functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
In a conformal field theory the conditions on correlators can be divided into two steps
for a fixed cobordism the correlators need to depend in a certain way on the choice of conformal structure, they need to satisfy the Ward identities (e.g. Gawedzki 99, around p. 30);
the correlators need to glue correctly underly composition of cobordisms.
The spaces of functionals that satisfy the first of these conditions are called conformal blocks . The second condition is called the sewing constraint on conformal blocks.
So conformal blocks are something like “precorrelators” or “potential correlators” of a CFT.
The assignment of spaces of conformal blocks to surfaces and their isomorphisms under diffeomorphisms of these surfaces together constitutes the modular functor. Under CS/WZW holography this is essentially the data also given by the Hitchin connection, see at quantization of 3d Chern-Simons theory for more on this.
From a point of view closer to number theory and geometric Langlands correspondence elements of conformal blocks are naturally thought of (Beauville-Laszlo 93) as generalized theta functions (see there for more).
The conformal blocks at least of the WZW model are by a holographic correspondence given by the space of quantum states of 3d Chern-Simons theory. See at AdS3-CFT2 and CS-WZW correspondence.
For the $G$-WZW model the assignment of spaces of conformal blocks, hence by the above equivalently modular functor for $G$-Chern-Simons theory restricted to genus-1 surfaces (elliptic curves) is essentially what is encoded in the universal $G$-equivariant elliptic cohomology (equivariant tmf). In fact equivariant elliptic cohomology remembers also the pre-quantum incarnation of the modular functor as a systems of prequantum line bundles over Chern-Simons phase spaces (which are moduli stacks of flat connections) and remembers the quantization-process from there to the actual space of quantum states by forming holomorphic sections. See at equivariant elliptic cohomology – Idea – Interpretation in Quantum field theory for more on this.
holographic principle in quantum field theory
bulk field theory | boundary field theory |
---|---|
dimension $n+1$ | dimension $n$ |
field | source |
wave function | correlation function |
space of quantum states | conformal blocks |
A review is around p. 30 of
On the Knizhnik-Zamolodchikov connection on configuration spaces of point? and conformal blocks:
Detailed discussion in terms of conformal nets is in
See also
A. Tsuchiya, Kenji Ueno, Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Adv. Studies in Pure Math. 19, 459–566, Academic Press (1989) MR92a:81191
Kenji Ueno, Conformal field theory with gauge symmetry, Fields Institute Monographs 2008 book page
Relation to theta functions:
Arnaud Beauville, Yves Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385 - 419, euclid, alg-geom/9309003, MR1289330
Arnaud Beauville, Conformal blocks, fusion rings and the Verlinde formula, Proc. of the Hirzebruch 65 Conf. on Algebraic Geometry, Israel Math. Conf. Proc. 9, 75-96 (1996) pdf
Krzysztof Gawędzki, Lectures on CFT (from Park City, published in QFT and strings for mathematicians, Dijkgraaf at al editors, site, source, dvi, ps
A.A. Beilinson, Yu.I. Manin, V.V. Schechtman, Sheaves of Virasoro and Neveu-Schwarz algebras, Lecture Notes in Math. 1289, Springer 1987, 52–66
A.Mironov, A.Morozov, Sh.Shakirov, Conformal blocks as Dotsenko-Fateev integral discriminants, arxiv/1001.0563
The “hypergeometric” 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$]$
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$]$
(for $\mathfrak{sl}(2)$)
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)
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, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39, 575–598 (2014). (doi:10.1007%2Fs40306-014-0088-6, 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)
See also:
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$]$
reviewed in:
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:
Conformal blocks for self-dual higher gauge theory are discussed in
Last revised on May 11, 2022 at 14:39:45. See the history of this page for a list of all contributions to it.