INTRODUCTION 9 up to homotopy in derived geometry [Lur09a]. The resulting derived covariant phase space is known in physics in terms of its Batalin-Vilkovisky–Becchi-Rouet- Stora-Tyutin (BV-BRST) complex. This article reviews the powerful description of variational calculus and the construction of the BV-BRST complex in terms of D-geometry [BeDr04] – the geometry over de Rham spaces – and uses this to analyze subtle finiteness conditions on the BV-construction. 2. Orientifold pr´ ecis – by Jacques Distler, Daniel Freed, and Gregory Moore. The consistent quantization of the sigma model for the (super-)string famously requires the target space geometry to satisfy the Euler-Lagrange equations of an effective supergravity theory on target space. In addition there are subtle cohomo- logical conditions for the cancellation of fermionic worldsheet anomalies. This article discusses the intricate conditions on the differential cohomology of the background fields – namely the Neveu-Schwarz B-field in ordinary differential cohomology (or a slight variant, which the authors discuss) and the RR-field in differential K-theory twisted by the B-field – in particular if target space is allowed to be not just a smooth manifold but more generally an orbifold and even more generally an orientifold. Among other things, the result shows that the “landscape of string theory vacua” – roughly the moduli space of consistent perturbative string backgrounds (cf. [Do10]) – is more subtle an object than often assumed in the literature. III. Two-dimensional Quantum Field Theories 1. Surface operators in 3d TFT and 2d Rational CFT – by Anton Kapustin and Natalia Saulina. Ever since Witten’s work on 3-dimensional Chern-Simons theory it was known that by a holographic principle this theory induces a 2d CFT on 2-dimensional boundary surfaces. This article amplifies that if one thinks of the 3d Chern- Simons TQFT as a topological QFT with defects, then the structures formed by codimension-0 defects bounded by codimension-1 defects naturally reproduce, holo- graphically, the description of 2d CFT by Frobenius algebra objects in modular tensor categories [FRS06]. 2. Conformal field theory and a new geometry – by Liang Kong. While the previous article has shown that the concept of TQFT together with the holographic principle naturally imply that 2-dimensional CFT is encoded by monoid objects in modular tensor categories, this article reviews a series of strong results about the details of this encoding. In view of these results and since every 2d CFT also induces an effective target space geometry – as described in more detail in the following contribution – the author amplifies the fact that stringy geometry is thus presented by a categorified version of the familiar duality between spaces and algebras: now for algebra objects internal to suitable monoidal categories.

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