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160 |
161 \dropcap{T}he aim of this paper is to describe a derived category version of TQFTs. |
161 \dropcap{T}he aim of this paper is to describe a derived category analogue of topological quantum field theories. |
162 |
162 |
163 For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of |
163 For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of |
164 invariants of manifolds of dimensions 0 through $n+1$. |
164 invariants of manifolds of dimensions 0 through $n+1$. In particular, |
165 The TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category. |
165 the TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category. |
166 If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford |
166 If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford |
167 a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$. |
167 a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$. |
168 (See \cite{1009.5025} and \cite{kw:tqft}; |
168 (See \cite{1009.5025} and \cite{kw:tqft}; |
169 for a more homotopy-theoretic point of view see \cite{0905.0465}.) |
169 for a more homotopy-theoretic point of view see \cite{0905.0465}.) |
170 |
170 |
171 We now comment on some particular values of $k$ above. |
171 We now comment on some particular values of $k$ above. |
172 By convention, a linear 0-category is a vector space, and a representation |
172 A linear 0-category is a vector space, and a representation |
173 of a vector space is an element of the dual space. |
173 of a vector space is an element of the dual space. |
174 So a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, |
174 Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, |
175 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
175 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
176 In fact we will be mainly be interested in so-called $(n{+}\epsilon)$-dimensional |
176 For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional |
177 TQFTs which have nothing to say about $(n{+}1)$-manifolds. |
177 TQFTs, which are slightly weaker structures and do not assign anything to general $(n{+}1)$-manifolds, but only to mapping cylinders. |
178 For the remainder of this paper we assume this case. |
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179 |
178 |
180 When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
179 When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
181 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
180 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
182 The gluing rule for the TQFT in dimension $n$ states that |
181 The TQFT gluing rule in dimension $n$ states that |
183 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
182 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
184 where $Y_1$ and $Y_2$ and $n$-manifolds with common boundary $S$. |
183 where $Y_1$ and $Y_2$ and $n$-manifolds with common boundary $S$. |
185 |
184 |
186 When $k=0$ we have an $n$-category $A(pt)$. |
185 When $k=0$ we have an $n$-category $A(pt)$. |
187 This can be thought of as the local part of the TQFT, and the full TQFT can be constructed of $A(pt)$ |
186 This can be thought of as the local part of the TQFT, and the full TQFT can be reconstructed from $A(pt)$ |
188 via colimits (see below). |
187 via colimits (see below). |
189 |
188 |
190 We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$ |
189 We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$ |
191 and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. |
190 and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. |
192 Examples of semisimple TQFTS include Witten-Reshetikhin-Turaev theories, |
191 Examples of semisimple TQFTs include Witten-Reshetikhin-Turaev theories, |
193 Turaev-Viro theories, and Dijkgraaf-Witten theories. |
192 Turaev-Viro theories, and Dijkgraaf-Witten theories. |
194 These can all be given satisfactory accounts in the framework outlined above. |
193 These can all be given satisfactory accounts in the framework outlined above. |
195 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories in order to be |
194 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak dependence on interiors in order to be |
196 extended all the way down to 0 dimensions.) |
195 extended all the way down to dimension 0.) |
197 |
196 |
198 For other TQFT-like invariants, however, the above framework seems to be inadequate. |
197 For other TQFT-like invariants, however, the above framework seems to be inadequate. |
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198 |
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199 \nn{kevin's rewrite stops here} |
199 |
200 |
200 However new invariants on manifolds, particularly those coming from |
201 However new invariants on manifolds, particularly those coming from |
201 Seiberg-Witten theory and Ozsv\'{a}th-Szab\'{o} theory, do not fit the framework well. |
202 Seiberg-Witten theory and Ozsv\'{a}th-Szab\'{o} theory, do not fit the framework well. |
202 In particular, they have more complicated gluing formulas, involving derived or |
203 In particular, they have more complicated gluing formulas, involving derived or |
203 $A_\infty$ tensor products \cite{1003.0598,1005.1248}. |
204 $A_\infty$ tensor products \cite{1003.0598,1005.1248}. |