# Fully extended TQFT and lattice models

I often read that fully extended TQFTs are supposed to classify topological phases of matter. So I would like to understand the formal nature of fully extended TQFTs on a more direct physical level (without having to read up on a huge amount of category theory language):

1) Many algebraic structures are given by a finite set of tensors (linear maps if you wish) which fulfil a finite set of tensor-network equations (equations between different compositions of the linear maps if you wish). For example, fusion categories are given by a 10-index $$F$$-tensor satisfying the pentagon equation, or (ordinary axiomatic) 2-dimensional TQFTs are given by a bunch of tensors associated to the pair of pants and a few other cobordisms, satisfying the axioms of (something like) a Frobenius algebra.

Can $$n$$-dimensional fully extended TQFTs be formulated as a finite set of tensors obeying a finite set of tensor-network axioms? Is it known what these tensors and axioms are?

2) Is there any idea for a construction of a local partition function (similar to the Turaev-Viro construction) that takes data related to a fully extended TQFT as input? I guess for Turaev-Viro models (and any other topological state-sum construction) it's possible to find a corresponding extended TQFT. Are there any examples of extended TQFTs that are conjectured to correspond to phases without known state-sum constructions (such as chiral topological phases in 2+1D)?

It may take a bit of extraction, but positive answers to both of your questions follow from my results joint with Gaiotto in Condensations in higher categories (arXiv:1905.09566). In that paper we build a $$\mathbb{C}$$-linear $$(d+1)$$-category that we call $$\Sigma^d\mathrm{Vect}_{\mathbb{C}}$$. ($$d$$ is arbitrary, and so is the ground field, but $$\mathbb{C}$$ is natural for bosonic physics.) One of our main theorems is that there is a natural equivalence of $$(d+1)$$-categories between $$\Sigma^n\mathrm{Vect}_{\mathbb{C}}$$ and the fully dualizable subcategory of the $$(d+1)$$-category $$n\mathrm{Cat}_{\mathbb C}$$ of all $$\mathbb{C}$$-linear $$d$$-categories [1]. Thus, given the cobordism hypothesis, $$\Sigma^d\mathrm{Vect}_{\mathbb{C}}$$ is the universal target for $$(d+1)$$-dimensional TQFTs, and in particular every object in $$\Sigma^d\mathrm{Vect}_{\mathbb{C}}$$ determines a (framed) TQFT [2, 3].

Second, our construction of $$\Sigma^d\mathrm{Vect}_{\mathbb{C}}$$ is sufficiently explicit so that, for each object of $$\Sigma^d\mathrm{Vect}_{\mathbb{C}}$$, we are able to build a (gapped topological) commuting projector Hamiltonian lattice system, which is, more or less, a very special type of the tensor-network models that you ask about. (Turaev–Viro is, essentially, the $$d=2$$ case of our construction.) Similarly, to each $$k$$-morphism, associate a (gapped topological) interface which is again commuting projector Hamiltonian. All together, this gives a realization of $$\Sigma^d\mathrm{Vect}_{\mathbb{C}}$$ in terms of lattice models. Conversely, we argue that any $$(d+1)$$-dimensional gapped topological system which can be "condensed from the vacuum" is (in the same gapped phase as) an object of $$\Sigma^d\mathrm{Vect}_{\mathbb{C}}$$.

There are systems, e.g. Kitaev's honecomb model of $$E_{8,1}$$, which are gapped topological and have commuting projector Hamiltonian models, but which cannot be condensed from the vacuum, and are not realized in our $$\Sigma^d\mathrm{Vect}_{\mathbb{C}}$$. Such systems correspond to TQFTs (e.g. TQFTs of Reshetikhin–Turaev type), but those TQFTs (probably) are not "fully extended" in any reasonable sense [4]. Indeed, the value a fully extended $$(d+1)$$-dimensional TQFT assigns to a point is (probably) the $$d$$-category of (local) gapped topological boundary conditions for the TQFT, which is (probably) empty unless the TQFT can be condensed from the vacuum.

In any case, our theorems do provide an isomorphism between some large families of TQFTs and lattice models, which I think includes all the ones you are asking about.

Footnotes.

[1] I take arbitrary linear functors as my morphisms in $$n\mathrm{Cat}_{\mathbb C}$$. This choice picks out the "naive tensor product" as the symmetric monoidal structure. There are other natural $$(d+1)$$-categories of $$n$$-linear categories. One expects, given the appendix to Bartlett–Douglas–Schommer-Pries–Vicary, Modular categories as representations of the 3-dimensional bordism 2-category (arXiv:1509.06811), that all these different $$(n+1)$$-categories should have equivalent fully dualizable subcategories, but this is not known. Scheimbauer calls this the "bestiary hypothesis".

[2] We conjecture that in fact each object of $$\Sigma^d\mathrm{Vect}_{\mathbb{C}}$$ determines an oriented TQFT in an essentially-canonical way. If this conjecture is true, then it is special to the complex numbers $$\mathbb{C}$$ — I expect it to fail over other fields. Specifically, the cobordism hypothesis determines an action of $$O(d+1)$$ on the $$(d+1)$$-groupoid of objects in $$\Sigma^d\mathrm{Vect}_{\mathbb{C}}$$. Using the usual topology on $$\mathbb{C}$$, this $$(d+1)$$-groupoid is in fact naturally a topological groupoid. Our conjecture is that the $$SO(d+1)$$-action is canonically homotopy-trivial on this topological groupoid (but not on the groupoid that only uses the algebraic structure of $$\mathbb{C}$$).

[3] (Added in response to comments discussion.) We work with weak $$n$$-categories, and manipulate them "synthetically", meaning model independently. Weak $$n$$-categories are not the same as $$(\infty,n)$$-categories, but there is an adjunction between the two worlds. The cobordism hypothesis is not known to be satisfied in any model of weak $$n$$-categories.

[4] (Added in response to Henriques' comment below.) One could also imagine looking at categories of boundary conditions which are not gapped. For instance, the category of all boundary conditions, or the category of conformal boundary conditions. My understanding is that this is how to understand the TQFT aspects of the work of Bartels–Douglas–Henriques culminating with Conformal nets V: dualizability (arXiv:1905.03393).

• I am tempted to disagree with your statement: "Such systems correspond to TQFTs (e.g. TQFTs of Reshetikhin–Turaev type), but those TQFTs (probably) are not 'fully extended' in any reasonable sense". See my recent paper arxiv.org/pdf/1905.03393.pdf for a way to (conjecturally) fit the Reshetikhin–Turaev invariants into the framework of fully extended TQFTs (specifically the section called "Manifold invariants" in the intro of the paper). – André Henriques Jun 12 at 4:21
• @AndréHenriques very good point. I felt it important enough to add a footnote to my answer. – Theo Johnson-Freyd Jun 13 at 7:19
• The cobordism hypothesis is "model dependent" in the following sense. (I learned this from Gwilliam and Scheimbauer.) Lurie, improved by Calaque–Scheimbauer, constructed a $\Gamma$-object in (incomplete!) $n$-fold Segal spaces which clearly deserves the name "$Bord_n$" — it's built out of spaces of smooth manifolds — and Lurie outlined (in great detail) how to show that $Bord_n$ (after completion!) is the symmetric monoidal $(\infty,n)$-category free on an $n$-dualizable object. You can transfer this $Bord_n$ to other models by using Barwick–Schommer-Pries, – Theo Johnson-Freyd Jun 13 at 8:22
• but the transfer doesn't deserve to be called "$Bord_n$", because it isn't necessarily very manifold-y. Each model (presumably) has its preferred manifold-y object called "$Bord_n$", and a cobordism hypothesis for that model would imply that these different $Bord_n$s are related by the Barwick–Schommer-Pries transfer. – Theo Johnson-Freyd Jun 13 at 8:23
• I took the liberty of adding the paper titles and authors. – David Roberts Jun 14 at 10:58