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\author{Gabriel C. Drummond-Cole}
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\title{Deformation Theory Workshop}
\begin{document}\maketitle
\section{Soibelman}
Let $k$ be a field.  A Calabi Yau manifold over $k$ is a somooth projectiv variety $X$ with trivial canonical class.

\begin{thm} ($k=\cC$) Let $X$ be a compact Kahler manifold with trivial determinant bundle.  Let $\omega_X$ be the symplectic form corresponding to the Kahler metric.  Then there exists a unique Ricci flat metric $g^{CY}$ with the same Kahler class.
\end{thm}

[missing a few lines]

There will be two main references.  One will be our paper with Kontsevich 06006241 on the arxiv, and the first volme of the still unfinished book on deformation theory.  The draft is available on my homepage.

I start with the background information of deformation theory.  We will specify a class $\Xi$ of structures like algebras, vector bundles with connections, operads, and then a class of spaces, typically local Artin algebras or schemes of complex manifolds.  Maybe I should also consider these to be pointed spaces.  This is just intuition, but next we consider families of structures $\Xi\to S$ so that the fiber over the marked point $x_{s_0}$ is fixed.  This gives rise to a functor which gives for a space $(S,s_0)$ to the category of sets, families up to isomorphism.  This should correspond to a space which should be called the moduli space of this structure.  What is this space, and what are the properties if it's represented by a space.  The generality in which I'm going to use this approach is the following.  So, I will work over the ground field $k$ with characteristic zero, and the class of spaces will be formal schemes in one of three tensor categories.  This is not maximal generality, but I don't want to work in maximal generality.  These three categories are $k$-vector spaces, $\zZ$-graded $k$-vector spaces, and $\zZ_2$-graded vector spaces.  In each of these tensor categories, we have notions of algebra, coalgebra, and so on, so we can do simple linear algebra.  I will denote any of these categories by $\Cc$, and then obvious notation for algebras, $Alg_\Cc$ and $Alg_{\Cc^f}$ (finite) and similarly.  

Let me introduce some things that will get us to $L_\infty$, $A_\infty$, and so on.  \begin{thm}
\begin{enumerate}\item Let $F:Alg_{\Cc^f}\to Sets$ be a covariant functor commuting with finite projective limits.  Then it is isomorphic to a functor $A\to Hom(A^*, B)$ where $B$ is a coalgebra and this is a homomorphism of coalegbras.  $B$ is a counital coalgebra

[Can you say what finite projective limits are?]

I don't want to discuss it, do you need a definition or an example.  Here's an infinite projcective limit:  $k[t]/\langle t^i\rangle$, so there is the obvious arrow, so you have a projective limit.  A finite limit is just a cutoff.
\item $B$ is a uniton of finite dumensional coalgebras
\item the same is true if $F$ is a functor from the category of finite dimensional commutative algebras.  Then $B$ will be a cocommutative coalgebra.
\end{enumerate}\end{thm}

All right, so, what are the examples of coalgebras which appear in this theory?
\begin{example}
We see that we have two areas in which this will be applied, one when we have not necessarily commutative algebras and one when we have commutative (corresponding to $A_\infty$ and $L_\infty$).  In the noncommutative case, the main example will be $B=T(V)=\bigoplus_{n\ge 0}V^{\otimes n}$, the cofree coalgebra, with the obvious coproduct $\Delta(v_1\otimes\cdots v_n)=\sum (v_1\otimes \cdots v_i)\bigotimes (v_{i+1}\otimes \cdots v_{n})$.  In the commutative case we have the same thing $B=S(V)=\bigoplus Sym^n(V)$ with the obvious modification to the coproduct formula.  If the object I'm introducing in smooth, then $B$ will be one of these.  I did not say an example of what but it will be clear in a minute.
\end{example}
I'd like to think of this as the coalgebra of distributions on a space, and I'd like to denote $B^*$ by $\Oo(X)$.  Since it's a union of finite things, it's a topological algebra.  It's not an arbitrary space, since it came from a special class of coalgebra.  I'll call it a thin space.  Either it's commutative because the coalgebra is cocommutative, and in the noncocommutative space I will call it a noncommutative thin space.  These form symmetric monoidal categories with inner Homs because we can take tensor products of coalgebras.

I'll denote inner homs as $Maps(X,Y)$.  So then I'm trying to switch to geometric language.  Here in the category of thin spaces, or maybe thin schemes, there is a notion of smoothness which is defined more or less in the same way as in commutative algebraic geometry.  These are examples of manifolds in this category.  

Now I can introduce the main objects, namely $A_\infty$ and $L_\infty$ algebras.  There is a notion of smoothness, you can talk about vector fields, which correspond to derivations of functions on a smooth commutative scheme.  Let me give definitions in the case, the category of $\zZ$-graded vector spaces.

\begin{defi}A non-commutative differential graded manifold is a pair $((X,x_0),d_x)$ which consists of a formal pointed manifold, the same as a formal pointed non-commutative thin scheme, and $d_X$ is a derivation of the coalgebra and in the geometric language a vector field.  I assume that $d_X$ has degree $+1$, $d_X(x_0)=0$, and the main thing is that $[d_X,d_X]=0$
\end{defi}
In the language of coalgebras, $x_0$ is a homomorphism of coalgebras $k\to B_X$.

I didn't say what formal meant.  A formal manifold is a noncommutative thin scheme which is isomorphic to $T(V)$.  The commutative case is isomorphic to $S(V)$.

Some people call these coderivations.  The idea of coderivation, well, derivation is a very general construction.

de Rham is like $\Oo(\Pi TX)$ and the differential $d_X=\sum \chi_i \frac{\partial}{\partial x_i}$.  This is not formal.  I won't use this.

All right, so algebraic terminology.  In the noncommutative case, $X$ corresponds to $A_\infty$ algebra, and in the commutative to $L_\infty$ algebra.  In order to define this structure algebraically, we should choose an isomorphism with one of these examples.  So $V$ will correspond to the tangent space $T_{x_0}X$, maybe shifted by one.  In particular, we can go back and spell out these structures algebraically.  Then what we have in both cases is a graded vector space $V$, in the noncommutative case a collection of linear maps $m_n:V^{\otimes n}\to V[2-n]$, which correspond to Taylor coefficients at $x_0$ of $d_X$, and in the commutative case a collection $b_n:\wedge^n V\to V[2-n]$.  These satisfy a system of quadratic equations.  I won't repeat them, but $m_1$ and $b_1$ square to zero so these are complexes.  For the purposes of deformation theory, I will work with $L_\infty$ algebras, but the theories are very much parallel.  The theory of some commutative or noncommutative local schemes.  The algebras are algebras of formal series in variables $x_n$ or the same thing with graded commuting variables.  So you consider graded derivations of either of these.  They are very similar.  I should not have said local, formal, local.

One more thing which I need is called the minimal model theorem, says that any noncommutative formal pointed $dg$-manifold can be decomposed into a direct product in the corresponding category into two simpler objects $(X, x_0),d_X \cong ((X^{min},x_0').d_X')\times ((X^{contr},x_0''),d_X'')$ where in $X^{min}$ the differential on the tangent complex is trivial where on the contractible part the tangent complex is quasiisomorphic to $0$.  This means in particular you have a projection and embedding to and from $X^{min}$.

Okay.  So then finally I will return to the deformation theory and we will use this language in order to, well, okay, $L_\infty$ algebra $g$.  In a special case you could have a differential graded Lie algebra $b_1$ and $b_2$.  If you, geometrically, $L_\infty$ algebra is the same as formal pointed differential manifold, but we can consider inside the subscheme of zeros of $d_X$.  It's nonempty because of $x_0$.  This is a functor.  Then basically this is up to some equivalence relation a moduli space associated to the $L_\infty$ algebra $g$.  It can be spelled out algebraically.  The same functor is isomorphic to another one.  Let me denote $b_1$ by $d$ because it's a differential, and $b_n(\gamma_1,\ldots,\gamma_n)$ as $[\gamma_1,\ldots, \gamma_n]$.

I can look to solutions to $d\gamma+\sum \frac{1}{n!}[\gamma,\ldots, \gamma]=0$.  Let me give it in the special case of differential graded Lie algebras, $d\gamma+\frac{1}{2}[\gamma,\gamma]=0$.  That's a set of solutions,  modulo the gauge action, in this case $g$ is a graded vector space, $g_0$ is a Lie algebra, and then the gauge group is $exp(g_0\otimes M)$.  In this case this corresponds to a differential graded Lie algebra but the same thing works in general for a $L_\infty$ algebra.  \begin{thm}The main theorem is that if you have two formal manifolds and a map between them that's a quasiisomorphism, then the functors $def_g$ are isomorphic.  So now this \end{thm}
Now let's go back to the beginning, in this case, back to Tian-Todorow, in this case we are dealing with deformation theory of complex structures, and we have $g=\Gamma(X,T_x^{1,0}\otimes \wedge (T_x^{0,1})^*),\bar{\partial})$ which is a differential graded Lie algebra which produces a deformation functor isomorphic to the one for deforming complex structures:  $(g,d)$ is quasiisomorphic to a differential graded Lie algebra with trivial differential and trivial bracket (homologically Abelian).  The differential is trivial so that the Lie algebra is formal, and the bracket is trivial.  The same theorem is true for any Calabi-Yau algebra.  According to Deligne's conjecture, the Hochschild complex is an algebra over chains on the little disks.  The $L_\infty$ part corresponds to chains in the top degree.  I need one more condition to impose, the degeneration of Hodge to de Rham.  You look for a negative cyclic chain complex equipped with double differential, Hochschild plus parameter times Connes.  This should be trivial.  This is equivalent to Hochschild complex is an algebra on the chains of framed little disks, the triviality of this action.  The top degree chains, this is trivial for degree reasons.  If you include, assume the degeneration then this is a theorem.  
\section{Kajiura}
What I want to say today can be stated mathematically.  On a quantum open-closed correspondence.  This is joint with Stasheff, Tradler, Zenalian.  When I say vector space, I mean a finite dimensional $\zZ$-graded vector space over $k$ of characteristic zero.

I want to discuss a version of a classical structure relating to string theory.  The classical case, the open-closed string theory, you have an open closed homotopy algebra and this gives an $L_\infty$ map from an $L_\infty$ algebra $(L,l)$ to the differential graded Lie algebra $(Hom(TA,A))$, the Hochschild complex.  The differential is given by $[m,],[,]$ where $m$ is the $A_\infty$ structure on $A$.    This $L_\infty$ map will be equivalent to the following: a weak $L_\infty$ map $F:(L,l)\to (Hom(TA,A),[,])$.  This is just a graded Lie algebra.  This is a set of maps $f_k:\wedge^k \to Hom(TA,A)$.  If $f_0$ satisfies the Maurer Cartan equation, this recreates the earlier situation.

The parallel statement holds for $A$ with a cyclic $A_\infty$ structure.  Then we can just replace $Hom(TA,A)$ with $Hom(TA_{cyc},k)$ which is equivalent to $TA^*_{cyc}$.  

So we have an $L_\infty$ structure, and this from the string theory viewpoint, corresponds to closed strings interacting.  If we discuss classical open closed strings, then we have the following [picture]

This gives a weak $L_\infty$ map.  This is a deformation quantization, but this is not the one I want today.  I want to quantize this picture in the sense of string theory.  This is the motivational problem.  What is the quantization?  By quantization, today I mean including loops, by which I mean closed string loops, which are just $S^1$ but we obtain tori, so we get genus.  On the other hand, an open string loop means an annulus, with two boundaries.  By Riemann surface with loops I mean that the genus is more than one and the number of boundaries is greater than 2.  The closed string interaction has genus zero and no boundary.  We want quantum theory.  The classical theory has only genus zero with $0$ or $1$ open boundary.

\begin{defi}$(V,\Delta,\{,\},\cdot)$ is a $BV$-algebra when $\cdot$ is a degree zero commutative product on $V$ and $\{,\}$ a degree one Poisson bracket with respect to $\cdot$.  Then $\Delta$ is a differential of degree one.  The relation is that $\Delta(AB)=\Delta A\cdot B+(-1)^{|A|}A\Delta(B)+(-1)^{|A|}\{A,B\}$.  Today, the algebra corresponding to quantum string theory, I want to define a quantum open string algebraic structure by using Zwieback's string field theory.  Now I want to consider the following complicated graded vector space: $Sym(L^*/\hbar^{1/2})\otimes Sym(T(A^*)_{cyc}/\hbar)[[\hbar^{1/2}]]$.

I should define the BV operator as the sum of $\Delta_c$ and $\Delta_o$.  Here $\Delta_c$ is $\omega_c^{ij}\frac{\partial}{\partial \psi_i}\frac{\partial}{\partial_j}$.  The open case is a sum over cyclic permutations of second order differential operators.  So $\psi_i$ and $\phi_i$ are bases of $L^*$ and $A^*$.  Then a fact is that $V,\Delta$ forms a $BV$-algebra, where the product is defined in the usual way.  

Now I can find the algebraic structure corresponding to quantum open closed strings.

\begin{defi}cf Zwiebach 95
A quantum open closed string algebra is a pair $(V,K=\hbar\Delta+\{S,\})$, and here $S$ is a solution to the Maurer Cartan equation.  We also need a technical condition that I will explain very roughly.
\end{defi}
I want to give an interpretation of $S$.  This is the sum of the polynomials and then we have $\psi$s.  \wh.

So $S$ is given like this, and we have $\hbar$ here, and we have it here, and then each term corresponds to some Riemann surface

[I am too fatigued to continue.]


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