-manifold.  You add a differential $\Omega^2_-$ to itself by $c \Id$.  When $c=0$ then you get the self-dual Yang Mills.

Many theories have a large volume limit that looks like a cotangent theory, and then you add small perturbations.

The aim is to say something about quantization.  If $U$ is a subset of $X$ then I can think of my moduli space $M(U)$, and functions on that, $\Oo(M(U))$ I'll define to be the Chevalley cochains of the Lie algebra $C^*\Ll_M(U)$.

If my moduli problem has a symplelctic form, then functions on it have a Poisson bracket of degree $1$.  We'd like to talk about a quantization of this structure.  What is quantization of a degree $1$ bracket?  That has a nice, uniform theory, but the most \wh case, let me remind you, is ordinary deformation quantization.

$A$ is a commutative algebra, and $\{\ \}$ is a degree $0$ Poisson bracket, then a quantization of $A$ is a product $a\times_\hbar b$, $\hbar$ dependent, on $A[[\hbar]]$, which is $\hbar$-linear, such that $a\times_\hbar - b\times_\hbar a=\hbar\{a,b\}\mod \hbar^2$.

We'd like a similar description when we have a degree $1$ Poisson bracket.  There's a very natural degree $1$ Poisson bracket. [Is it the BV bracket?] Yes.  If $A$ has a degree $1$ Poisson bracket, and again is a commutative dga, then 
\begin{defi}
A quantization of $A$ is an algebraic structure on $A[[\hbar]]$, for deformation quantization it was an associative structure, here it will be something else, a commutative product and Poisson bracket of degree $1$ and a differential on $A[[\hbar]]$, lifting the structure we already had, such that $d(ab)=d(a)b+(-1)^{|a|}a d(b)+\hbar\{a,b\}$.  When we deform, $d$ is not a derivation, and the failure is the bracket.

The failure of $A[[\hbar]]$ ot be an dga at all (that is, to have a product compatible with $d$), is measured by $\{ \}$.  You know as a topologist that commutativity is a lot more information than associativity.

[Is that a BV algebra with an $\hbar$?]  I call this a BD algebra, for Beilinson-Drinfeld.  In the mid-90s, this was shifted and given degree $-1$, it's kind of annoying.\end{defi}

I can see from the audience reaction that this is confusing.  Let $M$ be a manifold.  Then $\Oo(T^*[-1]M)$ has a Poisson bracket of degree $1$.

[What is equivalence?] It's homotopy equivalence.  This should be a family where the base is \wh on the $n$-simplex, like in John's talk.

In this situation, what does it mean to quantize? In degree $0$ it's smooth functions, in degree $-1$ it's vector fields and so on.
\[\Gamma(M,\wedge^2 TM)\quad \Gamma(M,TM)\quad C^\infty(M)\]
A volume form on $M$ gives one of these.  This is an operadic definition of a volume form.  Let me explain why this is true.

If I choose an $n$ form on $M$, $\omega$, then we get an isomorphism like $C^\infty(M)$ I can associate with top forms, vector fields with $n-1$-forms, and so on.  I'll write down the quantization from this.  I can choose to only deform the differential.  How can I do this?  I have the de Rham differential on $\Omega$.  We can do this using our isomorphism.  That's the divergence for the volume form $\hbar Div_\omega$

The point is that polyvector fields with this differential, $\Oo(T^*[-1]M),\hbar Div_\omega$ is a quantization, so that $Div_\omega(\alpha,\beta)=Div_\omega(\alpha)\beta+\alpha Div_\omega(\beta)+\{\alpha,\beta\}$.  

There is a natural set of deformations equivalent to volume forms.  Observe that $\Oo(T^*[-1]M)$ has an $\rR^*$ action by scaling the cotangent fiber $\{\ \}\to \lambda\{\ \}$ where we scale by $\lambda$.  Now, $\hbar$ and the Poisson bracket are next to each other.  Let $\rR^*$ act on functions on $\hbar$ by sending $\hbar$ to $\lambda^{-1}\hbar$, and then our axioms are $\rR^*$-invariant and we can ask for an $\rR^*$-invariant quantization.

The lemma is, that here, volume forms give quantizations, invariant quantizations, and these are all you find.  There is a bijection between volume forms on $M$ up to multiplication by a scalar and $\rR^*$-invariant quantizations.

In particular, we can talk about volume forms for these infinite dimensional things.

[Can you think of this as the \wh?] Yes, exactly. I am doing quantum field theory, so there's a measure, formally $e^{S/\hbar}d\mu$.

Let's talk about the infinite dimensional situation.  So $M$ is an elliptic moduli problem.  $T^*[-1]M$ has a degree $-1$ symplectic form which correpsonds to $\Ll_M\oplus \Ll'_M[-3]=\Ll_{T^*M[1-]}$.

If $U\subset X$, I'll define, again, $\Oo(T^*[-1](M(U)))$ to be $C^*(\Ll_{T^*[-1]M}(U))$, the Chevalley cochains on the Lie algebra.  This has a Poisson bracket of degree $1$ (it's nonessential that this be a cotangent).

\begin{defi}
A quantization is a quantization of each of these $\Oo(T^*[-1]M(U))$, $\rR^*$-invariant, compatible with the natural maps (let me explain these.
\end{defi}
There are maps $\Oo(T^*[-1]M(U_1))\otimes \Oo(T^*[-1]M(U_2))\otimes\cdots$ which maps to $\Oo(T^*[-1]MV)$ where $U_i$ is in $V$, all disjoint, maps of Poisson algebras.  If $U_i$ are not disjoint, this is not a Poisson map.

So we want quantization to be compatible with this structure.  This leads to what we call a ``factorization'' algebra, where $Obs(U)$ is the cochain complex $\Oo(T^*[-1]MU[[\hbar]]$, with maps as above.

If $X$ is compact, then this moduli space $M(X)$ is finite dimensional.  The upshot is that a quantization of the field theory leads to, on the whole space, a volume form on the finite dimensional dg manifold $M(X)$.  In good situations this might be the germ of a smooth manifold.  This is a volume form given by local data on $X$.

Let me explain how you might construct these and indicate why the volume form is unique.

\begin{thm}(Gwilliam)\\
There is a deformation complex, a sheaf you associate by a simple procedure, obstructions on $X$ associated to $M$, a sheaf of cochain complexes such that $H^1(X,Obstr^n(M))$ are the obstructions to quantization, and if the obstructions vanish, then the space is described by $H^0$, but this describes the deformations of a given quantization.  In many examples, these cohomology groups vanish.
\end{thm}
Here is an example.  Let $E$ be an elliptic curve, $Y$ a complex manifold, and $M$ the holomorphic maps $E\to Y$ near constant maps.  What you get from the cotangent theory then, $T^*[-1]M$ is what Witten calls the $``N=(0,2) \text{ susy twisted }\sigma\text{-model}$.  Maybe I'll tell you, this global moduli space is $T[-1]Y$, and $H^{\le 0}(Obstr^n)=0$, $H^1(Obstr)=\cC$, and the obstruction to quantization is $ch_2(Y)$, so if that vanishes there is a unique quantization (this gets to $Y$ by pushforward).  Then $M(E)$ has a volume form, $Vol(Y^E)=Wit(Y,E)$.

Grady and Gwilliam have this result: $Y$ is copmlex, lecally constant maps $S^1\to Y$, then there is a unique quantization $Vol(Y^{S^1})=\int_Y Td(Y)$.  A last one, if $M$ is the instanton moduli space, $X$ a $4$-manifold, $G$ a simple group, thene there is a unique quantization which gives a canonical, conformally invariant measure on the instanton moduli space.

I think I'll stop there.

[\wh]  [Are these obstructions related to what physicists call anomalies?]  Yes, they're the same.
\end{document}

