Schreiber Obstruction theory for parameterized higher WZW terms

Two talks that I have given:

• Obstruction theory for parameterized higher WZW terms

  talk at AMS/EMS meeting 2015 in Porto, June 9-13

  special session Higher Dimensional Algebra in Geometry and Quantum Field Theory

  (talk slides pdf)

• Obstruction theory for parameterized higher WZW terms

  talk at German Mathematical Society meeting 2015 in Hamburg, September 21-25

  section 1: Algebra und Zahlentheorie

  (talk slides pdf)

on parameterized WZW models and, more generally, the globalization of ∞-Wess-Zumino-Witten theory from higher Klein geometries to higher Cartan geometry; specifically applied to globalization of the Green-Schwarz sigma-models of the brane bouquet from extended super Minkowski spacetimes to general curved supergravity backgrounds.

(Using the theory of differential cohomology in a cohesive topos, based on results in Higher geometric prequantum theory.)

All details are in

• differential cohomology in a cohesive topos.

Course notes are at

• Structure Theory for Higher WZW Terms.

Abstract

We compute obstructions to globalizing WZW terms on Klein geometries $G/H$ to global WZW terms on Cartan geometries locally modeled on $G/H$. We do this in the generality of higher degree WZW terms on higher orbispaces (i.e. in higher differential geometry on higher cohesive stacks). As a first example, the Green-Schwarz anomaly cancellation (String structure) is recovered as the obstruction to parameterization of the traditional WZW-term on the group $\mathrm{Spin}\times E_8$; similarly Fivebrane structure is found to lift the obstruction to parameterization of the 6-dimensional WZW term on the group stack $\mathrm{String}$; and Ninebrane structure is found for parameterization of the 10-dimensional WZW term on the group 5-stack $\mathrm{Fivebrane}$. We then obtain obstructions to globalizing the full $\kappa$-symmetry WZW terms (not just their curvature/flux forms) of all the super p-brane sigma models from extended super Minkowski spacetimes to curved supergravity backgrounds. These obstructions are generally higher and super analogs of symplectic-, quaternionic-Kähler-, $G_2$- and $\mathrm{Spin}(7)$-manifold structures, namely they are infinitesimally integrable higher G-structures for $G$ the higher Heisenberg supergroup stack of the given higher WZW term. For the case of the M5-brane with its tensor multiplet fields, already the model space $G/H$ is a higher super-3-stack extension of the 11-dimensional super-Poincaré group, and for phenomenologically realistic models the Cartan geometry is furthermore an orbifold modeled on this super 3-stack; while the globalization obstruction is the existence of a certain super-6-group $G$-structure on that higher orbifold. So the generality of higher differential cohesive geometry is inevitable. We find in this case a forgetful $\infty$-functor from definite globalizations of the M5-brane WZW term to solutions of the Einstein equations of motion of 11-dimensional supergravity.

Related projects

• differential cohomology in a cohesive topos

  • Synthetic Quantum Field Theory

    • Quantum gauge field theory in Cohesive homotopy type theory

    • Type-semantics for quantization

    • Higher toposes of laws of motion

  • cohomology

    • Principal ∞-bundles – models and general theory

    • Fivebrane structures

    • Cosimplicial C-infinity rings and the de Rham complex of Euclidean space

  • ∞-Chern-Weil theory

    • L-∞ algebra connections

    • Cech cocycles for differential characteristic classes

    • Twisted differential structures

      • Twisted Differential String and Fivebrane Structures

      • The moduli 3-stack of the C-field

  • ∞-Chern-Simons theory

    • A higher stacky perspective on Chern-Simons theory

    • Higher Chern-Weil Derivation of AKSZ Sigma-Models

    • 7d Chern-Simons theory and the 5-brane

    • Extended higher cup-product Chern-Simons theories

  • ∞-Wess-Zumino-Witten theory

    • The brane bouquet

    • Obstruction theory for parameterized higher WZW terms

  • Higher geometric prequantum theory

    • Classical field theory via Cohesive homotopy types

    • Local prequantum field theory

    • L-∞ algebras of local observables from higher prequantum bundles

    • Higher extensions of mapping class groups

  • Motivic quantization of local prequantum field theory

    • Geometric quantization of symplectic and Poisson manifolds

    • Cohomological quantization of local prequantum boundary field theory