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:
Therefore the classical equations of motion for function are
The space of solutions to these equations is small. However, discussion of the quantization of the theory (below) suggests to consider these equations with the real Lie group replaced by its complexification to the complex Lie group . Then the general solution to the equations of motion above has the form
where hence is any holomorphic function and similarly any anti-holomorphic function.
(e.g. Gawedzki 99 (3.18), (3.19))
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