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arXiv:hep-th/9412199

Affine and Yangian Symmetries in $SU(2)_1$ Conformal Field Theory

Peter Bouwknegt, Andreas Ludwig, Kareljan Schoutens

Published Dec 23, 1994• 9 citations
arXivHTMLPDF

Abstract

In these lectures, we study and compare two different formulations of $SU(2)$, level $k=1$, Wess-Zumino-Witten conformal field theory. The first, conventional, formulation employs the affine symmetry of the model; in this approach correlation functions are derived from the so-called Knizhnik-Zamolodchikov equations. The second formulation is based on an entirely different algebraic structure, the so-called Yangian $Y(sl_2)$. In this approach, the Hilbert space of the theory is obtained by repeated application of modes of the so-called spinon field, which has $SU(2)$ spin $j=\thalf$ and obeys fractional (semionic) statistics. We show how this new formulation, which can be generalized to many other rational conformal field theories, can be used to compute correlation functions and to obtain new expressions for the Virasoro and affine characters in the theory. [Lectures given at the 1994 Trieste Summer School on High Energy Physics and Cosmology, Trieste, July 1994.]

DOI
10.48550/arxiv.hep-th/9412199

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Affine and Yangian Symmetries in SU(2)1SU(2)_1SU(2)1​ Conformal Field Theory
In these lectures, we study and compare two different formulations of SU(2)SU(2)SU(2), level k=1k=1k=1, Wess-Zumino-Witten conformal field theory. The first, conventional, formulation employs the affine symmetry of the model; in this approach correlation functions are derived from the so-called Knizhnik-Zamolodchikov equations. The second formulation is based on an entirely different algebraic structure, the so-called Yangian Y(sl2)Y(sl_2)Y(sl2​). In this approach, the Hilbert space of the theory is obtained by repeated application of modes of the so-called spinon field, which has SU(2)SU(2)SU(2) spin j=\thalfj=\thalfj=\thalf and obeys fractional (semionic) statistics. We show how this new formulation, which can be generalized to many other rational conformal field theories, can be used to compute correlation functions and to obtain new expressions for the Virasoro and affine characters in the theory. [Lectures given at the 1994 Trieste Summer School on High Energy Physics and Cosmology, Trieste, July 1994.]
The relation between the chiral-spin-liquid state and the correlators of the Wess-Zumino-Witten model is shown. I argue that in the Kalmeyer-Laughlin state there is a hidden symmetry, associated with the conformal invariance in the continuum limit. The fractional statistics of the spinons naturally follows from the conformal weight for the primary fields. The relation between the Kalmeyer-Laughlin and the spin-singlet quantum-Hall-effect (QHE) wave functions is shown and leads to the possibility of the spinon excitations in the spin-singlet QHE.
These lecture notes provide a comprehensive introduction to 2d conformal field theory, covering foundational topics such as free fields, minimal models, Zamolodchikov's ccc-theorem, Wess-Zumino-Witten models, modular invariant partition functions, and the connections to entanglement entropy. The material presented is based on lectures at the Southeast University Summer School and the course at Fudan University.
The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of nnn point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.
This article presents a systematic development of conformal field theory from first principles in quantum field theory, emphasizing complete derivations of the conformal group in arbitrary dimensions from the conformal Killing equation, proving its isomorphism to SO(d+1,1)\mathrm{SO}(d+1,1)SO(d+1,1), and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra [Lm,Ln]=(m−n)Lm+n+c12(m3−m)δm+n,0[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}[Lm​,Ln​]=(m−n)Lm+n​+12c​(m3−m)δm+n,0​ from infinitesimal generators ℓn=−zn+1∂z\ell_n = -z^{n+1}\partial_zℓn​=−zn+1∂z​, computing the central charge explicitly for free theories (boson c=1c=1c=1, Majorana fermion c=1/2c=1/2c=1/2, bcbcbc-ghosts c=1−3(2λ−1)2c = 1-3(2\lambda-1)^2c=1−3(2λ−1)2) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states ∣h,hˉ⟩|h,\bar{h}\rangle∣h,hˉ⟩, deriving the transformation law T′(w)=(dw/dz)−2[T(z)−(c/12){w;z}]T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]T′(w)=(dw/dz)−2[T(z)−(c/12){w;z}] that reveals the geometric origin of the conformal anomaly.
We employ the SU(n)k\mathrm{SU}{(n)}_{k}SU(n)k​ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions. The spins on all lattice sites are chosen to transform under the SU(n)\mathrm{SU}(n)SU(n) irreducible representation with a single row and kkk boxes in the Young tableau. It is demonstrated that the wave functions can be reinterpreted as parton states, which enables efficient conversion to matrix product states, such that many physical properties can be evaluated directly. In one dimension, these wave functions describe critical spin chains whose universality classes are in one-to-one correspondence with the WZW models used in the construction. In two dimensions, our constructions yield model wave functions for chiral spin liquids, and we show how to find all topological sectors of them in a systematic way. Using the null vectors of Kac-Moody algebras, parent Hamiltonians of the SU(3)k\mathrm{SU}{(3)}_{k}SU(3)k​ series are derived. The SU(3)k\mathrm{SU}{(3)}_{k}SU(3)k​ chiral spin liquids are lattice analogs of non-Abelian spin-singlet fractional quantum Hall states, and the k=2k=2k=2 member hosts Fibonacci anyons.