Duality Theory and Practice

Ramon Lopez and Rulon Pope have succeeded in providing rigorous and relatively comprehensive summaries of duality applications and theory, respectively, in admirably concise presentations. The timeliness of this topic among agricultural economists is indicated by the publication of at least five studies using duality theory in the first three issues of the AJAE this year [Babin, Willis, and Clyde; Chambers; Heien; Ray; Lopez, 1982b]. Although both duality theory and agricultural applications of the theory have been with us several years, the recent surge of interest among agricultural economists in duality has been so enthusiastic that some cautionary notes are in order. Consequently, I will attempt in this discussion to supplement, and perhaps further clarify, some of the pros and cons of duality approaches discussed by the two major contributors to this session. This discussion draws upon my review of Lopez and Pope's work, of other recent duality applications to agriculture, and some recent personal experience with empirical duality analysis at the aggregate level [Rostamizadeh et al.]. Most of my more specific remarks will deal with analysis of

To gain insight in the quantum nature of the big bang, we study the dual field theory description of asymptotically anti-de Sitter solutions of supergravity that have cosmological singularities. The dual theories do not appear to have a stable ground state. One regularization of the theory causes the cosmological singularities in the bulk to turn into giant black holes with scalar hair. We interpret these hairy black holes in the dual field theory and use them to compute a finite temperature effective potential. In our study of the field theory evolution, we find no evidence for a "bounce" from a big crunch to a big bang. Instead, it appears that the big bang is a rare fluctuation from a generic equilibrium quantum gravity state.

In recent work Kachru, Kumar, and Silverstein introduced a special class of nonsupersymmetric type II string theories in which the cosmological constant vanishes at the first two orders of perturbation theory. Heuristic arguments suggest the cosmological constant may vanish in these theories to all orders in perturbation theory leading to a flat potential for the dilaton. A slight variant of their model can be described in terms of a dual heterotic theory. The dual theory has a nonzero cosmological constant which is nonperturbative in the coupling of the original type II theory. The dual theory also predicts a mismatch between Bose and Fermi degrees of freedom in the nonperturbative D-brane spectrum of the type II theory.

The Sachdev-Ye-Kitaev (SYK) model is a quantum mechanical many body system with random all-to-all interactions on fermionic N sites (N>>1). This model is shown to saturate the known maximal chaos bound of many body system and then based on this observation it is conjectured to be dual to a quantum black hole in the sense of the AdS/CFT correspondence. In this dissertation, we show that the large N physics of the SYK model is systematically described by a single bi-local field. In particular, we emphasize the appearance of the emergent conformal reparametrization symmetry at the critical IR fixed point and the corresponding divergent contribution of the symmetry modes in the propagator of the bi-local field. We discuss non-linear-level derivation of the zero modes effective action, which is given by the Schwarzian derivative for finite reparametrization symmetry. Besides the symmetry modes, which correspond to the dilaton-gravity sector in the dual AdS theory, the SYK model also predicts an infinite tower of matter fields in AdS_2. We demonstrate that this infinite spectrum can be nicely packaged into a single field in 3-dimensional space-time. Finally, we consider the question of identifying the dual space-time of the SYK model. Focusing on the signature of emergent space-time of the (Euclidean) model, we explain the need for non-local (Radon-type) transformations on external legs of n-point Green's functions. This results in a dual theory with Euclidean AdS signature with additional leg-factors. We speculate that these factors incorporate the coupling of additional bulk states similar to the discrete states of 2D string theory.

We investigate backgrounds of Type IIB string theory with null singularities and their duals proposed in S. R. Das, J. Michelson, K. Narayan, S. P. Trivedi, hep-th/0602107. The dual theory is a deformed $\mathcal{N}=4$ Yang-Mills theory in $3+1$ dimensions with couplings dependent on a lightlike direction. We concentrate on backgrounds which become ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ at early and late times and where the string coupling is bounded, vanishing at the singularity. Our main conclusion is that in these cases the dual gauge theory is nonsingular. We show this by arguing that there exists a complete set of gauge invariant observables in the dual gauge theory whose correlation functions are nonsingular at all times. The two-point correlator for some operators calculated in the gauge theory does not agree with the result from the bulk supergravity solution. However, the bulk calculation is invalid near the singularity where corrections to the supergravity approximation become important. We also obtain pp-waves which are suitable Penrose limits of this general class of solutions, and construct the matrix membrane theory which describes these pp-wave backgrounds.

Given two quantum states of N q-bits we are interested to find the shortest quantum circuit consisting of only one- and two- q-bit gates that would transfer one state into another. We call it the quantum maze problem for the reasons described in the paper. We argue that in a large N limit the quantum maze problem is equivalent to the problem of finding a semiclassical trajectory of some lattice field theory (the dual theory) on an N+1 dimensional space-time with geometrically flat, but topologically compact spatial slices. The spatial fundamental domain is an N dimensional hyper-rhombohedron, and the temporal direction describes transitions from an arbitrary initial state to an arbitrary target state. We first consider a complex Klein-Gordon field theory and argue that it can only be used to study the shortest quantum circuits which do not involve generators composed of tensor products of multiple Pauli Z matrices. Since such situation is not generic we call it the Z-problem. On the dual field theory side the Z-problem corresponds to massless excitations of the phase (Goldstone modes) that we attempt to fix using Higgs mechanism. The simplest dual theory which does not suffer from the massless excitation (or from the Z-problem) is the Abelian-Higgs model which we argue can be used for finding the shortest quantum circuits. Since every trajectory of the field theory is mapped directly to a quantum circuit, the shortest quantum circuits are identified with semiclassical trajectories. We also discuss the complexity of an actual algorithm that uses a dual theory prospective for solving the quantum maze problem and compare it with a geometric approach. We argue that it might be possible to solve the problem in sub-exponential time in 2^N, but for that we must consider the Klein-Gordon theory on curved spatial geometry and/or more complicated (than N-torus) topology.

Small perturbations of a black brane are interpreted as small deviations from thermodynamic equilibrium in a dual theory with the AdS/CFT correspondence. In this paper, we calculate hydrodynamics of the dual Yang-Mills theory in the gravity side using membrane paradigm. This method is different from the usual AdS/CFT correspondence and evaluate classical solutions not at boundaries but at the place slightly away from a horizon. There are sound modes or shear modes for gravity perturbation. For sound modes, such calculation at the horizon has not yet been done. Then, we find that boundary stress tensor at the horizon satisfies conservation law in flat space and can represent dissipative parts of stress tensor in the dual theory by holography. Using them, we can read off directly shear and bulk viscosity of the dual theory. Quasinormal modes are solutions to linearized equations obeyed by classical fluctuations of a gravitational background subject to specific boundary conditions and are also gauge-invariant quantities. We use solutions for each fluctuation that compose such quantities and show that quasinormal modes are consistent with the membrane paradigm.

We use localization techniques to study several duality proposals for supersymmetric gauge theories in three dimensions reminiscent of Seiberg duality. We compare the partition functions of dual theories deformed by real mass terms and FI parameters. We find that Seiberg-like duality for mathcal{N} = 3 Chern-Simons gauge theories proposed by Giveon and Kutasov holds on the level of partition functions and is closely related to level-rank duality in pure Chern-Simons theory. We also clarify the relationship between the Giveon-Kutasov duality and a duality in theories of fractional M2 branes and propose a generalization of the latter. Our analysis also confirms previously known results concerning decoupled free sectors in mathcal{N} = 4 gauge theories realized by monopole operators.

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Yang Gao David 2001Complementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systemsPhil. Trans. R. Soc. A.3592347–2367http://doi.org/10.1098/rsta.2001.0855SectionRestricted accessComplementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systems David Yang Gao David Yang Gao Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA () Google Scholar Find this author on PubMed Search for more papers by this author David Yang Gao David Yang Gao Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA () Google Scholar Find this author on PubMed Search for more papers by this author Published:15 December 2001https://doi.org/10.1098/rsta.2001.0855AbstractThis paper presents a unified critical–point theory in non–smooth, non–convex and dissipative Hamilton systems. The canonical dual/polar transformation methods and the associated bi–duality and triality theories proposed recently in non–convex variational problems are generalized into fully nonlinear dissipative dynamical systems governed by non–smooth constitutive laws and boundary conditions. It is shown that, by this method, non–smooth and non–convex Hamilton systems can be reformulated into certain smooth dual, complementary and polar variational problems. Based on a newly proposed polar Hamiltonian, a nice bipolarity variational principle is established for three–dimensional non–smooth elastodynamical systems, and a potentially powerful complementary variational principle can be used for solving unilateral variational inequality problems governed by non–smooth boundary conditions. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Qiu Z and Xia H (2021) Symplectic perturbation series methodology for non-conservative linear Hamiltonian system with damping, Acta Mechanica Sinica, 10.1007/s10409-021-01076-0, 37:6, (983-996), Online publication date: 1-Jun-2021. Gao D and Ali E (2019) A Novel Canonical Duality Theory for Solving 3-D Topology Optimization Problems Advances in Mathematical Methods and High Performance Computing, 10.1007/978-3-030-02487-1_13, (209-246), . Gao D (2019) Canonical Duality-Triality Theory: Unified Understanding for Modeling, Problems, and NP-Hardness in Global Optimization of Multi-Scale Systems Advances in Mathematical Methods and High Performance Computing, 10.1007/978-3-030-02487-1_1, (3-50), . Gao D (2018) On topology optimization and canonical duality method, Computer Methods in Applied Mechanics and Engineering, 10.1016/j.cma.2018.06.027, 341, (249-277), Online publication date: 1-Nov-2018. Gao D, Ruan N and Latorre V (2017) Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex System Canonical Duality Theory, 10.1007/978-3-319-58017-3_1, (1-47), . Gao D (2015) Analytical solutions to general anti-plane shear problems in finite elasticity, Continuum Mechanics and Thermodynamics, 10.1007/s00161-015-0412-y, 28:1-2, (175-194), Online publication date: 1-Mar-2016. Gao D (2009) Canonical duality theory: Unified understanding and generalized solution for global optimization problems, Computers & Chemical Engineering, 10.1016/j.compchemeng.2009.06.009, 33:12, (1964-1972), Online publication date: 1-Dec-2009. Gao D and Sherali H (2009) Canonical Duality Theory: Connections between Nonconvex Mechanics and Global Optimization Advances in Applied Mathematics and Global Optimization, 10.1007/978-0-387-75714-8_8, (257-326), . Penot J (2009) Ekeland Duality as a Paradigm Advances in Applied Mathematics and Global Optimization, 10.1007/978-0-387-75714-8_10, (349-376), . Yuan Y (2008) Optimal solutions to a class of nonconvex minimization problems with linear inequality constraints, Applied Mathematics and Computation, 10.1016/j.amc.2008.04.016, 203:1, (142-152), Online publication date: 1-Sep-2008. Gao D and Yu H (2008) Multi-scale modelling and canonical dual finite element method in phase transitions of solids, International Journal of Solids and Structures, 10.1016/j.ijsolstr.2007.08.027, 45:13, (3660-3673), Online publication date: 1-Jun-2008. Gao D (2007) Duality-Mathematics Wiley Encyclopedia of Electrical and Electronics Engineering, 10.1002/047134608X.W2412.pub2 Gao D (2016) Complementary Principle, Algorithm, and Complete Solutions to Phase Transitions in Solids Governed by Landau-Ginzburg Equation, Mathematics and Mechanics of Solids, 10.1177/1081286504038455, 9:3, (285-305), Online publication date: 1-Jun-2004. Gao D (2003) Perfect duality theory and complete solutions to a class of global optimization problems*, Optimization, 10.1080/02331930310001611501, 52:4-5, (467-493), Online publication date: 1-Aug-2003. Gao D (2003) Nonconvex Semi-Linear Problems and Canonical Duality Solutions Advances in Mechanics and Mathematics, 10.1007/978-1-4613-0247-6_5, (261-312), . This Issue15 December 2001Volume 359Issue 1789Theme Issue ‘Non-smooth mechanics’ compiled by F. G. Pfeiffer Article InformationDOI:https://doi.org/10.1098/rsta.2001.0855Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Published online15/12/2001Published in print15/12/2001 License: Citations and impact Keywordsnon–convex variational problemsnon–smooth elastodynamicstrialitypolaritynon–conservative Hamilton systemscomplementarity

Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other high-level search techniques such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex problems, including discrete and mixed-integer problems where the feasible sets are generally nonconvex. In this paper, we propose an extended duality theory for nonlinear optimization in order to overcome some limitations of previous dual methods. Based on a new dual function, the extended duality theory leads to zero duality gap for general nonconvex problems defined in discrete, continuous, and mixed spaces under mild conditions. Comparing to recent developments in nonlinear Lagrangian functions and exact penalty functions, the proposed theory always requires lesser penalty to achieve zero duality. This is very desirable as the lower function value leads to smoother search terrains and alleviates the ill conditioning of dual optimization. Based on the extended duality theory, we develop a general search framework for global optimization. Experimental results on engineering benchmarks and a sensor-network optimization application show that our algorithm achieves better performance than searches based on conventional duality and Lagrangian theory.

Several important problems in control theory can be reformulated as semidefinite programming problems (SDPs), i.e., as convex optimization problems with linear matrix inequality (LMI) constraints. From duality theory in convex optimization, dual problems can be derived for these SDPs. These dual problems can in turn be reinterpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. We explore such connections for a few problems associated with linear time-invariant systems. Specifically, we discuss the following three applications of SDP duality. Theorems of alternatives provide systematic and unified proofs of necessary and sufficient conditions for solvability of LMIs. As an example, we present a simple new proof of the KYP lemma. The dual problem associated with an SDP can be used to derive lower bounds on the optimal value. As an example, we give a duality-based proof of the Enns-Glover lower bound. The optimal solution of an SDP is characterized by necessary and sufficient optimality conditions that involve the dual variables. As an example, we show that the properties of the solution of the LQR problem can be derived directly from the SDP optimality conditions. Several of the results that we use from convex duality require technical conditions (so-called constraint qualifications). We show that for problems involving Riccati inequalities these constraint qualifications are related to controllability and observability. In particular, this leads us to a new criterion for controllability. We also point out some implications of these results for computational methods for large-scale SDPs arising in control.

We present how to construct elliptically fibered K3 surfaces via Weierstrass models which can be parametrized in terms of Wilson lines in the dual heterotic string theory. We work with a subset of reflexive polyhedras that admit two fibers whose moduli spaces contain the ones of the E_{8}times E_{8} or frac{Spin(32)}{{mathbb {Z}}_{2}} heterotic theory compactified on a two torus without Wilson lines. One can then interpret the additional moduli as a particular Wilson line content in the heterotic strings. A convenient way to find such polytopes is to use graphs of polytopes where links are related to inclusion relations of moduli spaces of different fibers. We are then able to map monomials in the defining equations of particular K3 surfaces to Wilson line moduli in the dual theories. Graphs were constructed developing three different programs which give the gauge group for a generic point in the moduli space, the Weierstrass model as well as basic enhancements of the gauge group obtained by sending coefficients of the hypersurface equation defining the K3 surface to zero.

We propose that the state represented by the Nariai black hole inside de Sitter space is the ground state of the de Sitter gravity, while the pure de Sitter space is the maximal energy state. With this point of view, we investigate thermodynamics of de Sitter space, we find that if there is a dual field theory, this theory can not be a CFT in a fixed dimension. Near the Nariai limit, we conjecture that the dual theory is effectively an 1+1 CFT living on the radial segment connecting the cosmic horizon and the black hole horizon. If we go beyond the de Sitter limit, the "imaginary" high temperature phase can be described by a CFT with one dimension lower than the spacetime dimension. Below the de Sitter limit, we are approaching a phase similar to the Hagedorn phase in 2+1 dimensions, the latter is also a maximal energy phase if we hold the volume fixed.

We generate a class of string backgrounds by a sequence of TsT transformations of the NS1-NS5 system that we argue are holographically dual to states in a single-trace Toverline{T} + Joverline{T} + Toverline{J} -deformed CFT2. The new string backgrounds include general rotating black hole solutions with two U(1) charges as well as a smooth solution without a horizon that is interpreted as the ground state. As evidence for the correspondence we (i) derive the long string spectrum and relate it to the spectrum of the dual field theory; (ii) show that the black hole thermodynamics can be reproduced from single-trace Toverline{T} + Joverline{T} + Toverline{J} -deformed CFTs; and (iii) show that the energy of the ground state matches the energy of the vacuum in the dual theory. We also study geometric properties of these new spacetimes and find that for some choices of the parameters the three-dimensional Ricci scalar in the Einstein frame can become positive in a region outside the horizon before reaching closed timelike curves and singularities.

<div class="page" title="Page 1"><div class="section"><div class="layoutArea"><div class="column"><p><span>In this paper, we study the correspondence between a field theory in de Sitter space in D-dimensions and a dual conformal feld theory in a euclidean space in (D - 1)-dimensions. In particular, we investigate the form in which this correspondence is established for a system of interacting scalar and a vector fields propagating in de Sitter space. We analyze some necessary (but not sucient) conditions for which conformal symmetry is preserved in the dual theory in (D - 1)-dimensions, making possible the establishment of the correspondence. The discussion that we address in this paper is framed on the context of <em>inationary cosmology</em>. Thusly, the results obtained here pose some relevant possibilities of application to the calculation of the fields’s correlation functions and of the <em>primordial curvature perturbation</em> \zeta, in inationary models including coupled scalar and vector fields.</span></p></div></div></div></div>