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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