### AQFT from Lattice Models (?)

#### Posted by Urs Schreiber

In *AQFT from $n$-functorial QFT* (blog, arXiv) I had discussed how $n$-functors on $n$-paths in pseudo-Riemannian spaces give rise to *local nets of algebras* (or rather, more generally, of monoids) which are taken in algebraic quantum field theory (AQFT) as the definition of the local observables of QFT and indeed of QFT itself.

Now I am thinking about more examples in 2 dimensions.

A good approach would be to try to first handle all known lattice models of AQFT, then understanding the continuum limit. But work in this direction is scarce. One relevant article I am aware of is

Florian Nill, Kornél Szalachányi
*Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry*

arXiv:hep-th/9509100

They are considering a 1-dimensional lattice where to every second site is assigned a finite dimensional $C^*$-Hopf algebra $H$ and to every other site its dual $C^*$-Hopf algebra $\hat H$.

A possible 2-functorial interpretation would be to read this as the Heisenberg picture version of what in the Schrödinger picture is a 2-functor $Z : P_2(X) \to D$

on a causal lattice 2-path 2-category $P_2(X)$ of the form

$\array{ && \nearrow \searrow && \nearrow \searrow && \nearrow \searrow \\ \cdots & x_i &\Downarrow& x_{i+1} &\Downarrow& x_{i+2} &\Downarrow& x_{i+3} & \cdots \\ && \searrow \nearrow && \searrow \nearrow && \searrow \nearrow & }$

such that the 1-morphisms assigned to the lower paths

$\array{ \cdots & Z(x_i) && Z(x_{i+1}) && Z(x_{i+2}) && Z(x_{i+3}) & \cdots \\ && {}_{Z(i,d)}\searrow \nearrow_{Z({i,u})} && {}_{Z({i+1,d})}\searrow \nearrow_{Z({i+1,u})} && {}_{Z({i+2,d})}\searrow \nearrow_{Z({i+2,u})} & }$

are such that $H = End_D(Z({i,u})\circ Z(i,d))$ and $\hat H = End_D(Z({i+1,d})\circ Z(i,u)) \,.$

Given that, if we furthermore assume the time evolution propagator

$Z\left( \array{ & \nearrow \searrow \\ x_i &\Downarrow& x_{i+1} \\ & \searrow \nearrow } \right)$

on every elementary 2-path be in $End(Z(i,d)) \times End(Z(i,u))$, the local net obtained from this 2-functor restricted to a horizontal zig-zag path would be a net of the kind discussed by the above authors in section 2.2.

I wasn’t entirely sure how I had to choose the codomain 2-category $D$ and the action of the 2-functor $Z$ on 1-morphisms to achieve that. But luckily, Pasquale Zito kindly provided very helpful comments. He has written

Pasquale A. Zito,
*2-$C*$-categories with non-simple units*

arXiv:math/0509266

First of all, quite generally for many common classes of applications the right choice of codomain 2-category $D$ should be such that the endomorphism monoid of any 1-morphism is a $C^*$-algebra. This strongly suggests that the codomain be a *2-$C^*$-category*: a category *enriched* over $C^*$-categories. You can find the definition of $C^*$-category for instance in the entry Spaceoids. A $C^*$-category is a $C^*$-algebroid. The definition of a 2-$C^*$-category is for instance on p. 7 of Pasquale Zito’s article.

So that much is clear. Less immediate is the following nice fact, which Pasquale Zito discusses in section 5, from p. 39 on:

Let $D$ be a 2-$C^*$-category and let
$a \stackrel{\rho}{\to} b$
be a 1-morphism in there which has a *conjugate*
$b \stackrel{\bar \rho}{\to} a$
i.e. such that we have an *ambidextrous adjunction*.

Then $H := End_D( \bar \rho\circ\rho)$ is a Hopf algebra and $\hat H := End_D( \rho\circ \bar\rho)$ is its dual. Together they satisfy a bunch of nice properties, for which the source seems to be

Michael Mueger
*From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories*

arXiv:math/0111204

around p. 45.

So it seems that the “Hopf spin chain” nets discussed by Nill and Szalachányi can be regarded as the nets obtained from 2-functor $Z$ as above which come from a choice of ambidextrous adjunction $\rho, \bar \rho$ in a $2-C^*$-category under the assignment $Z \left( \array{ x_i \\ & \searrow } \right) = a \stackrel{\rho}{\to} b$ and $Z \left( \array{ & x_{i+ 1} \\ \nearrow } \right) = b \stackrel{\bar \rho}{\to} a \,.$

## Re: AQFT from Lattice Models (?)

Thanks for the belated birthday present. June 26 was my birthday :)

“Everything is proceeding as I have foreseen.”- The EmperorIt has always been my experience that taking the discrete theories seriously generally leads to improved understanding and if you do things right, the continuum limit (if one is desired) should be guaranteed.