Duality Mapping Research Articles

Convergence of an explicit scheme for nonexpansive semigroups in Banach spaces

In this work, we introduce a composite iterative scheme with Meir–Keeler contractions for nonexpansive semigroups in a q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping. We also prove strong convergence theorems for the composite iterative schemes. Our results improve and extend several recent results.

Algorithm for the generalized Φ-strongly monotone mappings and application to the generalized convex optimization problem

Let E be a uniformly smooth and uniformly convex real Banach space and E∗ be its dual space. We consider a multivalued mapping A : E → 2E∗ which is bounded, generalized Φ-strongly monotone and such that for all t > 0, the range R(Jp+tA) = E∗, where Jp (p > 1) is the generalized duality mapping from E into 2E∗ . Suppose A−1(0) = ∅, we construct an algorithm which converges strongly to the solution of 0 ∈ Ax. The result is then applied to the generalized convex optimization problem.

Dualities and 5-brane webs for 5d rank 2 SCFTs

We consider Type IIB 5-brane configurations for 5d rank 2 superconformal theories which are classified recently by geometry in [1]. We propose all the 5-brane web diagrams for these rank 2 theories and show dualities between some of different gauge theories with explicit duality map of mass parameters and Coulomb branch moduli. In particular, we explicitly construct 5-brane configurations for G2 gauge theory with six flavors and its dual Sp(2) and SU(3) gauge theories. We also present 5-brane webs for SU(3) theories of Chern-Simons level greater than 5.

Bose-Fermi Chern-Simons dualities in the Higgsed phase

It has been conjectured that fermions minimally coupled to a Chern-Simons gauge field define a conformal field theory (CFT) that is level-rank dual to Chern-Simons gauged Wilson-Fisher Bosons. The CFTs in question admit relevant deformations parametrized by a real mass. When the mass deformation is positive, the duality of the two deformed theories has previously been checked in detail in the large N limit by comparing explicit all orders results on both sides of the duality. In this paper we perform a similar check for the case of negative mass deformations. In this case the bosonic field condenses triggering the Higgs mechanism. The effective excitations in this phase are massive W bosons. By summing all leading large N graphs involving these W bosons we find an all orders (in the ’t Hooft coupling) result for the thermal free energy of the bosonic theory in the condensed phase. Our final answer perfectly matches the previously obtained fermionic free energy under the conjectured duality map.

Majorana stripe order on the surface of a three-dimensional topological insulator

The issue on the effect of interactions in topological states concerns not only interacting topological phases but also novel symmetry-breaking phases and phase transitions. Here we study the interaction effect on Majorana zero modes (MZMs) bound to a square vortex lattice in two-dimensional topological superconductors. Under the neutrality condition, where single-body hybridization between MZMs is prohibited by an emergent symmetry, a minimal square-lattice model for MZMs can be faithfully mapped to a quantum spin model, which has no sign problem in the world-line quantum Monte Carlo simulation. Guided by an insight from a further duality mapping, we demonstrate that the interaction induces a Majorana stripe state, a gapped state spontaneously breaking lattice translational and rotational symmetries, as opposed to the previously conjectured topological quantum criticality. Away from neutrality, a mean-field theory suggests a quantum critical point induced by hybridization.

Wilson loops in 5d mathcal{N}=1 theories and S-duality

We study the action of S-duality on half-BPS Wilson loop operators in 5d mathcal{N}=1 theories. The duality is the statement that different massive deformations of a single 5d SCFT are described by different gauge theories, or equivalently that the SCFT points in parameter space of two gauge theories coincide. The pairs of dual theories that we study are realized by brane webs in type IIB string theory that are S-dual to each other. We focus on SU(2) SQCD theories with Nf ≤ 4 flavors, which are self-dual, and on SU(3) SQCD theories, which are dual to SU(2)2 quiver theories. From string theory engineering we predict that Wilson loops are mapped to dual Wilson loops under S-duality. We confirm the predictions with exact computations of Wilson loop VEVs, which we extract from the 5d half-index in the presence of auxiliary loop operators (also known as higher qq-characters) sourced by D3 branes placed in the brane webs. A special role is played by Wilson loops in tensor products of the (anti)fundamental representation, which provide a natural basis to express the S-duality action. The exact computations also reveal the presence of additional multiplicative factors in the duality map, in the form of background Wilson loops.

Non-perturbative aspects of galileon duality

We study quantum aspects of the galileon duality, especially in the case of a particular interacting galileon theory that is said to be dual to a free theory through the action of a simultaneous field and coordinate transformation. This would appear to map a theory with multiple vacua to one with a unique vacuum state. However, by regulating the duality transformation using external sources, we are able to preserve the full vacuum structure in the dual frame. By explicitly calculating the one-particle irreducible effective action on a maximally symmetric background, we identify a semi-classical contribution to the Wightman functions that has not been taken into account in previous analyses due to the singular point in the duality map. This may affect its spectral properties at high energy scales. These observations cast doubt on the main evidence in support of a non-local UV structure for galileons.

3d Coulomb branch and 5d Higgs branch at infinite coupling

The Higgs branch of minimally supersymmetric five dimensional SQCD theories increases in a significant way at the UV fixed point when the inverse gauge coupling is tuned to zero. It has been a long standing problem to figure out how, and to find an exact description of this Higgs branch. This paper solves this problem in an elegant way by proposing that the Coulomb branches of three dimensional mathcal{N}=4 supersymmetric quiver gauge theories, named “Exceptional Sequences”, provide the solution to the problem. Thus, once again, 3d mathcal{N}=4 Coulomb branches prove to be useful tools in solving problems in higher dimensions. Gauge invariant operators on the 5d side consist of classical objects such as mesons, baryons and gaugino bilinears, and non perturbative objects such as instanton operators with or without baryon number. On the 3d side we have classical objects such as Casimir invariants and non perturbative objects such as monopole operators, bare or dressed. The duality map works in a very interesting way.

Strange Duality on Rational Surfaces II: Higher-Rank Cases

Abstract We study Le Potier’s strange duality conjecture on a rational surface. We focus on the strange duality map $SD_{c_n^r,L}$ that involves the moduli space of rank $r$ sheaves with trivial 1st Chern class and 2nd Chern class $n$, and the moduli space of one-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show there is an exact sequence relating the map $SD_{c_r^r,L}$ to $SD_{c^{r-1}_{r},L}$ and $SD_{c_r^r,L\otimes K_X}$ for all $r\geq 1$ under some conditions on $X$ and $L$ that applies to a large number of cases on $\mathbb{P}^2$ or Hirzebruch surfaces. Also on $\mathbb{P}^2$ we show that for any $r>0$, $SD_{c^r_r,dH}$ is an isomorphism for $d=1,2$, injective for $d=3,$ and moreover $SD_{c_3^3,rH}$ and $SD_{c_3^2,rH}$ are injective. At the end we prove that the map $SD_{c_n^2,L}$ ($n\geq 2$) is an isomorphism for $X=\mathbb{P}^2$ or Fano rational-ruled surfaces and $g_L=3$, and hence so is $SD_{c_3^3,L}$ as a corollary of our main result.

From spinning primaries to permutation orbifolds

We carry out a systematic study of primary operators in the conformal field theory of a free Weyl fermion. Using SO(4, 2) characters we develop counting formulas for primaries constructed using a fixed number of fermion fields. By specializing to particular classes of primaries, we derive very explicit formulas giving the generating functions for the number of primaries in these classes. We present a duality map between primary operators in the fermion field theory and polynomial functions. This allows us to construct the primaries that were counted. Next we show that these classes of primary fields correspond to polynomial functions on certain permutation orbifolds. These orbifolds have palindromic Hilbert series.

The exponential pencil of conics

The exponential pencil G_lambda :=G_1(G_0^{-1}G_1)^{lambda -1}, generated by two conics G_0,G_1, carries a rich geometric structure: It is closed under conjugation, it is compatible with duality and projective mappings, it is convergent for lambda rightarrow pm infty or periodic, and it is connected in various ways with the linear pencil g_lambda =lambda G_1+(1-lambda )G_0. The structure of the exponential pencil can be used to characterize the position of G_0 and G_1 relative to each other.

Realizing topological surface states in a lower-dimensional flat band

The anomalous surface states of symmetry protected topological (SPT) phases are usually thought to be only possible in conjunction with the higher dimensional topological bulk. However, it has recently been realized that a class of anomalous SPT surface states can be realized in the same dimension if symmetries are allowed to act in a nonlocal fashion. An example is the particle-hole symmetric half filled Landau level, which effectively realizes the anomalous surface state of a 3D chiral Topological Insulator (class AIII). A dual description in terms of Dirac composite fermions has also been discussed. Here we explore generalizations of these constructions to multicomponent quantum Hall states. Our results include a duality mapping of the bilayer case to composite bosons with Kramers degeneracy and the possibility of a particle hole symmetric integer quantum Hall state when the number of components is a multiple of eight. Next, we make a further extension by half filling other classes of topological bands and imposing particle hole symmetry. When applied to time-reversal invariant topological insulators we realize a different chiral class (CII) topological surface state. Notably, half-filling a 3D TI band allows for the realization of the surface of the otherwise inaccessible 4D topological insulator, which supports an anomalous 3D Dirac cone. Surface topological orders equivalent to the 3D Dirac cone (from the global anomaly standpoint) are constructed and connections to Witten's SU(2) anomaly are made. These observations may also be useful for numerical simulations of topological surface states and of Dirac fermions without fermion doubling.

A bifurcation result involving Sobolev trace embedding and the duality mapping of W 1 ,p

Abstract We consider the perturbed nonlinear boundary condition problem { - Δ p u = | u | p - 2 u + f ( λ , x , u ) in Ω | ∇ u | p - 2 ∇ u . ν = λ ρ ( x ) | u | p - 2 u on Γ . $$\left\{ {\matrix{ { - \Delta _p u} \hfill & = \hfill & {\left| u \right|^{p - 2} u + f\left( {\lambda ,x,u} \right)\,{\rm{in}}\,\Omega } \hfill \cr {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill & = \hfill & {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$ Using the Sobolev trace embedding and the duality mapping defined on W 1, p (Ω), we prove that this problem bifurcates from the principal eigenvalue λ 1 of the eigenvalue problem { - Δ p u = | u | p - 2 u in Ω | ∇ u | p - 2 ∇ u . ν = λ ρ ( x ) | u | p - 2 u on Γ . $$\left\{ {\matrix{ { - \Delta _p u} \hfill & = \hfill & {\left| u \right|^{p - 2} u\,{\rm{in}}\,\Omega } \hfill \cr {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill & = \hfill & {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$

Dark solitons, D-branes and noncommutative tachyon field theory

In this paper we discuss the boson/vortex duality by mapping the (3[Formula: see text]+[Formula: see text]1)D Gross–Pitaevskii theory into an effective string theory in the presence of boundaries. Via the effective string theory, we find the Seiberg–Witten map between the commutative and the noncommutative tachyon field theories, and consequently identify their soliton solutions with D-branes in the effective string theory. We perform various checks of the duality map and the identification of soliton solutions. This new insight between the Gross–Pitaevskii theory and the effective string theory explains the similarity of these two systems at quantitative level.

Simple Z2 lattice gauge theories at finite fermion density

Lattice gauge theories are a powerful language to theoretically describe a variety of strongly correlated systems, including frustrated magnets, high-$T_c$ superconductors, and topological phases. However, in many cases gauge fields couple to gapless matter degrees of freedom and such theories become notoriously difficult to analyze quantitatively. In this paper we study several examples of $\mathbb{Z}_2$ lattice gauge theories with gapless fermions at finite density, in one and two spatial dimensions, that are either exactly soluble or whose solution reduces to that of a known problem. We consider complex fermions (spinless and spinful) as well as Majorana fermions, and study both theories where Gauss' law is strictly imposed and those where all background charge sectors are kept in the physical Hilbert space. We use a combination of duality mappings and the $\mathbb{Z}_2$ slave-spin representation to map our gauge theories to models of gauge-invariant fermions that are either free, or with on-site interactions of the Hubbard or Falicov-Kimball type that are amenable to further analysis. In 1D, the phase diagrams of these theories include free-fermion metals, insulators, and superconductors; Luttinger liquids; and correlated insulators. In 2D, we find a variety of gapped and gapless phases, the latter including uniform and spatially modulated flux phases featuring emergent Dirac fermions, some violating Luttinger's theorem.

Strong-weak coupling duality between two perturbed quantum many-body systems: Calderbank-Shor-Steane codes and Ising-like systems

Graphs and recently hypergraphs have been known as an important tool for considering different properties of quantum many-body systems. In this paper, we study a mapping between an important class of quantum systems, namely quantum Calderbank-Shor-Steane (CSS) codes, and Ising-like systems by using hypergraphs. We show that the Hamiltonian corresponding to a CSS code on a hypergraph $H$ which is perturbed by a uniform magnetic field is mapped to Hamiltonian of a Ising-like system on dual hypergraph $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ in a transverse field. Interestingly, we show that a strong regime of couplings in one of the systems is mapped to a weak regime of couplings in another one. We also give some applications for such a mapping where we study robustness of different topological CSS codes against a uniform magnetic field including Kitaev's toric codes defined on graphs and color codes in different dimensions. We show that a perturbed Kitaev's toric code on an arbitrary graph is mapped to an Ising model in a transverse field on the same graph and a perturbed color code on a $D$ colex is mapped to a Ising-like model on a $D$-simplicial lattice in a transverse field. In particular, we use these results to explicitly compare the robustness of toric codes to uniform magnetic-field perturbations on different graphs. Interestingly, our results show that the robustness of such topological codes defined on graphs decreases with increasing dimension. Furthermore, we also use the duality mapping for some self-dual models where we exactly derive the point of phase transition.

Absence of Ergodicity without Quenched Disorder: From Quantum Disentangled Liquids to Many-Body Localization.

We study the time evolution after a quantum quench in a family of models whose degrees of freedom are fermions coupled to spins, where quenched disorder appears neither in the Hamiltonian parameters nor in the initial state. Focusing on the behavior of entanglement, both spatial and between subsystems, we show that the model supports a state exhibiting combined area and volume-law entanglement, being characteristic of the quantum disentangled liquid. This behavior appears for one set of variables, which is related via a duality mapping to another set, where this structure is absent. Upon adding density interactions between the fermions, we identify an exact mapping to an XXZ spin chain in a random binary magnetic field, thereby establishing the existence of many-body localization with its logarithmic entanglement growth in a fully disorder-free system.

Free Quantum Fields in 4D and Calabi-Yau Spaces.

We develop general counting formulas for primary fields in free four dimensional (4D) scalar conformal field theory (CFT). Using a duality map between primary operators in scalar field theory and multivariable polynomial functions subject to differential constraints, we identify a sector of holomorphic primary fields corresponding to polynomial functions on a class of permutation orbifolds. These orbifolds have palindromic Hilbert series, which indicates they are Calabi-Yau orbifolds. We construct the unique top-dimensional holomorphic form expected from the Calabi-Yau property. This sector includes and extends previous constructions of infinite families of primary fields. We sketch the generalization of these results to free 4D vector and matrix CFTs.

On the consistency of choice

Consistency of choice is a fundamental and recurring theme in decision theory, social choice theory, behavioral economics, and psychological sciences. The purpose of this paper is to study consistency of choice independent of the particular decision model at hand. Consistency is viewed as an inherently logical concept that is fundamentally void of connotation and is thus disentangled from traditional rationality or consistency conditions imposed on decision models. The proposed formalization of consistency takes two forms: internal consistency, which refers to the property that a choice model does not generate contradictory statements; and semantic consistency, which refers to the idea that a theory's predictions are valid with respect to some observed data. In addressing semantic consistency, the relationship between theory and data is analyzed in terms of so-called duality mappings, which allow a passage between the two universes in a way that consistency is preserved. The formalization of consistency concepts relies on adapting revealed preference theory to the context-dependent setting. The paper concludes by discussing the implications of the proposed framework and how it relates to classical revealed preference theory and other formalizations of the relationship between the theory and reality of choice.

Dual little strings from F-theory and flop transitions

A particular two-parameter class of little string theories can be described by M parallel M5-branes probing a transverse affine AN − 1 singularity. We previously discussed the duality between the theories labelled by (N, M) and (M, N). In this work, we propose that these two are in fact only part of a larger web of dual theories. We provide evidence that the theories labelled by (N, M) and left(frac{NM}{k},kright) are dual to each other, where k = gcd(N,M). To argue for this duality, we use a geometric realization of these little string theories in terms of F-theory compactifications on toric, non-compact Calabi-Yau threefolds XN,M which have a double elliptic fibration structure. We show explicitly for a number of examples that XNM/k,k is part of the extended moduli space of XN,M, i.e. the two are related through symmetry transformations and flop transitions. By working out the full duality map, we provide a simple check at the level of the free energy of little string theories.