blob: comparison text/tqftreview.tex

equal deleted inserted replaced

-:98b8559b0b7a
+:87b1507ebc56
 of points in the interior of $S$);
 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)
 by an object (0-morphism) of $C$;
 \item a transverse orientation of each 0-cell, thought of as a choice of
 ``domain" and ``range" for the two adjacent 1-cells; and
-\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
+\item a labeling of each 0-cell by a 1-morphism of $C$, with
 domain and range determined by the transverse orientation and the labelings of the 1-cells.
 \end{itemize}
-If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
+We want fields on 1-manifolds to be enriched over Vect, so we also allow formal linear combinations
+of the above fields on a 1-manifold $X$ so long as these fields restrict to the same field on $\bd X$.
+In addition, we mod out by the relation which replaces
+a 1-morphism label $a$ of a 0-cell $p$ with $a^*$ and reverse the transverse orientation of $p$.
+If $C$ is a *-algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
 interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
 of the algebra.
 \medskip
 \item a labeling of each 1-cell by a 1-morphism of $C$, with
 domain and range determined by the transverse orientation of the 1-cell
 and the labelings of the 2-cells;
 \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
 of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped
-to $\pm 1 \in S^1$; and
+to $\pm 1 \in S^1$
+(this amounts to splitting of the link of the 0-cell into domain and range); and
 \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
 determined by the labelings of the 1-cells and the parameterizations of the previous
 bullet.
 \end{itemize}
-\nn{need to say this better; don't try to fit everything into the bulleted list}
+As in the $n=1$ case, we allow formal linear combinations of fields on 2-manifolds,
+so long as their restrictions to the boundary coincide.
+In addition, we regard the labelings as being equivariant with respect to the * structure
+on 1-morphisms and pivotal structure on 2-morphisms.
+That is, we mod out my the relation which flips the transverse orientation of a 1-cell
+and replaces its label $a$ by $a^*$, as well as the relation which changes the parameterization of the link
+of a 0-cell and replaces its label by the appropriate pivotal conjugate.
+\medskip
 For general $n$, a field on a $k$-manifold $X^k$ consists of
 \begin{itemize}
 \item A cell decomposition of $X$;
 \item an explicit general position homeomorphism from the link of each $j$-cell
 to the boundary of the standard $(k-j)$-dimensional bihedron; and
 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
 domain and range determined by the labelings of the link of $j$-cell.
 \end{itemize}
-%\nn{next definition might need some work; I think linearity relations should
-%be treated differently (segregated) from other local relations, but I'm not sure
-%the next definition is the best way to do it}
-\medskip
 \subsection{Local relations}
 \label{sec:local-relations}
-Local relations are certain subspaces of the fields on balls, which form an ideal under gluing.
+For convenience we assume that fields are enriched over Vect.
+Local relations are subspaces $U(B; c)\sub \cC(B; c)$ of the fields on balls which form an ideal under gluing.
 Again, we give the examples first.
 \addtocounter{prop}{-2}
 \begin{example}[contd.]
 For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$,
 \subsection{Constructing a TQFT}
 \label{sec:constructing-a-tqft}
 In this subsection we briefly review the construction of a TQFT from a system of fields and local relations.
-(For more details, see \cite{kw:tqft}.)
+As usual, see \cite{kw:tqft} for more details.
 Let $W$ be an $n{+}1$-manifold.
 We can think of the path integral $Z(W)$ as assigning to each
 boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$.
 In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear
 maps $\lf(\bd W)\to \c$.
+(We haven't defined a path integral in this context; this is just for motivation.)
 The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace
 $Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections.
 The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$,
 can be thought of as finite linear combinations of fields modulo local relations.
 \label{defn:TQFT-invariant}
 The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is
 	$$A(X) \deq \lf(X) / U(X),$$
 where $\cU(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$;
 $\cU(X)$ is generated by things of the form $u\bullet r$, where
-$u\in \cU(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
+$u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$.
 \end{defn}
 (The blob complex, defined in the next section,
 is in some sense the derived version of $A(X)$.)
 If $X$ has boundary we can similarly define $A(X; c)$ for each
 boundary condition $c\in\cC(\bd X)$.

changeset 412	87b1507ebc56
parent 411	98b8559b0b7a
child 426	8aca80203f9d