Noncommutative Minimal Surfaces

Arnlind, Joakim; Choe, Jaigyoung; Hoppe, Jens

doi:10.1007/s11005-016-0861-7

Cited by 6 publications

(13 citation statements)

References 17 publications

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“…A detailed analysis of general abelian perturbations will be given below. 12 Although such a relation holds very generally in the higher-dimensional case [19,20], it is restricted to e σ = const in 2 dimensions; for a general formula in 2 dimensions see [37]. Here we need the Laplacian only for the unperturbed backgrounds, where (4.12) is sufficient.…”

Section: Jhep01(2014)100mentioning

confidence: 99%

2D fuzzy anti-de Sitter space from matrix models

Jurman

Steinacker

2014

J. High Energ. Phys.

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Abstract:We study the fuzzy hyperboloids AdS 2 and dS 2 as brane solutions in matrix models. The unitary representations of SO(2, 1) required for quantum field theory are identified, and explicit formulae for their realization in terms of fuzzy wavefunctions are given. In a second part, we study the (A)dS 2 brane geometry and its dynamics, as governed by a suitable matrix model. In particular, we show that trace of the energy-momentum tensor of matter induces transversal perturbations of the brane and of the Ricci scalar. This leads to a linearized form of Henneaux-Teitelboim-type gravity, illustrating the mechanism of emergent gravity in matrix models.

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Section: Jhep01(2014)100mentioning

confidence: 99%

2D fuzzy anti-de Sitter space from matrix models

Jurman

Steinacker

2014

J. High Energ. Phys.

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“…This operator satisfies, as usual, the relationCΓ M C −1 = −(Γ M ) T . (A.5)Then the Majorana condition in 9+1 dimensions is20 Ψ C = CΨ T = Ψ, hence Ψ = Ψ, Ψ = Ψ T C (A.6)since C = C (4) = γ 0 in the Majorana representation with real γ µ . Thus the spinor entries are Hermitian matrices in a MW basis.…”

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confidence: 99%

An extended standard model and its Higgs geometry from the matrix model

Steinacker

Zahn

2014

Progress of Theoretical and Experimental Physics

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We find a simple brane configuration in the IKKT matrix model which resembles the standard model at low energies, with a second Higgs doublet and right-handed neutrinos. The electroweak sector is realized geometrically in terms of two minimal fuzzy ellipsoids, which can be interpreted in terms of four point-branes in the extra dimensions. The electroweak Higgs connects these branes and is an indispensable part of the geometry. Fermionic would-be zero modes arise at the intersections with two larger branes, leading precisely to the correct chiral matter fields at low energy, along with right-handed neutrinos which can acquire a Majorana mass due to a Higgs singlet. The larger branes give rise to SU (3) c , extended by U (1) B and another U (1) which are anomalous at low energies and expected to disappear. At higher energies, mirror fermions and additional fields arise, completing the full N = 4 supersymmetry. The brane configuration is a solution of the model, assuming a suitable effective potential and a non-linear stabilization of the singlet Higgs. The basic results can be carried over to N = 4 SU (N ) super-Yang-Mills on ordinary Minkowski space with sufficiently large N . 1 harold.steinacker@univie.ac.at 2 jochen.zahn@univie.ac.at N = 4 U (N ) SYM, with sufficiently large N . In fact, most of the results apply also to N = 4 SU (N ) super-Yang-Mills on ordinary Minkowski space, with sufficiently large N . The main difference lies in the U (1) sector, which acquires a special role in the matrix model, related to the effective gravity [6,7]; however we largely ignore this issue in the present paper.At first sight, it may seem hopeless to obtain anything resembling the standard model from a maximally supersymmetric gauge theory. However, at low and intermediate energies this can be achieved. We establish certain backgrounds of the matrix model, interpreted as intersecting branes in 6 extra dimensions, which lead to fermionic and bosonic low-energy excitations governed by an effective action which is close to the standard model, with all the correct quantum numbers. This is a very remarkable result, given the non-chiral nature of N = 4 SYM. The price to pay are mirror fermions which arise at higher energies, along with Kaluza-Klein towers of massive fields, which ultimately complete the full N = 4 spectrum. There is indeed no way to obtain the standard model without Higgs: If we switch off the Higgs, some of these mirror modes become (quasi-) massless, and combine with the standard model fermions to form non-chiral multiplets. In that respect the Higgs sector differs from the standard model: It arises from two doublets which are an intrinsic part of two minimal fuzzy spheres. The spontaneous symmetry breaking (SSB) pattern is thus more intricate than in the standard model, but this does not rule out the possibility that its fluctuations realize the physical Higgs. The remarkable point is that the separation into chiral standard-model fields and the mirror sector arises quite naturally on simple geometrical backgro...

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“…Examples of a generalized King's equation include ADHM equations, noncommutative instantons, vortex equations (in particular Hitchin and Vafa-Witten equations), as well as Bogomolny and Nahm equations for the gauge group U (k). Furthermore, we discuss Nekrasov's suggestion to reinterpret noncommutative instantons as infinitedimensional versions of King's equation, also related to Quantum minimal surfaces considered recently in [3].…”

Section: Introductionmentioning

confidence: 99%

A generalization of King’s equation via noncommutative geometry

Bhattacharya

Kontsevich

2021

IRMA Lectures in Mathematics and Theoretical Physics

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We introduce a framework in noncommutative geometry consisting of a * -algebra, a bimodule endowed with a derivation ("1-forms") and a Hermitian structure (a "noncommutative Kähler form"), and a cyclic 1-cochain whose coboundary is determined by the previous structures. This data leads to moment map equations on the space of connections on arbitrary finitely-generated projective Hermitian module. As particular cases, we obtain a large class of equations in algebra (King's equations for representations of quivers, including ADHM equations), in classical gauge theory (Hermitian Yang-Mills equations, Hitchin equations, Bogomolny and Nahm equations, etc.), as well as in noncommutative gauge theory by Connes, Douglas and Schwarz. We also discuss Nekrasov's beautiful proposal for re-interpreting noncommutative instantons on C n R 2n as an infinite-dimensional solution of King's equationwhere H is a Hilbert space completion of a finitely-generated C[T 1 , . . . , Tn]-module (e.g. an ideal of finite codimension).

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Noncommutative Minimal Surfaces

Cited by 6 publications

References 17 publications

2D fuzzy anti-de Sitter space from matrix models

2D fuzzy anti-de Sitter space from matrix models

An extended standard model and its Higgs geometry from the matrix model

A generalization of King’s equation via noncommutative geometry

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