344 research outputs found

    Normal bundles to Laufer rational curves in local Calabi-Yau threefolds

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    We prove a conjecture by F. Ferrari. Let X be the total space of a nonlinear deformation of a rank 2 holomorphic vector bundle on a smooth rational curve, such that X has trivial canonical bundle and has sections. Then the normal bundle to such sections is computed in terms of the rank of the Hessian of a suitably defined superpotential at its critical points

    The K\ue4hler quotient resolution of C3/\u393 singularities, the McKay correspondence and D=3 N = 2 Chern-Simons gauge theories

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    We advocate that the generalized Kronheimer construction of the Ka \u308hler quotient crepant resolution M\u3b6 12\u2192 C3/\u393 of an orbifold singularity where \u393 82 SU(3) is a finite subgroup naturally defines the field content and the interaction structure of a superconformal Chern-Simons Gauge Theory. This latter is suppos- edlythedualofanM2-branesolutionofD=11supergravitywithC 7M\u3b6 astransversespace.Weillustrate and discuss many aspects of this type of constructions emphasizing that the equation p 27 p = 0 which provides the Ka \u308hler analogue of the holomorphic sector in the hyperKa \u308hler moment map equations canonically defines the structure of a universal superpotential in the CS theory. Furthermore the kernel D\u393 of the above equation can be described as the orbit with respect to a quiver Lie group G\u393 of a special locus L\u393 82 Hom\u393 (Q 97 R, R) that has also a universal definition. We provide an extensive discussion of the relation between the coset manifold G\u393/F\u393, the gauge group F\u393 being the maximal compact subgroup of the quiver group, the moment map equations and the first Chern classes of the so named tautological vector bundles that are in one-to-one correspondence with the nontrivial irreps of \u393. These first Chern classes are represented by (1,1)-forms on M\u3b6 and provide a basis for the cohomology group H2(M\u3b6 ). We also discuss the relation with conjugacy classes of \u393 and we provide the explicit construction of several examples emphasizing the role of a general- ized McKay correspondence. The case of the ALE manifold resolution of C2/\u393 singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited

    Semistable Higgs bundles on Calabi-Yau manifolds

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    We provide a partial classification of semistable Higgs bundles over a simply connected Calabi-Yau manifolds. Applications to a conjecture about a special class of semistable Higgs bundles are given. In particular, the conjecture is proved for K3 and Enriques surfaces, and some related classes of surfaces

    Gauge fixing and equivariant cohomology

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    The supersymmetric model developed by Witten to study the equivariant cohomology of a manifold with an isometric circle action is derived from the BRST quantization of a simple classical model. The gauge-fixing process is carefully analysed, and demonstrates that different choices of gauge-fixing fermion can lead to different quantum theories.Comment: 18 pages LaTe

    Parafermionic Liouville field theory and instantons on ALE spaces

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    In this paper we study the correspondence between the su^(n)ksu^(n)p/su^(n)k+p\hat{\textrm{su}}(n)_{k}\oplus \hat{\textrm{su}}(n)_{p}/\hat{\textrm{su}}(n)_{k+p} coset conformal field theories and N=2\mathcal{N}=2 SU(n) gauge theories on R4/Zp\mathbb{R}^{4}/\mathbb{Z}_{p}. Namely we check the correspondence between the SU(2) Nekrasov partition function on R4/Z4\mathbb{R}^{4}/\mathbb{Z}_{4} and the conformal blocks of the S3S_{3} parafermion algebra (in SS and DD modules). We find that they are equal up to the U(1)-factor as it was in all cases of AGT-like relations. Studying the structure of the instanton partition function on R4/Zp\mathbb{R}^4/\mathbb{Z}_p we also find some evidence that this correspondence with arbitrary pp takes place up to the U(1)-factor.Comment: 21 pages, 6 figures, misprints corrected, references added, version to appear in JHE

    Yang-Mills-Higgs connections on Calabi-Yau manifolds

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    Let XX be a compact connected K"ahler--Einstein manifold with c1(TX),geq,0c_1(TX), geq, 0. If there is a semistable Higgs vector bundle (E,,heta)(E, , heta) on XX with heta,ot=,0 heta, ot=, 0, then we show that c1(TX)=0c_1(TX)=0; any XX satisfying this condition is called a Calabi--Yau manifold, and it admits a Ricci--flat K"ahler form cite{Ya}. Let (E,,heta)(E, , heta) be a polystable Higgs vector bundle on a compact Ricci--flat K"ahler manifold XX. Let hh be an Hermitian structure on EE satisfying the Yang--Mills--Higgs equation for (E,,heta)(E, , heta). We prove that hh also satisfies the Yang--Mills--Higgs equation for (E,,0)(E, ,0). A similar result is proved for Hermitian structures on principal Higgs bundles on XX satisfying the Yang--Mills--Higgs equation. \ua9 2016 International Press

    On semistable principal bundles over a complex projective manifold, II

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    Let (X, \omega) be a compact connected Kaehler manifold of complex dimension d and E_G a holomorphic principal G-bundle on X, where G is a connected reductive linear algebraic group defined over C. Let Z (G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P of G and a holomorphic reduction of the structure group of E_G to P (say, E_P) such that the bundle obtained by extending the structure group of E_P to L(P)/Z(G) (where L(P) is the Levi quotient of P) admits a flat connection; (2) The adjoint vector bundle ad(E_G) is numerically flat; (3) The principal G-bundle E_G is pseudostable, and the degree of the charateristic class c_2(ad(E_G) is zero.Comment: 15 page
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