FQFT and cohomology
Types of quantum field thories
The Wess-Zumino-Witten model (or WZW model for short, also called Wess-Zumino-Novikov-Witten model, or short WZNW model) is a 2-dimensional sigma-model quantum field theory whose target space is a Lie group.
This may be regarded as the boundary theory of Chern-Simons theory for Lie group .
The construction of the canonical morphism goes as follows. Consider, inside the substack of trivialized principal -bundles with flat connections. The morphism , restricted to , factors as
where is identified with the 3-stack of trivialized circle bundles with connection whose connection form lives entirely in degree 3. Since factors through the stack of principal -bundles with flat conenctions, we have a homotopy commutative diagram
and so a canonically induced morphism between the homotopy fibers over the distinguished points. Since we have homotopy pullbacks
the morphism can naturally be seen as a morphism from (as a smooth manifold) to the 2-stack of circle 2-bundles with connection. In other words, if is a compact simply connected simple Lie group, the differential refinement of the degree 4 characteristic class provided by Chern-Simons theory naturally induces a circle 2-bundle with connection over the smooth manifold underlying the Lie group .
The surface holonomy of this is the topological part of the WZW action functional:
In fact a rigorous constructions of the -WZW model as a rational 2d CFT is via the FRS-theorem on rational 2d CFT, which constructs the model as a boundary field theory of the -Chern-Simons theory as a 3d TQFT incarnated via a Reshetikhin-Turaev construction.
For the open string CFT to still have current algebra worldsheet symmetry, hence for half the current algebra symmetry of the closed WZW string to be preserved, the D-brane submanifolds need to be conjugacy classes of the group manifold (see e.g. Alekseev-Schomerus for a brief review and further pointers). These conjugacy classes are therefore also called the symmetric D-branes.
Notice that these conjugacy classes are equivalently the leaves of the foliation induced by the canonical Cartan-Dirac structure on , hence (by the discussion at Dirac structure), the leaves induced by the Lagrangian sub-Lie 2-algebroids of the Courant Lie 2-algebroid which is the higher gauge groupoid (see there) of the background B-field on .(It has been suggested by Chris Rogers that such a foliation be thought of as a higher real polarization.)
In order for the Kapustin-part of the Freed-Witten-Kapustin anomaly of the worldsheet action functional of the open WZW string to vanish, the D-brane must be equipped with a Chan-Paton gauge field, hence a twisted unitary bundle (bundle gerbe module) of some rank for the restriction of the ambient B-field to the brane.
For simply connected Lie groups only the rank-1 Chan-Paton gauge fields and their direct sums play a role, and their existence corresponds to a trivialization of the underlying -principal 2-bundle (-bundle gerbe) of the restriction of the B-field to the brane. There is then a discrete finite collection of symmetric D-branes = conjugacy classes satisfying this condition, and these are called the integral symmetric D-branes. (Alekseev-Schomerus, Gawedzki-Reis). As observed in Alekseev-Schomerus, this may be thought of as identifying a D-brane as a variant kind of a Bohr-Sommerfeld leaf.
More generally, on non-simply connected group manifolds there are nontrivial higher rank twisted unitary bundles/Chan-Paton gauge fields over conjugacy classes and hence the cohomological “integrality” or “Bohr-Sommerfeld”-condition imposed on symmetric D-branes becomes more refined (Gawedzki 04).
In summary, the D-brane submanifolds in a Lie group which induce an open string WZW model that a) has one current algebra symmetry and b) is Kapustin-anomaly-free are precisely the conjugacy class-submanifolds equipped with a twisted unitary bundle for the restriction of the background B-field to the conjugacy class.
on quantization of the WZW model, see at
The Wess-Zumino gauge-coupling term goes back to
and was understood as yielding a 2-dimensional conformal field theory in
Edward Witten, Non-Abelian bosonization in two dimensions Commun. Math. Phys. 92, 455 (1984)
The WZ term on was understood in terms of an integral of a 3-form over a cobounding manifold in
Krzysztof Gawędzki, Topological actions in two-dimensional quantum field theories. In: Nonperturbative quantum field theory. ‘tHooft, G. et al. (eds.). London: Plenum Press 1988
An survey of and introduction to the topic is in
A classical textbook account in the general context of 2d CFT is
A basic introduction also to the super-WZW model (and with an eye towards the corresponding 2-spectral triple) is in
This starts in section 2 as a warmup with describing the 1d QFT of a particle propagating on a group manifold. The Hilbert space of states is expressed in terms of the Lie theory of and its Lie algebra .
In section 4 the quantization of the 2d WZW model is discussed in analogy to that. In lack of a full formalization of the quantization procedure, the author uses the loop algebra – the affine Lie algebra – of as the evident analog that replaces and discusses the Hilbert space of states in terms of that. He also indicates how this may be understood as a space of sections of a (prequantum) line bundle over the loop group.
Discussion of how this 2-bundle arises from the Chern-Simons circle 3-bundle is in
and related discussion is in
See also Section 2.3.18 and section 4.7 of
A characterization of D-branes in the WZW model as those conjugacy classes that in addition satisfy an integrality (Bohr-Sommerfeld-type) condition missed in other parts of the literature is stated in
The observation that this is just the special rank-1 case of the more general structure provided by a twisted unitary bundle of some rank on the D-brane (gerbe module) which is twisted by the restriction of the B-field to the D-brane – the Chan-Paton gauge field – is due to
The observation that the “multiplicative” structure of the WZW-B-field (induced from it being the transgression of the Chern-Simons circle 3-connection over the moduli stack of -principal connections) induces the Verlinde ring fusion product structure on symmetric D-branes equipped with Chan-Paton gauge fields is discussed in
Relation to extended TQFT is discussed in
The formulation of the Green-Schwarz action functional for superstrings (and other branes of string theory/M-theory) as WZW-models (and ∞-WZW models) on (super L-∞ algebra L-∞ extensions of) the super translation group is in