Three-dimensional Mirror Symmetry Research Articles

Characterizing chiroptical properties of 2D/3D structures based on an improved coupled dipole theory.

To reveal the difference/connection between two-dimensional and three- dimensional (2D and 3D) chiroptical properties and their relation with 2D/3D symmetry/breaking, we develop an improved coupled dipole theory (ICDT) based on a model system of nanorod (NR) dimer. Our analytical ICDT can overcome the shortcoming of the traditional coupled dipole theory and points out the important role of scattering circular dichroism (SCD) in characterizing 2D chirality. The ICDT, supported by finite-difference time-domain (FDTD) simulation, reveals the physical origin of 2D chiroptical response: the interaction induced asymmetric effective polarizability for two identical NRs in a symmetry broken configuration. By tuning the NR's position/inter-particle distance, we find an optimal structure of maximum SCD due to the competition between geometric symmetry breaking and interaction. In addition, the interplay between 2D in-plane mirror symmetry breaking and three-dimensional (3D) mirror symmetry breaking leads to a symmetry broken system with zero SCD. The relation between chirality and reciprocity has also been addressed.

3D Mirror Symmetry for Instanton Moduli Spaces

We prove that the Hilbert scheme of k points on {mathbb {C}}^2 (hbox {Hilb}^k[{mathbb {C}}^2]) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the {mathbb {C}}^times _hbar -action. First, we find a two-parameter family X_{k,l} of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of hbox {Hilb}^k[{mathbb {C}}^2] is obtained via direct limit llongrightarrow infty and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted hbar -opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-N sheaves on {mathbb {P}}^2 with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

Vortex counting and the quantum Hall effect

We provide evidence for conjectural dualities between nonrelativistic Chern-Simons-matter theories and theories of (fractional, nonAbelian) quantum Hall fluids in 2 + 1 dimensions. At low temperatures, the dynamics of nonrelativistic Chern-Simons-matter theories can be described in terms of a nonrelativistic quantum mechanics of vortices. At critical coupling, this may be solved by geometric quantisation of the vortex moduli space. Using localisation techniques, we compute the Euler characteristic χ( mathcal{L} λ) of an arbitrary power λ of a quantum line bundle mathcal{L} on the moduli space of vortices in U(Nc) gauge theory with Nf fundamental scalar flavours on an arbitrary closed Riemann surface. We conjecture that this is equal to the dimension of the Hilbert space of vortex states when the area of the metric on the spatial surface is sufficiently large. We find that the vortices in theories with Nc = Nf = λ behave as fermions in the lowest nonAbelian Landau level, with strikingly simple quantum degeneracy. More generally, we find evidence that the quantum vortices may be regarded as composite objects, made of dual anyons. We comment on potential links between the dualities and three-dimensional mirror symmetry. We also compute the expected degeneracy of local Abelian vortices on the Ω-deformed sphere, finding it to be a q-analog of the undeformed case.

Three-Dimensional Mirror Symmetry and Elliptic Stable Envelopes

Abstract We consider a pair of quiver varieties $(X;X^{\prime})$ related by 3D mirror symmetry, where $X =T^*{Gr}(k,n)$ is the cotangent bundle of the Grassmannian of $k$-planes of $n$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $X \times X^{\prime} $ (the mother function) whose restrictions to $X$ and $X^{\prime} $ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $X$ and $X^{\prime}$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.

Non-connected gauge groups and the plethystic program

We present in the context of supersymmetric gauge theories an extension of the Weyl integration formula, first discovered by Robert Wendt [1], which applies to a class of non-connected Lie groups. This allows to count in a systematic way gauge-invariant chiral operators for these non-connected gauge groups. Applying this technique to O(n), we obtain, via the ADHM construction, the Hilbert series for certain instanton moduli spaces. We validate our general method and check our results via a Coulomb branch computation, using three-dimensional mirror symmetry.

On three-dimensional quiver gauge theories of type B

We study three-dimensional supersymmetric quiver gauge theories with a nonsimply laced global symmetry primarily focusing on framed affine BN quiver theories. Using a supersymmetric partition function on a three sphere, and its transformation under S-duality, we study the three-dimensional ADHM quiver for SO(2N + 1) instantons with a half-integer Chern-Simons coupling. The theory after S-duality has no Lagrangian, and can not be represented by a single quiver, however its partition function can be conveniently described by a collection of framed affine BN quivers. This correspondence can be conjectured to generalize three-dimensional mirror symmetry to theories with nontrivial Chern-Simons terms. In addition, we propose a formula for the superconformal index of a theory described by a framed affine BN quiver.

Comments on twisted indices in 3d supersymmetric gauge theories

We study three-dimensional ${\mathcal N}=2$ supersymmetric gauge theories on ${\Sigma_g \times S^1}$ with a topological twist along $\Sigma_g$, a genus-$g$ Riemann surface. The twisted supersymmetric index at genus $g$ and the correlation functions of half-BPS loop operators on $S^1$ can be computed exactly by supersymmetric localization. For $g=1$, this gives a simple UV computation of the 3d Witten index. Twisted indices provide us with a clean derivation of the quantum algebra of supersymmetric Wilson loops, for any Yang-Mills-Chern-Simons-matter theory, in terms of the associated Bethe equations for the theory on ${\mathbb R}^2 \times S^1$. This also provides a powerful and simple tool to study 3d ${\mathcal N}=2$ Seiberg dualities. Finally, we study A- and B-twisted indices for ${\mathcal N}=4$ supersymmetric gauge theories, which turns out to be very useful for quantitative studies of three-dimensional mirror symmetry. We also briefly comment on a relation between the $S^2 \times S^1$ twisted indices and the Hilbert series of ${\mathcal N}=4$ moduli spaces.

Holomorphic blocks in three dimensions

We decompose sphere partition functions and indices of three-dimensional N=2 gauge theories into a sum of products involving a universal set of "holomorphic blocks". The blocks count BPS states and are in one-to-one correspondence with the theory's massive vacua. We also propose a new, effective technique for calculating the holomorphic blocks, inspired by a reduction to supersymmetric quantum mechanics. The blocks turn out to possess a wealth of surprising properties, such as a Stokes phenomenon that integrates nicely with actions of three-dimensional mirror symmetry. The blocks also have interesting dual interpretations. For theories arising from the compactification of the six-dimensional (2,0) theory on a three-manifold M, the blocks belong to a basis of wavefunctions in analytically continued Chern-Simons theory on M. For theories engineered on branes in Calabi-Yau geometries, the blocks offer a non-perturbative perspective on open topological string partition functions.

Tangles, generalized Reidemeister moves, and three-dimensional mirror symmetry

Three-dimensional $ \mathcal{N} $ = 2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted submanifold of the ultraviolet geometry. A choice of branch sheet for this cover yields a Lagrangian for the theory, and varying the branch sheet provides dual descriptions. Massless matter arises from vanishing size M2-branes and appears as singularities of the tangle where branch lines collide. Massive deformations of the field theory correspond to resolutions of singularities resulting in distinct smooth manifolds connected by geometric transitions. A generalization of Reidemeister moves for singular tangles captures mirror symmetries of the underlying theory yielding a geometric framework where dualities are manifest.

On three-dimensional mirror symmetry

Mirror Symmetry for a large class of three dimensional $\mathcal{N}=4$ supersymmetric gauge theories has a natural explanation in terms of M-theory compactified on a product of $\text{ALE}$ spaces. A pair of such mirror duals can be described as two different deformations of the eleven-dimensional supergravity background $\mathcal{M}=\mathbb{R}^{2,1} \times \text{ALE}_{1} \times \text{ALE}_{2}$, to which they flow in the deep IR. Using the $A-D-E$ classification of $\text{ALE}$ spaces, we present a neat way to catalogue dual quiver gauge theories that arise in this fashion. In addition to the well-known examples studied in \cite{Intriligator:1996ex}, \cite{deBoer:1996mp}, this procedure leads to new sets of dual theories. For a certain subset of dual theories which arise from the aforementioned M-theory background with an $A$-type $\text{ALE}_{1}$ and a $D$-type $\text{ALE}_2$, we verify the duality explicitly by a computation of partition functions of the theories on $S^3$, using localization techniques . We derive the relevant mirror map and discuss its agreement with predictions from the Type IIB brane construction for these theories.

Monopole operators in three-dimensionalN=4 SYM and mirror symmetry

We study non-abelian monopole operators in the infrared limit of three-dimensional SU(N_c) and N=4 SU(2) gauge theories. Using large N_f expansion and operator-state isomorphism of the resulting superconformal field theories, we construct monopole operators which are (anti-)chiral primaries and compute their charges under the global symmetries. Predictions of three-dimensional mirror symmetry for the quantum numbers of these monopole operators are verified.

Mirror symmetry in 2+1 and 1+1 dimensions

We study the Coulomb-Higgs duality of N=2 supersymmetric Abelian Chern-Simons theories in 2+1 dimensions, by compactifying dual pairs on a circle of radius R and comparing the resulting N=(2,2) theories in 1+1 dimensions. Below the compactification scale, the theory on the Higgs branch reduces to the non-linear sigma model on a toric manifold. In the dual theory on the Coulomb branch, the Kaluza-Klein modes generate an infinite tower of contributions to the superpotential. After resummation, in the limit R->0 the superpotential becomes that of the Landau-Ginzburg model which is the two-dimensional mirror of the toric sigma model. We further examine the conjecture of all-scale three-dimensional mirror symmetry and observe that it is consistent with mirror symmetry in 1+1 dimensions.

Dualities in integrable systems andN= 2 SUSY theories

We discuss dualities in the integrable dynamics behind the exact solution to the N = 2 SUSY Yang-Mills theory. It is shown that T duality in the string theory is related to the separation of variables procedure in the dynamical system. We argue that there are analogues of S duality as well as three-dimensional mirror symmetry in the many-body systems of Hitchin type governing the low-energy effective actions.

Calabi-Yau mirror symmetry as a gauge theory duality

Using brane set-ups we construct dual gauge theories in two dimensions with = (2,2) supersymmetry. Two different dualities are realized. One is basically a consequence of three-dimensional mirror symmetry. The nonlinear sigma model with a Calabi-Yau target space on the Higgs branch of the gauge theory is mapped into an equivalent non-linear sigma model on the Coulomb branch on the dual, realizing a T-dual target space with torsion. The second dual is genuine to two dimensions. In addition to swapping Higgs and Coulomb branches it trades twisted for untwisted multiplets, implying a sign flip of the left-moving U(1)R charge. Successive application of both dualities leads to geometric mirror symmetry for the Calabi-Yau target space.

The matrix formulation of type IIB five-branes

We present a matrix model which interpolates between type IIA and type IIB NS five-branes. The matrix description involves a three-dimensional bulk quantum field theory interacting with impurities localized in one spatial direction. We obtain a dual matrix formulation for the exotic six-dimensional theory on coincident type IIB NS five-branes by studying the T-dual description in terms of Kaluza-Klein monopoles in type IIA string theory. After decoupling the bulk physics, the matrix description reduces to the conformal field theory of the Coulomb branch for the type IIA matrix string propagating on certain singular spaces. In many ways, this dual realization of superconformal theories is the two-dimensional analog of three-dimensional mirror symmetry.

Duality in matrix theory and three-dimensional mirror symmetry

Certain limits of the duality between M-theory on T 5/ Z 2 and 1113 on K3 are analyzed in Matrix theory, The correspondence between M-theory five-branes and ALE backgrounds is realized as three-dimensional mirror symmetry, Non-critical strings dual to open membranes are explicitly described as gauge theory excitations. We also comment on Type IIA on K3 and the appearance of gauge symmetry enhancement at special points in the moduli space.

Processing of asymmetrical noise in two- and three-dimensional global symmetry

We investigated how the visual system processes asymmetrical noise in globally symmetrical patterns, and how much noise the visual system needs to discriminate between the perfectly symmetrical and imperfectly symmetrical patterns. Two experiments were carried out in which subjects were required to discriminate between the above patterns. This task was equivalent to a kind of the visual search task because subjects had to search for some targets asymmetrical dot pairs in two-dimensional (Experiment 1) or in three-dimensional mirror symmetry of dot patterns (Experiment 2). The results showed that the visual system processes noise largely in parallel in the two-dimensional symmetry, but largely in serial in the three-dimensional symmetry. In addition, analyses of the percent correct data suggested that the mechanism involved in detecting both the two- and three-dimensional symmetries in activated even by the patterns with a small amount of noise.

３次元対称構造の検出

We conducted two experiments in order to examine abilities of the human visual system to detect three-dimensional mirror-symmetry. In Experiment 1, three kinds of three-dimensional stimulus patterns were presented: (a) three-dimensionally mirror-symmetrical patterns (3-D symmetry), (b) patterns with mirror-symmetry in the frontal plane, but not symmetrical with regard to depth (2-D symmetry), and (c) random patterns. Subjects were required to discriminate among them. Both the reaction time and the discriminability index (d') were measured to estimate the detectability. The results indicate that the visual system can detect three-dimensional mirror-symmetry less easily than two-dimensional symmetry. In Experiment 2, we examined the effects of noise (i.e., asymmetrical constituents of the stimulus display) on the detection of three-dimensional mirror-symmetry. The results show that the detectability increases non-linearly with increase of the SN ratio.