Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Fermionic integral on loop space and the Pfaffian line bundle
64
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the top degree component of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible top degree component to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical 2-form on the loop space. The result is a section on the Pfaffian line bundle on the loop space. We then identify this with a section of the line bundle obtained by transgression of the spin lifting gerbe. These results are a crucial ingredient for defining the fermionic part of the supersymmetric path integral on the loop space.
rate research
Read More
We shall give a twisted Dirac structure on the space of irreducible connections on a SU(n)-bundle over a three-manifold, and give a family of twisted Dirac structures on the space of irreducible connections on the trivial SU(n)-bundle over a four-manifold. The twist is described by the Cartan 3-form on the space of connections. It vanishes over the subspace of flat connections. So the spaces of flat connections are endowed with ( non-twisted ) Dirac structures. The Dirac structure on the space of flat connections over the three-manifold is obtained as the boundary restriction of a corresponding Dirac structure over the four-manifold. We discuss also the action of the group of gauge transformations over these Dirac structures.
Let $M$ be a smooth closed orientable manifold and $mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on $mathcal{P}(M)$ by volume-preserving diffeomorphism of $M$. We then define an invariant of a Poisson structure that describes fixed points of the flow equation and compute it for regular Poisson 3-manifolds, where it detects unimodularity. For unimodular Poisson structures we define a further, related Poisson bracket and show that for symplectic structures the associated invariant counting fixed points of the flow equation is given in terms of the $d d^Lambda$ and $d+ d^Lambda$ symplectic cohomology groups defined by Tseng and Yau.
This is a survey based on joint work with Florian Hanisch and Batu Guneysu reporting on a rigorous construction of the supersymmetric path integral associated to compact spin manifolds.
In this short note, we investigate some features of the space $Inject{d}{m}$ of linear injective maps from $bbR^d$ into $bbR^m$; in particular, we discuss in detail its relationship with the Stiefel manifold $V_{m,d}$, viewed, in this context, as the set of orthonormal systems of $d$ vectors in $bbR^m$. Finally, we show that the Stiefel manifold $V_{m,d}$ is a deformation retract of $Inject{d}{m}$. One possible application of this remarkable fact lies in the study of perturbative invariants of higher-dimensional (long) knots in $bbR^m$: in fact, the existence of the aforementioned deformation retraction is the key tool for showing a vanishing lemma for configuration space integrals {`a} la Bott--Taubes (see cite{BT} for the 3-dimensional results and cite{CR1}, cite{C} for a first glimpse into higher-dimensional knot invariants).
We address the construction of smooth bundles of fermionic Fock spaces, a problem that appears frequently in fermionic gauge theories. Our main motivation is the spinor bundle on the free loop space of a string manifold, a structure anticipated by Killingback, with a construction outlined by Stolz-Teichner. We develop a general framework for constructing smooth Fock bundles, and obtain as an application a complete and well-founded construction of spinor bundles on loop space.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
