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June 27, 2008

AQFT from Lattice Models (?)

Posted by Urs Schreiber

In AQFT from nn-functorial QFT (blog, arXiv) I had discussed how nn-functors on nn-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 *C^*-Hopf algebra HH and to every other site its dual C *C^*-Hopf algebra H^\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)D Z : P_2(X) \to D

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

x i x i+1 x i+2 x i+3 \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

Z(x i) Z(x i+1) Z(x i+2) Z(x i+3) Z(i,d) Z(i,u) Z(i+1,d) Z(i+1,u) Z(i+2,d) Z(i+2,u) \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)Z(i,d)) H = End_D(Z({i,u})\circ Z(i,d)) and H^=End D(Z(i+1,d)Z(i,u)). \hat H = End_D(Z({i+1,d})\circ Z(i,u)) \,.

Given that, if we furthermore assume the time evolution propagator

Z( x i x i+1 ) 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))×End(Z(i,u))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 DD and the action of the 2-functor ZZ on 1-morphisms to achieve that. But luckily, Pasquale Zito kindly provided very helpful comments. He has written

Pasquale A. Zito,
2-C*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 DD should be such that the endomorphism monoid of any 1-morphism is a C *C^*-algebra. This strongly suggests that the codomain be a 2-C *C^*-category: a category enriched over C *C^*-categories. You can find the definition of C *C^*-category for instance in the entry Spaceoids. A C *C^*-category is a C *C^*-algebroid. The definition of a 2-C *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 DD be a 2-C *C^*-category and let aρb a \stackrel{\rho}{\to} b be a 1-morphism in there which has a conjugate bρ¯a b \stackrel{\bar \rho}{\to} a i.e. such that we have an ambidextrous adjunction.

Then H:=End D(ρ¯ρ) H := End_D( \bar \rho\circ\rho) is a Hopf algebra and H^:=End D(ρρ¯) \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 ZZ as above which come from a choice of ambidextrous adjunction ρ,ρ¯\rho, \bar \rho in a 2C *2-C^*-category under the assignment Z(x i )=aρb Z \left( \array{ x_i \\ & \searrow } \right) = a \stackrel{\rho}{\to} b and Z( x i+1 )=bρ¯a. Z \left( \array{ & x_{i+ 1} \\ \nearrow } \right) = b \stackrel{\bar \rho}{\to} a \,.

Posted at June 27, 2008 11:42 PM UTC

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4 Comments & 2 Trackbacks

Re: AQFT from Lattice Models (?)

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

“Everything is proceeding as I have foreseen.” - The Emperor

It 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.

Posted by: Eric on June 28, 2008 3:20 PM | Permalink | Reply to this

Re: AQFT from Lattice Models (?)

alternative: consider Milnor’s Lie groups made discrete

Posted by: jim stasheff on June 28, 2008 3:33 PM | Permalink | Reply to this

Re: AQFT from Lattice Models (?)

I think Eric’s comment is best interpreted as: ‘make the domain discrete, then take the limit’. Compare with early algebraic topology methods. Cohomology is generally described by a passage to the limit of some ‘chunkified’ description of the space in question, be that using subdivision of simplicial complexes, or anafunctors.

Posted by: David Roberts on June 29, 2008 4:50 PM | Permalink | Reply to this

Re: AQFT from Lattice Models (?)

I think Eric’s comment is best interpreted as: ‘make the domain discrete, then take the limit’.

Yes, I think so. I was trying to find literature which goes into detail for how local nets of vonNeumann algebra factors are obtained by such a method. For that case such a process ought to exists. But I had trouble finding such literature.

The article I quote above does take some kind of limit by the following procedure:

they form the inducive limit algebra of their Hopf spin chain algebra – so something arising a a direct limit of finite dimensional algebras – and then embed that into a “field algebra” defined to be a bigger context in which what used to be amplimorphisms become true endomorphisms.

Originally I thought that passing to this “field algebra” might correspond to passing to the continuum limit. But in private communication with somebody I was told that possibly this just corresponds to passing to an unbounded lattice.

I wish I understood this continuum limit issue in AQFT better. If any reader here does, please drop me a comment!

Posted by: Urs Schreiber on June 29, 2008 5:35 PM | Permalink | Reply to this
Read the post Talk on AQFT from FQFT and Applications
Weblog: The n-Category Café
Excerpt: On local nets constructed from transport 2-functors and examples relating to lattice models, Hopf spin chains, asymptotic inclusion of subfactors. And some remarks on the relation between conformal nets and vertex operator algebras.
Tracked: July 10, 2008 2:10 PM
Read the post Planar Algebras, TFTs with Defects
Weblog: The n-Category Café
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Tracked: September 11, 2008 3:50 PM

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