Quantum mechanics on moduli spaces

Moss, Ian G.; Shiiki, Noriko

doi:10.1016/s0550-3213(99)00650-1

Cited by 14 publications

(21 citation statements)

References 25 publications

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“…In particular, we choose the time evolution of h such that the dynamical degrees of freedom of A i lie transverse to gauge orbits, and as such the resulting model describes only these physical degrees of freedom. This amounts to requiring that at any fixed time, 15) for any iΛ : R 1,4 → su(2) that vanishes at spatial infinity. This stipulation ensures that g(t) remains intact as a physical degree of freedom [26].…”

Section: )mentioning

confidence: 99%

“…The study of the low-energy scattering of solitons began with the work of Manton [9] on BPS monopoles, where it was first argued that the dynamics of slowly moving solitons can be captured by geodesic motion on the moduli space of static solutions. Later work explored many different avenues, including applying similar methods to Yang-Mills solitons of other codimension [10,11], and most relevant to this work, the study of a supersymmetric extension to this approximation [12][13][14][15] to determine the supersymmetric effective theory for both the bosonic and fermionic soliton zero modes.…”

Section: Introductionmentioning

confidence: 99%

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Supersymmetric soliton σ-models from non-Lorentzian field theories

Mouland

2020

J. High Energ. Phys.

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Recently, a class of non-Lorentzian supersymmetric Lagrangian field theories was considered, in some cases describing M-theory brane configurations, but more generally found as fixed points of non-Lorentzian RG flows induced upon Lorentzian theories. In this paper, we demonstrate how the dynamics of such theories can be reduced to motion on the supersymmetric moduli space of BPS solitons of the parent theory. We focus first on the N = (1, 1) σ-model in (1 + 1)-dimensions with potential, where we produce a supersymmetric extension to the standard geodesic approximation for slow kink motion. We then revisit the (4 + 1)-dimensional Yang-Mills-like theory with 24 supercharges describing a null compactification of M5-branes. We show that the theory reduces to a σ-model on instanton moduli space, extended by couplings to additional fields from the parent theory, and possessing (8+8) super(conformal) symmetries along with 8 further fermionic shift symmetries. We derive this model explicitly for the single SU (2) instanton.

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Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Supersymmetric soliton σ-models from non-Lorentzian field theories

Mouland

2020

J. High Energ. Phys.

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“…Additionally there can be extrinsic curvature terms that encode how the soliton moduli space is embedded in the infinite-dimensional field configuration space. See [115,126,127] for discussions of this phenomenon in the field theory context. Other operators in addition to the Hamiltonian can suffer such ambiguities.…”

Section: Collective Coordinate Ansatzmentioning

confidence: 99%

Semiclassical framed BPS states

Moore

Royston

Bleeken

2016

J. High Energ. Phys.

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We provide a semiclassical description of framed BPS states in fourdimensional N = 2 super Yang-Mills theories probed by 't Hooft defects, in terms of a supersymmetric quantum mechanics on the moduli space of singular monopoles. Framed BPS states, like their ordinary counterparts in the theory without defects, are associated with the L 2 kernel of certain Dirac operators on moduli space, or equivalently with the L 2 cohomology of related Dolbeault operators. The Dirac/Dolbeault operators depend on two Cartan-valued Higgs vevs. We conjecture a map between these vevs and the SeibergWitten special coordinates, consistent with a one-loop analysis and checked in examples. The map incorporates all perturbative and nonperturbative corrections that are relevant for the semiclassical construction of BPS states, over a suitably defined weak coupling regime of the Coulomb branch. We use this map to translate wall crossing formulae and the no-exotics theorem to statements about the Dirac/Dolbeault operators. The no-exotics theorem, concerning the absence of nontrivial SU(2) R representations in the BPS spectrum, implies that the kernel of the Dirac operator is chiral, and further translates into a statement that all L 2 cohomology of the Dolbeault operator is concentrated in the middle degree. Wall crossing formulae lead to detailed predictions for where the Dirac operators fail to be Fredholm and how their kernels jump. We explore these predictions in nontrivial examples. This paper explains the background and arguments behind the results announced in the short note [1].

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“…However, there seems to be general agreement that, following De Witt [5], one should include (a positive multiple of) κ in V . For a recent discussion of this subject, specifically in the context of σ-models, see [23]. So the relevance of κ to quantum lump dynamics, as well as simple geometric curiosity, motivate us to calculate it.…”

Section: Scalar Curvaturementioning

confidence: 99%

The L2 geometry of spaces of harmonic maps S2→S2 and

Speight

2003

Journal of Geometry and Physics

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Department of Pure Mathematics, University of LeedsLeeds LS2 9JT, Abstract Harmonic maps from S 2 to S 2 are all weakly conformal, and so are represented by rational maps. This paper presents a study of the L 2 metric γ on M n , the space of degree n harmonic maps S 2 → S 2 , or equivalently, the space of rational maps of degree n. It is proved that γ is Kähler with respect to a certain natural complex structure on M n . The case n = 1 is considered in detail: explicit formulae for γ and its holomorphic sectional, Ricci and scalar curvatures are obtained, it is shown that the space has finite volume and diameter and codimension 2 boundary at infinity, and a certain class of Hamiltonian flows on M 1 is analyzed. It is proved that M n , the space of absolute degree n (an odd positive integer) harmonic maps RP 2 → RP 2 , is a totally geodesic Lagrangian submanifold of M n , and that for all n ≥ 3, M n is geodesically incomplete. Possible generalizations and the relevance of these results to theoretical physics are briefly discussed.

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Quantum mechanics on moduli spaces

Cited by 14 publications

References 25 publications

Supersymmetric soliton σ-models from non-Lorentzian field theories

Supersymmetric soliton σ-models from non-Lorentzian field theories

Semiclassical framed BPS states

The L2 geometry of spaces of harmonic maps S2→S2 and

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