Search

The separability of the Hamilton-Jacobi equation has a well-known connection to the existence of Killing vectors and rank-two Killing tensors. This paper combines this connection with the detailed knowledge of the compactification metrics of consistent truncations on spheres. The fact that both the inverse metric of such compactifications, as well as the rank-two Killing tensors can be written in terms of bilinears of Killing vectors on the underlying "round metric," enables us to perform a detailed analyses of the separability of the Hamilton-Jacobi equation for consistent truncations. We introduce the idea of a separating isometry and show that when a consistent truncation, without reduction gauge vectors, has such an isometry, then the Hamilton-Jacobi equation is always separable. When gauge vectors are present, the gauge group is required to be an abelian subgroup of the separating isometry to not impede separability. We classify the separating isometries for consistent truncations on spheres, $S^n$, for $n=2, \dots, 7$, and exhibit all the corresponding Killing tensors. These results may be of practical use in both identifying when supergravity solutions belong to consistent truncations and generating separable solutions amenable to scalar probe calculations. Finally, while our primary focus is the Hamilton-Jacobi equation, we also make some remarks about separability of the wave equation., Comment: 58 pages, 2 tables

On a 4-manifold of conformal torsion-free connection with zero signature (−−++) we found conditions under which the conformal curvature matrix is dual (self-dual or anti-self-dual). These conditions are 5 partial differential equations of the 2nd order on 10 coefficients of the angular metric and 4 partial differential equations of the 1st order, containing also 3 coefficients of external 2-form of charge. (External 2-form of charge is one of the components of the conformal curvature matrix.) Duality equations for a metric of a diagonal type are composed. They form a system of five second-order differential equations on three unknown functions of all four variables. We found several series of solutions for this system. In particular, we obtained all solutions for a logarithmically polynomial diagonal metric, that is, for a metric whose coefficients are exponents of polynomials of four variables.

We generalize the conformally invariant topological quantum mechanics of a particle propagating on a punctured plane by introducing a potential that breaks both the rotational and the conformal invariance down to a ${\bf Z}_2$ angular-dependent discrete symmetry. We derive a topological quantum mechanics whose localization gauge functions give interesting self-dual equations. The model contains an order parameter and exhibits a spontaneous symmetry breaking with two ground states above a critical scale. Unlike the ordinary $O(2)$-invariant Higgs potential, an angular-dependence is found and saddle points, instead of local maxima, appear, posing subtle questions about the existence of instantons. The supersymmetric quantum mechanical model is constructed in both the path integral and the operatorial frameworks., 19 pages; final version to appear in Nucl. Phys. B

Compactifications of the heterotic string on special T^d/Z_2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d+8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II_{(d)}, which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d=1 and 2, and give a list of maximally enhanced points where the U(1)^{d+8} enhances to a rank d+8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E_{10}. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings ofof lattices into the dual of II_{(2)}. Our results easily generalize to d > 2., 65 pages. v2: corrected global groups

We present a simple compact formula for a topologically nontrivial map $S^7 \to Spin(7)$ associated with the fiber bundle $Spin(7) \stackrel{G_2}{\to} S^7$. The homotopy group $\pi_7[Spin(7)] = \mathbb{Z}$ brings about the topologically nontrivial 8-dimensional gauge field configurations that belong to the algebra $spin(7)$. The instantons are special such configurations that minimize the functional $\int {\rm Tr} \{F\wedge F \wedge \star(F \wedge F)\} $ and satisfy non-linear self-duality conditions, $ F \wedge F \ =\ \pm \star (F\wedge F)$. $Spin(7) \subset SO(8)$, and $Spin(7)$ instantons represent simultaneously $SO(8)$ instantons of a new type. The relevant homotopy is $\pi_7[SO(8)] = \mathbb{Z} \times \mathbb{Z}$, which implies the existence of two different topological charges. This also holds for all groups $SO(4n)$ with integer $n$. We present explicit expressions for two topological charges and calculate their values for the conventional 4-dimensional and 8-dimensional instantons and also for the 8-dimensional instantons of the new type. Similar constructions for other algebras in different dimensions are briefly discussed., Comment: New material concerning generalized SO(8) instantons and the second topological charge added

It has long been understood that non-trivial Conformal Field Theories (CFTs) with vanishing central charge ($c=0$) are logarithmic. So far however, the structure of the identity module -- the (left and right) Virasoro descendants of the identity field -- had not been elucidated beyond the stress-energy tensor $T$ and its logarithmic partner $t$ (the solution of the "$c\to 0$ catastrophe"). In this paper, we determine this structure together with the associated OPE of primary fields up to level $h=\bar{h}=2$ for polymers and percolation CFTs. This is done by taking the $c\to 0$ limit of $O(n)$ and Potts models and combining recent results from the bootstrap with arguments based on conformal invariance and self-duality. We find that the structure contains a rank-3 Jordan cell involving the field $T\bar{T}$, and is identical for polymers and percolation. It is characterized in part by the common value of a non-chiral logarithmic coupling $a_0=-{25\over 48}$., Comment: 24 pages. v2: comments added, typos corrected. v3: comments in footnote 7 improved

We investigate exotic supergravity theories in 6D with maximal (4,0) and (3,1) supersymmetry, which were conjectured by C. Hull to exist and to describe strong coupling limits of ${\cal N}=8$ theories in 5D. These theories involve exotic gauge fields with non-standard Young tableaux representations, subject to (self-)duality constraints. We give novel actions in a 5+1 split of coordinates whose field equations reproduce those of the free bosonic (4,0) and (3,1) theory, respectively, including the (self-)duality relations. Evidence is presented for a master exceptional field theory formulation with an extended section constraint that, depending on the solution, produces the (4,0), (3,1) or the conventional (2,2) theory. We comment on the possible construction of a fully non-linear master exceptional field theory., Comment: 30 pages

Inspired by the works in [1] and [8] we introduce what we call $k$-th-order fluctuation fields and study their scaling limits. This construction is done in the context of particle systems with the property of orthogonal self-duality. This type of duality provides us with a setting in which we are able to interpret these fields as some type of discrete analogue of powers of the well-known density fluctuation field. We show that the weak limit of the $k$-th order field satisfies a recursive martingale problem that formally corresponds to the SPDE associated with the $k$th-power of a generalized Ornstein-Uhlenbeck process., Comment: 32 pages

We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A Kähler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction, the Zak's magnetic translations in the Bravais lattice generate a non-commutative quantum torus. We concentrate on the case when the magnetic flux density is a rational number. The Bloch wavefunctions form a finite-dimensional module of the noncommutative torus of magnetic translations as well as of its commutant, which is the non-commutative torus of magnetic translations in the dual Bravais lattice. The bi-module structure of the Bloch waves is shown to be the connecting link between two Morita equivalent non-commutative tori. The main focus of our review is the Kähler structure on the Hilbert space of Bloch waves and its inherent quantum toric geometry. We reveal that the metaplectic group Mp(2,R) of the automorphisms of magnetic translation algebras is represented by the quantum optics squeezing operators., Scientific and Technological Research Council of Turkey (TÜBİTAK); Bulgarian National Science Fund Research; Turkish Academy of Sciences

�� ������������ ������������������������������ ���������������������� �������������������� �������������������� Logic �������������� ������������������������ �������������� Maple �� �������������� �������������������������� �������������������������� �������������������� �������� �� ������������������ ��������������. �������������������������� ���������������� �������������������� ���������������� �� Maple. ���� �������������������� �������������� ���������������� ���������������������� ���������������� �������������������� �������������������� ���������� �� ������������������ ��������������., The article discusses the possibilities of using the Logic library of the Maple computer algebra system in the aspect of computer modeling of logic circuits in various bases. Basic logic gates are modeled in Maple. On a specific example, an algorithm for constructing a logical circuit in various bases is presented in detail., �������������� ������������. ������������-���������������������������� ����������, ������������ 3 2021, Pages 155-164

International audience; The simplest topologically ordered phase in 2+1D is the deconfined phase of Z2 gauge theory (realized in the toric code, for example). This phase permits a duality that exchanges electric and magnetic excitations (“e” and “m” particles). The phase transition where one of these particles condenses, while the other remains gapped, has 3D Ising exponents. But the transition out of the deconfined phase when self-duality symmetry is preserved is more mysterious. It has so far been unclear whether this transition is continuous, but if continuous, it may be the simplest critical point for which a useful continuum Lagrangian is still lacking. These questions are relevant to soft matter, too, since the gauge theory also describes classical membranes in 3D. Here, we study the self-dual transition with Monte Carlo simulations of the Z2 gauge-Higgs model on cubic lattices of linear size L≤96. Our results indicate a continuous transition, for example via a striking parameter-free scaling collapse. We use duality symmetry to distinguish the leading duality-odd and duality-even scaling operators A and S. We explain why standard techniques for locating the critical point are ineffective, and we develop an alternative using “renormalization group trajectories” of cumulants. We check that two- and three-point functions are scale invariant, with scaling dimensions xA and xS (autocorrelations in the Monte Carlo dynamics also yield a dynamical exponent z). Separately, we propose a general picture for emergent 1-form symmetries, in terms of “patching” of membranes or world surfaces. We relate this to the percolation of anyon worldlines in spacetime. The latter yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.

We construct the first complete exceptional field theory that is based on an infinite-dimensional symmetry group. E$_9$ exceptional field theory provides a unified description of eleven-dimensional and type IIB supergravity covariant under the affine Kac-Moody symmetry of two-dimensional maximal supergravity. We present two equivalent formulations of the dynamics, which both rely on a pseudo-Lagrangian supplemented by a twisted self-duality equation. One formulation involves a minimal set of fields and gauge symmetries, which uniquely determine the entire dynamics. The other formulation extends $\mathfrak{e}_9$ by half of the Virasoro algebra and makes direct contact with the integrable structure of two-dimensional supergravity. Our results apply directly to other affine Kac-Moody groups, such as the Geroch group of general relativity., Comment: 111 pages, typos corrected, JHEP version

We study a class of interacting particle systems with asymmetric interaction showing a self-duality property. The class includes the ASEP(q, θ), asymmetric exclusion process, with a repulsive interaction, allowing up to θ ∈ N particles in each site, and the ASIP(q, θ), θ ∈ R+, asymmetric inclusion process, that is its attractive counterpart. We extend to the asymmetric setting the investigation of orthogonal duality properties done in [8] for symmetric processes. The analysis leads to multivariate q−analogues of Krawtchouk polynomials and Meixner polynomials as orthogonal duality functions for the generalized asymmetric exclusion process and its asymmetric inclusion version, respectively. We also show how the q-Krawtchouk orthogonality relations can be used to compute exponential moments and correlations of ASEP(q, θ).

Publisher's version (útgefin grein), Large classes of AdSpsupergravity backgrounds describing the IR dynamicsofp-branes wrapped on a Riemann surface are determined by a solution to the Liouvilleequation. The regular solutions of this equation lead to the well-known wrapped branesupergravity solutions associated with the constant curvature metric on a compact Riemannsurface. We show that some singular solutions of the Liouville equation have a physicalinterpretation as explicit point-like brane sources on the Riemann surface. We uncover thedetails of this picture by focusing onN= 1theories of classSarising from M5-branes on apunctured Riemann surface. We present explicit AdS5solutions dual to these SCFTs andcheck the holographic duality by showing the non-trivial agreement of ’t Hooft anomalies.

We compute the four-dimensional gaugino mass for a Dp-brane extended in spacetime and wrapping a cycle on the internal geometry in a warped compactification with fluxes. Motivated by the backreaction of gaugino bilinear VEVs, we use Generalized Complex Geometry to characterize the internal geometry as well as the cycle wrapped by the brane. We find that the RR fluxes and the non-closure of the generalized complex structures combine in the gaugino mass terms in the same form as they do in the bulk superpotential, while for the NSNS fluxes there is a crucial minus sign in the component normal to the brane. Our expression extends the known result for D3 and D7-branes in Calabi-Yau manifolds, where the gaugino masses are induced respectively by the imaginary anti-self dual and imaginary self-dual components of the complex 3-form flux $G_3$., 23 pages. v2: version published in JHEP with minor modifications

The construction of morphological filters by iteration of an arbitrary increasing operator is described. The role of continuity is emphasized. It is shown that the finite window operators are suitable for this. All translation-invariant operators that use finite structuring elements belong to this family.

Exact field theory dualities can be implemented by duality domain walls such that passing any operator through the interface maps it to the dual operator. This paper describes the S-duality wall of four-dimensional ${\cal N}=2$ SU(N) SQCD with 2N hypermultiplets in terms of fields on the defect, namely three-dimensional ${\cal N}=2$ SQCD with gauge group U(N-1) and 2N flavours, with a monopole superpotential. The theory is self-dual under a duality found by Benini, Benvenuti and Pasquetti, in the same way that T[SU(N)] (the S-duality wall of ${\cal N}=4$ super Yang-Mills) is self-mirror. The domain-wall theory can also be realized as a limit of a USp(2N-2) gauge theory; it reduces to known results for N=2. The theory is found through the AGT correspondence by determining the braiding kernel of two semi-degenerate vertex operators in Toda CFT., v2: correct superpotential to fix symmetries, better introduction and references, 42 pages

The most detailed constructions of microstate geometries, and particularly of superstrata, are done using $\mathcal{N} = (1,0)$ supergravity coupled to two anti-self-dual tensor multiplets in six dimensions. We show that an important sub-sector of this theory has a consistent truncation to a particular gauged supergravity in three dimensions. Our consistent truncation is closely related to those recently laid out by Samtleben and Sarioglu (arXiv:1907.08413 [hep-th]), which enables us to develop complete uplift formulae from the three-dimensional theory to six dimensions. We also find a new family of multi-mode superstrata, indexed by two arbitrary holomorphic functions of one complex variable, that live within our consistent truncation and use this family to provide extensive tests of our consistent truncation. We discuss some of the future applications of having an intrinsically three-dimensional formulation of a significant class of microstate geometries., Comment: 41 pages

We establish an explicit correspondence between two–dimensional projective structures admitting a projective vector field, and a class of solutions to the SU (∞) Toda equation.We give several examples of new, explicit solutions of the Toda equation, and construct their mini–twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which givesa neutral signature Einstein metric on a cotangent bundle T∗ N of any projective structure (N,[∇]). We show that there is a canonical Einstein of metric on an R∗–bundle over T∗ N, with a connection whose curvature is the pull–back of the natural symplectic structure from T∗ N

Different versions of consistent canonical realizations of hypersurface deformations of spherically symmetric space-times have been derived in models of loop quantum gravity, modifying the classical dynamics and sometimes also the structure of space-time. Based on a canonical version of effective field theory, this paper provides a unified treatment, showing that modified space-time structures are generic in this setting. The special case of Euclidean gravity demonstrates agreement also with existing operator calculations., Comment: 37 pages

The purpose of this paper is to develop landscape model applicable to the scenery through video media. At first, the self-duality in the scenery experience is pointed out referring the knowledge of psychology. Secondly, basic landscape theories are explained by different aspect of the self; the ecological self and the remembered self. At last, referring knowledge of virtual reality studies, the new model was proposed that the scenery through video media is the experience of moving spatiotemporally with reproducing or expanding body function.