# HG changeset patch # User Kevin Walker # Date 1290014227 28800 # Node ID 77154439205822e5f0efd05b5072a709d9ce6ef8 # Parent da5077cae33ce2f701a5c3cac1e11146fa235299 more intro diff -r da5077cae33c -r 771544392058 pnas/pnas.tex --- a/pnas/pnas.tex Tue Nov 16 16:55:55 2010 -0800 +++ b/pnas/pnas.tex Wed Nov 17 09:17:07 2010 -0800 @@ -162,46 +162,41 @@ For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of invariants of manifolds of dimensions 0 through $n+1$. - - -\dropcap{T}opological quantum field theories (TQFTs) provide local invariants of manifolds, which are determined by the algebraic data of a higher category. +The TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category. +If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford +a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$. +(See \cite{1009.5025} and \cite{kw:tqft}; +for a more homotopy-theoretic point of view see \cite{0905.0465}.) -An $n+1$-dimensional TQFT $\cA$ associates a vector space $\cA(M)$ -(or more generally, some object in a specified symmetric monoidal category) -to each $n$-dimensional manifold $M$, and a linear map -$\cA(W): \cA(M_0) \to \cA(M_1)$ to each $n+1$-dimensional manifold $W$ -with incoming boundary $M_0$ and outgoing boundary $M_1$. -An $n+\epsilon$-dimensional TQFT provides slightly less; -it only assigns linear maps to mapping cylinders. - -There is a standard formalism for constructing an $n+\epsilon$-dimensional -TQFT from any $n$-category with sufficiently strong duality, -and with a further finiteness condition this TQFT is in fact $n+1$-dimensional. -\nn{not so standard, err} +We now comment on some particular values of $k$ above. +By convention, a linear 0-category is a vector space, and a representation +of a vector space is an element of the dual space. +So a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, +and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. +In fact we will be mainly be interested in so-called $(n{+}\epsilon)$-dimensional +TQFTs which have nothing to say about $(n{+}1)$-manifolds. +For the remainder of this paper we assume this case. -These invariants are local in the following sense. -The vector space $\cA(Y \times I)$, for $Y$ an $n-1$-manifold, -naturally has the structure of a category, with composition given by the gluing map -$I \sqcup I \to I$. Moreover, the vector space $\cA(Y \times I^k)$, -for $Y$ and $n-k$-manifold, has the structure of a $k$-category. -The original $n$-category can be recovered as $\cA(I^n)$. -For the rest of the paragraph, we implicitly drop the factors of $I$. -(So for example the original $n$-category is associated to the point.) -If $Y$ contains $Z$ as a codimension $0$ submanifold of its boundary, -then $\cA(Y)$ is natually a module over $\cA(Z)$. For any $k$-manifold -$Y = Y_1 \cup_Z Y_2$, where $Z$ is a $k-1$-manifold, the category -$\cA(Y)$ can be calculated via a gluing formula, -$$\cA(Y) = \cA(Y_1) \Tensor_{\cA(Z)} \cA(Y_2).$$ +When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, +and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. +The gluing rule for the TQFT in dimension $n$ states that +$A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, +where $Y_1$ and $Y_2$ and $n$-manifolds with common boundary $S$. + +When $k=0$ we have an $n$-category $A(pt)$. +This can be thought of as the local part of the TQFT, and the full TQFT can be constructed of $A(pt)$ +via colimits (see below). -In fact, recent work of Lurie on the `cobordism hypothesis' \cite{0905.0465} -shows that all invariants of $n$-manifolds satisfying a certain related locality property -are in a sense TQFT invariants, and in particular determined by -a `fully dualizable object' in some $n+1$-category. -(The discussion above begins with an object in the $n+1$-category of $n$-categories. -The `sufficiently strong duality' mentioned above corresponds roughly to `fully dualizable'.) +We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$ +and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. +Examples of semisimple TQFTS include Witten-Reshetikhin-Turaev theories, +Turaev-Viro theories, and Dijkgraaf-Witten theories. +These can all be given satisfactory accounts in the framework outlined above. +(The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories in order to be +extended all the way down to 0 dimensions.) -This formalism successfully captures Turaev-Viro and Reshetikhin-Turaev invariants -(and indeed invariants based on semisimple categories). +For other TQFT-like invariants, however, the above framework seems to be inadequate. + However new invariants on manifolds, particularly those coming from Seiberg-Witten theory and Ozsv\'{a}th-Szab\'{o} theory, do not fit the framework well. In particular, they have more complicated gluing formulas, involving derived or @@ -228,6 +223,8 @@ \nn{perhaps say something explicit about the relationship of this paper to big blob paper. like: in this paper we try to give a clear view of the big picture without getting bogged down in details} +\nn{diff w/ lurie} + \section{Definitions} \subsection{$n$-categories} \mbox{} @@ -532,7 +529,7 @@ that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. -\subsubsection{Homotopy colimits} +\subsubsection{Colimits} \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} \nn{Explain codimension colimits here too}