## March 7, 2011

### Liang Kong on Levin-Wen Models

#### Posted by John Baez

Liang Kong gave what was probably the first talk at the Centre for Quantum Technologies to explicitly mention tricategories:

But as his talk shows, tricategories are a quite natural formalism for studying models of 2d condensed matter physics. Two dimensions of space, one dimension of time: a tricategory!

The relation to the work of Fjelstad–Fuchs–Runkel–Schweigert is visible, but the focus on lattice models of condensed matter physics — in particular, the so-called Levin–Wen models — gives Liang Kong’s work a somewhat different flavor.

If you’re in a big rush, see the two-part “dictionary” on pages 41 and 42. Also see the diagram on page 46 — it’s probably quite cryptic taken out of context, but it means that Kong is working with a tricategory having:

• unitary tensor categories $C, D,\dots$ as objects,
• $C-D$ bimodule categories as 1-morphisms,
• maps between $C-D$ bimodule categories as 2-morphisms,
• natural transformations between these maps as 3-morphisms.

And, each bit of this structure corresponds to something people see in the Levin–Wen models:

Posted at March 7, 2011 9:47 AM UTC

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### Re: Liang Kong on Levin-Wen Models

That tricategory looks like it’s secretly what I would call a $(1\times 2)$-category. A lot of the tricategories that I see people studying nowadays are secretly either $(1\times 2)$-categories or $(2\times 1)$-categories. Chris Douglas also likes to talk about this distinction; I think he draws rectangular grids of dots to indicate how the cells are arranged.

In general an $(n\times k)$-category is an $n$-category (weakly) internal to $k$-categories. If it is sufficiently “fibrant,” then it has an underlying $(n+k)$-category, which is sometimes all one needs, but sometimes it’s better to work with $(n\times k)$-categories. In particular, they can be easier to construct, and they have different and sometimes better-behaved higher transfors.

For instance, a $(1\times 1)$-category is an internal category in $Cat$, which is a double category, and any double category has an underlying 2-category obtained by discarding the vertical (or the horizontal) arrows. So in that case no fibrancy is required.

A $(1\times 2)$-category is an internal category in 2-Cat. So it has a 2-category of objects, and a 2-category of 1-cells. In Kong’s case, the 2-category of objects would be the 2-category of unitary tensor categories with an appropriate notion of functor and natural transformation. The 2-category of 1-cells would be the 2-category of bimodule categories, with morphisms being functors that are equivariant relative to a couple of tensor functors on either side, and similarly for transformations. By forgetting the morphisms and 2-morphisms in the 2-category of objects, and relabeling those in the 2-category of 1-cells as “2-cells” and “3-cells,” one obtains a tricategory; no fibrancy is required here either (that seems to be the case with $(1\times k)$-categories for any $k$).

By contrast, a $(2\times 1)$-category is an internal 2-category in Cat, so it has a category of objects, a category of 1-cells, and a category of 2-cells. In the prototypical example here, the objects are commutative rings, the 1-cells $R\to S$ are $(R\otimes S^{op})$-algebras, and the 2-cells are bimodules for such algebras; each with a suitable notion of morphism. By forgetting the ring and algebra morphisms and relabeling the bimodule morphisms as “3-cells” one obtains a tricategory, now using the “fibrancy” fact that any ring morphism gives rise to a bimodule.

Posted by: Mike Shulman on March 7, 2011 1:09 PM | Permalink | PGP Sig | Reply to this

### Re: Liang Kong on Levin-Wen Models

Mike, your comment is very nice. I did not know that it is secretly a (1x2)-category.

Your prototypical example of (2x1)-category reminds me of another categorical structure appeared in the Levin-Wen model. It is too late for me to say more today. I will say something about it tomorrow.

Posted by: Liang Kong on March 7, 2011 4:23 PM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

Nice observation, Mike!

For the nonexperts in the crowd, it may help to observe that Liang Kong is working with a categorified analogue of the $(1 \times 1)$-category, or double category, where:

• objects are rings,
• vertical arrows are ring homomorphisms,
• horizontal arrows are bimodules,
• squares encode bimodule homomorphisms.

People often think of this as a bicategory, but Mike likes to think of it as a $(1 \times 1)$-category, so it’s not surprising that he likes to think of Liang Kong’s tricategory as a $(1 \times 2)$-category. This exhibits more of the structure in the situation.

Posted by: John Baez on March 8, 2011 6:12 AM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

The tricategory mentioned in John’s original writing consists of all the data that define the lattice model.

There is another categorical structure which consists of all excitations which determine the physical phases. It is described in the commutative diagram on the page 47. This diagram is in the image of $Z$ in a conjectured 3- or 4-category, in which

0-cells are braided tensor categories;

1-cells are cospans in the category of monoidal categories: $L_M$, $R_M$ are monoidal functors satisfying certain properties;

2-cells are again some sorts of cospans with two arrows given by a right $Z(M)$-module functor and a left $Z(N)$-module functor;

3-cells are bimodule functors between two $Z(N)-Z(M)$-bimodules;

4-cells are natural transformations between bimodule functors. I did not draw 4-cells in the diagram.

It seems natural to guess that this structure give arise to a 4-category. But I don’t know the physical meaning of the 4-cells. So by replacing 3-cells by their equivalent classes and then throwing away all 4-cells, we obtain a 4-layered structure which is conjectured to be a tricategory. This “tricategory” should be much more interesting than the first one. I wonder if it can be understood as a $(2\times 1)$-category as pointed out by Mike. Moreover, we conjecture that the assignment $Z$ is a functor.

Posted by: Liang Kong on March 9, 2011 12:25 PM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

That conjectural 4-category looks to me like a $(2\times 2)$-category. That may be the first time I’ve seen one of those appearing in nature.

Posted by: Mike Shulman on March 9, 2011 3:53 PM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

Hi, John, thank you for being interested in my joint work with Kitaev!

Let me add a few remarks:

Levin-Wen’s original work is about the bulk theory. Alexei and I worked out the boundary and defect theories. This generalization is quite simple. It is, however, a good example of the physical meaning behind extended topological field theories.

In the special case of Kitaev’s Toric code model, the boundary theories has already been worked out by Sergey B. Bravyi and Alexei Yu. Kitaev in quant-ph/9811052.

This work can be viewed as a categorified version of Fjelstad-Fuchs-Runkel-Schweigert’s theory of defects in RCFT.

Posted by: Liang Kong on March 7, 2011 3:28 PM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

Very cool stuff - thanks for posting the notes! Fascinating to see the condensed matter story taken so far in this direction. I recently heard a very interesting talk by Chris Schommer-Pries on joint work with Chris Douglas and Noah Snyder, relating these kinds of TFTs to the cobordism hypothesis – more precisely they show that fusion categories are fully dualizable objects in the 3-category of tensor categories, bimodule categories etc that is discussed here (hence define framed extended 3d TFTs). Moreover spherical categories form $SO(3)$-invariant objects, so define oriented extended 3d TFTs.

Posted by: David Ben-Zvi on March 9, 2011 2:03 AM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

This sounds like information worth adding to nLab: spherical categories. Does

spherical categories form SO(3)-invariant objects

mean that there is an action on them of the 2-group with SO(3)’s worth of 1-morphisms and trivial 2-morphisms?

Posted by: David Corfield on March 9, 2011 8:55 AM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

Thanks a lot! David.

Actually, Chris Douglas briefly mentioned (as a prize question if you remember) Turaev-Viro theory as extended TQFT in May 2009 at the TFT conference held in Northwestern University. I also announced this work there. I am glad that they are writing it down. Alexander Kirillov is also considering it but from different angle.

As far as I know, many condensed matter physicists are interested in tensor category theory now. Maybe higher categories as well in the future. It is desirable to see that these lattice models can eventually be manufactured in physics labs (via perhaps some cold atoms technology). Some physicists has proposed to manufacture another Kitaev’s model (“slightly” different from Toric code model) by using optical lattice. See for example:
http://arxiv.org/abs/cond-mat/0210564.
So there is a hope.

Your comment on spherical category is very interesting. I am also wondering the meaning of SO(3)-invariant objects as David Corfield.

Posted by: Liang Kong on March 9, 2011 1:33 PM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

There is another interesting thing.

If we forget about extended TQFT, just ask a naive algebraic question: is the notion of (monoidal) center functorial? Then you realize that one need at least 4 layers of structures to do it if it is indeed possible. This gives a pure algebraic support of the role of n-category played in an n-dimensional extended TQFT.

Of course, one can check even simpler cases. Is the notion of the center of an algebra functorial? A naive guess is negative because an algebra map $f: A \to B$ does not give an algebra map $f: Z(A) \to Z(B)$ between centers. But if we change the 1-morphisms in the target category (a category of commutative algebras) to cospans, together with proper 2-cells, then the center is functorial (a lax functor). Alexei Davydov, Ingo Runkel and myself are writing another paper on this topic. In particular, we show that the notion of center for algebras in tensor category is functorial (a functor between two bicategories). The physics behind it is 2-dimensional rational conformal field theories with defects of codimension 1,2. That is why one need bicategories. In that case, it is also the content of the so-called open-closed duality. So maybe a better name for such functor is a boundary-to-bulk functor or Holographic functor.

It also suggests that boundary-to-bulk functor (by taking center) is an dispensable ingredients of extended TQFTs. I believe that this phenomenon is universal. One can check it in other contexts such as A-infinity algebras, tensor-infinity categories, etc. It is desirable to understand it in more general contexts.

Posted by: Liang Kong on March 9, 2011 12:57 PM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

This is indeed a very interesting aspect. Maybe I can explain another perspective on such issues (that I learned in working with David Nadler). Drinfeld centers (or if you prefer, their derived versions) are an example of Hochschild cohomology (applied to a monoidal category), and as you say their functoriality is a little complicated. However there’s a dual notion, namely Hochschild homology, whose functoriality is very easy, and is very often identified with Hochschild cohomology (eg for Frobenius algebras or for pivotal tensor categories etc), explaining functoriality of the latter in various instances. Moreover I claim it is the natural target of boundary-to-bulk maps in great generality.

The Hochschild homology (or derived abelianization) of an algebra (or generally an algebra object - eg a tensor category) is the target of the universal trace map out of the algebra (ie universally coequalizes $a\cdot b$ and $b\cdot a$). For any homomorphism of algebra objects there’s a canonical map (covariant) of the Hochschild homologies. Moreover for any module over your algebra satisfying a finiteness (dualizability) condition, there’s a canonical “character” class in Hochschild homology, generalizing the character of a representation, the Chern character of a vector bundle, the charge (or boundary state) associated to a D-brane (boundary condition) in a TFT, etc. In fact what we assign to a circle in a TFT is canonically the Hochschild homology of what we assign to a point, and one can express the boundary state as a special case of the cobordism hypothesis with singularities (though of course it’s much easier than that).

In any case in a 2d framed TFT you have two different framed circles, one giving [Drinfeld] center, aka Hochschild cohomology, another giving Hochschild homology. In an oriented theory the two are identified. This is the case eg for pivotal tensor categories, we can canonically identify the two, and hence we find boundary states/characters in the Drinfeld center as well as a covariant functoriality for the Drinfeld center.

Posted by: David Ben-Zvi on March 9, 2011 4:05 PM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

It is a very nice observation. I have thought about the funtoriality of the universal trace map. From functorial QFT (FQFT) point of view, Hochschild homology is more natural as pointed out by David. But it is unclear to me its physical meaning.

Although the functoriality of center looks complicated, its content exactly encode all the information of excitations, nothing more and nothing less.

I am wondering if there is an exact dual picture between physical meaning and mathematical FQFT. For example, what we associate to a 0-cell in FQFT is exactly what we associate to a 3-cell on the physics side (2+1 cell in Levin-Wen models). Perhaps there is still some interesting structure missing from our understanding.

Posted by: Liang Kong on March 11, 2011 3:00 AM | Permalink | Reply to this

### natural Hochschild (co)homology

Hochschild homology is more natural as pointed out by David.

In fact, it is remarkable just how natural Hochschild homology is: in full abstract generality (generalized to arbitrary notions of algebras and to “higher order” Hochschild homology over spaces more general than the circle), it is nothing but the canonical powering of any ambient $\infty$-topos over $\infty$-groupoids.

Last semester for our seminar on derived differential geometry I had occasion to write out a fair bit of details of what this statement amounts to at $n$Lab:Hochschild cohomology.

Now here is something concerning Hochschild co-homology that might be a noteworthy observation (which materialized in the form that I present it now in discussion with Domenico Fiorenza).

Hochschild homology of $\mathcal{O}(X)$ over the circle is the algebra of functions $\mathcal{O}(\mathcal{L}X)$ on the derived self-intersection of $X$ via the diagonal

$\array{ \mathcal{L}X &\to& X \\ \downarrow && \downarrow \\ X &\to & X \times X }$

But (at least if $X$ is an underived 0-truncated space) this derived self-intersection takes place only in the formal neighbourhood of the diagonal

$\array{ \mathcal{L}X &\to& X \\ \downarrow && \downarrow \\ X &\to & T_f X } \,,$

where now we think of the two morphisms being pulled back as being the 0-sections of the tangent bundle.

This perspective has an evident dual: consider the derived self-intersection of the vanishing 1-form on $X$. More interestingly, consider a function

$S : X \to \mathbb{A}^1$

and its derived critical locus : the derived intersection of $d S$ with the 0-form

$\array{ X_{\{d S = 0\}} &\to& X \\ \downarrow && \downarrow^{\mathrlap{0}} \\ X &\stackrel{d S}{\to} & T^*_f X } \,.$

I think one can show that $\mathcal{O}(X_{\{d S = 0\}})$ is the BRST-BV complex corresponding to the “BRST complex” $\mathcal{O}(X)$ and the “action functional” $S$.

This is the $S$-twisted Hochschild-cohomology of $\mathcal{O}(X)$ (at least if $X$ is “smooth”). The fact that there is BV-algebra structure on the BV-complex should be precisely the dual argument that $\mathcal{O}\mathcal{L}X$ carries a framed little disk action for $X$ $\mathcal{O}$-perfect, as pointed out by David Ben-Zvi and David Nadler.

I have a short preliminary note on this at derived critical locus. But I am still thinking about this. Would be glad to hear comments (you see, I am trying to answer my own MO-question).

Posted by: Urs Schreiber on March 11, 2011 9:35 AM | Permalink | Reply to this

### Re: natural Hochschild (co)homology

Urs - I’m not sure I follow your picture about Hochschild homology being defined quite in that generality. Commutative algebras (by which I mean comm alg objects in a symmetric monoidal $\infty$-category) are tensored over simplicial sets, in other words it makes sense to integrate a commutative algebra over any simplicial set (giving functions on the mapping space to the spectrum of the commutative algebra), and this generalizes Hochschild homology of commutative algebras (when the simplicial set is the circle).

However I don’t believe say associative algebras can be integrated over arbitrary simplicial sets. In fact Lurie explains that $E_n$ algebras are things you can integrate over manifolds of dimension at most n (topological chiral homology, or the cobordism hypothesis, or blob homology, etc) and not over more general things. So for a general associative algebra eg I don’t know of any kind of Hochschild homology other than the usual one. But maybe you’re talking about a very different construction, which I’d be happy to learn.

Posted by: David Ben-Zvi on March 11, 2011 3:49 PM | Permalink | Reply to this

### Re: natural Hochschild (co)homology

Ah sorry I looked more at n-lab and I think I understand — you are talking about commutative algebras (in the forms of functions on spaces - obviously in a very very general setting), and they are indeed tensored in simplicial sets (or $\infty$-groupoids) as are spaces. I like this picture very much. I guess my point though is that it doesn’t include HH for noncommutative algebras..

Posted by: David Ben-Zvi on March 11, 2011 3:57 PM | Permalink | Reply to this

### Re: natural Hochschild (co)homology

Hi David,

thanks for your comment. Yes, the entry currently only looks at Hochschild homology in the presence of $\infty$-tensoring of $\infty$-algebras over $\infty$-groupoids, and specifically at the case where we have function algebras on $\infty$-stacks.

I think in the formal part of the discussion this is stated explicitly and correctly, but I realize that the Idea-section of the entry probably suggested too broad an applicability of this case (as maybe did my mentioning of it in the context of the present discussio here). I have rewritten the Idea-section now in an attempt to fix this.

And the entry deserves to be expanded to include more general cases of Hochschild homology. I had written it when I was preparing seminar talks on the cdg-algebraic version and found that I needed to fill in a few technical details over what I could see made explicit in the literature.

So I’d be happy to hear about mistakes in the formal definition/proposition/proof-discussion.

And hopefully eventually somebody (maybe me) finds the time and energy to provide discussion there of more general situations of Hochschild (co)homology.

Posted by: Urs Schreiber on March 14, 2011 9:32 AM | Permalink | Reply to this

### Re: Liang Kong on Levin-Wen Models

The idea assembling the possible defects of an $d$-dimensional TQFT into a $d+1$-dimensional TQFT is developed in Section 6.7 of my recent paper with Scott Morrison (math.AT/1009.5025, but a more frequently updated version can be found on my web page). In that paper we call them “sphere modules” rather than “defects”, but the idea is the same. Turaev-Viro type TQFTs (a.k.a. Levin-Wen models) were one of the motivating examples.

The main construction is as follows. (For simplicity, assume that all $n$-categories mentioned are an $n$-dimensional version strict pivotal). Given an $n$-category $C$, we can form an $n-k$-category $C(S^k)$ whose $j$-morphisms are shaped like $B^j\times S^k$. ($B^j$ is a $j$-dimensional ball, $S^k$ is a $k$-dimensional sphere.) A representation of $C(S^k)$ is (by definition) a family of $(n-k-1)$-categories with some additional structure. We call such a representation a $k$-sphere module, but “codimension $k+1$ defect” would also be an appropriate name. More generally, one can replace the unadorned sphere $S^k$ with a pair $(S^k, K)$, where $K$ is a cell complex embedded in $S^k$, and the codimension-$m$ cells of $K$ are labeled by $m-1$-sphere modules. One can again form an $n-k$ category $C(S^k, K)$, and representations of such a $n-k$-category are also called $k$-sphere modules.

We can now form an $n+1$-category whose objects are $n$-categories and whose $k$-morphisms are $k-1$-sphere modules. The $n+1$-morphisms are ordinary intertwiners between representations of 1-categories.

We can also restrict to various subcategories of this big category of all sphere modules. For example, given a pivotal 2-category $A$, we can consider the sub-3-category of sphere modules with

• only a single object, $A$
• only a single 1-morphism, $_A A_A$ (i.e. $A$ though of as the trivial 0-sphere module (categorified bimodule) over itself)
• 2-morphisms are all possible representations of the “annular” 1-category A(S^1)
• 3-morphsims are all possible intertwiners

It’s not hard to see that this 3-category is isomorphic to the Drinfeld center of $A$. Variations on this construction give various higher centers of $n$-categories.

Posted by: Kevin Walker on March 9, 2011 3:04 PM | Permalink | Reply to this

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