Central extensions of mapping class groups from characteristic classes

Tangential structures on smooth manifolds, and the extension of mapping class groups they induce, admit a natural formulation in terms of higher (stacky) differential geometry. This is the literal translation of a classical construction in differential topology to a sophisticated language, but it has the advantage of emphasizing how the whole construction naturally emerges from the basic idea of working in slice categories. We characterize, for every higher smooth stack equipped with tangential structure, the induced higher group extension of the geometric realization of its higher automor- phism stack. We show that when restricted to smooth manifolds equipped with higher degree topological structures, this produces higher extensions of homotopy types of diffeomorphism groups. Passing to the groups of connected components, we obtain abelian extensions of mapping class groups and we derive sufficient conditions for these being central. We show as a special case that this provides an elegant re-construction of Segal’s approach to $\mathbb{Z}$ -extensions of mapping class groups of surfaces that provides the anomaly cancellation of the modular functor in Chern-Simons theory. Our construction generalizes Segal’s approach to higher central extensions of mapping class groups of higher dimensional manifolds with higher tangential structures, expected to provide the analogous anomaly cancellation for higher dimensional TQFTs

T-Duality from super Lie n-algebra cocycles for super p-branes

We compute the $L_\infty$-theoretic dimensional reduction of the F1/D$p$-brane super $L_\infty$-cocycles with coefficients in rationalized twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to 9d. We show that the two resulting coefficient $L_\infty$-algebras are naturally related by an $L_\infty$-isomorphism which we find to act on the super $p$-brane cocycles by the infinitesimal version of the rules of topological T-duality and inducing an isomorphism between $K^0$ and $K^1$, rationally. In particular this is a derivation of the Buscher rules for RR-fields (Hori's formula) from first principles. Moreover, we show that these $L_\infty$-algebras are the homotopy quotients of the RR-charge coefficients by the "T-duality Lie 2-algebra". We find that the induced L_\infty$-extension is a gerby extension of a 9+(1+1) dimensional (i.e. "doubled") T-duality correspondence super-spacetime, which serves as a local model for T-folds. We observe that this still extends, via the D0-brane cocycle of its type IIA factor, to a 10+(1+1)-dimensional super Lie algebra. Finally we observe that this satisfies expected properties of a local model space for F-theory elliptic fibrations

"Discrepant hardenings" in cosmic ray spectra: a first estimate of the effects on secondary antiproton and diffuse gamma-ray yields

Recent data from CREAM seem to confirm early suggestions that primary cosmic ray (CR) spectra at few TeV/nucleon are harder than in the 10-100 GeV range. Also, helium and heavier nuclei spectra appear systematically harder than the proton fluxes at corresponding energies. We note here that if the measurements reflect intrinsic features in the interstellar fluxes (as opposed to local effects) appreciable modifications are expected in the sub-TeV range for the secondary yields, such as antiprotons and diffuse gamma-rays. Presently, the ignorance on the origin of the features represents a systematic error in the extraction of astrophysical parameters as well as for background estimates for indirect dark matter searches. We find that the spectral modifications are appreciable above 100 GeV, and can be responsible for ~30% effects for antiprotons at energies close to 1 TeV or for gamma's at energies close to 300 GeV, compared to currently considered predictions based on simple extrapolation of input fluxes from low energy data. Alternatively, if the feature originates from local sources, uncorrelated spectral changes might show up in antiproton and high-energy gamma-rays, with the latter ones likely dependent from the line-of-sight.Comment: 6 pages, 3 figures. Clarifications and references added, conclusions unchanged. Matches published versio

Endpoint estimates and weighted norm inequalities for commutators of fractional integrals

We prove that the commutator [b, Iα], b ∈ BMO, Iα the fractional integral operator, satisfies the sharp, modular weak-type inequality f(x) tdx, where B(t) = tlog(e + t) and Ψ(t)=[tlog(e + tα/n)]n/(n−α). These commutators were first considered by Chanillo, and our result complements his. The heart of our proof consists of the pointwise inequality, M#([b, Iα]f)(x) ≤ CbBMO [Iαf(x) + Mα,Bf(x)], where M# is the sharp maximal operator, and Mα,B is a generalization of the fractional maximal operator in the scale of Orlicz spaces. Using this inequality we also prove one-weight inequalities for the commutator; to do so we prove one and two-weight norm inequalities for Mα,B which are of interest in their own right.[b, Iα]f(x

L∞L_\infty-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism

We review in detail the Batalin-Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an $L_\infty$-algebra and how quasi-isomorphisms between $L_\infty$-algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern-Simons theories and give some useful shortcuts in usually rather involved computations.Comment: v3: 131 pages, minor improvements, published versio

Kodaira-Spencer formality of products of complex manifolds

We shall say that a complex manifold $X$ is emph{Kodaira-Spencer formal} if its Kodaira-Spencer differential graded Lie algebra $A^{0,*}_X(Theta_X)$ is formal; if this happen, then the deformation theory of $X$ is completely determined by the graded Lie algebra $H^*(X,Theta_X)$ and the base space of the semiuniversal deformation is a quadratic singularity.. Determine when a complex manifold is Kodaira-Spencer formal is generally difficult and we actually know only a limited class of cases where this happen. Among such examples we have Riemann surfaces, projective spaces, holomorphic Poisson manifolds with surjective anchor map $H^*(X,Omega^1_X) o H^*(X,Theta_X)$ and every compact K"{a}hler manifold with trivial or torsion canonical bundle. In this short note we investigate the behavior of this property under finite products. Let $X,Y$ be compact complex manifolds; we prove that whenever $X$ and $Y$ are K"{a}hler, then $X imes Y$ is Kodaira-Spencer formal if and only if the same holds for $X$ and $Y$. A revisit of a classical example by Douady shows that the above result fails if the K"{a}hler assumption is droppe

Sums over Graphs and Integration over Discrete Groupoids

We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pull-back or push-forward formulas for integrals over suitable groupoids.Comment: 27 pages, 4 eps figures; LaTeX2e; uses Xy-Pic. Some ambiguities fixed, and several proofs simplifie

T Cell Leukemia/Lymphoma 1A is essential for mouse epidermal keratinocytes proliferation promoted by insulin-like growth factor 1

T Cell Leukemia/Lymphoma 1A is expressed during B-cell differentiation and, when overexpressed, acts as an oncogene in mouse (Tcl1a) and human (TCL1A) B-cell chronic lymphocytic leukemia (B-CLL) and T-cell prolymphocytic leukemia (T-PLL). Furthermore, in the murine system Tcl1a is expressed in the ovary, testis and in pre-implantation embryos, where it plays an important role in blastomere proliferation and in embryonic stem cell (ESC) proliferation and self-renewal. We have also observed that Tcl1-/-adult mice exhibit alopecia and deep ulcerations. This finding has led us to investigate the role of TCL1 in mouse skin and hair follicles. We have found that TCL1 is expressed in the proliferative structure (i.e.The secondary hair germ) and in the stem cell niche (i.e.The bulge) of the hair follicle during regeneration phase and it is constitutively expressed in the basal layer of epidermis where it is required for the correct proliferative-differentiation program of the keratinocytes (KCs). Taking advantage of the murine models we have generated, including the Tcl1-/-and the K14-TCL1 transgenic mouse, we have analysed the function of TCL1 in mouse KCs and the molecular pathways involved. We provide evidence that in the epidermal compartment TCL1 has a role in the regulation of KC proliferation, differentiation, and apoptosis. In particular, the colony-forming efficiency (CFE) and the insulin-like growth factor 1 (IGF1)-induced proliferation are dramatically impaired, while apoptosis is increased, in KCs from Tcl1-/-mice when compared to WT. Moreover, the expression of differentiation markers such as cytokeratin 6 (KRT6), filaggrin (FLG) and involucrin (IVL) are profoundly altered in mutant mice (Tcl1-/-). Importantly, by over-expressing TCL1A in basal KCs of the K14-TCL1 transgenic mouse model, we observed a significant rescue of cell proliferation, differentiation and apoptosis of the mutant phenotype. Finally, we found TCL1 to act, at least in part, via increasing phospho-ERK1/2 and decreasing phospho-P38 MAPK. Hence, our data demonstrate that regulated levels of Tcl1a are necessary for the correct proliferation and differentiation of the interfollicular KC

Graph complexes in deformation quantization

Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an $L_\infty$-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.Comment: 39 pages; 3 eps figures; uses Xy-pic. Final version. Details added, mainly concerning the tree-level approximation. Typos corrected. An abridged version will appear in Lett. Math. Phy