functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
physics, mathematical physics, philosophy of physics
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Axiomatizations
Tools
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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 $G$.
The vertex operator algebras corresponding to the WZW model are current algebras.
For $G$ a Lie group, the configuration space of the WZW over a 2-dimensional manifold $\Sigma$ is the space of smooth functions $g : \Sigma \to G$.
The action functional of the WZW sigma-model is the sum of two terms, a kinetic term and a topological term
The Lie group canonically carries a Riemannian metric and the kinetic term is the standard one for sigma-models on Riemannian target spaces.
In higher differential geometry, then given any closed differential (p+2)-form $\omega \in \Omega^{p+2}_{cl}(X)$, it is natural to ask for a prequantization of it, namely for a circle (p+1)-bundle with connection $\nabla$ (equivalently: cocycle in degree-$(p+2)$-Deligne cohomology) on $X$ whose curvature is $F_\nabla = \omega$. In terms of moduli stacks this means asking for lifts of the form
in the homotopy theory of smooth homotopy types.
This immediately raises the question for natural classes of examples of such prequantizations.
One such class arises in infinity-Lie theory, where $\omega$ 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 $G$, those closed $(p+2)$-forms $\omega$ which are left invariant forms may be identified, via the general theory of Chevalley-Eilenberg algebras, with degree $(p+2)$-cocycles $\mu$ in the Lie algebra cohomology of the Lie algebra $\mathfrak{g}$ corresponding to $G$. These in turn may arise, via the van Est map, as the Lie differentiation of a degree-$(p+2)$-cocycle $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^{p+2}U(1)$ in the Lie group cohomology of $G$ itself, with coefficients in the circle group $U(1)$.
This happens to be the case notably for $G$ a simply connected compact semisimple Lie group such as SU or Spin, where $\mu = \langle -,[-,-]\rangle$ is the Lie algebra 3-cocycle in transgression with the Killing form invariant polynomial $\langle -,-\rangle$. This is, up to normalization, a representative of the de Rham image of a generator $\mathbf{c}$ of $H^3(\mathbf{B}G, U(1)) \simeq H^4(B G, \mathbb{Z}) \simeq \mathbb{Z}$.
Generally, by the discussion at geometry of physics – principal bundles, the cocycle $\mathbf{c}$ modulates an infinity-group extension which is a circle p-group-principal infinity-bundle
whose higher Dixmier-Douady class class $\int \Omega \mathbf{c} \in H^{p+2}(X,\mathbb{Z})$ is an integral lift of the real cohomology class encoded by $\omega$ under the de Rham isomorphism. In the example of Spin and $p = 1$ this extension is the string 2-group.
Such a Lie theoretic situation is concisely expressed by a diagram of smooth homotopy types of the form
where $\flat_{dR}\mathbf{B}^{p+2}\mathbb{R} \simeq \flat_{dR}\mathbf{B}^{p+2}U(1)$ is the de Rham coefficients (see also at geometry of physics – de Rham coefficients) and where the homotopy filling the diagram is what exhibits $\omega$ as a de Rham representative of $\Omega \mathbf{c}$.
Now, by the very homotopy pullback-characterization of the Deligne complex $\mathbf{B}^{p+1}U(1)_{conn}$ (here), such a diagram is equivalently a prequantization of $\omega$:
For $\omega = \langle -,[-,-]\rangle$ as above, we have $p= 1$ and so $\nabla$ here is a circle 2-bundle with connection, often referred to as a bundle gerbe with connection. As such, this is also known as the WZW gerbe or similar.
This terminology arises as follows. In (Wess-Zumino 71) the sigma-model for a string propagating on the Lie group $G$ 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 $G$ 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 $\omega$. 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 $\nabla$ on $G$. This is equivalently the $\nabla$ that we just motivated above.
Later WZW terms, or at least their curvature forms $\omega$, 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 $p+2$.
In general WZW terms are “gauged” which means, as we will see, that they are not defined on the give smooth infinity-group $G$ itself, but on a bundle $\tilde G$ of differential moduli stacks over that group, such that a map $\Sigma \to \tilde G$ is a pair consisting of a map $\Sigma \to G$ and of a higher gauge field on $\Sigma$ (a “tensor multiplet” of fields).
The following (FSS 12, dcct) is a general axiomatization of WZW terms in cohesive homotopy theory.
In an ambient cohesive (∞,1)-topos $\mathbf{H}$, let $\mathbb{G}$ be a sylleptic ∞-group, equipped with a Hodge filtration, hence in particular with a chosen morphism
to its de Rham coefficients
Given an ∞-group object $G$ in $\mathbf{H}$ and given a group cocycle
then a refinement of the Hodge filtration of $\mathbb{G}$ along $\mathbf{c}$ is a completion of the cospan formed by $\flat_{dR}\mathbf{c}$ and by $\iota$ above to a diagram of the form
We write $\tilde G$ for the homotopy pullback of this refinement along the Maurer-Cartan form $\theta_G$ of $G$
Let $\mathbf{H} =$ Smooth∞Grpd and $\mathbb{G} = \mathbf{B}^p U(1)$ the circle (p+1)-group.
For $G$ an ordinary Lie group, then $\mu$ may be taken to be the Lie algebra cocycle corresponding to $\mathbf{c}$ and $\tilde G \simeq G$.
On the opposite extreme, for $G = \mathbf{B}^p U(1)$ itself with $\mathbf{c}$ the identity, then $\tilde G = \mathbf{B}^p U (1)_{conn}$ is the coefficients for ordinary differential cohomology (the Deligne complex under Dold-Kan correspondence and infinity-stackification).
Hence a more general case is a fibered product of these two, where $\tilde G$ is such that a map $\Sigma \longrightarrow \tilde G$ is equivalently a pair consisting of a map $\Sigma \to G$ and of differential $p$-form data on $\Sigma$. 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.
In the situation of def. there is an essentially unique prequantization
of the closed differential form
whose underlying $\mathbb{G}$-principal ∞-bundle is modulated by the looping $\Omega \mathbf{c}$ of the original cocycle.
This we call the WZW term of $\mathbf{c}$ with respect to the chosen refinement of the Hodge structure.
The variational derivative of the WZW action functional is
Therefore the classical equations of motion for function $g \colon \Sigma \to G$ 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 $G$ replaced by its complexification to the complex Lie group $G({\mathbb{C}})$. Then the general solution to the equations of motion above has the form
where hence $g_{\ell} \colon \Sigma \to G(\mathbb{C})$ is any holomorphic function and $g_r$ similarly any anti-holomorphic function.
(e.g. Gawedzki 99 (3.18), (3.19))
By the AdS3-CFT2 and CS-WZW correspondence (see there for more details) the 2d WZW CFT on $G$ is related to $G$-Chern-Simons theory in $3d$.
In fact a rigorous constructions of the $G$-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 $G$-Chern-Simons theory as a 3d TQFT incarnated via a Reshetikhin-Turaev construction.
The “hypergeometric integral” construction of conformal blocks for affine Lie algebra/WZW model-2d CFTs and of more general solutions to the Knizhnik-Zamolodchikov equation, via twisted de Rham cohomology of configuration spaces of points, originates with:
Vadim Schechtman, Alexander Varchenko, Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$cds:1044951$]$
Etsuro Date, Michio Jimbo, Atsushi Matsuo, Tetsuji Miwa, Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation, International Journal of Modern Physics B 04 05 (1990) 1049-1057 $[$doi:10.1142/S0217979290000528$]$
Atsushi Matsuo, An application of Aomoto-Gelfand hypergeometric functions to the $SU(n)$ Knizhnik-Zamolodchikov equation, Communications in Mathematical Physics 134 (1990) 65–77 $[$doi:10.1007/BF02102089$]$
Vadim Schechtman, Alexander Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 $[$doi:10.1007/BF00626523$]$
Vadim Schechtman, Alexander Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones mathematicae 106 1 (1991) 139-194 $[$dml:143938, pdf$]$
following precursor observations due to:
Vladimir S. Dotsenko, Vladimir A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nuclear Physics B 240 3 (1984) 312-348 $[$doi:10.1016/0550-3213(84)90269-4$]$
Philippe Christe, Rainald Flume, The four-point correlations of all primary operators of the $d = 2$ conformally invariant $SU(2)$ $\sigma$-model with Wess-Zumino term, Nuclear Physics B
282 (1987) 466-494 $[$doi:10.1016/0550-3213(87)90693-6$]$
The proof that for rational levels this construction indeed yields conformal blocks is due to:
Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by correlators in Wess-Zumino-Witten models, Lett Math Phys 20 (1990) 291–297 $[$doi:10.1007/BF00626525$]$
Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. I, Commun. Math. Phys. 163 (1994) 173–184 $[$doi:10.1007/BF02101739$]$
Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 1 (1995) 219-247 $[$euclid:cmp/1104272957$]$
Review:
Alexander Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Advanced Series in Mathematical Physics 21, World Scientific 1995 (doi:10.1142/2467)
Ivan Cherednik, Section 8.2 of: Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Mathematical Society of Japan Memoirs 1998 (1998) 1-96 $[$doi:10.2969/msjmemoirs/00101C010$]$
Pavel Etingof, Igor Frenkel, Alexander Kirillov, Lecture 7 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998) $[$ISBN:978-1-4704-1285-2, review pdf$]$
Toshitake Kohno, Homological representations of braid groups and KZ connections, Journal of Singularities 5 (2012) 94-108 $[$doi:10.5427/jsing.2012.5g, pdf$]$
Toshitake Kohno, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39 (2014) 575–598 $[$doi:10.1007%2Fs40306-014-0088-6, pdf$]$
Toshitake Kohno, Introduction to representation theory of braid groups, Peking 2018 $[$pdf, pdf$]$
(motivation from braid representations)
See also:
Alexander Varchenko, Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base, Comm. Math. Phys. 171 1 (1995) 99-137 $[$arXiv:hep-th/9403102, doi:10.1007/BF02103772$]$
Edward Frenkel, David Ben-Zvi, Section 14.3 in: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, AMS 2004 $[$ISBN:978-1-4704-1315-6, web$]$
This “hypergeometric” construction uses results on the twisted de Rham cohomology of configuration spaces of points due to:
Peter Orlik, Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent Math 56 (1980) 167–189 $[$doi:10.1007/BF01392549$]$
Kazuhiko Aomoto, Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan 39 2 (1987) 191-208 $[$doi:10.2969/jmsj/03920191$]$
Hélène Esnault, Vadim Schechtman, Eckart Viehweg, Cohomology of local systems on the complement of hyperplanes, Inventiones mathematicae 109.1 (1992) 557-561 $[$pdf$]$
Vadim Schechtman, H. Terao, Alexander Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, Journal of Pure and Applied Algebra 100 1–3 (1995) 93-102 $[$arXiv:hep-th/9411083, doi:10.1016/0022-4049(95)00014-N$]$
also:
reviewed in:
Discussion for the special case of level$=0$ (cf. at logarithmic CFT – Examples):
Fedor A. Smirnov, Remarks on deformed and undeformed Knizhnik-Zamolodchikov equations, $[$arXiv:hep-th/9210051$]$
Fedor A. Smirnov, Form factors, deformed Knizhnik-Zamolodchikov equations and finite-gap integration, Communications in Mathematical Physics 155 (1993) 459–487 $[$doi:10.1007/BF02096723, arXiv:hep-th/9210052$]$
S. Pakuliak, A. Perelomov, Relation Between Hyperelliptic Integrals, Mod. Phys. Lett. 9 19 (1994) 1791-1798 $[$doi:10.1142/S0217732394001647$]$
Interpretation of the hypergeometric construction as happening in twisted equivariant differential K-theory, showing that the K-theory classification of D-brane charge and the K-theory classification of topological phases of matter both reflect braid group representations as expected for defect branes and for anyons/topological order, respectively:
The characterization of D-brane submanifolds for the open string WZW model on a Lie group $G$ comes from two consistency conditions:
geometrical condition:
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 $G$, 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 $G$.(It has been suggested by Chris Rogers that such a foliation be thought of as a higher real polarization.)
cohomological condition:
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 $n \in \mathbb{N}$ 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 $\mathbf{B}U(1)$-principal 2-bundle ($U(1)$-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 $G$ 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
Textbook accounts:
Philippe Di Francesco, Pierre Mathieu, David Sénéchal, Part C of: Conformal field theory, Springer (1997) [doi:10.1007/978-1-4612-2256-9]
Bojko Bakalov, Alexander Kirillov, Wess-Zumino-Witten model, chapter 7 of: Lectures on tensor categories and modular functors, University Lecture Series 21, Amer. Math. Soc. (2001) [pdf, web, ams:ulect/21]
Ralph Blumenhagen, Erik Plauschinn, Chapter 3 of: Introduction to Conformal Field Theory – With Applications to String Theory, Lecture Notes in Physics 779, Springer (2009) [doi:10.1007/978-3-642-00450-6]
Lecture notes:
Patrick Meessen, Strings Moving on Group Manifolds: The WZW Model [pdf, pdf]
Lorenz Eberhardt, Wess-Zumino-Witten models, lecture notes at YRISW 2019: A modern primer for 2D CFT, Vienna (2019) [pdf, pdf]
A basic introduction also to the super-WZW model (and with an eye towards the corresponding 2-spectral triple) is in
A useful account of the WZW model that encompasses both its action functional and path integral quantization as well as the current algebra aspects of the QFT 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 $G$ and its Lie algebra $\mathfrak{g}$.
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 $\mathcal{l} \mathfrak{g}$ – the affine Lie algebra – of $\mathfrak{g}$ as the evident analog that replaces $\mathfrak{g}$ 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.
See also
L. Fehér, L. O’Raifeartaigh, P. Ruelle, I. Tsutsui, A. Wipf, On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories, Phys. Rep. 222 (1992), no. 1, 64 pp. MR93i:81225, doi
Krzysztof Gawedzki, Rafal Suszek, Konrad Waldorf, Global gauge anomalies in two-dimensional bosonic sigma models (arXiv:1003.4154)
Paul de Fromont, Krzysztof Gawędzki, Clément Tauber, Global gauge anomalies in coset models of conformal field theory (arXiv:1301.2517)
The gauged WZW term of chiral perturbation theory/quantum hadrodynamics which reproduces the chiral anomaly of QCD in the effective field theory of mesons and Skyrmions:
The original articles:
Julius Wess, Bruno Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95-97 (spire:67330, doi:10.1016/0370-2693(71)90582-X)
Edward Witten, Global aspects of current algebra, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 422-432 (doi:10.1016/0550-3213(83)90063-9)
See also:
Corrections and streamlining of the computations:
Chou Kuang-chao, Guo Han-ying, Wu Ke, Song Xing-kang, On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 67-69 (doi:10.1016/0370-2693(84)90986-9))
H. Kawai, S.-H. H. Tye, Chiral anomalies, effective lagrangians and differential geometry, Physics Letters B Volume 140, Issues 5–6, 14 June 1984, Pages 403-407 (doi:10.1016/0370-2693(84)90780-9)
J. L. Mañes, Differential geometric construction of the gauged Wess-Zumino action, Nuclear Physics B Volume 250, Issues 1–4, 1985, Pages 369-384 (doi:10.1016/0550-3213(85)90487-0)
Tomáš Brauner, Helena Kolešová, Gauged Wess-Zumino terms for a general coset space, Nuclear Physics B Volume 945, August 2019, 114676 (doi:10.1016/j.nuclphysb.2019.114676)
See also
Interpretation as Skyrmion/baryon current:
Jeffrey Goldstone, Frank Wilczek, Fractional Quantum Numbers on Solitons, Phys. Rev. Lett. 47, 986 (1981) (doi:10.1103/PhysRevLett.47.986)
Edward Witten, Current algebra, baryons, and quark confinement, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 433-444 (doi:10.1016/0550-3213(83)90064-0)
Gregory Adkins, Chiara Nappi, Stabilization of Chiral Solitons via Vector Mesons, Phys. Lett. 137B (1984) 251-256 (spire:194727, doi:10.1016/0370-2693(84)90239-9)
(beware that the two copies of the text at these two sources differ!)
Mannque Rho et al., Introduction, In: Mannque Rho et al. (eds.) The Multifaceted Skyrmion, World Scientific 2016 (doi:10.1142/9710)
Concrete form for $N$-flavor quantum hadrodynamics in 2d:
Concrete form for 2 flavors in 4d:
Concrete form for 2-flavor quantum hadrodynamics in 4d with light vector mesons included (omega-meson and rho-meson):
Ulf-G. Meissner, Ismail Zahed, equation (6) in: Skyrmions in the Presence of Vector Mesons, Phys. Rev. Lett. 56, 1035 (1986) (doi:10.1103/PhysRevLett.56.1035)
Ulf-G. Meissner, Norbert Kaiser, Wolfram Weise, equation (2.18) in: Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (doi:10.1016/0375-9474(87)90463-5)
Ulf-G. Meissner, equation (2.45) in: Low-energy hadron physics from effective chiral Lagrangians with vector mesons, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (doi:10.1016/0370-1573(88)90090-7)
Roland Kaiser, equation (12) in: Anomalies and WZW-term of two-flavour QCD, Phys. Rev. D63:076010, 2001 (arXiv:hep-ph/0011377, spire:537600)
Including heavy scalar mesons:
specifically kaons:
Curtis Callan, Igor Klebanov, equation (4.1) in: Bound-state approach to strangeness in the Skyrme model, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (doi10.1016/0550-3213(85)90292-5)
Igor Klebanov, equation (99) of: Strangeness in the Skyrme model, in: D. Vauthrin, F. Lenz, J. W. Negele, Hadrons and Hadronic Matter, Plenum Press 1989 (doi:10.1007/978-1-4684-1336-6)
N. N. Scoccola, D. P. Min, H. Nadeau, Mannque Rho, equation (2.20) in: The strangeness problem: An $SU(3)$ skyrmion with vector mesons, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (doi:10.1016/0375-9474(89)90029-8)
specifically D-mesons:
(…)
specifically B-mesons:
Inclusion of heavy vector mesons:
specifically K*-mesons:
Including electroweak fields:
J. Bijnens, G. Ecker, A. Picha, The chiral anomaly in non-leptonic weak interactions, Physics Letters B Volume 286, Issues 3–4, 30 July 1992, Pages 341-347 (doi:10.1016/0370-2693(92)91785-8)
Gerhard Ecker, Helmut Neufeld, Antonio Pich, Non-leptonic kaon decays and the chiral anomaly, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (doi:10.1016/0550-3213(94)90623-8)
Discussion for the full standard model of particle physics:
Interpretation of the 3d WZW term as defining a 2d CFT
Edward Witten, Non-Abelian bosonization in two dimensions Commun. Math. Phys. 92, 455 (1984)
Vadim Knizhnik, Alexander Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nucl. Phys. B 247, 83-103 (1984)
and hence as part of a perturbative string theory vacuum/target space
The WZ term on $\Sigma_2$ was understood in terms of an integral of a 3-form over a cobounding manifold $\Sigma_3$ in
for the case that $\Sigma_2$ is closed, and generally, in terms of surface holonomy of bundle gerbes/circle 2-bundles with connection in
Krzysztof Gawedzki, Topological Actions in two-dimensional Quantum Field Theories, in Gerard 't Hooft et. al (eds.) Nonperturbative quantum field theory Cargese 1987 proceedings, (web)
Giovanni Felder , Krzysztof Gawedzki, A. Kupianen, Spectra of Wess-Zumino-Witten models with arbitrary simple groups. Commun. Math. Phys. 117, 127 (1988)
Krzysztof Gawedzki, Topological actions in two-dimensional quantum field theories. In: Nonperturbative quantum field theory. ‘tHooft, G. et al. (eds.). London: Plenum Press 1988
as the surface holonomy of a circle 2-bundle with connection. See also the references at B-field and at Freed-Witten anomaly cancellation.
See also
Pierre Deligne, Daniel Freed, chapter 6 of Classical field theory (1999) (pdf)
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)
For the fully general understanding as the surface holonomy of a circle 2-bundle with connection see the references below.
See also
Discussion of circle 2-bundles with connection (expressed in terms of bundle gerbes) and discussion of the WZW-background B-field (WZW term) in this language is in
Krzysztof Gawedzki, Nuno Reis, WZW branes and gerbes, Rev.Math.Phys. 14 (2002) 1281-1334 (arXiv:hep-th/0205233)
Christoph Schweigert, Konrad Waldorf, Gerbes and Lie Groups, in Trends and Developments in Lie Theory, Progress in Math., Birkhäuser (arXiv:0710.5467)
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 refined interpretation of the integrality condition as a choice of trivialization of the underling principal 2-bundle/bundle gerbe of the B-field over the brane was then noticed in section 7 of
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 $n$ 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 $G$-principal connections) induces the Verlinde ring fusion product structure on symmetric D-branes equipped with Chan-Paton gauge fields is discussed in
The image in K-theory of these Chan-Paton gauge fields over conjugacy classes is shown to generate the Verlinde ring/the positive energy representations of the loop group in
Formalization of WZW terms in cohesive homotopy theory is in
One can also obtain the WZW-model by KK-reduction from Chern-Simons theory.
E.g.
A discussion in higher differential geometry via transgression in ordinary differential cohomology is in
Relation to extended TQFT is discussed in
For a formulation of the WZW term in the presence of D-branes as an open-closed smooth functorial field theory:
Severin Bunk, Konrad Waldorf, Transgression of D-branes, Adv. Theor. Math. Phys. 25 5 (2021) 1095-1198 [arXiv:1808.04894, doi:10.4310/ATMP.2021.v25.n5.a1]
Severin Bunk, Konrad Waldorf, Smooth functorial field theories from B-fields and D-branes, J. Homot. Rel. Struc. 16 1 (2021) 75-153 [doi:10.1007/s40062-020-00272-2, arXiv:1911.09990]
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.
Discussion of symmetry protected topological order phases of matter in solid state physics via higher dimensional WZW models is in
On WZW models at fractional level:
Sunil Mukhi, Sudhakar Panda, Fractional-level current algebras and the classification of characters, Nuclear Physics B 338 1 (1990) 263-282 $[$doi:10.1016/0550-3213(90)90632-N$]$
Hidetoshi Awata, Yasuhiko Yamada, Fusion rules for the fractional level $\widehat{\mathfrak{sl}(2)}$ algebra, Mod. Phys. Lett. A 7 (1992) 1185-1196 $[$spire:332974, doi:10.1142/S0217732392003645$]$
P. Furlan, A. Ch. Ganchev, R. Paunov, Valentina B. Petkova, Solutions of the Knizhnik-Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models, Nucl. Phys. B394 (1993) 665-706 (arXiv:hep-th/9201080, doi:10.1016/0550-3213(93)90227-G)
J. L. Petersen, J. Rasmussen, M. Yu, Fusion, Crossing and Monodromy in Conformal Field Theory Based on $SL(2)$ Current Algebra with Fractional Level, Nucl. Phys. B481 (1996) 577-624 (arXiv:hep-th/9607129, doi:10.1016/S0550-3213(96)00506-8)
Boris Feigin, Feodor Malikov, Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level, p. 357-405 in: Loday, Stasheff, Voronov (eds.) Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202, AMS (1997) $[$arXiv:q-alg/9511011, ams:conm-202$]$
(an $\mathfrak{osp}(1\vert2)$-factor appears)
with a good review in:
On braided fusion categories formed by affine Lie algebra-representations at admissible fractional level:
On interpreting fractional level WZW models as logarithmic CFTs:
Matthias R. Gaberdiel, Fusion rules and logarithmic representations of a WZW model at fractional level, Nucl. Phys. B 618 (2001) 407-436 $[$arXiv:hep-th/0105046, doi:10.1016/S0550-3213(01)00490-4)$]$
Matthias R. Gaberdiel, Section 5 of: An algebraic approach to logarithmic conformal field theory, Int. J. Mod. Phys. A 18 (2003) 4593-4638 $[$arXiv:hep-th/0111260, doi:10.1142/S0217751X03016860$]$
David Ridout, $\widehat{\mathfrak{sl}}(2)_{-1/2}$: A Case Study, Nucl. Phys. B 814 (2009) 485-521 $[$arXiv:0810.3532, doi:10.1016/j.nuclphysb.2009.01.008$]$
Thomas Creutzig, David Ridout, Modular Data and Verlinde Formulae for Fractional Level WZW Models I, Nuclear Physics B 865 1 (2012) 83-114 $[$arXiv:1205.6513, doi:10.1016/j.nuclphysb.2012.07.018$]$
Thomas Creutzig, David Ridout, Modular Data and Verlinde Formulae for Fractional Level WZW Models II, Nuclear Physics B 875 2 (2013) 423-458 $[$arXiv:1306.4388, doi:10.1016/j.nuclphysb.2013.07.008$]$
Thomas Creutzig, David Ridout, Section 4 of: Logarithmic conformal field theory: beyond an introduction, J. Phys. A: Math. Theor. 46 (2013) 494006 (doi:10.1088/1751-8113/46/49/494006, arXiv:1303.0847)
Kazuya Kawasetsu, David Ridout, Relaxed highest-weight modules I: rank 1 cases, Commun. Math. Phys. 368 (2019) 627–663 $[$arXiv:1803.01989, doi:10.1007/s00220-019-03305-x$]$
Kazuya Kawasetsu, David Ridout, Relaxed highest-weight modules II: classifications for affine vertex algebras, Communications in Contemporary Mathematics, 24 05 (2022) 2150037 $[$arXiv:1906.02935, doi:10.1142/S0219199721500371$]$
Reviewed in:
David Ridout, Fractional Level WZW Models as Logarithmic CFTs (2010) $[$pdf, pdf$]$
David Ridout, Fractional-level WZW models (2020) $[$pdf, pdf$]$
In particular, the logarithmic $c = -2$ model is essentially an admissible-level WZW model (namely at level $k = 0$):
with a comprehensive account in:
On the $c = -1$ model as the $\mathfrak{su}(2)$ WZW model at fractional level $-1/2$ and relation to the beta-gamma system:
and its lift to a logarithmic CFT:
On quasi-characters at fractional level:
Identification of would-be fractional level $\mathfrak{su}(2)$ conformal blocks in twisted equivariant K-theory of configuration spaces of points:
Last revised on July 27, 2023 at 08:26:20. See the history of this page for a list of all contributions to it.