Monopoles and Baker functions

Ercolani, Nicholas M.; Sinha, Aninda

doi:10.1007/bf01218409

Cited by 40 publications

(83 citation statements)

References 18 publications

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“…Modulo a base change of sl(2) these are the same equations which were found later in the description of self-dual magnetic monopoles [7] [8] [9].…”

Section: Schmid's Work On Large Complex Structuresmentioning

confidence: 72%

Mirror symmetry and self-duality equations

Nahm¹

2000

Proceedings of Non-Perturbative Quantum Effects 2000 — PoS(tmr2000)

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A central problem of mirror symmetry is to understand Calabi-Yau moduli spaces at large complex structure. The basic techniques for this problem have been developped by Griffiths, Deligne and Schmid. The moduli space for complex structures is mapped into Griffiths' classifying space for Hodge structures, and the image is approximated by a nilpotent orbit up to exponentially small terms. Schmid described the geometry of the nilpotent orbit by equations which later were found by the speaker in the context of magnetic monopoles. The nilpotent orbit case corresponds to the special case of a maximally degenerate spectral curve.Keywords: large complex structure, moduli, conformal field theory. Boundaries of the M-theory moduli spaceThere are two limits in which non-perturbative string theory (alias M-theory) is accessible to calculations, namely small spacetime curvature and small string coupling. In the first case, in particular for strings on the product of a flat spacetime with a compact internal space of large radius, one obtains in the limit a conventional quantum field theory on an Einstein spacetime. The behaviour at small string coupling, on the other hand, is described by a superconformal theory on the string worldsheet. If the present indications of a low Higgs mass will hold true, one even can hope for low energy supersymmetry and consequently for extended supersymmetry in the worldsheet theory. Together with T-and S-duality, the two limits might give sufficient control over the boundaries of the (hypothetical) M-theory moduli space to get a handle on more realistic situations. At present, even the understanding of the boundaries is far from complete. Even in the favorable case of extended supersymmetry, the conformal field theories are only solvable for special parameters, and there is still much to be learned from the case where the two limits are performed together. But at least a complete understanding of the geometry and metric of their moduli spaces seems feasible. This means that it will be much easier to classify the theories then to describe them in detail. Moreover, for the special physical quantities described by meromorphic functions the description of the moduli space takes us very close to an explicit computation, since for a conventional vacuum one expects compactifiable spaces. Classical and quantum moduli spacesThe moduli space M of a conformal theory with extended supersymmetry is closely related to certain moduli spaces of classical geometrical structures. In the limiting case when both the string coupling and the spacetime curvature tend to zero, it is clear that the conformal theory is given by a non-linear sigma model on a Calabi-Yau manifold X, such that M can be projected to the moduli space M (X) of complex structures of the underlying differentiable manifold. For IIA theories with (exactly) N = (2, 2) supersymmetry, in particular for central charges c = 3 and c = 9, one also obtains a projection to the moduli space of the mirror manifold X , and M = M(X) × M(X ), at least...

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“…Modulo a base change of sl(2) these are the same equations which were found later in the description of self-dual magnetic monopoles [7] [8] [9].…”

Section: Schmid's Work On Large Complex Structuresmentioning

confidence: 72%

Mirror symmetry and self-duality equations

Nahm¹

2000

Proceedings of Non-Perturbative Quantum Effects 2000 — PoS(tmr2000)

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“…Thus the integration of the Adjoint Weyl equation reduces to the matrix spectral problem (4.2). The same problem appeared in [1] and [3] when focussing on the algebro-geometric integration of the Nahm equation and we shall use the results of our recent paper [1] for the integration of the Weyl equation. Let θ be the canonical θ -function of the curve C and let τ be its period matrix.…”

Section: The Baker-akhiezer Functionmentioning

confidence: 99%

“…Although the associated spectral problem is equivalent to that appearing in the algebro-geometric integration of the Nahm equation, we do not use solutions of Nahm's equation directly in our development. To achieve our result we implement the θ -functional integration of the Nahm equation by Ercolani and Sinha [3] and our recent analysis [1]. The limitations of space in this volume prevent detailed examples being given and a fuller exposition will be given elsewhere.…”

Section: Introductionmentioning

confidence: 99%

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FINITE-GAP INTEGRATION OF THE SU(2) BOGOMOLNY EQUATIONS

Braden¹,

Enolskii²

2009

Glasgow Math. J.

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Abstract. The Atiyah-Drinfeld-Hitchin-Manin-Nahm (ADHMN) construction of magnetic monopoles is given in terms of the (normalizable) solutions of an associated Weyl equation. We focus here on solving this equation directly by algebrogeometric means. The (adjoint) Weyl equation is solved using an ansatz of Nahm in terms of Baker-Akhiezer functions. The solution of Nahm's equation is not directly used in our development.

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“…Moreover, based on the integrability one can systematically construct the Nahm data, i.e. the solutions of the Nahm equations [17,18]. It is natural to expect that the present BPS equations are also integrable and one may be able to construct solutions systematically making use of the integrability.…”

Section: Introductionmentioning

confidence: 99%

Integrability of BPS equations in ABJM theory

Sakai

Terashima

2013

J. High Energ. Phys.

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We investigate BPS equations which determine the configuration of an M2-M5 bound state preserving half of the supersymmetries in the ABJM theory. We argue that the BPS equations are classically integrable, showing that they admit a Lax representation. The integrable structure of the BPS equations is closely related to that of the Nahm equations. Using this relation we formulate an efficient way of constructing solutions of the BPS equations from those of the Nahm equations. As an illustration of our method, we construct explicitly the most general solutions describing two M2-branes suspended between two parallel M5-branes as well as two semi-infinite M2-branes ending on an M5-brane. These include previously unknown new solutions. We also discuss a reduction of the BPS equations in connection with the periodic Toda chain.

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Monopoles and Baker functions

Cited by 40 publications

References 18 publications

Mirror symmetry and self-duality equations

Mirror symmetry and self-duality equations

FINITE-GAP INTEGRATION OF THE SU(2) BOGOMOLNY EQUATIONS

Integrability of BPS equations in ABJM theory

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