# Cappelli, Itzykson, Zuber (a) ## Modular invariant partition functions in two dimensions \[Links: [DOI](https://doi.org/10.1016/0550-3213(87)90155-6)\] \[Abstract: We present a systematic study of [[0612 Modular invariance|modular invariance]] of partition functions, relevant both for two-dimensional minimal conformal invariant theories and for string propagation on a $SU(2)$ group manifold. We conjecture that all solutions are labelled by simply laced Lie algebras.\] # Cappelli, Itzykson, Zuber (b) ## The A-D-E classification of minimal and $A^{(1)}_1$ conformal invariant theories \[Links: [DOI](https://link.springer.com/article/10.1007/BF01221394)\] \[Abstract: We present a detailed and complete proof of our earlier conjecture on the classification of minimal conformal invariant theories. This is based on an exhaustive construction of all modular invariant sesquilinear forms, with positive integral coefficients, in the characters of the [[0596 Virasoro TQFT|Virasoro]] or of the $A^{(1)}_1$ [[0069 Kac-Moody algebra|Kac-Moody algebras]], which describe the corresponding partition functions on a torus. A remarkable correspondence emerges with simply laced Lie algebras.\] # Friedan, Shenker ## The Analytic Geometry of Two-Dimensional Conformal Field Theory \[Links: [Inspire](https://inspirehep.net/literature/230038)\] \[Abstract: Two-dimensional conformal field theory is formulated as analytic geometry on the universal moduli space of Riemann surfaces.\] # Kirillov, Neretin ## The variety $A_n$ of $n$-dimensional Lie algebra structures \[Links: [Scholar](https://scholar.google.com/citations?view_op=view_citation&hl=en&user=VNy9-bYAAAAJ&citation_for_view=VNy9-bYAAAAJ:nb7KW1ujOQ8C), [DOI](https://www.ams.org/books/trans2/137/), [PDF](https://www.mat.univie.ac.at/~neretin/kiri87a.pdf)\] # Metsaev, Tseytlin ## Curvature Cubed Terms in String Theory Effective Actions \[Links: [Inspire](https://inspirehep.net/literature/20940), [PRB](https://www.sciencedirect.com/science/article/abs/pii/0370269387915279?via%3Dihub)\] \[Abstract: Starting from the three- and four-point amplitudes we determine the ${R}^3$-terms in the gravitational [[0329 String effective action|effective actions]] in the Bose, heterotic and type II superstring theories. We prove the absence of unambiguous $R^3$-terms (including the Gauss-Bonnet $R^3$-invariant) in the heterotic and superstring cases. As a by-product, we check the presence of the square of the Lorentz Chern-Simons term in the heterotic string effective action and obtain the three-loop contribution to the $\beta$-function of the bosonic $\sigma$-model. We also discuss the structure of the field redefinition ambiguity in a gravitational effective action.\] ## Refs - [[0006 Higher-derivative gravity]] - [[0329 String effective action]] # Narain, Sarmadi, Witten ## A Note on Toroidal Compactification of Heterotic String Theory \[Links: [Inspire](https://inspirehep.net/literature/18664)\] \[Abstract: The connection of recently constructed lower dimensional heterotic strings with conventional toroidal compactification is clarified.\] # Zamolodchikov ## Conformal symmetry in 2d space: recursion representation of conformal block \[Links: [PDF](http://cftconf.itp.ac.ru/seminars/AlZamolodchikov/TMP_1987_73_01088.pdf)\] \[Abstract: The four-point [[0031 Conformal block|conformal block]] plays an important part in the analysis of the conformally invariant operator algebra in two-dimensional space. The behavior of the conformal block is calculated in the present paper in the limit in which the dimension $\Delta$ of the intermediate operator tends to infinity. This makes it possible to construct a recursion relation for this function that connects the conformal block at arbitrary $\Delta$ to the blocks corresponding to the dimensions of the zero vectors in the degenerate representations of the [[0032 Virasoro algebra|Virasoro algebra]]. The relation is convenient for calculating the expansion of the [[0031 Conformal block|conformal block]] in powers of the uniformizing parameter $q = \exp i\pi\tau$.\]