Cohomology of the moduli of parabolic vector bundles

Nitsure, Nitin

doi:10.1007/bf02837250

Cited by 26 publications

(31 citation statements)

References 5 publications

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“…More important for us is the generalization of the above result to allow parabolic structure [77]. We will state this theorem for any number of ramification points with arbitrary choices of parabolic subgroups.…”

Section: Scaling Symmetrymentioning

confidence: 93%

Gauge theory, ramification, and the geometric Langlands program

Gukov¹,

Witten²

2006

Current Developments in Mathematics

412

1,287

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In the gauge theory approach to the geometric Langlands program, ramification can be described in terms of "surface operators," which are supported on two-dimensional surfaces somewhat as Wilson or 't Hooft operators are supported on curves. We describe the relevant surface operators in N = 4 super Yang-Mills theory, and the parameters they depend on, and analyze how S-duality acts on these parameters. Then, after compactifying on a Riemann surface, we show that the hypothesis of S-duality for surface operators leads to a natural extension of the geometric Langlands program for the case of tame ramification. The construction involves an action of the affine Weyl group on the cohomology of the moduli space of Higgs bundles with ramification, and an action of the affine braid group on A-branes or B-branes on this space.

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Section: Scaling Symmetrymentioning

confidence: 93%

Gauge theory, ramification, and the geometric Langlands program

Gukov¹,

Witten²

2006

Current Developments in Mathematics

412

1,287

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“…In case (a) the corresponding critical subvariety can obviously be identified with the moduli space of ordinary parabolic bundles. Its Betti numbers can be computed from a formula given by Nitsure [34] and Holla [25]. In Section 10 below we work out explicitly what their formula gives for the Poincaré polynomial in our situation of rank three parabolic bundles.…”

Section: Morse Indicesmentioning

confidence: 99%

“…The Betti numbers of the moduli space of parabolic vector bundles were computed by Nitsure [34] and Holla [25]. Here we work out Holla's formula for the special case when the rank is 3 and all flags at the parabolic points are full.…”

Section: Betti Numbers Of the Moduli Space Of Rank Three Parabolic Bumentioning

confidence: 99%

Betti numbers of the moduli space of rank 3 parabolic Higgs bundles

Garcı́a-Prada¹,

Gothen²,

Muñoz³

2007

Memoirs of the AMS

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Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and non-fixed determinant, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles.

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“…It was reproved by Atiyah and Bott [1] using gauge theory. The methods of Nitsure [9] generalize the approach of Atiyah and Bott [1] to parabolic bundles.) Now, if i : SU X (n, L, ∆)−→SU X (n, L, ∆) × J X denotes the map E → (E, O X ), and if j : SU X (n, L, ∆)→U X (n, d, ∆) denotes the inclusion, then j = π • i.…”

Section: Proofsmentioning

confidence: 98%

Canonical generators of the cohomology of moduli of parabolic bundles on curves

Biswas

Raghavendra

1996

Math. Ann.

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Let X be a compact Riemann surface, fix integers n and d with n positive, and let ∆ be a parabolic datum of rank n on X (see Section 3 below). Denote by U X (n, d, ∆) the moduli space of parabolic vector bundles of rank n and degree d, which are parabolic semistable with respect to ∆. Fix a holomorphic line bundle L of degree d on X, and let SU X (n, L, ∆) be the subvariety of U X (n, d, ∆) consisting of vector bundles with determinant isomorphic to L. We make the following two hypotheses on the parameters n, d and ∆.Assumption 1.1 Every parabolic vector bundle of rank n and degree d on X which is parabolic semistable with respect to the parabolic datum ∆ is in fact parabolic stable.Assumption 1.2 There exists a universal parabolic bundle (or briefly, a universal bundle) on U X (n, d, ∆) × X.Recall that a universal bundle on U X (n, d, ∆) × X is a vector bundle U on U X (n, d, ∆) × X together with a flag of subbundles j *x U for each x ∈ J (J being the set of parabolic points), where j x : U X (n, d, ∆)→U X (n, d, ∆)×X is the map E → (E, x), such that for each E ∈ U X (n, d, ∆), the restriction of U to {E}×X is parabolically isomorphic to E. We use the same symbols to denote the restrictions of U and U x,i etc. to SU X (n, L, ∆). Notation 1.3 • If S and T are topological spaces, and if α ∈ H * (S × T, Q), then σ(α) : H * (T, Q) → H * (S, Q) is the map σ(α)z = α/z, where / denotes the slant product.

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Cohomology of the moduli of parabolic vector bundles

Cited by 26 publications

References 5 publications

Gauge theory, ramification, and the geometric Langlands program

Gauge theory, ramification, and the geometric Langlands program

Betti numbers of the moduli space of rank 3 parabolic Higgs bundles

Canonical generators of the cohomology of moduli of parabolic bundles on curves

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