Abstract:
An interpretation of the gauge anomaly of the two-dimensional multi-phase sigma model is presented in terms of an obstruction to the existence of a topological defect network implementing a local trivialisation of the gauged sigma model.

Abstract:
This is the second in a series of papers discussing in the framework of gerbe theory canonical and geometric aspects of the 2d nonlinear sigma model in the presence of conformal defects in the worldsheet. Employing the formal tools worked out in the first paper of the series, 1101.1126 [hep-th], a thorough analysis of rigid symmetries of the sigma model is carried out with emphasis on algebraic structures on generalised tangent bundles over the target space of the theory and over its state space that give rise to a realisation of the symmetry algebra on states. The analysis leads to a proposal for a novel differential-algebraic construct extending the original definition of the (gerbe-twisted) Courant algebroid on the generalised tangent bundles over the target space in a manner codetermined by the structure of the 2-category of abelian bundle gerbes with connection over it. The construct admits a neat interpretation in terms of a relative Cartan calculus associated with the hierarchy of manifolds that compose the target space of the multiphase sigma model. The paper also discusses at length the gauge anomaly for the rigid symmetries, derived and quantified cohomologically in a previous work of Gaw\c{e}dzki, Waldorf and the author. The ensuing reinterpretation of the small gauge anomaly in terms of the twisted rel. Courant algebroid modelling the Poisson algebra of Noether charges of the symmetries is elucidated through an equivalence between a category built from data of the gauged sigma model and that of principal bundles over the worldsheet with a structural action groupoid based on the target space. Finally, the large gauge anomaly is identified with the obstruction to the existence of topological defect networks implementing the action of the gauge group of the gauged sigma model and those giving a local trivialisation of a gauge bundle of an arbitrary topology over the worldsheet.

Abstract:
We define the sigma-model action for world-sheets with embedded defect networks in the presence of a three-form field strength. We derive the defect gluing condition for the sigma-model fields and their derivatives, and use it to distinguish between conformal and topological defects. As an example, we treat the WZW model with defects labelled by elements of the centre Z(G) of the target Lie group G; comparing the holonomy for different defect networks gives rise to a 3-cocycle on Z(G). Next, we describe the factorisation properties of two-dimensional quantum field theories in the presence of defects and compare the correlators for different defect networks in the quantum WZW model. This, again, results in a 3-cocycle on Z(G). We observe that the cocycles obtained in the classical and in the quantum computation are cohomologous.

Abstract:
We give a geometric description of the fusion rules of the affine Lie algebra su(2)_k at a positive integer level k in terms of the k-th power of the basic gerbe over the Lie group SU(2). The gerbe can be trivialised over conjugacy classes corresponding to dominant weights of su(2)_k via a 1-isomorphism. The fusion-rule coefficients are related to the existence of a 2-isomorphism between pullbacks of these 1-isomorphisms to a submanifold of SU(2) x SU(2) determined by the corresponding three conjugacy classes. This construction is motivated by its application in the description of junctions of maximally symmetric defect lines in the Wess-Zumino-Witten model.

Abstract:
The prevention of chronic organic damage and complete inhibition of inflammatory activity of the disease are the main goals in the treatment of systemic lupus erythematosus (SLE). Current therapies of SLE are not effective enough and they may cause various serious side effects. Biological therapies, affecting important pathogenetic disturbances in the immunological system of SLE patients, give hope for the development of a new treatment for SLE. Currently the most advanced clinical trials are being conducted with anti-lymphocyte B drugs, such as rituximab, belimumab and epratuzumab. Belimumab as the first biological agent was registered for treatment of the active, seropositive form of SLE. The advances in immunology and rheumatology nowadays raise the hope of finding effective and safe treatment for SLE. In our article we present an overview of data concerning perspectives of biological treatment in SLE.

Abstract:
Jedn z postaci seronegatywnych spondyloartropatii zapalnych(SpA) jest SpA towarzysz ca nieswoistym zapaleniom jelit (NZJ).Obie grupy chorób maj prawdopodobnie wspóln patogenez .W przebiegu NZJ u 20–50% chorych obserwuje si zapaleniestawów obwodowych lub osiowych, a w przypadku SpA w 30–60%przypadków stwierdza si w badaniu histopatologicznym cechyzapalenia jelit. Mo e to powodowa trudno ci diagnostycz ne,które zosta y przedstawione na przyk adzie 25-letniej chorejz obci aj cym wywiadem rodzinnym w kierunku NZJ, wywiademzapalenia stawów obwodowych i z objawami zapalnego bólu kr gos upa.W badaniach radiologicznych stwierdzono obustronnezapalenie stawów krzy owo-biodrowych (ryc. 1 i 2), brak istotnychzmian w zdj ciach kr gos upa (ryc. 3 i 4) i stawów biodrowych(ryc. 5), a w badaniu endoskopowym zapalenie jelit (bezobjawowe).Ostatecznie u chorej rozpoznano SpA towarzysz c NZJ.Diagnostyka w kierunku NZJ u chorych na SpA ma istotne znaczeniez uwagi na rokowanie w przebiegu choroby oraz dostosowanydo rozpoznania schemat terapeutyczny.

Abstract:
Bundle gerbes with connection and their modules play an important role in the theory of two-dimensional sigma models with a background Wess-Zumino flux: their holonomy determines the contribution of the flux to the Feynman amplitudes of classical fields. We discuss additional structures on bundle gerbes and gerbe modules needed in similar constructions for orientifold sigma models describing closed and open strings.

Abstract:
We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. The obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.

Abstract:
The simplest orientifolds of the WZW models are obtained by gauging a Z_2 symmetry group generated by a combined involution of the target Lie group G and of the worldsheet. The action of the involution on the target is by a twisted inversion g \mapsto (\zeta g)^{-1}, where \zeta is an element of the center of G. It reverses the sign of the Kalb-Ramond torsion field H given by a bi-invariant closed 3-form on G. The action on the worldsheet reverses its orientation. An unambiguous definition of Feynman amplitudes of the orientifold theory requires a choice of a gerbe with curvature H on the target group G, together with a so-called Jandl structure introduced in hep-th/0512283. More generally, one may gauge orientifold symmetry groups \Gamma = Z_2 \ltimes Z that combine the Z_2-action described above with the target symmetry induced by a subgroup Z of the center of G. To define the orientifold theory in such a situation, one needs a gerbe on G with a Z-equivariant Jandl structure. We reduce the study of the existence of such structures and of their inequivalent choices to a problem in group-\Gamma cohomology that we solve for all simple simply-connected compact Lie groups G and all orientifold groups \Gamma = Z_2 \ltimes Z.

Abstract:
We extend our analysis of the gauging of rigid symmetries in bosonic two-dimensional sigma models with Wess-Zumino terms in the action to the case of world-sheets with defects. A structure that permits a non-anomalous coupling of such sigma models to world-sheet gauge fields of arbitrary topology is analysed, together with obstructions to its existence, and the classification of its inequivalent choices.