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The Uncertainty of Fluxes
 Commun. Math. Phys
"... Abstract. In the ordinary quantum Maxwell theory of a free electromagnetic field, formulated on a curved 3manifold, we observe that magnetic and electric fluxes cannot be simultaneously measured. This uncertainty principle reflects torsion: fluxes modulo torsion can be simultaneously measured. We a ..."
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Cited by 9 (2 self)
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Abstract. In the ordinary quantum Maxwell theory of a free electromagnetic field, formulated on a curved 3manifold, we observe that magnetic and electric fluxes cannot be simultaneously measured. This uncertainty principle reflects torsion: fluxes modulo torsion can be simultaneously measured. We also develop the Hamilton theory of selfdual fields, noting that they are quantized by Pontrjagin selfdual cohomology theories and that the quantum Hilbert space is Z/2Zgraded, so typically contains both bosonic and fermionic states. Fluxes in the classical theory of electromagnetism and its generalizations are realvalued and Poissoncommute. Our main result is a Heisenberg uncertainty principle in the quantum theory: magnetic and electric fluxes cannot be measured simultaneously. This observation applies to any abelian gauge field, including the standard Maxwell field theory in four spacetime dimensions as well as the Bfield and RamondRamond fields in string theories. It is the torsion part of the fluxes which experience uncertainty—the nontrivial commutator of torsion fluxes is computed by the link pairing on the cohomology of space, and there are always nontrivial commutators if torsion is present. We remark that torsion fluxes arise from Dirac charge/flux quantization. This Heisenberg uncertainty relation goes against the conventional wisdom that the quantum Hilbert
Setting the quantum integrand of Mtheory
"... Abstract. In anomalyfree quantum field theories the integrand in the bosonic functional integral— the exponential of the effective action after integrating out fermions—is often defined only up to a phase without an additional choice. We term this choice “setting the quantum integrand”. In the low ..."
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Cited by 7 (2 self)
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Abstract. In anomalyfree quantum field theories the integrand in the bosonic functional integral— the exponential of the effective action after integrating out fermions—is often defined only up to a phase without an additional choice. We term this choice “setting the quantum integrand”. In the lowenergy approximation to Mtheory the E8model for the Cfield allows us to set the quantum integrand using geometric index theory. We derive mathematical results of independent interest about pfaffians of Dirac operators in 8k+3 dimensions, both on closed manifolds and manifolds with boundary. These theorems are used to set the quantum integrand of Mtheory for closed manifolds and for compact manifolds with either temporal (global) or spatial (local) boundary conditions. In particular, we show that Mtheory makes sense on arbitrary 11manifolds with spatial boundary, generalizing the construction of heterotic Mtheory on cylinders. The lowenergy approximation to Mtheory is a refinement of classical 11dimensional supergravity. It has a simple field content: a metric g, a 3form gauge potential C, and a gravitino. The Mtheory action contains rather subtle “ChernSimons ” terms which, on a topologically nontrivial manifold Y, raise delicate issues in the definition of the (exponentiated) action. Some aspects of the problem were resolved by Witten [W1]. The key ingredients are: a quantization law for C
Homotopy Batalin–Vilkovisky algebras
"... This paper provides an explicit cofibrant resolution of the operad encoding BatalinVilkovisky algebras. Thus it defines the notion of homotopy BatalinVilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads ..."
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Cited by 6 (3 self)
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This paper provides an explicit cofibrant resolution of the operad encoding BatalinVilkovisky algebras. Thus it defines the notion of homotopy BatalinVilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin– Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a PoincaréBirkhoffWitt Theorem for such an operad and to give an explicit small quasifree resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BValgebras and of homotopy BValgebras. We show that any topological conformal field theory carries a homotopy BValgebra structure which lifts the BValgebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian–Zuckerman, showing that certain vertex algebras have an explicit homotopy BValgebra structure.
Physics, Topology, Logic and Computation: A Rosetta Stone
, 2009
"... Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objec ..."
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Cited by 5 (1 self)
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Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps
The integration problem for complex Lie algebroids
, 2006
"... A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie algebroids by using analytic con ..."
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Cited by 4 (1 self)
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A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie algebroids by using analytic continuation to a complexification of M and integration to a holomorphic groupoid. A collection of diverse examples reveal that the holomorphic stacks presented by these groupoids tend to coincide with known objects associated to structures in complex geometry. This suggests that the object integrating a complex Lie algebroid should be a holomorphic stack.
I.Kriz: Laplaza sets, or how to select coherence diagrams for pseudo algebras
"... Abstract. We define a general concept of pseudo algebras over theories and 2theories. A more restrictive such notion was introduced in [5], but as noticed by M. Gould, did not capture the desired examples. The approach taken in this paper corrects the mistake by introducing a more general concept, ..."
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Cited by 3 (1 self)
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Abstract. We define a general concept of pseudo algebras over theories and 2theories. A more restrictive such notion was introduced in [5], but as noticed by M. Gould, did not capture the desired examples. The approach taken in this paper corrects the mistake by introducing a more general concept, allowing more flexibility in selecting coherence diagrams for pseudo algebras. 1.
AQFT from nfunctorial QFT
"... There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended ” functoria ..."
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Cited by 3 (1 self)
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There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended ” functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra “of observables”, the latter uses nfunctors which assign to each patch a “propagator of states”. In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2dimensional extended Minkowskian QFT 2functor (”parallel surface transport”) naturally yields a local net. This is obtained by postcomposing the propagation 2functor with an operation that mimics the passage from the Schrödinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general pseudo