8 HISHAM SATI AND URS SCHREIBER
We now outline the contents of the volume, highlighting how the various articles
are related and emphasizing how they ﬁt into the big picture that we have drawn
I. Foundations of Quantum Field Theory
1. Models for (∞, n)-Categories and the Cobordism Hypothesis – by Julia
The Schr¨ odinger picture of extended topological quantum ﬁeld theory of dimen-
sion n is formalized as being an (∞, n)-functor on the (∞, n)-category of cobordisms
of dimension n. This article reviews the deﬁnition and construction of the ingredi-
ents of this statement, due to [Lur09b].
This picture is the basis for the formulation of QFTs on cobordisms with struc-
ture. Contributions below discuss cobordisms with defect structure, with conformal
structure and with flat Riemannian structure.
2. From operads to dendroidal sets – by Ittay Weiss.
The higher algebra that appears in the algebraic description of QFT – by local
nets of observables, factorization algebra or chiral algebras – is in general operadic.
For instance the vertex operator algebras appearing in the description of CFT
(see Liang Kong’s contribution below) are algebras over an operad of holomorphic
This article reviews the theory of operads and then discusses a powerful pre-
sentation in terms of dendroidal sets – the operadic analog of what simplicial sets
are for (∞, 1)-categories. This provides the homotopy theory for (∞, 1)-operads,
closely related to the traditional model by topological operads.
3. Field theories with defects and the centre functor – by Alexei Davydov,
Liang Kong and Ingo Runkel.
This article gives a detailed discussion of cobordism categories for cobordisms
with defects/domain walls. An explicit construction of a lattice model of two-
dimensional TQFT with defects is spelled out. The authors isolate a crucial aspect
of the algebraic structure induced by defect TQFTs on their spaces of states: as
opposed to the algebra of ordinary bulk states, that of defect states is in general
non-commutative, but certain worldsheet topologies serve to naturally produce the
centre of these algebras.
Below in Surface operators in 3d TQFT topological ﬁeld theories with defects
are shown to induce, by a holographic principle, algebraic models for 2-dimensional
CFT. In Topological modular forms and conformal nets conformal ﬁeld theories
with defects are considered.
II. Quantization of Field Theories
1. Homotopical Poisson reduction of gauge theories – by Fr´ ed´ eric Paugam.
The basic idea of quantization of a Lagrangian ﬁeld theory is simple: one forms
the covariant phase space given as the critical locus of the action functional, then
forms the quotient by gauge transformations and constructs the canonical symplec-
tic form. Finally, one applies deformation quantization or geometric quantization
to the resulting symplectic manifold.
However, to make this naive picture work, care has to be taken to form both
the intersection (critical locus) and the quotient (by symmetries) not naively but