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$\newcommand\Cob{\mathrm{Cob}}\newcommand\Vect{\mathrm{Vect}}\DeclareMathOperator\Rep{Rep}$The ordinary definition of a TQFT is:

Defnition: A $d$-dimensional TQFT is a symmetric monoidal functor $\Cob^d\to \Vect$.

This means: you have a vector space for every $d-1$ dimensional topological manifold, and every cobordism gives you a map. The empty manifold is sent to $\mathbf{C}$; more generally disjoint unions are sent to tensor products. In particular we get a number $$Z(M^d)\ \in\ \mathbf{C}$$ for every $d$-manifold without boundary $M^d$.

There is also an extended version, where we use $n$-categories instead of ordinary $1$-categories:

Defnition: An $n$-extended $d$-dimensional TQFT is a symmetric monoidal functor $Z:\Cob^d_n\to \Vect_n$ (the $n$-category versions of these functors).

Thus we supply data for $d-2,d-3,\dots,d-n$ categories too.

This extension is not what my question is about. In physics, e.g. according to Tachikawa's "On 'Categories' of Quantum Field Theories", for each QFT we should also get

  1. a vector space $V_{\{x_1,\dotsc,x_n\}}$ for every collection of points $x_1,\dotsc,x_n\subseteq M^d$, and for every $\phi(x_1,\dotsc,x_n)\in V_{\{x_1,\dotsc,x_n\}}$ a number $Z(M^d; \phi(x_1,\dotsc,x_n))\in\mathbf{C}$.
  2. a tensor category $V_\ell$ for every one dimensional submanifold $\ell\subseteq M^d$, and for every object $\phi(\ell)\in V_\ell$ a number $Z(M^d;\phi(\ell))$,
  3. for every $r<d$ dimensional submanifold, some sort of "$r$ category with algebra structure".

Presumably there is also structure capturing firstly how this interacts with the functor $Z:\Cob^d\to \Vect$ (e.g. what do point/line/… operators have to do with $d-3$ dimensional manifolds $M^{d-3}$ and $Z(M^{d-3})$?), and secondly how they interact with each other (e.g. if $\partial \ell =\{x,y\}$ then is $V_{\{x,y\}}$ something like endomorphisms of the identity inside $V_\ell$?), and thirdly what sort of algebra structure the point/line/… operators are meant to have in the TQFT case (in general this third question is too complicated, e.g. for $2$d CFTs the structure is at least as complicated as vertex algebras).

My question is:

  • is there an axiomatisation of "TQFT with point/line/… operators"?
  • (sanity check) is Chern–Simons obviously a dimension $3$ example of this (where I'm told there are no point operators $V_x=0$ and line operators $V_\ell\approx\Rep U_q(\mathfrak{g})$ have something to do with quantum groups)?
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  • $\begingroup$ Is the $(\ell)$ in $\phi(\ell)$ just punctuation? That is, is there a function $\phi$ that we are evaluating at $\ell$, or is $\phi$ just a generic notation for an element of a vector space, and we are using functional notation $\phi(\ell)$ instead of subscript notation $\phi_\ell$ to specify where $\phi$ lives? $\endgroup$ Commented May 1, 2023 at 22:55
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    $\begingroup$ The latter, though it should presumably also be possible to define "family of line operators", in which case $\phi(\ell)$ will be an element of $V_\ell$ for each line $\ell$. $\endgroup$ Commented May 2, 2023 at 8:51

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Yes there's a very natural way to incorporate defects of arbitrary dimension into the formalism of extended topological field theory, and this is a vital part of the structure of TQFT. For example in the TQFT point of view on the Langlands program much of the structure of interest is captured by defects (this is explained e.g. in Ben-Zvi and Nadler - Betti Geometric Langlands; a discussion somewhere between math and physics is in Beem, Ben-Zvi, Bullimore, Dimofte, and Neitzke - Secondary products in supersymmetric field theory and an excellent introduction to some of these ideas is in Kapustin's ICM address Topological Field Theory, Higher Categories, and Their Applications).

You can define a (fully local) codimension $k$ defect in a TFT $Z$ as (roughly speaking) the extra data needed to define the functor $Z$ on manifolds equipped with an embedded codimension $k$ submanifold. Lurie's Cobordism Hypothesis with Singularities (Section 4.3 of On the Classification of Topological Field Theories) then identifies these data with states of the theory on the linking of the defect — i.e., defect theories are identified with suitable (dualizable) objects of the value $Z(S^{n-k-1})$ of the theory.

The easiest example of this is local operators: in an $n$-dimensional TQFT, the value $Z(S^{n-1})$ gives a vector space; given any closed $n$-manifold $M$ with distinct points $\{x_i\}$ we get a bordism from the disjoint union of linking spheres to the empty manifold, which gives a map from the tensor product of spaces of local operators to $\mathbb C$. These are the correlation functions in your 1. Moreover for a closed $n-1$ manifold $N$ and a point $x\in N$ you get an action of $Z(S^{n-1})$ on $Z(N)$ depending locally constantly on $x$, whence "local operator". Moreover the TFT structure makes $Z(S^{n-1})$ into an $E_n$ algebra and this makes $Z(N)$ a left module (in fact a module over factorization homology of the algebra of local operators; correlation functions also descend to factorization homology, a version of the Ward identities in QFT).

Similarly line operators form an $E_{n-1}$-category $Z(S^{n-2})$ — in Chern–Simons theory this is the modular tensor category obtained from quantum group representations at a root of unity. A dualizable object in this tensor category defines e.g. a state on $Z(S^{n-2}\times S^1)$, the boundary of the tubular neighborhood of an embedded loop, and these give the Wilson line observables of knots in the Chern–Simons case. Surface defects $Z(S^{n-3})$ form an $E_{n-2}$-2-category and so on.

There are lots more defects than these though — for example (as hinted in your questions) given two line operators you can talk about an interface between them, i.e. a codimension one defect on an embedded line separating two different line operators. This captures the category structure of line operators. You can use this reasoning to give one (to me the most convincing) picture of why you might think of TQFT evaluated on a point as defining something like a higher category — $Z(\mathrm{pt})$ gives interfaces between $Z$ and the trivial theory, AKA boundary conditions in $Z$. But there are interfaces between boundary conditions (codim 2), interfaces between those (codim 3), all the way up, whence an ($n-1$)-category — a physics version of this picture is in Kapustin's ICM address (cf. above).

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  • $\begingroup$ Markdown can be a bit unpredictable; a bare link like https://arxiv.org/abs/1606.08523; sometimes (e.g., in your original post, but not in this comment—I don't know why, and don't have the patience to dig into regexes to find out) gobbles up the following ;. To avoid this, you can use <https://arxiv.org/abs/1606.08523>; to force the link parsing to stop. (I edited in the names, which has the side effect of explicitly indicating the end of the link in another way.) $\endgroup$ Commented May 2, 2023 at 3:06

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