't Hooft anomalies of discrete gauge theories and non-abelian group cohomology

We study discrete symmetries of Dijkgraaf-Witten theories and their gauging in the framework of (extended) functorial quantum field theory. Non-abelian group cohomology is used to describe discrete symmetries and we derive concrete conditions for such a symmetry to admit 't Hooft anomalies in terms of the Lyndon-Hochschild-Serre spectral sequence. We give an explicit realization of a discrete gauge theory with 't Hooft anomaly as a state on the boundary of a higher-dimensional Dijkgraaf-Witten theory. This allows us to calculate the 2-cocycle twisting the projective representation of physical symmetries via transgression. We present a general discussion of the bulk-boundary correspondence at the level of partition functions and state spaces, which we make explicit for discrete gauge theories.Comment: 46 pages, 1 figure; v2: minor corrections and clarifying comments added, references updated; Final version to appear in Communications in Mathematical Physic

Extended quantum field theory, index theory and the parity anomaly

We use techniques from functorial quantum field theory to provide a geometric description of the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We give an explicit construction of a geometric cobordism bicategory which incorporates general background fields in a stack, and together with the theory of symmetric monoidal bicategories we use it to provide the concrete forms of invertible extended quantum field theories which capture anomalies in both the path integral and Hamiltonian frameworks. Specialising this situation by using the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners due to Loya and Melrose, we obtain a new Hamiltonian perspective on the parity anomaly. We compute explicitly the 2-cocycle of the projective representation of the gauge symmetry on the quantum state space, which is defined in a parity-symmetric way by suitably augmenting the standard chiral fermionic Fock spaces with Lagrangian subspaces of zero modes of the Dirac Hamiltonian that naturally appear in the index theorem. We describe the significance of our constructions for the bulk-boundary correspondence in a large class of time-reversal invariant gauge-gravity symmetry-protected topological phases of quantum matter with gapless charged boundary fermions, including the standard topological insulator in 3+1 dimensions.Comment: 63 pages, 3 figures; v2: clarifying comments and references added; Final version to be published in Communications in Mathematical Physic

The distinguished invertible object as ribbon dualizing object in the Drinfeld center

We prove that the Drinfeld center $Z(\mathcal{C})$ of a pivotal finite tensor category $\mathcal{C}$ comes with the structure of a ribbon Grothendieck-Verdier category in the sense of Boyarchenko-Drinfeld. Phrased operadically, this makes $Z(\mathcal{C})$ into a cyclic algebra over the framed $E_2$-operad. The underlying object of the dualizing object is the distinguished invertible object of $\mathcal{C}$ appearing in the well-known Radford isomorphism of Etingof-Nikshych-Ostrik. Up to equivalence, this is the unique ribbon Grothendieck-Verdier structure on $Z(\mathcal{C})$ extending the canonical balanced braided structure that $Z(\mathcal{C})$ already comes equipped with. The duality functor of this ribbon Grothendieck-Verdier structure coincides with the rigid duality if and only if $\mathcal{C}$ is spherical in the sense of Douglas-Schommer-Pries-Snyder. The main topological consequence of our algebraic result is that $Z(\mathcal{C})$ gives rise to an ansular functor, in fact even a modular functor regardless of whether $\mathcal{C}$ is spherical or not. In order to prove the aforementioned uniqueness statement for the ribbon Grothendieck-Verdier structure, we derive a seven-term exact sequence characterizing the space of ribbon Grothendieck-Verdier structures on a balanced braided category. This sequence features the Picard group of the balanced version of the M\"uger center of the balanced braided category.Comment: 21 pages, diagrams partly in color; v2: minor changes, Cor. 3.1 strengthene

The Little Bundles Operad

Hurwitz spaces are homotopy quotients of the braid group action on the moduli space of principal bundles over a punctured plane. By considering a certain model for this homotopy quotient we build an aspherical topological operad that we call the little bundles operad. As our main result, we describe this operad as a groupoid-valued operad in terms of generators and relations and prove that the categorical little bundles algebras are precisely Turaev's crossed categories. Moreover, we prove that the evaluation on the circle of a homotopical two-dimensional equivariant topolological field theory yields a little bundles algebra up to coherent homotopy.Comment: 33 pages, 6 figures, lots of diagrams; small changes; final version to appear in Algebr. Geom. Topo

Cyclic framed little disks algebras, Grothendieck-Verdier duality and handlebody group representations

We characterize cyclic algebras over the associative and the framed little disks operad in any symmetric monoidal bicategory. The cyclicity is appropriately treated in a homotopy coherent way. When the symmetric monoidal bicategory is specified to be a certain symmetric monoidal bicategory of linear categories subject to finiteness conditions, we prove that cyclic associative and cyclic framed little disks algebras, respectively, are equivalent to pivotal Grothendieck-Verdier categories and balanced braided Grothendieck-Verdier categories, a type of category that was introduced by Boyarchenko-Drinfeld based on Barr's notion of a $*$-autonomous category. We use these results and Costello's modular envelope construction to obtain two applications to quantum topology: I) We extract a consistent system of handlebody group representations from any balanced braided Grothendieck-Verdier category inside a certain symmetric monoidal bicategory of linear categories and show that this generalizes the handlebody part of Lyubashenko's mapping class group representations. II) We establish a Grothendieck-Verdier duality for the category extracted from a modular functor by evaluation on the circle (without any assumption on semisimplicity), thereby generalizing results of Tillmann and Bakalov-Kirillov.Comment: 60 pages, lots of figures and diagrams (some in color); v2: presentation improved, details and examples added in several place

Spread and establishment of Aedes albopictus in southern Switzerland between 2003 and 2014 : an analysis of oviposition data and weather conditions

The Asian tiger mosquito, Aedes albopictus, is a highly invasive mosquito species of public health importance. In the wake of its arrival in neighbouring Italy the authorities of the canton of Ticino in southern Switzerland initiated a surveillance programme in 2000 that is still on-going. Here we explored the unique data set, compiled from 2003 to 2014, to analyse the local dynamic of introduction and establishment of Ae. albopictus, its relative density in relation to precipitation and temperature, and its potential distribution at the passage from southern to northern Europe.; The presence of Ae. albopictus was recorded by ovitraps placed across Ticino. In addition to presence-absence, the relationship between relative egg densities and year, month, temperature and precipitation was analysed by a generalised linear mixed model.; Since its first detection in 2003 at Ticino's border with Italy Ae. albopictus has continuously spread north across the lower valleys, mainly along the trans-European motorway, E35. Detailed local analysis showed that industrial areas were colonised by the mosquito before residential areas and that, afterwards, the mosquito was more present in residential than in industrial areas. Ae. albopictus appeared sporadically and then became more present in the same places the following years, suggesting gradual establishment of locally reproducing populations that manage to overwinter. This trend continues as witnessed by both a growing area being infested and increasing egg counts in the ovitraps. There was a clear South-North gradient with more traps being repeatedly positive in the South and fewer eggs laid during periods of intensive precipitation. In the North, the mosquito appeared repeatedly through the years, but never managed to establish, probably because of unfavourable weather conditions and low road traffic.; Given the present results we assume that additional areas may still become infested. While the current study provides good estimates of relative egg densities and shows the local and regional dynamics of Ae. albopictus invasion, additional parameters ought to be measured to make an objective risk assessment for epidemic disease transmission. The likelihood of Ae. albopictus to further spread and increase in densities calls for continued surveillance

Probing the Constituent Structure of Black Holes

Based on recent ideas, we propose a framework for the description of black holes in terms of constituent graviton degrees of freedom. Within this formalism a large black hole can be understood as a bound state of N longitudinal gravitons. In this context black holes are similar to baryonic bound states in quantum chromodynamics which are described by fundamental quark degrees of freedom. As a quantitative tool we employ a quantum bound state description originally developed in QCD that allows to consider black holes in a relativistic Hartree like framework. As an application of our framework we calculate the cross section for scattering processes between graviton emitters outside of a Schwarzschild black hole and absorbers in its interior, that is gravitons. We show that these scatterings allow to directly extract structural observables such as the momentum distribution of black hole constituents.Comment: Extended version, accepted for publication in JHE

Hard-switched switched capacitor converter design

Switched capacitor (SC) converters are becoming quite popular for use in DC-DC power conversion. The concept of equivalent resistance in SC converters is frequently used to determine the conduction losses due to the load current. A variety of methodologies have been presented in the literature to predict the equivalent resistance in hard-switched SC converters. However, a majority of the methods described are difficult to apply to general SC converter topologies. Additionally, previous works have not considered all nonidealities in their analysis, such as switching losses or stray inductances. This work presents a generalized and easy to use model to determine the equivalent resistance of any high-order SC converter. The presented concepts are combined to derive a complete loss model for SC converters. The challenges of implementing output voltage regulation are addressed as well. A current-fed SC topology is presented in this work that overcomes the problems associated with voltage regulation. The new topology opens up a variety of additional operating modes, such as power sharing. These additional operating modes are explored as well. The presented concepts are verified using digital simulation tools and prototype converters. --Abstract, page iii