Gaussian free field and conformal field theory (Q2851018)

The first part of these lectures (Lecture 1---Lecture 12) is an elementary exposition on conformal field theory (CFT) of a simply connected domain with a marked point on the boundary from the point of view of random or statistical fields. Then the chordal Schramm-Loewner evolution (SLE) theory [\textit{G. Lawler}, IAS/Parc City Math. Ser. 16, 231--295 (2009; Zbl 1180.82002)] is discussed from the point of view Conformal Field Theory in the second part (Appendix 13 -- Lecture 15: cf. \textit{M. Bauer} and \textit{D. Bernard} [Commun. Math. Phys. 239, No. 3, 493--521 (2003; Zbl 1046.81091)], \textit{R. Friedrich} and \textit{W. Werner} [Commun. Math. Phys. 243, No. 1, 105--122 (2003; Zbl 1030.60095)]).NEWLINENEWLINENEWLINEThe outline of these lectures is as follows: In Lecture 1, the Gaussian free filed \(\Phi\) in \(D\) with boundary condition, where \(D\) is a planar domain with the Green's function \(G= G_D(\zeta, z)\), Wick's product \(f\odot g\), Fock space of Gaussian free fields, etc. are defined. The authors say all fields in this lecture are constructed from the Gaussian free field and its derivatives by means of Wick's calculus. These fields are called Fock space fields. Expositions of Fock space functionals and fields (distributional fields) are given in Appendix 2. In Lecture 3, the operator product expansion (OPE), the expansion of the tensor product of two fields, near diagonal, is explained. If NEWLINENEWLINE\[NEWLINEX(\zeta) Y(z)= \sum_n C_n(z)(\zeta- z)^n,NEWLINE\]NEWLINENEWLINE then the \(*_n\) product \(X*_n Y\) is defined to be \(C_n\). \(X*_0 y\) is denoted by \(X*Y\). Geometric (conformal) structures are equipped to Fock space fields in Lecture 3. The transformation law from one coordinate chart to another is complicated. For example, Schwarzian forms of order \(\mu\) are defined by the transformation law NEWLINE\[NEWLINEf= (h')^2\widetilde f\circ h+\mu S_h,NEWLINE\]NEWLINE where \(S_h\) is the Schwarz derivative of \(h\). Lie derivatives \({\mathcal L}_v\) is also defined in this lecture. Lecture 5 deals with stress tensor and Ward's identities. Ward's identities for finite Boltzmann-Gibbs ensembles are explained in Appendix 6. A pair \(W= (A^+, A^-)\) is said to be a stress tensor if \(A^+\) is holomorphic, \(A^-\) is anti holomorphic, and \({\mathcal L}_v X= A^+_v X+ A^-_v X\) (the residue form of Ward's identity). \({\mathcal F}(W)={\mathcal F}(A^+, A^-)\) is the linear space of all Fock space fields \(X\) such that \(W\) is a stress tensor of \(X\). Then a Fock space field \(T\) is said to be a Virasoro field for the family \({\mathcal F}(W)\) if \(T\in{\mathcal F}(W)\) and \(T-A\) is a non-random holomorphic Schwarzian form. The existence of Virasoro field is proved in Appendix 8. It is unique and a Schwarzian form. For the Gaussian free filed \(\Phi\), it takes the form NEWLINENEWLINE\[NEWLINET= {1\over 2} J* J,\quad J=d\Phi.NEWLINE\]NEWLINENEWLINE In Lecture 7, after explaining properties of Virasora field, Virasoro algebra is constructed and shows \(c= 12\mu\), where \(c\) is the central charge of the algebra and \(\mu\) is the order of the form. Then mainly focused on singular vectors, representation theory of the Virasoro algebra is reviewed [\textit{K. Iohara} and \textit{Y. Koga}, Representation theory of the Virasoro algebra. Berlin: Springer (2011; Zbl 1222.17001)]. In Appendix 9, relation between operator algebra formalism and theory of this lecture is outlined. Then algeraic modification of the Virasoro algebra, Fiarlie's construction [\textit{V. G. Kac} and \textit{A. K. Raina}, Bombay lecture on highest weight representations of infinite-dimensional Lie algebras. Singapore, etc.: World Scientific (1987; Zbl 0668.17012)] is derived. This is applied to the modification of the Gaussian free field in a simply connected domain \(D\) with a marked boundary point \(q\) in Lecture 10. The authors remark such modification appeared in the context of chordal SLE theory [\textit{I. Rushkin} et al., J. Phys. A, Math. Theor. 40, No. 9, 2165--2195 (2007; Zbl 1160.82324)]. As a result, vertex fields in \(\mathcal{F}_b\), the OPE families corresponding to the Gaussian free fields parametrized by real number \(b\), produce singular null vector. Combining the degeneracy equations with Ward's identities, the equations of Belavin-Polyakov-Zanolodehikov equations (BPZ equations) are derived. In Appendix 11, a characterization of level two degenerate vertex fields; NEWLINE\[NEWLINET*{\mathcal V}={1\over 2a^2} \partial^2{\mathcal V},NEWLINE\]NEWLINE is derived from the fact that vertex fields in \({\mathcal F}_b\) are primary fields of the corresponding current algebra. Then the author derive the Knizhnik-Zamolodchikov (KZ) equations for corresponding primary fields. To interpret chiral fields, which are described as elements of ``holomorphic part'' of conformal field theory, in the ``statistical'' setting, multivalued confromal Fock space fields are investigated in Lecture 12. Especially, chiral vertex fields exist only as bi-variant objects. As a result, degenerate equation is stated for this case Then, combining Ward's identity, it is shown the correlation functions of any fields in \({\mathcal F}_b\) under the insertion of \({\mathcal V}^{ia}_*(\xi)\) are annihilated by the operator NEWLINE\[NEWLINE{2\over 2a^2} \partial^2_\xi-{\mathcal L}_{v_\xi},NEWLINE\]NEWLINE which appears in Itô's calculus. Let \((D,p,q)\) be a simply connected domain with two marked points, and let \(g_t(z)\) be the solution of the chordal equation Schramm-Loewner Evolution (SLE) NEWLINE\[NEWLINE\partial_t g_t(zx)= {2\over g_t(z)- \sqrt{\kappa}B_t},\quad B_0= 0,NEWLINE\]NEWLINE where \(B_t\) is the Brownian motion from \(p\) to \(q\) and \(g_0:(D,p,q)\to(\mathbb{H},0,\infty)\) is a given conformal map. Then \(w_t(z)= g_t(z)- \sqrt{\kappa}B_t\) is a conformal map from \(D_t= \{z\in D:\tau_t> t\}\) onto the upper half plane \(\mathbb{H}\). Hence we can discuss SLE theory from the point of view of CFT, which is given in Lecture 14. Before to state SLE, the relation of parameter \(\kappa\) of SLE and central charge of the Virasoro algebra is investigated in Appendix 13. As a result, if NEWLINE\[NEWLINEa= \sqrt{{2\over\pi}},\quad b= a\Biggl({\kappa\over 4}- 1\Biggr),NEWLINE\]NEWLINE then under insertion of a boundary vertex \(V^{ia}_*(p)\), all fields in the theory \({\mathcal F}_b\) satisfy the field ``Markov property'' with respect to the SLE filtration. Interpreting the insertion \(V^{ia}_*(p)\) as a boundary condition changing operator, Cardy-type equations for correlation functions under this insertion, are derived [\textit{J. L. Cardy}, Nucl. Phys. B 270, No. 2, 186--204 (1986; Zbl 0689.17016)]. Further examples related to vertex fields are presented in Lecture 15, the last lecture.