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
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.