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\begin{document}

\title{Lecture 5: Generalizations of Operads}
\author{Alexander A. Voronov}
%\address{Department of Mathematics\\ M.I.T.,
%2-246\\ 77 Massachusetts Ave.\\ Cambridge, MA 02139-4307}
%\curraddr{}
%\email{voronov@math.mit.edu}
%\urladdr{http://www-math.mit.edu/~voronov/}
%\thanks{Research supported in part by an AMS Centennial Fellowship.}

%\subjclass{Primary 14H10; Secondary 32G15, 55P62}
\date{September 30, 1997}

\maketitle

\section{Generalizations of operads}

\subsection{PROP's and operads}
We have approached operads from the point of view of PROP's, tensor
categories whose set of objects is the set of nonnegative integers
with the sum as the tensor product, see Lecture 1. If we take the sets
$\Mor(n,1)$ of morphism for $n \ge 0$ in a PROP, we will obtain an
operad.

There is a converse construction of a minimal PROP, freely generated
by an operad. Suppose we have an operad $\OO$ of sets. Define a PROP
by defining the set of morphisms as
\[
\Mor (m,n) = \bigcup_{m_1 + \dots + m_n = m} 
\OO(m_1) \otimes \dots \otimes \OO(m_n) \times_G S_m,
\]
where the summation runs over all sequences $m_1, \dots, m_n \ge 0$
summing up to $m$, $G = S_{m_1} \times \dots \times S_{m_n}$,
naturally embedded in $S_m$. The product with $S_m$ over $G$ is meant
to allow all possible ways of labeling the inputs and provide an
action of $S_m$ on the labels.

Notice that starting from an operad, going to the corresponding PROP
and then getting back to an operad, returns the original operad. On
the other hand, if you start from a PROP, construct the corresponding
operad and then the corresponding PROP, then the new PROP will be in
general different from the old one. Therefore, the theory of PROP's is
richer than that of operads, but one can think of an operad as a more
basic object.

\subsection{Modular operads}
\begin{df}
A \emph{modular operad} is a collection of spaces $\OO(n)$, $n \ge 0$,
along with an $S_n$-action on each $\OO(n)$ and two types of
compositions:
\begin{align*}
\circ_{ij}: \OO(m) \otimes \OO(n) & \to \OO(m+n-2),   & 1 \le i \le m,  
1 \le j \le n,\\
\circ_{ij}: \OO(n) & \to \OO(n-2),   & 1 \le i < j \le n,
\end{align*}
satisfying natural associativity and equivariance properties.
\end{df}

Modular operads were introduced by Getzler and Kapranov in \cite{gk}
as more symmetric objects than PROP's. The main example they had in
mind was the moduli spaces (whence the term ``modular'') $\M(n) =
\bigcup_{g \ge 0} \Mc_{g,n}$ of stable complex compact algebraic
curves of an arbitrary genus $g$ with $n$ punctures. The $\circ_{ij}$
operations are the operations of joining two punctures on two different
curves or on a single curve to form a double point. The more familiar
to us space $\PP(n)$ of smooth complex compact algebraic curves (or
Riemann surfaces) of an arbitrary genus with $n$ holomorphic holes is
another example of a modular operad.

Getzler and Kapranov used a different definition of a modular operad,
which was particularly suitable to handle the moduli spaces of stable
curves. The problem with the moduli space of stable curves with
punctures is essentially that not all combinations of genus $g$ and
the number of punctures $n$ are allowed: the Euler characteristic
$2-2g_c - n_c$ of each irreducible component $c$ of a stable curve
must be negative. Just for completeness, we would like to define
modular operads in their sense, as well.

\begin{df}
A \emph{stable $($i.e., Getzler-Kapranov$)$ modular operad} is a
collection of spaces $\OO(g,n)$, $g, n \ge 0$, $2-2g-n < 0$, along
with an $S_n$-action on each $\OO(g,n)$ and two types of compositions:
\begin{align*}
\circ_{ij}: \OO(g_1,m) \otimes \OO(g_2,n) & \to \OO(g_1+g_2,m+n-2),
& 1 \le i \le m,  
1 \le j \le n,\\
\circ_{ij}: \OO(g,n) & \to \OO(g+1,n-2),   & 1 \le i < j \le n,
\end{align*}
satisfying natural associativity and equivariance properties.
\end{df}

\begin{xca}
Construct functors between operads, modular operads, and PROP's and
study their relationship.
\end{xca}

\section{Operads generalizing those of Riemann surfaces}

The previous section dealt with generalizations of operads, whereas
here we would like to consider (modular) operads generalizing those of
Riemann surfaces and yet relevant to 2d QFT's. First of all, those
relevant to super CFT's.

\subsection{The operad of super Riemann surfaces}

A \emph{super Riemann surface or a SUSY curve} is a complex
supermanifold of dimension $1|1$ with a subbundle $S \subset \T$
of dimension $0|1$ in the holomorphic tangent bundle $\T$,
satisfying the following nonintegrability condition: the morphism
\begin{align*}
S \otimes S & \to \T/S,\\
X_1 \otimes X_2 & \mapsto [X_1,X_2] \mod S,
\end{align*}
$[X_1,X_2]$ being the super commutator of vector fields, is an
isomorphism. Usually, it makes more sense to consider families $X \to
B$ of super Riemann surfaces --- then one just replaces the
holomorphic tangent bundle with the relative one.

A trivial example of a super Riemann surface is the standard unit disk
$D^{1|1} = \{ (z,\zeta) \in \nc^{1|1} \; | \; \abs{z} < 1\}$ with the
subbundle $S$ spanned by the odd vector field $X = \partial/\partial
\zeta + \zeta \partial/\partial z$ (Note that $[X,X] = 2 \partial/\partial
z$). Thus, it is clear what a super Riemann surface with a holomorphic
hole would be. The moduli spaces of such form a PROP (as well as an
operad and a modular operad), and an algebra over such PROP (operad)
would be an $N=1$ Super CFT (SCFT) of central charge 0 (at the tree
level, respectively).

Moving on to $N=2$ SCFT, it is believed that one has to consider
so-called semirigid Riemann surfaces of Distler and Nelson
\cite{dn}. Not much of operadic properties of such objects is studied.

\begin{prob}
Study the notion of $N=2$ SCFT in physics literature, define the
operad of semirigid super Riemann surfaces, and prove that an algebra
over this operad is the same as an $N=2$ SCFT. This will generalize
Huang's theorem \cite{huang} which deals with the usual CFT.
\end{prob}

\subsection{Universal Grassmannian}

This example belongs to A.~S. Schwarz \cite{as}, who suggested it as
an operad governing a universal CFT in a certain sense.

Let $H = H_+ \oplus H_-$ be a separable Hilbert space split into the
direct sum of two subspaces, along with a unitary involution $K$ on
$H$ interchanging $H_+$ with $H_-$ isomorphically. An example is the
space $H = L^2 (S^1)$ of square-integrable functions on the unit
circle $S^1$. $H_+$ is the closure of the space of functions which may
be extended holomorphically into the unit disk, and $H_-$ is the
closure of those functions which may be extended meromorphically
inside the unit disk with a single pole at the origin. The involution
$K$ takes a function $f(z)$ to $f(1/z)/z$. The \emph{universal
Grassmannian} (as a set, although it is in fact an infinite
dimensional complex manifold) is $\Gr (H) = \{ V \subset H \; | \;
\pi_+: V \to H_+ \text{ is Fredholm and } \pi_-: V \to H_- \text{ is
compact }\}$, where $\pi_{\pm}$ are the natural projections onto
$H_{\pm}$. Recall that an operator is Fredholm, if it is bounded and
has finite-dimensional kernel and cokernel, and compact, if the
closure of the image of the unit ball is compact. The idea of the
Grassmannian is to select a graspable set of subspaces $V$ in $H$
which differ not too much from $H_+$, but a lot from $H_-$.

For $n\ge 0$ there is a similar structure $H^n_+$, $H^n_-$, $K^{\oplus
n}$ on the space $H^n$, and therefore, one can define the universal
Grassmannian $\Gr(H^n)$ for each $n \ge 0$. For $n=0$, it is just a
point.

The observation of Schwarz is that the collection $\Gr(H^n)$, $n
\ge 0$, is a modular operad. Indeed, for $1 \le i < j \le n$, we can
define a mapping
\begin{align*}
\circ_{ij}: \Gr(H^n) & \to \Gr(H^{n-2}),\\
V & \mapsto p(f^{-1}(V)),
\end{align*}
where
\begin{align*}
f: H^{n-1} &\to H^n,\\
(x_1, \dots, x_i, \dots, \widehat x_j, \dots, x_n) & \mapsto (x_1, \dots, x_i, \dots, Kx_i, \dots, x_n),
\end{align*}
and
\begin{align*}
p: H^{n-1} &\to H^{n-2},\\ (x_1, \dots, x_i, \dots, \widehat x_j,
\dots, x_n) & \mapsto (x_1, \dots, \widehat x_i, \dots, \widehat x_i,
\dots, x_n).
\end{align*}

For $1 \le i \le m$ and $1 \le j \le n$, we can define a mapping
\[
\circ_{ij}: \Gr(H^m) \otimes \Gr(H^n) \to \Gr(H^{m+n-2})
\]
by composing the above $\circ_{ij}$ with the direct sum mapping
\begin{align*}
\Gr(H^m) \otimes \Gr(H^n) & \to \Gr(H^{m+n}),\\
(V_1 ,V_2) & \mapsto V_1 \oplus V_2.
\end{align*}

\begin{prop}
The above mappings $\circ_{ij}$ provide the collection $\Gr(H^n)$, $n
\ge 0 $, with the structure of a modular operad.
\end{prop}

Another remarkable fact, which makes this construction relevant to 2d
quantum field theory, is that this operad structure is compatible with
the Krichever map, i.e., there is a natural morphism from the modular
operad of Riemann surfaces with holomorphic holes to the universal
Grassmannian operad, see \cite{as} for more detail.

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