Full Duality Research Articles

The SL(2, ℤ) dualization algorithm at work

Recently an algorithm to dualize a theory into its mirror dual has been proposed, both for 3d mathcal{N} = 4 linear quivers and for their 4d mathcal{N} = 1 uplift. This mimics the manipulations done at the level of the Type IIB brane setup that engineers the 3d theories, where mirror symmetry is realized as S-duality, but it is enirely field-theoretic and based on the application of genuine infra-red dualities that implement the local action of S-duality on the quiver. In this paper, we generalize the algorithm to the full duality group, which is SL(2, ℤ) in 3d and PSL(2, ℤ) in 4d. This also produces dualities for 3d mathcal{N} = 3 theories with Chern-Simons couplings, some of which have enhanced mathcal{N} = 4 supersymmetry, and their new 4d mathcal{N} = 1 counterpart. In addition, we propose three ways to study the RG flows triggered by possible VEVs appearing at the last step of the algorithm, one of which uses a new duality that implements the Hanany-Witten move in field theory.

Duality between hydrogen atom and oscillator systems via hidden SO(d,2) symmetry and 2T-physics* * In memory of Peter George Oliver Freund.

The relation between motion in −1/r and r 2 potentials, known since Newton, can be demonstrated by the substitution r → r 2 in the classical/quantum radial equations of the Kepler/Hydrogen problems versus the harmonic oscillator. This suggests a duality-type relationship between these systems. However, when both radial and angular components of these systems are included the possibility of a true duality seems to be remote. Indeed, investigations that explored and generalized Newton’s radial relation, including algebraic approaches based on noncompact groups such as SO(4,2), have never exhibited a full duality consistent with Newton’s. On the other hand, 2T-physics predicts a host of dualities between pairs of a huge set of systems that includes Newton’s two systems. These dualities take the form of rather complicated canonical transformations that relate the full phase spaces of these respective systems in all directions. In this paper we focus on Newton’s case by imposing his radial relation to find an appropriate basis for 2T-physics dualities, and then construct the full duality. Using the techniques of 2T-physics, we discuss the hidden symmetry of the actions (beyond the symmetry of Hamiltonians) for the Hydrogen atom in D dimensions and the harmonic oscillator in dimensions. The symmetries lead us to find the one-to-one relation between the quantum states, including angular degrees of freedom, for specific values of , and construct the explicit quantum canonical transformation in those special cases. We find that the canonical transformation has itself a hidden gauge symmetry that is crucial for the respective phase spaces to be dual even when . In this way we display the surprising beautiful new symmetry of the full duality that generalizes Newton’s radial duality.

Distributive nearlattices with a necessity modal operator

Abstract The aim of this paper is to study the class of distributive nearlattices with a necessity modal operator. We develop a full duality to the category of distributive nearlattices whose morphisms are applications that preserving the infimum when exists and, as special case, we obtain a representation and duality for distributive nearlattices with a necessity modal operator. We study certain particular subclasses and give some applications.

New-from-old full dualities via axiomatisation

We study different full dualities based on the same finite algebra. Our main theorem gives conditions on two different alter egos of a finite algebra under which, if one yields a full duality, then the other does too. We use this theorem to obtain a better understanding of several important examples from the theory of natural dualities. We also clarify what it means for two full dualities based on the same finite algebra to be different. Throughout the paper, a fundamental role is played by the universal Horn theory of the dual categories.

Pathwise superhedging for time-dependent barrier options on càdlàg paths—Finite or infinite tradeable European, One-Touch, lookback or forward starting options

We establish pathwise duality using simple predictable trading strategies for the robust hedging problem associated with a barrier option whose payoff depends on the terminal level and the infimum of a càdlàg strictly positive stock price process, given tradeable European options at all strikes at a single maturity. The result allows for a significant dimension reduction in the computation of the superhedging cost, via an alternate lower-dimensional formulation of the primal problem as a convex optimization problem, which is qualitatively similar to the duality which was formally sketched using linear programming arguments in Duembgen and Rogers [10] for the case where we only consider continuous sample paths. The proof exploits a simplification of a classical result by Rogers (1993) which characterizes the attainable joint laws for the supremum and the drawdown of a uniformly integrable martingale (not necessarily continuous), combined with classical convex duality results from Rockefellar (1974) using paired spaces with compatible locally convex topologies and the Hahn–Banach theorem. We later adapt this result to include additional tradeable One-Touch options using the Kertz and Rösler (1990) condition. We also compute the superhedging cost when in the more realistic situation where there is only finite tradeable European options; for this case we obtain the full duality in the sense of quantile hedging as in Soner (2015), where the superhedge works with probability 1−ε where ε can be arbitrarily small), and we obtain an upper bound for the true pathwise superhedging cost. In Section 5, we extend our analysis to include time-dependent barrier options using martingale coupling arguments, where we now have tradeable European options at both maturities at all strikes and tradeable forward starting options at all strikes. This set up is designed to approximate the more realistic situation where we have a finite number of tradeable Europeans at both maturities plus a finite number of tradeable forward starting options.11The author would like to thank Professors Charles Akemann and Teemu Pennanen for useful discussions.

Finite affine algebras are fully dualizable

ABSTRACTWe show that every finite affine algebra A admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure of finite type. We give an explicit bound on the arities of the partial and total operations appearing in . In addition, we show that the enriched partial hom-clone of A is finitely generated as a clone.

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.

Convex duality for stochastic singular control problems

We develop a general theory of convex duality for certain singular control problems, taking the abstract results by Kramkov and Schachermayer [Ann. Appl. Probab. 9 (1999) 904–950] for optimal expected utility from nonnegative random variables to the level of optimal expected utility from increasing, adapted controls. The main contributions are the formulation of a suitable duality framework, the identification of the problem’s dual functional as well as the full duality for the primal and dual value functions and their optimizers. The scope of our results is illustrated by an irreversible investment problem and the Hindy–Huang–Kreps utility maximization problem for incomplete financial markets.

Duality group actions on fermions

In this short paper we look at the action of T-duality and string duality groups on fermions, in maximally-supersymmetric theories and related theories. Briefly, we argue that typical duality groups such as SL(2,Z) have sign ambiguities in their actions on fermions, and propose that pertinent duality groups be extended by Z_2, to groups such as the metaplectic group. Specifically, we look at duality groups arising from mapping class groups of tori in M theory compactifications, T-duality, ten-dimensional type IIB S-duality, and (briefly) four-dimensional N=4 super Yang-Mills, and in each case, propose that the full duality group is a nontrivial Z_2 extension of the duality group acting on bosonic degrees of freedom, to more accurately describe possible actions on fermions. We also walk through U-duality groups for toroidal compactifications to nine, eight, and seven dimensions, which enables us to perform cross-consistency tests of these proposals.

Recursive Spoke Darts: Local Hyperplane Sampling for Delaunay and Voronoi Meshing in Arbitrary Dimensions

We introduce Recursive Spoke Darts (RSD): a recursive hyperplane sampling algorithm that exploits the full duality between Voronoi and Delaunay entities of various dimensions. Our algorithm abandons the dependence on the empty sphere principle in the generation of Delaunay simplices providing the foundation needed for scalable consistent meshing. The algorithm relies on two simple operations: line-hyperplane trimming and spherical range search. Consequently, this approach improves scalability as multiple processors can operate on different seeds at the same time. Moreover, generating consistent meshes across processors eliminates the communication needed between them, improving scalability even more. We introduce a simple tweak to the algo- rithm which makes it possible not to visit all vertices of a Voronoi cell, generating almost-exact Delaunay graphs while avoiding the natural curse of dimensionality in high dimensions.

From U(1) to E8: soft theorems in supergravity amplitudes

It is known that for $$ \mathcal{N}=8 $$ supergravity, the double-soft-scalar limit of a n-point amplitude is given by a sum of local SU(8) rotations acting on a (n−2)-point amplitude. For $$ \mathcal{N}<8 $$ supergravity theories, complication arises due to the presence of a U(1) in the U( $$ \mathcal{N} $$ ) isotropy group, which introduces a soft-graviton singularity that obscures the action of the duality symmetry. In this paper, we introduce an anti-symmetrised extraction procedure that exposes the full duality group. We illustrate this procedure for tree-level amplitudes in 4 ≤ $$ \mathcal{N}<8 $$ supergravity in four dimensions, as well as $$ \mathcal{N}=16 $$ supergravity in three dimensions. In three dimensions, as all bosonic degrees of freedom transform under the E8 duality group, supersymmetry ensures that the amplitude vanishes in the single-soft limit of all particle species, in contrast to its higher dimensional siblings. Using recursive formulas and generalized unitarity cuts in three dimensions, we demonstrate the action of the duality group for any tree-level and one-loop amplitudes. Finally we discuss the implications of the duality symmetry on possible counter terms for this theory. As a preliminary application, we show that the vanishing of single-soft limits of arbitrary component fields in three-dimensional supergravity rules out the direct dimensional reduction of D 8 R 4 as a valid counter term.

Dualities and fivebrane instantons

We derive the fivebrane instanton corrections to the hypermultiplet moduli space $M_H$ of Calabi-Yau string vacua using S-duality symmetry of the type IIB formulation. The result is given in terms of a set of holomorphic functions on the twistor space of $M_H$. It contains not only all orders of the instanton expansion, but also takes into account the presence of D1-D(-1)-brane instantons. Furthermore, we provide a thorough study of the group of discrete isometries of $M_H$ and show that its closure requires a modification of certain symmetry transformations. After this modification, the fivebrane instantons are proven to be consistent with the full duality group.

Resummation and S-duality in N = 4 SYM

We consider the problem of resumming the perturbative expansions for anomalous dimensions of low twist, non-BPS operators in four dimensional N=4 supersymmetric Yang-Mills theories. The requirement of S-duality invariance imposes considerable restrictions on any such resummation. We introduce several prescriptions that produce interpolating functions on the upper half plane that are compatible with a subgroup of the full duality group. These lead to predictions for the anomalous dimensions at all points in the fundamental domain of the complex gauge coupling, and in particular at the duality-invariant values \tau=i and \tau=exp(i\pi/3). For low-rank gauge groups, the predictions are compatible with the bounds derived by conformal bootstrap methods for these anomalous dimensions; within numerical errors, they are in good agreement with the conjecture that said bounds are saturated at a duality-invariant point. We also find that the anomalous dimensions of the lowest twist operators lie within an extremely narrow window around a straight line as we vary the moduli of the theory over the two dimensional fundamental domain.

Full dualisability is independent of the generating algebra

Given a full duality based on a finite algebra M, we show how to create a full duality based on any other finite algebra N for which \({\mathbb{ISP}({\bf M}) = \mathbb{ISP}({\bf N})}\). So the full dualisability of a quasivariety is independent of the algebra chosen as the generator. We obtain this result by proving the corresponding result for multisorted full dualities.

Samuelson's Full Duality and the Use of Directed Acyclical Graphs

To date, mixed demand systems have been all but ignored in empirical work. A possible reason for the scarcity of such applications is that one needs to know a priori which prices and quantities are endogenous in the mixed demand system. By using a directed acyclical graph (DAG), causal relationships among price and quantity variables are identified giving rise to a causally identified mixed demand system. A statistical comparison is made of the traditional Rotterdam model, a synthetic demand system, which subsumes the traditional Rotterdam model, and a Rotterdam mixed demand system identified through the use of a DAG. In this analysis, the respective demand systems consist of five products: steak, ground beef, beef roasts, pork, and chicken.

Not every full duality is strong!

Is every full duality strong? We show that the answer is ‘no’, thereby answering a question that goes back to the earliest foundations of the theory of natural dualities.

Structural reducts and the full implies strong problem

Let \(\underline{\text{M}}\) be a finite algebra. We show that if there exists a particular dualising alter ego \({\mathop{{\rm M}}\limits_{\sim}}^{\flat}\) that satisfies a weak form of injectivity, then the notions of full duality and strong duality for \(\underline{\text{M}}\) are equivalent.

Generalized switch-setting problems

Switch-setting games like Lights Out are typically modelled as a graph, where the vertices represent switches and lamps, and the edges capture the switching rules. We generalize the concepts used for a mathematical description of Lights Out and its relatives. Our approach uses bipartite graphs and allows for the analysis of a broader class of switch-setting games, which is demonstrated at a new variant of the Lights Out puzzle. Our method exhibits full duality between switches and lamps, and we get rid of some insufficiencies inherent in the modelling with non-bipartite graphs. We present a detailed analysis of the new Lights Out variant, formulate solvability conditions, give graphically aesthetic interpretations and discuss aspects on minimum solutions. In our study of parity domination in bipartite graphs we incorporate methods from linear algebra and linear programming. We point out the close relations between graph theoretic terms and the language of algebra over Z 2 .

Duality in Infinite Graphs

The adaption of combinatorial duality to infinite graphs has been hampered by the fact that while cuts (or cocycles) can be infinite, cycles are finite. We show that these obstructions fall away when duality is reinterpreted on the basis of a ‘singular’ approach to graph homology, whose cycles are defined topologically in a space formed by the graph together with its ends and can be infinite. Our approach enables us to complete Thomassen's results about ‘finitary’ duality for infinite graphs to full duality, including his extensions of Whitney's theorem.

Full does not imply strong, does it?

We give a duality for the variety of bounded distributive lattices that is not full (and therefore not strong) although it is full but not strong at the finite level. While this does not give a complete solution to the “Full vs Strong” Problem, which dates back to the beginnings of natural duality theory in 1980, it does solve it at the finite level. One consequence of this result is that although there is a Duality Compactness Theorem, which says that if an alter ego of finite type yields a duality at the finite level then it yields a duality, there cannot be a corresponding Full Duality Compactness Theorem.