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I remember that in the beginning, there was an axiom for $(n+1)$-dimensional TQFT that said that the state space $V(\Sigma)$ assigned to an $n$-dimensional oriented manifold is spanned by the invariants of all $n+1$-dimensional oriented manifolds $M$ with $\partial M=\Sigma$. If we call the invariant of $M$, $Z(M)\in V(\Sigma)$, this just says that $V(\Sigma)$ is spanned by all $Z(M)$. Maybe it was in Segal's Swansea notes, maybe it was in an early version of Atiyah's axioms. It doesn't seem to have made it into "The Geometry and Physics of Knots"

For instance if you read "Topological Quantum Field Theories Derived from the Kauffman Bracket" by Blanchet, Habegger, Masbaum and Vogel, the vacuum hypothesis is implicit in their constructions.

There is a theorem that the category of Frobenius algebras is equivalent to the category of $1+1$-dimensional TQFT's. For instance, let $A=\mathbb{C}[x]/(x^3)$, with Frobenius map $\epsilon(1)=\epsilon(x)=0$ and $\epsilon(x^2)=-1$. This is the choice that Khovanov made to construct his $sl_3$-invariant of links. At this point, no one would deny that this gives rise to a TQFT.

The state space associated to a circle is just $A$. The $2$-manifolds with boundary the circle are classified by genus. Using TQFT to compute them, I get that a disk has invariant $1$, a surface of genus one with one boundary component has invariant $3x^2$, and any other surface has invariant $0$. The invariants don't span $A$.

The problem as I learned from Chris French is that $A$ is not semisimple. In fact, $A$ being spanned by the invariants of surfaces with one boundary component is equivalent to the semisimplicity of $A$.

Here is my question. At what point, and why was the vacuum hypothesis abandoned?

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    $\begingroup$ I don't know. Was it replaced by a "no invisible morphism" hypothesis? A morphism $f$ is invisible if Tr(fg)=0 for all g. $\endgroup$
    – Bruce Westbury
    Mar 7, 2010 at 16:59
  • $\begingroup$ @Bruce, these are called negligible morphisms -- try googling the two phrases. $\endgroup$
    – Scott Morrison
    Mar 7, 2010 at 17:34
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    $\begingroup$ @ScottMorrison, Bruce and John Barrett basically invented invisible/negligible morphisms, or more precisely the quotient by these morphisms. $\endgroup$
    – Manuel Bärenz
    Feb 4, 2017 at 12:07

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At least in your example about Khovanov's TQFT for $\mathfrak{sl}_3$ link homology, part of the answer is that that theory also allows 'singular surfaces', locally modelled on a book with 3 pages. Now you can take a "punctured torus with a disk stuck in its throat" (see my paper, for example), as a singular 2-manifold with boundary that circle, and you'll see that this gives the missing element of $A$.

I'm not sure, though, what the general story is here.

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    $\begingroup$ Good point. Of course network TQFT is even more uncharted than TQFT. Here is a motivating question. Suppose you had an invariant of closed oriented surfaces that assigned to every closed surface except the torus, the number $0$ and assigned to the torus $3$. How do you construct the TQFT? $\endgroup$
    – Charlie Frohman
    Mar 7, 2010 at 22:13

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