Nahm Equations Research Articles

On the moduli space of the octonionic Nahm's equations

On the moduli space of the octonionic Nahm's equations

Combinatorics of Nahm sums, quiver resultants and the K-theoretic condition

Algebraic Nahm equations, considered in the paper, are polynomial equations, governing the q → 1 limit of the q-hypergeometric Nahm sums. They make an appearance in various fields: hyperbolic geometry, knot theory, quiver representation theory, topological strings and conformal field theory. In this paper we focus primarily on Nahm sums and Nahm equations that arise in relation with symmetric quivers. For a large class of them, we prove that quiver A-polynomials — specialized resultants of the Nahm equations, are tempered (the so-called K-theoretic condition). This implies that they are quantizable. Moreover, we find that their face polynomials obey a remarkable combinatorial pattern. We use the machinery of initial forms and mixed polyhedral decompositions to investigate the edges of the Newton polytope. We show that this condition holds for the diagonal quivers with adjacency matrix C = diag(α, α, . . . , α), α ≥ 2, and provide several checks for non-diagonal quivers. Our conjecture is that the K-theoretic condition holds for all symmetric quivers.

Local Nahm Transform and Singularity Formation of ASD Connections

This paper develops a local analogue of the ADHM construction, which characterises ASD instantons defined over smooth bounded domains inside Euclidean mathbb {R}^4 diffeomorphic to the 4-ball, in terms of infinite dimensional Hilbert spaces and bounded Hermitian linear operators satisfying an analogue of the ADHM equation. Morever, we describe the degeneration of this construction when a family of instantons develops a curvature singularity at the origin.

Forecasting Laurent Polynomial in the Chern–Simons Current of V3 Loop Time Series

AbstractThe genetic variation in the V3 loop obstruction component of gp120 HIV glycoprotein is defined as a solitary standing wave function with the transition state between the interaction of wave function in the DNA and RNA hidden states. It is a parasitism co‐state generated from the coupling between two pairs of hidden 8 states of Laurent polynomial in the Chern–Simons current of HIV genotypes. A modified Nahm equation for a biology with Lax pairs is defined as the pairs of viral glycoprotein states with the transition between DNA and RNA in the reverse transcription process. The direction in the spinor field with genetic variation in 30 samples of the V3 loop time series with hidden 8 states of Laurent polynomial over the Chern–Simons current is predicted using a support spinor machine and convolutional neural network based on the imaging technique generated from tensor correlation network.

Holographic duals of 6d RG flows

A notable class of superconformal theories (SCFTs) in six dimensions is parameterized by an integer N , an ADE group G, and two nilpotent elements μL,R in G. Nilpotent elements have a natural partial ordering, which has been conjectured to coincide with the hierarchy of renormalization-group flows among the SCFTs. In this paper we test this conjecture for G = SU(k), where AdS7 duals exist in IIA. We work with a seven-dimensional gauged supergravity, consisting of the gravity multiplet and two SU(k) non-Abelian vector multiplets. We show that this theory has many supersymmetric AdS7 vacua, determined by two nilpotent elements, which are naturally interpreted as IIA AdS7 solutions. The BPS equations for domain walls connecting two such vacua can be solved analytically, up to a Nahm equation with certain boundary conditions. The latter admit a solution connecting two vacua if and only if the corresponding nilpotent elements are related by the natural partial ordering, in agreement with the field theory conjecture.

Three classes of quadratic vector fields for which the Kahan discretisation is the root of a generalised Manin transformation

We apply Kahan’s discretisation method to three classes of 2D quadratic vector fields with quadratic, resp. cubic, resp. quartic Hamiltonians. We show that the maps obtained in this way can be geometrically understood as the composition of two involutions, one of which is a (linear) symmetry switch, and the other is a generalised Manin involution. Applications to 2D Suslov and reduced Nahm equations are included.

Kempf–Ness type theorems and Nahm equations

We prove a version of the affine Kempf–Ness theorem for non-algebraic symplectic structures and shifted moment maps, and use it to describe hyperkähler quotients of T∗G, where G is a complex reductive group.

AdS5 black strings in the stu model of FI-gauged N = 2 supergravity

We analytically construct asymptotically AdS5 black string solutions starting from the four-dimensional domain wall black hole of [1]. It is shown that its uplift gives a black string in d = 5 minimal gauged supergravity, with momentum along the string. Applying instead the residual symmetries of N = 2, d = 4 Fayet-Iliopoulos-gauged super-gravity discovered in [2] to the domain wall seed leads, after uplifting, to a dyonic black string that interpolates between AdS5 and AdS3 × H2 at the horizon. A Kaluza-Klein reduction of the latter along an angular Killing direction ϕ followed by a duality transformation yields, after going back to five dimensions, a black string with both momentum along the string and rotation along ϕ. This is the first instance of using solution-generating techniques in gauged supergravity to add rotation to a given seed. These solutions all have constant scalar fields. As was shown in [3], the construction of supersymmetric static magnetic black strings in the FI-gauged stu model amounts to solving the SO(2, 1) spinning top equations, which descend from an inhomogeneous version of the Nahm equations. We are able to solve these in a particular case, which leads to a generalization of the Maldacena-Nuñez solution.

Spatially Periodic Instantons: Nahm Transform and Moduli

This paper establishes that the Nahm transform sending spatially periodic instantons (instantons on the product of the real line and a three-torus) to singular monopoles on the dual three-torus is indeed a bijection as suggested by the heuristic. In the process, we show how the Nahm transform intertwines to a Fourier–Mukai transform via Kobayashi–Hitchin correspondences. We also prove existence and non-existence results.

The Nahm–Schmid equations and hypersymplectic geometry

We explore the geometry of the Nahm-Schmid equations, a version of Nahm's equations in split signature. Our discussion ties up different aspects of their integrable nature: dimensional reduction from the Yang--Mills anti-self-duality equations, explicit solutions, Lax-pair formulation, conservation laws and spectral curves, as well as their relation to hypersymplectic geometry.

Discrete Nahm equations for SU(N) hyperbolic monopoles

In a paper of Braam and Austin, SU ( 2 ) magnetic monopoles in hyperbolic space H 3 with half-integer mass and maximal symmetry breaking, were shown to be the same as solutions to matrix-valued difference equations called the discrete Nahm equations. Here, I discover the ( N − 1 ) -interval discrete Nahm equations and show that their solutions are equivalent to SU ( N ) hyperbolic monopoles of integer or half-integer mass, and maximal symmetry breaking. These discrete time evolution equations on an interval feature a jump in matrix dimensions at certain points in the evolution, which are given by the mass data of the corresponding monopole. I prove the correspondence with higher rank hyperbolic monopoles using localisation and Chern characters. I then prove that the monopole is determined up to gauge transformations by its “holographic image” of U ( 1 ) fields at the asymptotic boundary of H 3 .

The Coulomb Branch of 3d {mathcal{N}= 4} Theories

We propose a construction for the quantum-corrected Coulomb branch of a general 3d gauge theory with {mathcal{N}=4} supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkähler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations.

A hyperholomorphic line bundle on certain hyperkähler manifolds not admitting an S1-symmetry

Generalizing work of Haydys [7] and Hitchin [8], we prove the existence of a hyperholomorphic line bundle on certain hyperkähler manifolds that do not necessarily admit an S1 action. As examples, we consider the moduli space of (non-strongly) parabolic Higgs bundles, the moduli space of solutions to Nahm's equations, and Nakajima quiver varieties.

Nahm transformation for parabolic Higgs bundles on the projective line—Case of non-semisimple residues

We extend our earlier construction of Nahm transformation for parabolic Higgs bundles on the projective line to solutions with not necessarily semisimple residues and show that it determines a holomorphic mapping on corresponding moduli spaces. The construction relies on suitable elementary modifications of the logarithmic Dolbeault complex.

M5-branes on S 2 × M 4: Nahm’s equations and 4d topological sigma-models

We study the 6d N=(0,2) superconformal field theory, which describes multiple M5-branes, on the product space S^2 x M_4, and suggest a correspondence between a 2d N=(0,2) half-twisted gauge theory on S^2 and a topological sigma-model on the four-manifold M_4. To set up this correspondence, we determine in this paper the dimensional reduction of the 6d N=(0,2) theory on a two-sphere and derive that the four-dimensional theory is a sigma-model into the moduli space of solutions to Nahm's equations, or equivalently the moduli space of k-centered SU(2) monopoles, where k is the number of M5-branes. We proceed in three steps: we reduce the 6d abelian theory to a 5d Super-Yang-Mills theory on I x M_4, with I an interval, then non-abelianize the 5d theory and finally reduce this to 4d. In the special case, when M_4 is a Hyper-Kahler manifold, we show that the dimensional reduction gives rise to a topological sigma-model based on tri-holomorphic maps. Deriving the theory on a general M_4 requires knowledge of the metric of the target space. For k=2 the target space is the Atiyah-Hitchin manifold and we twist the theory to obtain a topological sigma-model, which has both scalar fields and self-dual two-forms.

Nahm's Equations, Quiver Varieties and Parabolic Sheaves

Nahm's Equations, Quiver Varieties and Parabolic Sheaves

Nahm's Equations, Quiver Varieties and Parabolic Sheaves

We construct an isomorphism between moduli spaces of solutions of Nahm's equations over the circle and framed moduli spaces of locally free parabolic sheaves over \mathbb P^1 \times \mathbb P^1 through chainsaw quiver varieties.

Sasakian quiver gauge theories and instantons on Calabi–Yau cones

We consider SU(2)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form $M\times S^3/\Gamma$, where $M$ is a smooth manifold and $S^3/\Gamma$ is a three-dimensional Sasaki-Einstein orbifold. We obtain new quiver gauge theories on $M$ whose quiver bundles are based on the affine ADE Dynkin diagram associated to $\Gamma$. We relate them to those arising through translationally-invariant dimensional reduction over the associated Calabi-Yau cones $C(S^3/\Gamma)$ which are based on McKay quivers and ADHM matrix models, and to those arising through SU(2)-equivariant dimensional reduction over the leaf spaces of the characteristic foliations of $S^3/\Gamma$ which are Kahler orbifolds of $\mathbb{C} P^1$ whose quiver bundles are based on the unextended Dynkin diagram corresponding to $\Gamma$. We use Nahm equations to describe the vacua of SU(2)-equivariant quiver gauge theories on the cones as moduli spaces of spherically symmetric instantons. We relate them to the Nakajima quiver varieties which can be realized as Higgs branches of the worldvolume quiver gauge theories on D$p$-branes probing D$(p+4)$-branes which wrap an ALE space, and to the moduli spaces of spherically symmetric solutions in putative non-abelian generalizations of two-dimensional affine Toda field theories.

Berry’s connection, Kähler geometry and the Nahm construction of monopoles

We study supersymmetric deformations of N = 4 quantum mechanics with a Kahler target space admitting a holomorphic isometry. We show that the twisted mass deformation generalises to a deformation constructed from matrix-valued functions of the moment map, which obey the Nahm equations. We also explain how N = 4 supersymmetry implies that the Berry connection on the vacuum bundle for this theory satisfies the BPS monopole equations. In the case where the target space is a Riemann sphere, our analysis reduces to the standard Nahm construction of monopoles. This generalises an earlier result by Sonner and Tong to the case of monopoles of magnetic charge greater than one.

Instantons on Calabi–Yau cones

The Hermitian Yang–Mills equations on certain vector bundles over Calabi–Yau cones can be reduced to a set of matrix equations; in fact, these are Nahm-type equations. The latter can be analysed further by generalising arguments of Donaldson and Kronheimer used in the study of the original Nahm equations. Starting from certain equivariant connections, we show that the full set of instanton equations reduce, with a unique gauge transformation, to the holomorphicity condition alone.