160 |
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 version of TQFTs. |
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 $n+1$. |
165 |
165 The TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category. |
166 |
166 If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford |
167 \dropcap{T}opological quantum field theories (TQFTs) provide local invariants of manifolds, which are determined by the algebraic data of a higher category. |
167 a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$. |
168 |
168 (See \cite{1009.5025} and \cite{kw:tqft}; |
169 An $n+1$-dimensional TQFT $\cA$ associates a vector space $\cA(M)$ |
169 for a more homotopy-theoretic point of view see \cite{0905.0465}.) |
170 (or more generally, some object in a specified symmetric monoidal category) |
170 |
171 to each $n$-dimensional manifold $M$, and a linear map |
171 We now comment on some particular values of $k$ above. |
172 $\cA(W): \cA(M_0) \to \cA(M_1)$ to each $n+1$-dimensional manifold $W$ |
172 By convention, a linear 0-category is a vector space, and a representation |
173 with incoming boundary $M_0$ and outgoing boundary $M_1$. |
173 of a vector space is an element of the dual space. |
174 An $n+\epsilon$-dimensional TQFT provides slightly less; |
174 So a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, |
175 it only assigns linear maps to mapping cylinders. |
175 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
176 |
176 In fact we will be mainly be interested in so-called $(n{+}\epsilon)$-dimensional |
177 There is a standard formalism for constructing an $n+\epsilon$-dimensional |
177 TQFTs which have nothing to say about $(n{+}1)$-manifolds. |
178 TQFT from any $n$-category with sufficiently strong duality, |
178 For the remainder of this paper we assume this case. |
179 and with a further finiteness condition this TQFT is in fact $n+1$-dimensional. |
179 |
180 \nn{not so standard, err} |
180 When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
181 |
181 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
182 These invariants are local in the following sense. |
182 The gluing rule for the TQFT in dimension $n$ states that |
183 The vector space $\cA(Y \times I)$, for $Y$ an $n-1$-manifold, |
183 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
184 naturally has the structure of a category, with composition given by the gluing map |
184 where $Y_1$ and $Y_2$ and $n$-manifolds with common boundary $S$. |
185 $I \sqcup I \to I$. Moreover, the vector space $\cA(Y \times I^k)$, |
185 |
186 for $Y$ and $n-k$-manifold, has the structure of a $k$-category. |
186 When $k=0$ we have an $n$-category $A(pt)$. |
187 The original $n$-category can be recovered as $\cA(I^n)$. |
187 This can be thought of as the local part of the TQFT, and the full TQFT can be constructed of $A(pt)$ |
188 For the rest of the paragraph, we implicitly drop the factors of $I$. |
188 via colimits (see below). |
189 (So for example the original $n$-category is associated to the point.) |
189 |
190 If $Y$ contains $Z$ as a codimension $0$ submanifold of its boundary, |
190 We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$ |
191 then $\cA(Y)$ is natually a module over $\cA(Z)$. For any $k$-manifold |
191 and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. |
192 $Y = Y_1 \cup_Z Y_2$, where $Z$ is a $k-1$-manifold, the category |
192 Examples of semisimple TQFTS include Witten-Reshetikhin-Turaev theories, |
193 $\cA(Y)$ can be calculated via a gluing formula, |
193 Turaev-Viro theories, and Dijkgraaf-Witten theories. |
194 $$\cA(Y) = \cA(Y_1) \Tensor_{\cA(Z)} \cA(Y_2).$$ |
194 These can all be given satisfactory accounts in the framework outlined above. |
195 |
195 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories in order to be |
196 In fact, recent work of Lurie on the `cobordism hypothesis' \cite{0905.0465} |
196 extended all the way down to 0 dimensions.) |
197 shows that all invariants of $n$-manifolds satisfying a certain related locality property |
197 |
198 are in a sense TQFT invariants, and in particular determined by |
198 For other TQFT-like invariants, however, the above framework seems to be inadequate. |
199 a `fully dualizable object' in some $n+1$-category. |
199 |
200 (The discussion above begins with an object in the $n+1$-category of $n$-categories. |
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201 The `sufficiently strong duality' mentioned above corresponds roughly to `fully dualizable'.) |
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202 |
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203 This formalism successfully captures Turaev-Viro and Reshetikhin-Turaev invariants |
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204 (and indeed invariants based on semisimple categories). |
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205 However new invariants on manifolds, particularly those coming from |
200 However new invariants on manifolds, particularly those coming from |
206 Seiberg-Witten theory and Ozsv\'{a}th-Szab\'{o} theory, do not fit the framework well. |
201 Seiberg-Witten theory and Ozsv\'{a}th-Szab\'{o} theory, do not fit the framework well. |
207 In particular, they have more complicated gluing formulas, involving derived or |
202 In particular, they have more complicated gluing formulas, involving derived or |
208 $A_\infty$ tensor products \cite{1003.0598,1005.1248}. |
203 $A_\infty$ tensor products \cite{1003.0598,1005.1248}. |
209 It seems worthwhile to find a more general notion of TQFT that explain these. |
204 It seems worthwhile to find a more general notion of TQFT that explain these. |