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For a 2-group constructed from a finite group and 3-cocycle, we provide an explicit description of the bicategory of flat 2-group bundles on an oriented surface in terms of weak representations of the fundamental group. We show that this bicategory encodes (flat) string structures. Furthermore, we identify the space of isomorphism classes of objects with Freed and Quinn's line bundle appearing in Chern-Simons theory of a finite group.

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A bstract
In general quantum field theories (QFTs), ordinary (0-form) global symmetries and 1-form symmetries can combine into 2-group global symmetries. We describe this phenomenon in detail using the language of symmetry defects. We exhibit a simple procedure to determine the (possible) 2-group global symmetry of a given QFT, and provide a classification of the related ’t Hooft anomalies (for symmetries not acting on spacetime). We also describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory. Finally, we provide a variety of examples, ranging from TQFTs (gapped systems) to gapless QFTs. Along the way, we stress that the “obstruction to symmetry fractionalization” discussed in some condensed matter literature is really an instance of 2-group global symmetry.

We study the gerbal representations of a finite group $G$ or, equivalently,
module categories over Ostrik's category $Vec_G^\alpha$ for a 3-cocycle
$\alpha$. We adapt Bartlett's string diagram formalism to this situation to
prove that the categorical character of a gerbal representation is a module
over the twisted Drinfeld double $D^\alpha(G)$. We interpret this twisted
Drinfeld double in terms of the inertia groupoid of a categorical group.

We introduce certain relative differential characters which we call
\emph{Cheeger-Chern-Simons characters}. These combine the well-known
Cheeger-Simons characters with Chern-Simons forms. In the same way as the
Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed
manifolds, the Cheeger-Chern-Simons characters generalize Chern-Simons
invariants of oriented manifolds with boundary.
Using Cheeger-Chern-Simons characters, we introduce the notion of
differential trivializations of universal characteristic classes. Specializing
to the class $\frac{1}{2} p_1 \in H^4(B\mathrm{Spin}_n;\mathbb Z)$ this yields
a notion of differential String classes. Differential String classes turn out
to be stable isomorphism classes of geometric String structures.

The present paper is a contribution to categorial index theory. Its main result is the calculation of the Pfaffian line bundle
of a certain family of real Dirac operators as an object in the category of line bundles. Furthermore, it is shown how string
structures give rise to trivialisations of that Pfaffian.

This survey discusses our results and conjectures concerning supersymmetric
field theories and their relationship to cohomology theories. A careful
definition of supersymmetric Euclidean field theories is given, refining
Segal's axioms for conformal field theories. We state and give an outline of
the proof of various results relating field theories to cohomology theories.

The symposium held in honour of the 60th birthday of Graeme Segal brought together leading physicists and mathematicians. Its topics were centred around string theory, M-theory, and quantum gravity on the one hand, and K-theory, elliptic cohomology, quantum cohomology and string topology on the other. Geometry and quantum physics developed in parallel since the recognition of the central role of non-abelian gauge theory in elementary particle physics in the late seventies and the emerging study of super-symmetry and string theory. With its selection of survey and research articles these proceedings fulfil the dual role of reporting on developments in the field and defining directions for future research. For the first time Graeme Segal's manuscript 'The definition of Conformal Field Theory' is published, which has been greatly influential over more than ten years. An introduction by the author puts it into the present context.

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the relationship between $S^1$-gerbes and groupoid $S^1$-central extensions. We define connective structures, curvings for groupoid $S^1$-central extensions extending the corresponding notions of Brylinski and Hitchin for $S^1$-gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both $S^1$-bundles and $S^1$-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of $S^1$-central extensions with prescribed curvature-like data.

I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2-groups, and bundles with a suitable notion of 2-bundle. To link this with previous work, I show that certain 2-categories of principal 2-bundles are equivalent to certain 2-categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2-bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2-category of 2-bundles over a given 2-space under a given 2-group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2-transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2-space is the 2-space corresponding to a given space and the 2-group is the automorphism 2-group of a given group, then this 2-category is equivalent to the 2-category of gerbes over that space under that group (being described by the same cohomological data).

A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse": an object y such that x tensor y and y tensor x are isomorphic to 1. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse x* and isomorphisms i_x: 1 -> x tensor x* and e_x: x* tensor x -> 1 forming an adjunction. We describe 2-categories of weak and coherent 2-groups and an "improvement" 2-functor that turns weak 2-groups into coherent ones, and prove that this 2-functor is a 2-equivalence of 2-categories. We internalize the concept of coherent 2-group, which gives a quick way to define Lie 2-groups. We give a tour of examples, including the "fundamental 2-group" of a space and various Lie 2-groups. We also explain how coherent 2-groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G_hbar (for integral values of hbar) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2-groups are built using Chern-Simons theory, and are closely related to the Lie 2-algebras g_hbar (for real hbar) described in a companion paper.

In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v_2-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology.

Categorifying the concept of topological group, one obtains the notion of a “topological 2-group”. This in turn allows a theory of “principal 2-bundles” generalizing the usual theory of principal bundles. It is well known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the Čech cohomology Ĥ1(M,G) or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurčo, Baas–Bökstedt–Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and the Čech cohomology Ĥ1(M, G) with coefficients in a topological 2-group G, also known as “nonabelian cohomology”. Then we give an elementary proof that under mild conditions on M and G there is a bijection Ĥ1(M, G) ≡ [M,B|G|]G] where B|G| is the classifying space of the geometric realization of the nerve of G. Applying this result to the “string 2-group” String(G) of a simply-connected compact simple Lie groupG, it follows that principal String(G)-2-bundles have rational characteristic classes coming from elements of H*(BG,Q)/‹c›, where c is any generator of H
4(BG,Q).

This paper studies connections between generalized moonshine and elliptic
cohomology with a focus on the action of the Hecke correspondence and its
implications for the notion of replicability.

We present a calculation, which shows how the moduli of complex analytic elliptic curves arises naturally from the Borel cohomology of an extended moduli space of $U(1)$-bundles on a torus which admit flat connection. More generally, we show how the analogous calculation, applied to a moduli space of $U(1)^d$-bundles on a torus equipped with vanishing degree 4 characteristic class, give rise to Looijenga line bundles. We then speculate on the relation of these calculations to complex analytic equivariant elliptic cohomology.

The theory of principal bundles makes sense in any \(\infty \)-topos, such as the \(\infty \)-topos of topological, of smooth, or of otherwise geometric \(\infty \)-groupoids/\(\infty \)-stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure \(\infty \)-group \(G\) these \(G\)-principal
\(\infty \)-bundles reproduce various higher structures that have been considered in the literature and further generalize these to a full geometric model for twisted higher nonabelian sheaf cohomology. We discuss here this general abstract theory of principal \(\infty \)-bundles, observing that it is intimately related to the axioms that characterize \(\infty \)-toposes. A central result is a natural equivalence between principal \(\infty \)-bundles and intrinsic nonabelian cocycles, implying the classification of principal \(\infty \)-bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber \(\infty \)-bundles associated to principal \(\infty \)-bundles subsumes a theory of \(\infty \)-gerbes and of twisted
\(\infty \)-bundles, with twists deriving from local coefficient
\(\infty \)-bundles, which we define, relate to extensions of principal \(\infty \)-bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice \(\infty \)-topos.

The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras are also described. This book is suitable for second or third year graduate students. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. By making these tools more accessible, the book helps to break down the technological barrier between experts and casual users of homological algebra.

In this paper we introduce gerbes as semi-strict principal 2-bundles and show how they are parametrised by non-abelian ÿ Cech cohomology. Moreover, we derive their gauge 2-groups (or gauge stacks) from first principles and show that they can be described by group-valued functors, much like in classical bundle theory. Moreover, we show that these 2- groups of group valued functors possess a natural smooth structure (under some mild requirements on the structure group). In the last section we provide some explicit examples. MSC: 55R65, 22E65, 81T13

We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2Cat. Fibred bicategories correspond to trihomomorphisms from a bicategory into Bicat. We describe the Grothendieck construction for each kind of fibration and present a few examples of each. Fibrations in our sense, between bicategories, are closed under composition and are stable under equiv-comma. The free such fibration on a homomorphism is obtained by taking an oplax comma along an identity.

This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in Sect. 1 with an overview of the classical theory of elliptic cohomology. In Sect. 2 we review the theory of
E∞-ring spectra and introduce the language of derived algebraic geometry.We apply this theory in Sect. 3, where we introduce
the notion of an oriented group scheme and describe connection between oriented group schemes and equivariant cohomology theories. In Sect. 4 we sketch a proof
of our main result, which relates the classical theory of elliptic cohomology to the classification of oriented elliptic curves.
In Sect. 5 we discuss various applications of these ideas, many of which rely upon a special feature of elliptic cohomology
which we call 2-equivariance. The theory that we are going to describe lies at the intersection of homotopy theory and algebraic geometry. We have tried
to make our exposition accessible to those who are not specialists in algebraic topology; however, we do assume the reader
is familiar with the language of algebraic geometry, particularly with the theory of elliptic curves. In order to keep our
account readable, we will gloss overmany details, particularly where the use of higher category theory is required. A more
comprehensive account of the material described here, with complete definitions and proofs, will be given in [1].

Perturbative and global world-sheet anomalies are discussed in terms of an infinite dimensional geometry which generalizes the finite dimensional geometry of spin structures. This geometrical structure is related to the Dirac-Ramond operator and provides an insight into the symmetries of string theory.

We show that fermionic left-right level matching is necessary and sufficient to guarantee modular invariance to 1-loop order in orbifold compactification of heterotic string. We also discuss the possibility of turning on discrete torsion on orbifolds.

We recall and partially improve four versions of smooth, non-abelian gerbes:
Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We
prove that all these four versions are equivalent, and so establish new
relations between interesting recent developments. Prominent partial results we
prove are a bijection between continuous and smooth non-abelian cohomology, and
an explicit equivalence between bundle gerbes and principal 2-bundles as
2-stacks.

We present a finite-dimensional and smooth formulation of string structures
on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated
to this bundle. Our formulation is particularly suitable to deal with string
connections: it enables us to prove that every string structure admits a string
connection and that the possible choices form an affine space. Further we
discover a new relation between string connections, 3-forms on the base
manifold, and degree three differential cohomology. We also discuss in detail
the relation between our formulation of string connections and the original
version of Stolz and Teichner.

We show that three dimensional Chern-Simons gauge theories with a compact gauge group G (not necessarily connected or simply connected) can be classified by the integer cohomology group H (BG; Z). In a similar way, possible Wess-Zumino interactions of such a group G are classified by H (G; Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models involves a certain natural map from H (G; Z). We generalize this correspondence to topological `spin' theories, which are defined on three manifolds with spin structure, and are related to what might be called Z 2 graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.

The twisted Drinfeld double (or quasi-quantum double) of a finite group with a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3-cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties of the Drinfeld double, such as representations being classified by their characters. This is all motivated by gerbes and 3-dimensional topological quantum field theory. In particular the representation category of the twisted Drinfeld double is viewed as the `space of sections' associated to a transgressed gerbe over the loop groupoid. Comment: 25 pages, 10 pictures

These theories, which are surely some of the simplest possible quantum field theories, were introduced in a paper of Dijkgraaf and Witten. The path integral reduces to a finite sum, so it is quite amenable to direct mathematical study. Although the theory exisits in arbitrary dimensions, it is most interesting in $2+1$~dimensions, where it has a ``modular structure.'' This is related to quantum groups, and the precise details may give clues as to what happens in other contexts. This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). 1 encapsulated postscript file was submitted separately in uuencoded tar-compressed format. Comment: 44 pages + 1 figure (revised version, this revision fixes some mistakes, changes some notation, clarifies some arguments, redraws the figure, and generally improves the previous version.)

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Lifting problems and transgression for non-abelian gerbes

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