Stable triples, equivariant bundles and dimensional reduction

Bradlow, Steven B.; Garcı́a-Prada, Oscar

doi:10.1007/bf01446292

Cited by 98 publications

(169 citation statements)

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“…Now we move to the analysis of the moduli spaces of σ-polystable triples of rank (3,1). Let N σ = N σ (3, 1, d 1 , d 2 ).…”

Section: Critical Values For Triples Of Rank (3 1)mentioning

confidence: 99%

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Hodge polynomials of the moduli spaces of rank 3 pairs

Muñoz

2008

Geom Dedicata

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Abstract. Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple (E 1 , E 2 , φ) on X consists of two holomorphic vector bundles E 1 and E 2 over X and a holomorphic map φ : E 2 → E 1 . There is a concept of stability for triples which depends on a real parameter σ. In this paper, we determine the Hodge polynomials of the moduli spaces of σ-stable triples with rk(E 1 ) = 3, rk(E 2 ) = 1, using the theory of mixed Hodge structures. This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles.

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“…Now we move to the analysis of the moduli spaces of σ-polystable triples of rank (3,1). Let N σ = N σ (3, 1, d 1 , d 2 ).…”

Section: Critical Values For Triples Of Rank (3 1)mentioning

confidence: 99%

“…To study the dependence of the moduli spaces N σ on the parameter, we need to introduce the concept of critical value [1,11]. …”

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confidence: 99%

Hodge polynomials of the moduli spaces of rank 3 pairs

Muñoz

2008

Geom Dedicata

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“…We collect here some known results about the moduli spaces of pairs; the details can be found in [6], [7], [17], [15] and [18].…”

Section: Moduli Spaces Of Pairsmentioning

confidence: 99%

Rationality of the Moduli Space of Stable Pairs over a Complex Curve

Biswas

Logares

Muñoz

2012

Springer Optimization and Its Applications

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Abstract. Let X be a smooth complex projective curve of genus g ≥ 2. A pair on X is formed by a vector bundle E → X and a global non-zero section φ ∈ H 0 (E). There is a concept of stability for pairs depending on a real parameter τ , giving rise to moduli spaces M τ (r, Λ) of τ -stable pairs of rank r and fixed determinant Λ. In this paper we prove that the moduli spaces M τ (r, Λ) are in many cases rational.Dedicated to the 60th Anniversary of Themistocles M. Rassias

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“…In the context of model building, some physicists have applied for a long time the method of 'coset-space dimensional reduction' in the construction of gauge unified theories (see, e.g., [15,7,20,26]). In [16,17,11,1] the dimensional reduction techniques were brought into the context of holomorphic vector bundles over Kähler manifolds, to study the dimensional reduction of stable SL(2, C)-equivariant bundles over X × P 1 and the corresponding Hermitian-YangMills equations, where X is a compact Kähler manifold and P 1 is the Riemann sphere. In [2] this construction was generalised to G-equivariant vector bundles on X ×G/P , where G is a connected simply connected complex semisimple Lie group and P ⊂ G is a parabolic subgroup.…”

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confidence: 99%

A Tannakian approach to dimensional reduction of principal bundles

Álvarez-Cónsul

Biswas

Garcı́a-Prada

2017

Journal of Geometry and Physics

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Abstract. Let P be a parabolic subgroup of a connected simply connected complex semisimple Lie group G. Given a compact Kähler manifold X, the dimensional reduction of G-equivariant holomorphic vector bundles over X × G/P was carried out by the first and third authors [2]. This raises the question of dimensional reduction of holomorphic principal bundles over X × G/P . The method used for equivariant vector bundles does not generalize to principal bundles. In this paper, we adapt to equivariant principal bundles the Tannakian approach of Nori, to describe the dimensional reduction of Gequivariant principal bundles over X × G/P , and to establish a Hitchin-Kobayashi type correspondence. In order to be able to apply the Tannakian theory, we need to assume that X is a complex projective manifold. IntroductionDimensional reduction is a very powerful construction in the context of gauge theory, both in physics and geometry. Many important gauge-theoretic equations appear as symmetric solutions of fundamental equations for connections, like the instanton equations on Riemannian 4-manifolds. Examples of these are the Bogomol'nyi equations for magnetic monopoles in 3 dimensions and the vortex equation in 2 dimensions, as well as many other important integrable systems and soliton equations. In the context of model building, some physicists have applied for a long time the method of 'coset-space dimensional reduction' in the construction of gauge unified theories (see, e.g., [15,7,20,26]).In [16,17,11,1] the dimensional reduction techniques were brought into the context of holomorphic vector bundles over Kähler manifolds, to study the dimensional reduction of stable SL(2, C)-equivariant bundles over X × P 1 and the corresponding Hermitian-YangMills equations, where X is a compact Kähler manifold and P 1 is the Riemann sphere. In [2] this construction was generalised to G-equivariant vector bundles on X ×G/P , where G is a connected simply connected complex semisimple Lie group and P ⊂ G is a parabolic subgroup. These gave rise to quiver bundles with relations, that is representations of a quiver with relations in the category of vector bundles, where the quiver with relations (Q, K) is determined by P . In particular when X is a point, one has a description of 2000 Mathematics Subject Classification. 53C07, 14D21, 16G20.

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Stable triples, equivariant bundles and dimensional reduction

Cited by 98 publications

References 20 publications

Hodge polynomials of the moduli spaces of rank 3 pairs

Hodge polynomials of the moduli spaces of rank 3 pairs

Rationality of the Moduli Space of Stable Pairs over a Complex Curve

A Tannakian approach to dimensional reduction of principal bundles

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