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 .
In higher differential geometry, then given any closed differential (p+2)-form , it is natural to ask for a prequantization of it, namely for a circle (p+1)-bundle with connection (equivalently: cocycle in degree--Deligne cohomology) on whose curvature is . In terms of moduli stacks this means asking for lifts of the form
This immediately raises the question for natural classes of examples of such prequantizations.
One such class arises in infinity-Lie theory, where is a left invariant form on a smooth infinity-group given by a cocycle in L-infinity algebra cohomology. The prequantum n-bundles arising this way are the higher WZW terms discussed here.
In low degree of traditional Lie theory this appears as follows: On Lie groups , those closed -forms which are left invariant forms may be identified, via the general theory of Chevalley-Eilenberg algebras, with degree -cocycles in the Lie algebra cohomology of the Lie algebra corresponding to . These in turn may arise, via the van Est map, as the Lie differentiation of a degree--cocycle in the Lie group cohomology of itself, with coefficients in the circle group .
This happens to be the case notably for a simply connected compact semisimple Lie group such as SU or Spin, where is the Lie algebra 3-cocycle in transgression with the Killing form invariant polynomial . This is, up to normalization, a representative of the de Rham image of a generator of .
This terminology arises as follows. In (Wess-Zumino 84) the sigma-model for a string propagating on the Lie group was considered, with only the standard kinetic action term. Then in (Witten 84) it was observed that for this action functional to give a conformal field theory after quantization, a certain higher gauge interaction term has to the added. The resulting sigma-model came to be known as the Wess-Zumino-Witten model or WZW model for short, and the term that Witten added became the WZW term. In terms of string theory it describes the propagation of the string on the group subject to a force of gravity given by the Killing form Riemannian metric and subject to a B-field higher gauge force whose field strength is . In (Gawedzki 87) it was observed that when formulated properly and generally, this WZW term is the surface holonomy functional of a connection on a bundle gerbe on . This is equivalently the that we just motivated above.
Later WZW terms, or at least their curvature forms , were recognized all over the place in quantum field theory. For instance the Green-Schwarz sigma-models for super p-branes each have an action functional that is the sum of the standard kinetic action plus a WZW term of degree .
In general WZW terms are “gauged” which means, as we will see, that they are not defined on the give smooth infinity-group itself, but on a bundle of differential moduli stacks over that group, such that a map is a pair consisting of a map and of a higher gauge field on (a “tensor multiplet” of fields).
to its de Rham coefficients
Hence a more general case is a fibered product of these two, where is such that a map is equivalently a pair consisting of a map and of differential -form data on . This is the case of relevance for WZW models of super p-branes with “tensor multiplet” fields on them, such as the D-branes and the M5-brane.
of the closed differential form
This we call the WZW term of with respect to the chosen refinement of the Hodge structure.
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
this is a chapter in
P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison, E. Witten (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
An survey of and introduction to the topic is in
A classical textbook accounts include
P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal field theory, Springer 1997
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
Formalization of WZW terms in cohesive homotopy theory is 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
The low-energy physics of a Heisenberg antiferromagnetic spin chain is argued to be described by a WZW model in
See also section 7.10 of Fradkin’s book.