What is the algebraic property that corresponds to a topological term?

Question

Warning: This question will be fairly ill-posed. I have spent a lot of time trying to make it better posed without success, so please bear with me.

A single $SU(2)$ spin may be represented by the $0+1$ dimensional non-linear $\sigma$ model with target space $S^2$. This $\sigma$ model admits a Wess-Zumino-Witten term. Now WZW terms are very funny things. One way to think about is purely from the path integral side. The existence of the topological term comes from the topological structure of the path integral, and the quantization of the WZW term comes from the single valued-ness of the integral.

Another way to obtain the WZW term is to start from the Hilbert space of an $SU(2)$ spin (the "algebraic" side) and then construct the path integral. The fact that the WZW term is quantized and topological is totally natural from this perspective, since the WZW term has nothing to do with dynamics. It simply keeps track of which $SU(2)$ representation we have - so of course it doesn't renormalize or care about any details.

Further, instead of a single spin, we can consider a $d$ dimensional lattice spins. Under some conditions we can again write this as $\sigma$ model. There may be a WZW term depending on $d$. So there is a classification of possible WZW terms depending on dimension.

Here is the issue: The classification of WZW terms on the path integral side is very clear. It is just some appropriate dimensional homotopy group. What is being classified on the algebraic side? We are somehow counting distinct representations of the operator algebra - but what precisely?

Hopefully, that makes non-zero sense. What follows is a dump of my brain contents.

1) The question has nothing to do with symmetry. I will still have a topological term if add a nonzero Hamiltonian that breaks every symmetry. It is true that at long distances I may end up in different topological sector but this seems irrelevant.

2) A quantum system consists of two parts: an operator algebra (with a Hamiltonian) and a Hilbert space, which is a choice representation of this algebra - the WZW terms keeps track of this choice.

3) It is obviously sensitive to more information than the dimension of the Hilbert space since there are representations of $SU(2)$ and $SU(3)$ with the same dimension but the homotopy sequence of $S^2$ is different from that of $S^3$.

4) As far as I can tell there is nothing special about WZW terms. Any topological term should have some similar question, at least via the bulk-boundary correspondance. Since we can set $H=0$ the question is somehow about the algebraic analogue of Topological QFT. So someone who understands TQFT should be able to explain.

The closest I have been able to come to an answer is something like the following. To construct a coherent state path integral representation my Hamiltonian can only involve a certain algebra of operators. (I think.) For example, if I have a lattice with a four dimensional Hilbert space on each site, I can pretend that that is a lattice of spin $3/2$. But if I just have random matrix elements coupling neighboring sites then the coherent state path integral doesn't seem to come out well. It's only when the only couplings that appear in the Hamiltonian are the spin operators that I can perform the usual path integral manipulation.

So as I basically said before, the defining question is "what is the algebra of operators defined on a site that may appear in the Hamiltonian"? Again, as I said before there is no notion of symmetry in this definition, since the Hamiltonian does not have to lie in the center of this operator algebra. And again the spectrum of topological invariants is sensitive to the representations of this algebra.

Now to construct a path integral representation I would like to have coherent states. In the case that my algebra is just a finite Lie Algebra then this probably can be done using a root decomposition, and basically following the $SU(2)$ construction. So I get some geometric space, that I can probably read off the Dynkin diagram somehow. Then maybe by going backwards from the homotopy calculation I can figure whats happening on the algebraic side. So I guess its just the ADE classification of symmetric spaces, maybe?

In the case that my operator algebra is not a finite Lie algebra, I don't know, mostly because I know nothing about Algebra.

Answer 1

A modern geometric-algebraic tool presently being under active research in the context of quantization of field theories with topological terms is the theory of gerbes. The main reference for this application (quantization) is the book by Brylinski : "Loop Spaces, Characteristic Classes, and Geometric Quantization".

Gerbes are extensively used in string theory, please see the following introduction by Graeme Segal. (i.e., the stringy structures depending on $B$-fields in string theory are actually properties of gerbes).

One of the early physics references on this subject is the seminal article by Orlando Alvarez. (The word gerbe is not mentioned in the article, because it preceded the use of this terminology in physics.).

In this work the quantization condition of the coefficient of a WZW terms is achieved as a by-product of the construction of the isomorphism between the Cech and the de-Rham cohomology groups.

The work describes two examples, the first is the the translation of the Wu-Yang argument of a particle in the field of a magnetic monopole (in 0+1 d) to Cech cohomology. This example is actually an introduction to line bundles. The WZW coefficient quantization in 0+1 dimensions is just the Dirac quantization condition, associating (equivalence classes) of line bundles over the manifold with the first Chern class of the manifold, which is the globally defined magnetic field. The construction of the line bundle is the first step towards quantization (sometimes called prequantization). Basically, the quantization Hilbert space can be chosen to be the space zero modes of the (gauged) Laplacian on the sections of the bundle. The values of the magnetic charges determine the dimension of the quantization Hilbert space via the Atiyah-Singer index theorem.

The second (and the main) example is a two dimensional sigma model with a WZW term , which is the important contribution of the work. In the course of the process of the quantization of the WZW coefficient, the basic ingredients of the gerbe are constructed: The $H$-field: the global WZW three form, the $B$-field (two form) denoted in the article by $T$ and the $A$-field (one form) denoted in the article by $J$.

This work by Orlando Alvarez is very essential for the understanding of the following gerbe constructions.

The quantization of gerbes follows the same lines as the quantization of line bundles. The quantization condition, or the association of line bundles with the first Chern class is replaced by the association of the gerbe with a Dixmier-Douady class whose representative is just the WZW three form with the quantized coefficient. The line bundle holonomy is replaced by the gerbe surface holonomy. The quantization spaces are infinite dimensional, and associated with representations of Kac-Moody algebras, please see the following work by Juoko Mickelsson.

Gerbes enable the extension of the theory of geometric quantization to loop spaces of manifolds $L\mathcal{M}$ while working with finite dimensional objects. The transition between the two pictures is by means of the transgression map, in our case $H^3(\mathcal{M}) \cong H^2(L\mathcal{M})$. Thus the WZW three form is actually the Chern class of a line bundle over a loop space. Please see for example the following Work by Sämann and Szabo.

The use of gerbes is not confined to two dimensional field theories, although, the quantization of higher dimensional field theories, (with topological terms, for example when anomalies are present), constitutes of a much more difficullt challenge and the work is not concluded yet. Steps in this direction were taken by Juoko Mickelsson and his collaborators, please see references in the above Mickelsson work and other works by him in the Archiv.

Answer 2

We have an algebraic classification of WZW terms that works for any dimensions and for any symmetry groups (including discrete groups). In fact what we did is that we classified the so call SPT states for any on-site symmetries in any dimensions. The boundary excitations of a SPT states are describe by effective non-linear sigma-model of the symmetry group with a WZW term. The classification of the bulk SPT states, in turn, classify the WZW terms for the effective boundary non-linear sigma-model.

The classification can be stated as the following (arXiv:1106.4772):

Consider a non-linear sigma-model whose target space is the symmetry group $G$ in $d$ space-time dimensions, its WZW terms are classified by group-cohomology classes $H^{d+1}(G,R/Z)$. Here $G$ can be continuous or discrete.

Add: What we really classified is the WZW term in a non-linear sigma-model whose target space is the symmetry group $G$ in $d$ dimensional space-time lattice. That is we have a discrete space-time. Even for discrete space-time and discrete groups, a generalization of WZW term can be defined, and their classification is purely algebraic. This is why our classification involves co-homology theory rather than homotopy theory.

If we break the symmetry, the low energy effects of WZW term disappear (ie the low energy properties of the theory is the same with or without WZW term. This is why we require symmetry.