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71
Closed/open string diagrammatics
 Nucl. Phys. B
"... Abstract. We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. All of the predicted equations of string theory are derived in this model, and new equations can be discovered as well. In particular, several ne ..."
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Abstract. We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. All of the predicted equations of string theory are derived in this model, and new equations can be discovered as well. In particular, several new equations together with known transformations generate the combinatorial version of open/closed duality. On the topological and chain levels, the algebraic structure discovered is new, but it specializes to a modular bioperad on the level of homology.
2008) Classical and quantum structures
"... In recent work, symmetric daggermonoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to ..."
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In recent work, symmetric daggermonoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to Frobenius algebras with some additional properties. They express the distinguishing capabilities of classical data: in contrast with quantum data, classical data can be copied and deleted. The algebraic approach thus shifts the paradigm of ”quantization ” of a classical theory to ”classicization ” of a quantum theory. Remarkably, the simple SDM framework suffices not only for this conceptual shift, but even allows us to distinguish the deterministic classical operations (i.e. functions) from the nondeterministic classical operations (i.e. relations), and the probabilistic classical operations (stochastic maps). Moreover, a combination of some basic categorical constructions (due to Kleisli, resp. Grothendieck) with the categorical presentations of quantum states, provides a resource sensitive account of various quantumclassical interactions: of classical control of quantum data, of classical data arising from quantum measurements, as well as of the classical data processing inbetween controls and measurements. A salient feature here is the graphical calculus for categorical quantum mechanics, which allows a purely diagrammatic representation of classicalquantum interaction. 1
A double bicategory of cobordisms with corners arXiv:0611930
"... Abstract. Interest in cobordism categories arises in areas from topology to theoretical physics, and in particular in Topological Quantum Field Theories (TQFT’s). These categories have manifolds as objects, and cobordisms between them as morphisms, have that is, manifolds of one dimension higher wh ..."
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Abstract. Interest in cobordism categories arises in areas from topology to theoretical physics, and in particular in Topological Quantum Field Theories (TQFT’s). These categories have manifolds as objects, and cobordisms between them as morphisms, have that is, manifolds of one dimension higher whose boundary decomposes into the source and target. Since the boundary of a boundary is empty, this formulation cannot account for cobordisms between manifolds with boundary. This is needed to describe openclosed TQFT’s, and more generally, “extended TQFT’s”. We describe a framework for describing these, in the form of what we call a Verity double bicategory. This is similar to a double category, but with properties holding only up to certain 2morphisms. We show how a broad general class of examples arise from a construction involving spans (or cospans) in suitable settings, and how this gives cobordisms between cobordisms when we start with the category of manifolds.
The Feynman Legacy
"... The article is an overview of the role of graph complexes in the Feynman path integral quantization. The underlying mathematical language is that of PROPs and operads, and their representations. The sum over histories approach, the Feynman Legacy, is the bridge between quantum physics and quantum co ..."
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Cited by 5 (4 self)
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The article is an overview of the role of graph complexes in the Feynman path integral quantization. The underlying mathematical language is that of PROPs and operads, and their representations. The sum over histories approach, the Feynman Legacy, is the bridge between quantum physics and quantum computing, pointing towards a deeper understanding of the fundamental concepts of space, time and information. 1
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
Openclosed TQFTs extend Khovanov homology from links to tangles
, 2006
"... We use a special kind of 2dimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homol ..."
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We use a special kind of 2dimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant under Reidemeister moves. The terms of this chain complex are modules of a suitable algebra A such that there is one action of A or A op for every boundary point of the tangle. We give examples of such algebras A for which our tangle homology theory reduces to the link homology theories of Khovanov, Lee, and BarNatan if it is evaluated for links. As a consequence of the Cardy condition, Khovanov’s graded theory can only be extended to tangles if the underlying field has finite characteristic. In all cases in which the algebra A is strongly separable, i.e. for BarNatan’s theory in any characteristic and for Lee’s theory in characteristic other than 2, we also provide the required algebraic operation for the composition of oriented tangles. Just as Khovanov’s theory for links can be recovered from Lee’s or BarNatan’s by a suitable spectral sequence, we provide a spectral sequence in order to compute our tangle extension of Khovanov’s theory from that of BarNatan’s or Lee’s theory. Thus, we provide a tangle homology theory that is locally computable and still strong enough to recover characteristic p Khovanov homology for links.
TWIN TQFTs AND FROBENIUS ALGEBRAS
, 901
"... Abstract. We introduce the category of singular 2dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius ..."
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Abstract. We introduce the category of singular 2dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (C, W, z, z ∗ ) consists of a commutative Frobenius algebra C, a symmetric Frobenius algebra W, and an algebra homomorphism z: C → W with dual z ∗ : W → C with some extra conditions. We also introduce a special type of extended 2dimensional Topological Quantum Field Theory defined on singular 2dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.
A TQFT for intersection numbers on moduli spaces of admissible covers
, 2005
"... We construct a twolevel weighted TQFT whose structure coefficents are equivariant intersection numbers on moduli spaces of admissible covers. Such a structure is parallel (and strictly related) to the local GromovWitten theory of curves in [BP04]. We compute explicitly the theory using techniques ..."
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We construct a twolevel weighted TQFT whose structure coefficents are equivariant intersection numbers on moduli spaces of admissible covers. Such a structure is parallel (and strictly related) to the local GromovWitten theory of curves in [BP04]. We compute explicitly the theory using techniques of localization on moduli spaces of admissible covers of a parametrized P 1. The Frobenius Algebras we obtain are one parameter deformations of the class algebra of the symmetric group Sd. In certain special cases we are able to produce explicit closed formulas for such deformations in terms of the representation theory of Sd.
Curves in CalabiYau 3folds and topological quantum field theory
"... We continue our study of the local GromovWitten invariants of curves in CalabiYau 3folds. We define relative invariants for the local theory which give rise to a 1+1dimensional TQFT taking values in the ring Q[[t]]. The associated Frobenius algebra over Q[[t]] is semisimple. Consequently, we obt ..."
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We continue our study of the local GromovWitten invariants of curves in CalabiYau 3folds. We define relative invariants for the local theory which give rise to a 1+1dimensional TQFT taking values in the ring Q[[t]]. The associated Frobenius algebra over Q[[t]] is semisimple. Consequently, we obtain a structure result for the local invariants. As an easy consequence of our structure formula, we recover the closed formulas for the local invariants in case either the target genus or the degree equals 1. We prove there exist degree 2 rigid curves of any genus. Hence, our degree 2 theory agrees with the double cover contributions to the GromovWitten invariants of the ambient 3folds. 1 Notation, definitions and results A central problem in GromovWitten theory is to determine the structure of the GromovWitten invariants. Of special interest is the case where the