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We derive a family of matrix models which encode solutions to the Seiberg–Witten theory in 4 and 5 dimensions. Partition functions of these matrix models are equal to the corresponding Nekrasov partition functions, and their spectral curves are the Seiberg–Witten curves of the corresponding theories. In consequence of the geometric engineering, the 5-dimensional case provides a novel matrix model formulation of the topological string theory on a wide class of non-compact toric Calabi–Yau manifolds. This approach also unifies and generalizes other matrix models, such as the Eguchi–Yang matrix model, matrix models for bundles over P1, and Chern–Simons matrix models for lens spaces, which arise as various limits of our general result. 相似文献
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We explore a new connection between Seiberg–Witten theory and quantum statistical systems by relating the dual partition function of SU(2) Super Yang–Mills theory in a self-dual \(\Omega \) background to the spectral determinant of an ideal Fermi gas. We show that the spectrum of this gas is encoded in the zeroes of the Painlevé \(\mathrm{III}_3\) \(\tau \) function. In addition, we find that the Nekrasov partition function on this background can be expressed as an O(2) matrix model. Our construction arises as a four-dimensional limit of a recently proposed conjecture relating topological strings and spectral theory. In this limit, we provide a mathematical proof of the conjecture for the local \({\mathbb P}^1 \times {\mathbb P}^1\) geometry. 相似文献
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We analyze the noncommutative two-dimensional Wess–Zumino–Witten model and its properties under Seiberg–Witten transformations in the operator formulation. We prove that the model is invariant under such transformations even for the noncritical (non-chiral) case, in which the coefficients of the kinetic and Wess–Zumino terms are not related. The pure Wess–Zumino term represents a singular case in which this transformation fails to reach a commutative limit. We also discuss potential implications of this result for bosonization. 相似文献
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Let F be a differentiable manifold endowed with an almost K?hler structure (J,ω), α a J-holomorphic action of a compact Lie group on F, and K a closed normal subgroup of which leaves ω invariant.
The purpose of this article is to introduce gauge theoretical invariants for such triples (F,α,K). The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface Σ.
Our main results concern the special case of the triple
where αcan denotes the canonical action of on . We give a complex geometric interpretation of the corresponding moduli spaces of solutions in terms of gauge theoretical
quot spaces, and compute the invariants explicitly in the case r=1.
Proving a comparison theorem for virtual fundamental classes, we show that the full Seiberg–Witten invariants of ruled surfaces,
as defined in [OT2], can be identified with certain gauge theoretical Gromov–Witten invariants of the triple (Hom(ℂ,ℂ<
r
0),αcan, U(1)). We find the following formula for the full Seiberg–Witten invariant of a ruled surface over a Riemann surface of genus
g:
where [F] denotes the class of a fibre. The computation of the invariants in the general case r >1 should lead to a generalized Vafa-Intriligator formula for “twisted”Gromov–Witten invariants associated with sections
in Grassmann bundles.
Received: 22 February 2001 / Accepted: 16 January 2002 相似文献
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We construct a natural L2-metric on the perturbed Seiberg–Witten moduli spaces Mμ+ of a compact 4-manifold M, and we study the resulting Riemannian geometry of Mμ+. We derive a formula which expresses the sectional curvature of Mμ+ in terms of the Green operators of the deformation complex of the Seiberg–Witten equations. In case M is simply connected, we construct a Riemannian metric on the Seiberg–Witten principal U(1) bundle P→Mμ+ such that the bundle projection becomes a Riemannian submersion. On a Kähler surface M, the L2-metric on Mμ+ coincides with the natural Kähler metric on moduli spaces of vortices. 相似文献
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We construct and study a new topological field theory in three dimensions. It is a hybrid between Chern–Simons and Rozansky–Witten theory and can be regarded as a topologically-twisted version of the N=4d=3 supersymmetric gauge theory recently discovered by Gaiotto and Witten. The model depends on a gauge group G and a hyper-Kähler manifold X with a tri-holomorphic action of G. In the case when X is an affine space, we show that the model is equivalent to Chern–Simons theory whose gauge group is a supergroup. This explains the role of Lie superalgebras in the construction of Gaiotto and Witten. For general X, our model appears to be new. We describe some of its properties, focusing on the case when G is simple and X is the cotangent bundle of the flag variety of G. In particular, we show that Wilson loops are labeled by objects of a certain category which is a quantum deformation of the equivariant derived category of coherent sheaves on X. 相似文献
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Weiyue Ding Jürgen Jost Jiayu Li Xiaowei Peng Guofang Wang 《Communications in Mathematical Physics》2001,217(2):383-407
The abelian Chern–Simons–Higgs model of Hong-Kim-Pac and Jackiw–Weinberg leads to a Ginzburg–Landau type functional with
a 6th order potential on a compact Riemann surface. We derive the existence of two solutions with different asymptotic behavior
as the coupling parameter tends to 0, for any number of prescribed vortices. We also introduce a Seiberg–Witten type functional
with a 6th order potential and again show the existence of two asymptotically different solutions on a compact K?hler surface. The analysis
is based on maximum principle arguments and applies to a general class of scalar equations.
Received: 13 October 1998 / Accepted: 21 October 2000 相似文献
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X. Liu 《Communications in Mathematical Physics》2001,216(3):705-728
We study some necessary and sufficient conditions for the genus-1 Virasoro conjecture proposed by Eguchi–Hori–Xiong and S.
Katz.
Received: 22 August 1999 / Accepted: 7 October 2000 相似文献
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We study the enumerative significance of the s-pointed genus zero Gromov–Witten invariant on a homogeneous space X. For that, we give an interpretation in terms of rational curves on X. 相似文献
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