The nonabelian Szegő kernel and theta-divisor

Fay, John D.

doi:10.1090/conm/136/1188198

Cited by 13 publications

(25 citation statements)

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“…8.1). This is the parallel of a result of Fay (Theorem 2 in [10]). However the form that Fay introduced in loc.…”

Section: Introduction and Resultssupporting

confidence: 54%

“…Proof. [1] The proof follows along the lines of an argument by Fay [10] adapted to our setting. We start by observing that if r ∈ H 0 (E 0 (−∞)) and η ∈ H 0 (E * 0 ⊗ K) then Φ = ηr t is a Higgs field, see (5.1).…”

Section: Singularities Of the Malgrange-fay Form Along The Non-abelia...mentioning

confidence: 92%

“…In this last section we show that the Malgrange-Fay form Ξ on the bundle of connections A D has a first order pole with residue given by the index h 1 (E). This is the content of the following theorem, which is an analogue of Theorems 3 and 4 in [10]. More precisely, we consider a holomorphic family E ǫ , for |ǫ| < 1, such that E 0 ∈ (Θ).…”

Section: Singularities Of the Malgrange-fay Form Along The Non-abelia...mentioning

confidence: 99%

“…Remark 7.6 (Relationship with Fay's [10]) In [10] Fay introduced, in a slightly different context, a similar one-form. The main difference is that Fay works with flat vector bundles tensored by a root of the canonical bundle, and hence he works on vector bundles of degree n(g − 1) so that Riemann-Roch's theorem states h 0 = h 1 and both are generically zero.…”

Section: Monodromy Representationmentioning

confidence: 99%

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Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

Bertola,

Norton,

Ruzza

2021

Preprint

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We consider the moduli space of vector bundles of rank n and degree ng over a fixed Riemann surface of genus g ≥ 2. We use the explicit parametrization in terms of the Tyurin data. In the moduli space there is a "non-abelian" Theta divisor, consisting of bundles with h 1 ≥ 1. On the complement of this divisor we construct a non-abelian Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which is also dependent holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the GLn character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of affine connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by h 1 .

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“…8.1). This is the parallel of a result of Fay (Theorem 2 in [10]). However the form that Fay introduced in loc.…”

Section: Introduction and Resultssupporting

confidence: 54%

Section: Singularities Of the Malgrange-fay Form Along The Non-abelia...mentioning

confidence: 92%

Section: Singularities Of the Malgrange-fay Form Along The Non-abelia...mentioning

confidence: 99%

Section: Monodromy Representationmentioning

confidence: 99%

See 2 more Smart Citations

Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

Bertola,

Norton,

Ruzza

2021

Preprint

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“…Kernels similar to G(z, z ) were studied in Fay [19,61], who succeeded in expressing them explicitly in terms of θ-functions.…”

Section: The Metric and Three-formmentioning

confidence: 99%

Gauge Theory And Integrability, II

Costello

Witten

Yamazaki

2018

Notices of the International Congress of Chinese Mathematicians

125

223

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We study two-dimensional integrable field theories from the viewpoint of the four-dimensional Chern-Simons-type gauge theory introduced recently. The integrable field theories are realized as effective theories for the four-dimensional theory coupled with two-dimensional surface defects, and we can systematically compute their Lagrangians and the Lax operators satisfying the zero-curvature condition. Our construction includes many known integrable field theories, such as Gross-Neveu models, principal chiral models with Wess-Zumino terms and symmetric-space coset sigma models. Moreover we obtain various generalization these models in a number of different directions, such as trigonometric/elliptic deformations, multi-defect generalizations and models associated with higher-genus spectral curves, many of which seem to be new. 16 Gluing 95 16.

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[87] (with N. S. Hawley) Half-order differentials on Riemann surfaces

Fay

2013

Menahem Max Schiffer: Selected Papers Volume 2

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The nonabelian Szegő kernel and theta-divisor

Cited by 13 publications

References 0 publications

Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

Gauge Theory And Integrability, II

[87] (with N. S. Hawley) Half-order differentials on Riemann surfaces

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