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A holomorphic Casson invariant for CalabiYau 3folds, and bundles on K3 fibrations
 J. DIFFERENTIAL GEOM
, 2000
"... We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a CalabiYau 3fol ..."
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Cited by 199 (8 self)
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We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a CalabiYau 3fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of [LT], [BF] in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in P 3, and Donaldson – and GromovWitten – like invariants of Fano 3folds. It also allows us to define the holomorphic Casson invariant of a CalabiYau 3fold X, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general K3 fibration X, enabling us to compute the invariant for some ranks and Chern classes, and equate it to GromovWitten invariants of the “Mukaidual” 3fold for others. As an example the invariant is shown to distinguish Gross’ diffeomorphic 3folds. Finally the Mukaidual 3fold is shown to be CalabiYau and its cohomology is related to that of X.
The noncommutative and nonassociative geometry of octonionic spacetime, modified dispersion relations and grand unification
 J. Math. Phys
, 2007
"... The Octonionic Geometry (Gravity) developed long ago by Oliveira and Marques is extended to Noncommutative and Nonassociative Spacetime coordinates associated with octonionicvalued coordinates and momenta. The octonionic metric Gµν already encompasses the ordinary spacetime metric gµν, in addition ..."
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Cited by 8 (2 self)
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The Octonionic Geometry (Gravity) developed long ago by Oliveira and Marques is extended to Noncommutative and Nonassociative Spacetime coordinates associated with octonionicvalued coordinates and momenta. The octonionic metric Gµν already encompasses the ordinary spacetime metric gµν, in addition to the Maxwell U(1) and SU(2) YangMills fields such that implements the KaluzaKlein Grand Unification program without introducing extra spacetime dimensions. The color group SU(3) is a subgroup of the exceptional G2 group which is the automorphism group of the octonion algebra. It is shown that the flux of the SU(2) YangMills field strength ⃗ Fµν through the areamomentum ⃗ Σ µν in the internal isospin space yields corrections O(1/M 2 P lanck) to the energymomentum dispersion relations without violating Lorentz invariance as it occurs with Hopf algebraic deformations of the Poincare algebra. The known Octonionic realizations of the Clifford Cl(8), Cl(4) algebras should permit the construction of octonionic string actions that should have a correspondence with ordinary string actions for strings moving in a curved Cliffordspace target background associated with a Cl(3, 1) algebra.
The large N limit of exceptional Jordan matrix models and
 M, F theory, J. Geometry Phys
, 2007
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Twistor actions for gauge theory and gravity
"... Abstract: This is a review of recent developments in the study of perturbative gauge theory and gravity using action functionals on twistor space. It is intended to provide a userfriendly introduction to twistor actions, geared towards researchers or graduate students interested in learning somethi ..."
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Cited by 5 (2 self)
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Abstract: This is a review of recent developments in the study of perturbative gauge theory and gravity using action functionals on twistor space. It is intended to provide a userfriendly introduction to twistor actions, geared towards researchers or graduate students interested in learning something about the utility, prospects, and shortcomings of this approach. For those already familiar with the twistor approach, it should provide a condensed overview of the literature as well as several novel results of potential interest. This work is based primarily upon the author’s D.Phil. thesis. We first consider fourdimensional, maximally supersymmetric YangMills theory as a gauge theory in twistor space. We focus on the perturbation theory associated to this action, which in an axial gauge leads to the MHV formalism. This allows us to efficiently compute scattering amplitudes at treelevel (and beyond) in twistor space. Other gauge theory observables such as local operators and null polygonal Wilson loops can also be formulated twistorially, leading to proofs for several correspondences between correlation functions and Wilson loops, as well as a recursive formula for computing mixed Wilson loop / local operator correlators. We then apply the twistor action approach to general relativity, using the onshell equivalence between conformal and Einstein gravity. This can be extended to N = 4 supersymmetry. The perturbation theory of the twistor action leads to formulae for the MHV amplitude with and without cosmological constant, yields a candidate for the Einstein twistor action, and induces a MHV formalism on twistor space. Appendices include discussion of superconnections and Coulomb branch regularization on twistor space. ar
Topological symmetry of forms, N = 1 supersymmetry and Sduality on special manifolds
 J. Geom. Phys
"... Abstract: We study the quantization of a holomorphic two–form coupled to a Yang–Mills field on special manifolds in various dimensions, and we show that it yields twisted supersymmetric theories. The construction determines ATQFT’s (Almost Topological Quantum Field Theories), that is, theories with ..."
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Abstract: We study the quantization of a holomorphic two–form coupled to a Yang–Mills field on special manifolds in various dimensions, and we show that it yields twisted supersymmetric theories. The construction determines ATQFT’s (Almost Topological Quantum Field Theories), that is, theories with observables that are invariant under changes of metrics belonging to restricted classes. For Kähler manifolds in four dimensions, our topological model is related to N = 1 Super Yang–Mills theory. Extended supersymmetries are recovered by considering the coupling with chiral multiplets. We also analyse Calabi–Yau manifolds in six and eight dimensions, and seven dimensional G2 manifolds of the kind recently discussed by Hitchin. We argue that the two–form field could play an interesting rôle for the study of the conjectured S–duality in topological string. We finally show that in the case of real forms in six dimensions the partition function of our topological model is related to the square of that of the holomorphic Chern–Simons theory, and we discuss the uplift to seven dimensions and its relation with the recent proposals for the topological M–theory.
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, 2002
"... Abstract. We show how the sums over graphs of the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams are interpreted as pullback formulas for integrals over suitable groupoids. ..."
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Abstract. We show how the sums over graphs of the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams are interpreted as pullback formulas for integrals over suitable groupoids.