156 research outputs found
Z2 lattice Gerbe Theory
No full textDiscretized formulations of 2-form Abelian and non-Abelian gauge fields on d-dimensional hypercubiclattices have been discussed in the past by various authors and most recently by Lipstein and Reid-Edwards [J. High Energy Phys. 09 (2014) 034]. In this paper we recall that the Hamiltonian of a Z2 variant ofsuch theories is one of the family of generalized Ising models originally considered by Wegner. For such“Z2 lattice Gerbe theories” general arguments can be used to show that a phase transition for Wilsonsurfaces will occur for d > 3 between volume and area scaling behavior. In 3d the model is equivalentunder duality to an infinite coupling model and no transition is seen, whereas in 4d the model is dual to the 4d Ising model and displays a continuous transition. In 5d the Z 2 lattice Gerbe theory is self-dual in the presence of an external field and in 6d it is self-dual in zero external field
Transgression of the index gerbe
No full textThe holonomy of an unitary line bundle with connection over some base space B is a U(1)-valued function on the loop space LB. In a parallel manner, the holonomy of a gerbe with connection on B is a line bundle with connection over LB. Given a family of graded Dirac operators on B and some additional geometric data one can define the determinant line bundle with Quillen metric and Bismut-Freed connection. According to Witten, Bismut-Freed the holonomy of this determinant bundle can be expressed in terms of an adiabatic limit of eta invariants of an associated family of Dirac operators over LB. Recently, for a family of ungraded Dirac operators on B J. Lott constructed an index gerbe with connection., In the present paper we show, in analogy to the holonomy formula for the determinant bundle, that the holonomy of the index gerbe coincides with an adiabatic limit of determinant bundles of the associated family of Dirac operators over LB
Quantum mechanics as a spontaneously broken gauge theory on a U(1)gerbe
No full textAny quantum-mechanical system possesses a U(1) gerbe naturally defined on configuration space. Acting on Feynman's kernel exp(iS/ħ), this U(1) symmetry allows one to arbitrarily pick the origin for the classical action S, on a point-by-point basis on configuration space. This is equivalent to the statement that quantum mechanics is a U(1) gauge theory. Unlike Yang–Mills theories, however, the geometry of this gauge symmetry is not given by a fibre bundle, but rather by a gerbe. Since this gauge symmetry is spontaneously broken, an analogue of the Higgs mechanism must be present. We prove that a Heisenberg-like noncommutativity for the space coordinates is responsible for the breaking. This allows to interpret the noncommutativity of space coordinates as a Higgs mechanism on the quantum-mechanical U(1) gerbe
The 2-Hilbert space of a prequantum bundle gerbe
No full textWe construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier–Douady class is torsion. Analogously to usual prequantization, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Waldorf’s version of the 2-category of line bundle gerbes. We show that these morphism categories carry a monoidal structure under which they are semisimple and abelian. We introduce a dual functor on the sections, which yields a closed structure on the morphisms between bundle gerbes and turns the category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert spaces fit various expectations from higher prequantization. We then extend the transgression functor to the full 2-category of bundle gerbes and demonstrate its compatibility with the additional structures introduced. We discuss various aspects of Kostant–Souriau prequantization in this setting, including its dimensional reduction to ordinary prequantization.</p
On the Derivative of 2-Holonomy for a Non-Abelian Gerbe
Get PDFThe local 2-holonomy for a non abelian gerbe with connection is first studied via a local zig-zag Hochschild complex. Next, by locally integrating the cocycle data for our gerbe with connection, and then glueing this data together, an explicit definition is offered for a global version of 2-holonomy. After showing this definition satisfies the desired properties for 2-holonomy, its derivative is calculated whereby the only interior information added is the integration of the 3-curvature. Finally, for the case when the surface being mapped into the manifold is a sphere, the derivative of 2-holonomy is extended to an equivariant closed form in the spirit of the construction of Tradler-Wilson-Zeinalian for abelian gerbes
Wirydiana Fiszerowa : Dzieje moje wlasne i osob postronnych — Wiazanka spraw powanych, ciekawych i blahych (Ma propre histoire et celle des personnes de mon entourage — Une gerbe d'affaires sérieuses, intéressantes et sans conséquence). Traduction d'E. Raczynskî, préface de J. Jasnowski, 1975
No full textZoltowska Maria Evelina. Wirydiana Fiszerowa : Dzieje moje wlasne i osob postronnych — Wiazanka spraw powanych, ciekawych i blahych (Ma propre histoire et celle des personnes de mon entourage — Une gerbe d'affaires sérieuses, intéressantes et sans conséquence). Traduction d'E. Raczynskî, préface de J. Jasnowski, 1975. In: Dix-huitième Siècle, n°9, 1977. Le sain et le malsain. pp. 407-408
Higher U (1)-gerbe connections in geometric prequantization
No full textWe promote geometric prequantization to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show fairly generally how there is canonically a tower of higher gauge groupoids and Courant groupoids assigned to a higher prequantization, and establish the corresponding Atiyah sequence as an integrated Kostant-Souriau infinity-group extension of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the infinity-group cocycle which classifies this extension and discuss how its restrictions along Hamiltonian infinity-actions yield higher Heisenberg cocycles. In the special case of higher differential geometry over smooth manifolds, we find the L-infinity-algebra extension of Hamiltonian vector fields - which is the higher Poisson bracket of local observables - and show that it is equivalent to the construction proposed by the second author in n-plectic geometry. Finally, we indicate a list of examples of applications of higher prequantization in the extended geometric quantization of local quantum field theories and specifically in string geometry
On the constitution and development of the ovarian egg of the Sacculin\ue6
No full textVolume: 3Start Page: 321End Page: 32
Ressenyes
No full textIndex de les obres ressenyades: Teaching Geography for a better world. Edició a càrrec de J. Fien i R. Gerbe
Ressenyes
No full textIndex de les obres ressenyades: Teaching Geography for a better world. Edició a càrrec de J. Fien i R. Gerbe
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