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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