Conformal field theories associated to regular chiral vertex operator algebras. I: Theories over the projective line. (English) Zbl 1074.81065

Nagatomo, Kiyokazu; Tsuchiya, Akihiro

Duke Math. J. 128, No. 3, 393-471 (2005).

Chiral vertex operator algebra is the quartet \((V,J,T,|0\rangle)\) satisfying the conditions: \(V=\oplus_{\Delta=0\sim N}V_\Delta\) is a \(N\)-graded vector space such that \(\dim V_0=1\) with a basis \(|0 \rangle\) and \(\dim V_\Delta<\infty\). \(T \in V_2\). \(J_n:V\to\text{End}_{-n}V=\{\varphi|\varphi(V_\Delta)\subseteq V_{\Delta-n}\}\) is linear. \(T(n)=J_n(T)\), \(T(0)v=\Delta v\), and \(J_0(|0\rangle) =\text{Id}_v\) give rise to a representation of the Virasolo algebra on \(V\). The authors construct conformal field theories over the projective line \({\mathfrak P}^1 ={\mathfrak C}\cup\{\infty\}\). Let \(A=\{0,1,\dots,N,\infty\}\). \(W_A =(W_a)_{a\in A}\) is the set of distinct points in \({\mathfrak P}^1\) with \(w_\infty=\infty\), \(X=X_A\) is the set of \(W_A\)’s in \(({\mathfrak P}^1)^{N +2}\), and \(Z=Z_A=\{W_A\in X_A|W_0=0, W_1=1\}\). \(M_A=\oplus_{a \in A}M_a\) by \(V\)-modules \(M_a\), \({\mathcal O}_z\): sheaf of germs of holomorphic functions on \(Z\), and \(g(V)\): current Lie algebra. Sheaves of covacua over \(Z\): \({\mathcal V}_z(M_A)={\mathcal O}_z \otimes M_A/ g_z^{\text{out}} (V) ({\mathcal O}_z\otimes M_A)\). Next \(|A|\), \(|B|\geq 3\), \(\overline A=A-\{0_A\}\), \(\overline B=B-\{\infty_B\}\), and \(C= \overline A\cup \overline B\). \(0_A,1_A, \infty_A\); \(0_B,1_B,\infty_B\in B\); \(0_B,1_A,\infty_A\in C\): fixed points. They prove the theorem: \({\mathcal V}_D(M_c)\cong\oplus_{i=0\sim r}({\mathcal V}_D^A (M_{\overline A}\otimes L_i)\otimes{\mathcal V}_D^B(L^+_i\otimes M_{\overline B}))\). \({\mathcal V}_D(M_c)\) are defined by using the trivial \({\mathfrak P}^1\)-bundle \(D\times {\mathfrak P}^1\to D\) and cross sections: \(S_a:(W_A,W_B)\to (W_A,W_B,W_a)\), \(S_b:(W_A,W_B) \to(W_A,W_B,W_b)\).

Reviewer: Hideo Yamagata (Osaka)

MSC:

81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
17B69	Vertex operators; vertex operator algebras and related structures
17B81	Applications of Lie (super)algebras to physics, etc.
32K07	Formal and graded complex spaces
76R10	Free convection
18F20	Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)

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