Topological twist in four dimensions, R-duality and hyperinstantons

Anselmi, Damiano; Fré, Pietro

doi:10.1016/0550-3213(93)90481-4

Cited by 32 publications

(101 citation statements)

References 41 publications

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“…In general, although the starting N=2 theory is defined on a flat world-manifold M (when supersymmetry is global), yet the final theory can be defined on any M and it is independent of the choice of the metric on M. Recently [15,16,17,18], we have shown that any N=2 globally but also locally supersymmetric theory in four dimensions can be twisted, if the topological twist is suitably improved. One gets a topological version of the theories, namely topological gravity, topological Yang-Mills theory and topological σ-model, eventually coupled together.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, topological Yang-Mills theory is gauge-fixed by the equations of the usual Yang-Mills instantons. The novelty comes from the twist of the N=2 σ-model [17,18] describing the self-interaction of hypermultiplets [20,21]. The new instantons that gauge-fix this theory were called by us hyperinstantons and are the main subject of the present paper.…”

Section: Introductionmentioning

confidence: 99%

“…One gets a topological version of the theories, namely topological gravity, topological Yang-Mills theory and topological σ-model, eventually coupled together. In pure topological gravity [15,16] or topological gravity coupled to topological Yang-Mills theory [17,18], the gravitational instantons [19] are the solutions to the condition of selfduality for the spin connection ω ab , namely ω −ab = 0. On the other hand, topological Yang-Mills theory is gauge-fixed by the equations of the usual Yang-Mills instantons.…”

Section: Introductionmentioning

confidence: 99%

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Topological σ-models in four dimensions and triholomorphic maps

1994

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It is well-known that topological σ-models in two dimensions constitute a path-integral approach to the study of holomorphic maps from a Riemann surface Σ to an almost complex manifold K, the most interesting case being that where K is a Kähler manifold. We show that, in the same way, topological σ-models in four dimensions introduce a path integral approach to the study of triholomorphic maps q : M → N between a four dimensional Riemannian manifold M and an almost quaternionic manifold N . The most interesting cases are those where M, N are hyperKähler or quaternionic Kähler. BRST-cohomology translates into intersection theory in the moduli-space of this new class of instantonic maps, that are named by us hyperinstantons. The definition of triholomorphicity that we propose is expressed by the equation q * − J u • q * • j u = 0, where {j u , u = 1, 2, 3} is an almost quaternionic structure on M and {J u , u = 1, 2, 3} is an almost quaternionic structure on N . This is a generalization of the Cauchy-Fueter equations. For M, N hyperKähler, this generalization naturally arises by obtaining the topological σ-model as a twisted version of the N=2 globally supersymmetric σ-model. We discuss various examples of hyperinstantons, in particular on the torus and the K3 surface. We also analyse the coupling of the topological σ-model to topological gravity. The classification of triholomorphic maps and the analysis of their moduli-space is a new and fully open mathematical problem that we believe deserves the attention of both mathematicians and physicists.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Topological σ-models in four dimensions and triholomorphic maps

1994

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“…For this reason the maps satisfying eq. (2.28) were dubbed hyperinstantons in [22] and it was also observed that they are tri-holomorphic since eq. (2.28) can be interpreted as the statement that they are holomorphic with respect to the average of the three complex structures.…”

Section: Relation Of the Arnold-beltrami Fluxes With Hyperinstantonsmentioning

confidence: 99%

Minimal D = 7 supergravity and the supersymmetry of Arnold-Beltrami flux branes

Fré

Grassi

Ravera

et al. 2016

J. High Energ. Phys.

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In this paper we study some properties of the newly found Arnold-Beltrami flux-brane solutions to the minimal D = 7 supergravity. To this end we first single out the appropriate Free Differential Algebra containing both a gauge 3-form B [3] and a gauge 2-form B [2] : then we present the complete rheonomic parametrization of all the generalized curvatures. This allows us to identify two-brane configurations with Arnold-Beltrami fluxes in the transverse space with exact solutions of supergravity and to analyze the Killing spinor equation in their background. We find that there is no preserved supersymmetry if there are no additional translational Killing vectors. Guided by this principle we explicitly construct Arnold-Beltrami flux two-branes that preserve 0, 1 8 and 1 4 of the original supersymmetry. Two-branes without fluxes are instead BPS states and preserve 1 2 supersymmetry. For each two-brane solution we carefully study its discrete symmetry that is always given by some appropriate crystallographic group Γ. Such symmetry groups Γ are transmitted to the D = 3 gauge theories on the brane world-volume that would occur in the gauge/gravity correspondence. Furthermore we illustrate the intriguing relation between gauge fluxes in two-brane solutions and hyperinstantons in D = 4 topological sigma-models.

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“…In this section, following once again [11] and [10] we present the general form of a rigid N = 2 supersymmetric sigma-model in D = 4 and we perform its topological twist according to the 4D-algorithm developed by Anselmi and Fré in [12], [13], [14]. Given the base flat manifold M 4 , which has always a local HyperKähler structure, supersymmetry requires that the target manifold should be a HyperKähler manifold N 4m of real dimension 4m.…”

Section: Gamma Matricesmentioning

confidence: 99%

Hyperinstantons, the Beltrami equation, and triholomorphic maps

2015

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We consider the Beltrami equation for hydrodynamics and we show that its solutions can be viewed as instanton solutions of a more general system of equations. The latter are the equations of motion for an N = 2 sigma model on 4-dimensional worldvolume (which is taken locally HyperKähler) with a 4-dimensional HyperKähler target space. By means of the 4D twisting procedure originally introduced by Witten for gauge theories and later generalized to 4D sigmamodels by Anselmi and Fré, we show that the equations of motion describe triholomophic maps between the worldvolume and the target space. Therefore, the classification of the solutions to the 3-dimensional Beltrami equation can be performed by counting the triholomorphic maps. The counting is easily obtained by using several discrete symmetries. Finally, the similarity with holomorphic maps for N = 2 sigma on Calabi-Yau space prompts us to reformulate the problem of the enumeration of triholomorphic maps in terms of a topological sigma model.

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Topological twist in four dimensions, R-duality and hyperinstantons

Cited by 32 publications

References 41 publications

Topological σ-models in four dimensions and triholomorphic maps

Topological σ-models in four dimensions and triholomorphic maps

Minimal D = 7 supergravity and the supersymmetry of Arnold-Beltrami flux branes

Hyperinstantons, the Beltrami equation, and triholomorphic maps

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