Using the parent Lagrangian approach we construct a dual formulation, in the sense originally proposed by Curtright and Freund, of a massive spin two Fierz-Pauli theory in arbitrary dimensions $D$. This is achieved in terms of a mixed symmetry tensor $T_{A[B_{1}B_{2}... B_{D-2}]}$, without the need of auxiliary fields. The relation of this method with an alternative formulation based on a gauge symmetry principle proposed by Zinoviev is elucidated. We show that the latter formulation in four dimensions, with a given gauge fixing together with a definite sequence of auxiliary fields elimination via their equations of motion, leads to the parent Lagrangian already considered by West completed by a Fierz-Pauli mass term, which in turns yields the Curtright-Freund action. This motivates our generalization to arbitrary dimensions leading to the corresponding extension of the four dimensional result. We identify the transverse true degrees of freedom of the dual theory and verify that their number is in accordance with those of the massive Fierz-Pauli field.
Lectures given by C.G. in the Advanced School on Effective Theories (Almuñecar, Granada, 1995) on duality in N=2 supersymmetric Yang-Mills, and the coupling to gravity.
Topologically non trivial effects appearing in the discussion of duality transformations in higher genus manifolds are discussed in a simple example, and their relation with the properties of Topological Field Theories is established.
Duality symmetries are discussed for non-linear gauge theories of (n-1)-th rank antisymmetric tensor fields in general even dimensions d=2n. When there are M field strengths and no scalar fields, the duality symmetry groups should be compact. We find conditions on the Lagrangian required by compact duality symmetries and show an example of duality invariant non-linear theories. We also discuss how to enlarge the duality symmetries to non-compact groups by coupling scalar fields described by non-linear sigma models.
The quantum Yang-Mills theory describing dual ($\tilde g$) and non-dual ($g$) charges and revealing the generalized duality symmetry was developed by analogy with the Zwanziger formalism in QED.
The massive Schwinger and the massive sine-Gordon models are proved to be equivalent at finite temperature, using the path-integral framework. The well known relations among the parameters of these models to establish the duality at $T=0$, also remain valid at non zero temperature.
We suggest IR-dual descriptions for d=3 N=2 supersymmetric gauge theories with gauge groups USp(2N_c) and U(N_c) and matter in the fundamental representation. We relate this duality to the IR duality of d=4 N=1 SQCD theories, and in one case also to mirror symmetry.
Recently Seiberg has conjectured a duality symmetry connecting different theories of the supersymmetric QCD type. We provide support for this conjecture by analyzing a flat direction of the theory along which the two dual theories go over to the same theory in the IR.
We study the significance of T-duality in the context of the gravitational description of gauge theories. We found that T-duality relates the deferents points of the moduli of a given gauge theory always far from the conformal fixed point. Also the described gauge theories seems to flow naturally to the able conformal points, those that naturally saturate all the possible known examples of near horizon geometries. Supersymmetry properties and T-duality breaking of it are discuss.
An intriguing connection, based on duality symmetry, between ordinary (commutative) Born-Infeld type theory and non-commutative Maxwell type theory, is pointed out. Both discrete as well as continuous duality transformations are considered and their implications for self duality condition and Legendre transformations are analysed.
It is well known that the classical equations of motion of Maxwell and Born-Infeld theories are invariant under a duality symmetry acting on the field strengths. We review the implementation of the SL(2,Z) duality in these theories as linear but non-local transformations of the potentials.
This Ph.D. thesis was submitted to the Catholic University of Leuven. We discuss two main topics: various formulations of duality-symmetric theories and D=5 supergravity (both construction and supersymmetric solutions). The thesis is structured as follows: Ch. 1 Introduction; Ch. 2 String theory toolkit; Ch. 3 Free duality-symmetric theories; Ch. 4 Interacting duality-symmetric theories; Ch. 5 Conformal supergravity in D=5: Weyl multiplets; Ch. 6 Conformal supergravity in D=5: matter and gauge-fixing; Ch. 7 Supersymmetric solutions in D=5 gauged supergravity.
We propose an explicit description of duality walls which encode at low energy the global symmetry enhancement expected in the UV completion of certain five-dimensional gauge theories. The proposal is supported by explicit localization computations and implies that the instanton partition function of these theories satisfies novel and unexpected integral equations.
This is a review of some basic features on the relation between supergravity and pure gauge theories with special emphasis on the relation between T-duality and supersymmetry. Some new results concerning the interplay between T-duality and near horizon geometries are presented
In this work, we show that duality symmetry can be implemented for massive theories at the level of the action, whenever we can formulate appropriates gauge invariant actions. For a massive vectorial field, we use a known gauge invariant description, while for a massive graviton, we introduce a novel gauge invariant action in order to show duality invariance.
It has been conjectured that duality cascade occurs in the $\mathcal{N}=3$ supersymmetric Yang-Mills Chern-Simons theory with the gauge group $U(N )_k \times U(N+M )_{-k}$ coupled to two bi-fundamental hypermultiplets. The brane picture suggests that this duality cascade can be generalized to a class of 3d $\mathcal{N}=3$ supersymmetric quiver gauge theories coming from so-called Hanany-Witten type brane configurations. In this paper we perform non-perturbative tests of the duality cascades using supersymmetry localization. We focus on $S^3$ partition functions and prove predictions from the d…
We review the current status of the construction of unitary representations of U-duality groups of supergravity theories in five, four and three dimensions. We focus mainly on the maximal supergravity theories and on the N=2 Maxwell-Einstein supergravity (MESGT) theories defined by Jordan algebras of degree three in five dimensions and their descendants in four and three dimensions. Entropies of the extremal black hole solutions of these theories in five and four dimensions are given by certain invariants of their U-duality groups. The five dimensional U-duality groups admit extensions to spec…
We study the equivalence/duality between various non-commutative gauge models at the classical and quantum level. The duality is realised by a linear Seiberg-Witten-like map. The infinitesimal form of this map is analysed in more details.
We briefly review the basics of electric-magnetic duality symmetry and their geometric interpretation in the M-theory context. Then we recall some no-go theorems that prevent a simple extension of the duality symmetry to non-Abelian gauge theories.
We explore the dynamics of three-dimensional Chern-Simons gauge theories with N=2 supersymmetry and matter in the fundamental and adjoint representations of the gauge group. Realizing the gauge theories of interest in a setup of threebranes and fivebranes in type IIB string theory we argue for a Seiberg duality that relates Chern-Simons theories with non-trivial superpotentials.
We construct T-duality on K3 surfaces. The T-duality exchanges a 4-brane R-R charge and a 0-brane R-R charge. We study the action of the T-duality on the moduli space of 0-branes located at points of K3 and 4-branes wrapping it. We apply the construction to F-theory compactified on a Calabi-Yau 4-fold and study the duality of N=2 SU(N_c) gauge theories in four dimensions. We discuss the generalization to the N=1 duality scenario.
We analyze the Coulomb phase of theories of $N=2$ SQCD with $SU(N_c)$ gauge groups which are conjectured to have exact electric-magnetic duality. We discuss the duality transformation of the particle spectrum, emphasizing the differences between the general case and the $SU(2)$ case. Some difficulties associated with the definition of the duality transformation for a general gauge group are discussed. We compute the classical monopole spectrum of these theories, and when it is possible we use it to check the consistency of the duality. Generally these theories may have phase transitions betwee…
We consider new issues of duality in four-dimensional Lorentz-breaking field theories. In particular, we demonstrate that the arising of the aether-like Lorentz-breaking term is necessary in order for the 4D models to display the duality analog between the MCS and self-dual models in 3D. We further study the dispersion relations in both theories and discuss the physical contents of the models involved in this new dualilty.
Recently N. Seiberg has shown that certain N=1 supersymmetric Yang -- Mills theories have the property that their infrared physics can be equivalently described by {\it two different} gauge theories, with the strong coupling region of one corresponding to the weak coupling region of the other. This duality leads to interesting results about the infrared dynamics of these theories. We generalize Seiberg's ideas to a new class of models for which a similar duality allows one to obtain rather detailed dynamical information about the long distance physics.
We describe some of the novel 6d quantum field theories which have been discovered in studies of string duality. The role these theories (and their 4d descendants) may play in alleviating the vacuum degeneracy problem in string theory is reviewed. The DLCQ of these field theories is presented as one concrete way of formulating them, independent of string theory.
In this paper we propose a duality for non-holomorphic N=1 CS-matter theories living on M2 branes probing Spin(7) cones. We call this duality Spin(7) duality. Two theories are named Spin(7) dual if they have the same moduli space: a real Spin(7) cone with base a weak G_2 manifold, and they are hence holographic dual to the same AdS_4 x G_2 M theory solution. We provide a systematic way to generate these dualities, derived by combining toric duality for N=2 CS-matter theories and generalized non-holomorphic orientifold projections to N=1. Brane construction, AdS/CFT correspondence, and the comp…
The supersymmetric generalization of Poisson-Lie T-duality in superconformal WZNW models is considered. It is shown that the classical N=2 superconformal WZNW models posses a natural Poisson-Lie symmetry which allows to construct dual $σ$- models.
We study four N=1 SU(N)^6 gauge theories, with bi-fundamental chiral matter and a superpotential. In the infrared, these gauge theories all realize the low-energy world-volume description of N coincident D3-branes transverse to the complex cone over a del Pezzo surface dP_3 which is the blowup of P^2 at three generic points. Therefore, the four gauge theories are expected to fall into the same universality class--an example of a phenomenon that has been termed "toric duality." However, little independent evidence has been given that such theories are infrared-equivalent. In fact, we show that …
We present more examples of dual N=1 SUSY gauge theories. This set of theories is connected by flows to both Seiberg's and Kutasov's dual theories. This provides a unifying picture of the various dual theories. We investigate the dual theories, their flat directions and mass perturbations.
Evidence in favor of $SL(2,Z)$ S-duality in $N=4$ supersymmetric Yang-Mills theories in four dimensions and with general compact, simple gauge groups is presented. (Contribution to the Proceedings of the Strings '95 conference, March 13-18, 1995, USC, and the Proceedings of the Trieste Conference on S-Duality and Mirror Symmetry, June 5-9, 1995.)
We prove that a sum of free non-covariant duality-symmetric actions does not allow consistent, continuous and local self-interactions that deform the gauge transformations. For instance, non-Abelian deformations are not allowed, even in 4 dimensions where Yang-Mills type interactions of 1-forms are allowed in the non-manifestly duality-symmetric formulation. This suggests that non-Abelian duality should require to leave the standard formalism of perturbative local field theories. The analyticity of self-interactions for a single duality-symmetric gauge field in four dimensions is also analyzed…
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 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 …
We show that dualization of Stueckelberg-like massive gauge theories and $B\wedge F$ models, follows form a general p-dualization of interacting theories in d spacetime dimensions. This is achieved by a particular choice of the external current.
A massive version of T-duality in six dimensions is given, that maps the K3 compactification of Romans' theory onto the K3 compactification of Type IIB theory. This is done by performing a (standard) Kaluza-Klein reduction on six-dimensional massive Type IIA and a Scherk-Schwarz reduction on Type IIB, mapping both theories onto the same five-dimensional theory. We also comment shortly on the difficulties arising if one intends to construct a massive generalisation of the six-dimensional string-string duality.
We study the invariance under duality transformations in massless and massive $p-$form field theories and obtain the Noether generators of the infinitesimal transformations that correspond to this symmetry. These generators are realized in geometrical representations that generalize the Loop Representation of the Maxwell field, allowing for a geometrical interpretation which is studied.
We consider a general class of symmetries of hyper-Kahler quotients which can be interpreted as classical analogs of Seiberg duality for N=2 supersymmetric quiver gauge theories in the baryonic Higgs branch. Along the way we find that a limit application of this duality yields new exotic realizations of ALE spaces as hyper-Kahler quotients. We also discuss how these results admit in some particular cases a natural interpretation in string theory. We finally comment on a possible relationship with Fourier-Mukai tranforms.
We discuss the non--perturbative formulation for $c \leq 1$ string theory. The field theory like formulation of topological and non--topological models is presented. The integral representation for arbitrary $(p,q)$ solutions is derived which explicitly obeys $p-q$ duality of these theories. The exact solutions to string equation and various examples are also discussed.
Starting from Seiberg's electric-magnetic duality for supersymmetric QCD, we construct dual pairs of non-supersymmetric gauge theories. This is accomplished by first taking the large N limit of supersymmetric QCD and its dual partner and then performing a special ``orbifold projection'' recently introduced by Kachru and Silverstein. We argue that in the large N limit the two projected theories remain dual. The non-supersymmetric gauge theories which can be studied in this fashion have non-supersymmetric field content, chiral fermions and exactly massless scalar matter.
We present evidence for the target space duality symmetry associated with massive excited states of closed bosonic string. The evolution of string is considered in ${\hat D}$ spacetime dimensions; out of which d spacial dimensions are compactified on torus. The phase space Hamiltonian formulation is adopted to unveil the T-duality symmetry from the worldsheet perspective. The existence of this symmetry is verified for a few massive levels. A systematic procedure is presented to study T-duality symmetry of vertex operators for all massive levels of closed bosonic string. It is argued that all v…
Superconformal sigma models with Calabi--Yau target spaces described as complete intersection subvarieties in toric varieties can be obtained as the low-energy limit of certain abelian gauge theories in two dimensions. We formulate mirror symmetry for this class of Calabi--Yau spaces as a duality in the abelian gauge theory, giving the explicit mapping relating the two Lagrangians. The duality relates inequivalent theories which lead to isomorphic theories in the low-energy limit. This formulation suggests that mirror symmetry could be derived using abelian duality. The application of duality …
Non-supersymmetric d=4 gauge theories which arise from superstring duality on a manifold $AdS_5 \times S_5/Z_p$ are cataloged for a range $2 \leq p \leq 41$. A number have vanishing two-loop gauge β-function, a necessary but not sufficient condition to be a conformal field theory.
We analyze a recent conjecture regarding the perturbative construction of non-linear deformations of all classically duality invariant theories, including N=8 supergravity. Starting with an initial quartic deformation, we engineer a procedure that generates a particular non-linear deformation (Born-Infeld) of the Maxwell theory. This procedure requires the introduction of an infinite number of modifications to a constraint which eliminates degrees of freedom consistent with the duality and field content of the system. We discuss the extension of this procedure to N=1 and N=2 supersymmetric the…
In this paper, we generalize the duality between self-dual and Maxwell-Chern-Simons theories for the case of a CPT-even Lorentz-breaking extension of these theories. The duality is demonstrated with use of the gauge embedding procedure, both in free and coupled cases, and with the master action approach. The physical spectra of both Lorentz-breaking theories are studied. The massive poles are shown to coincide and to respect the requirements for unitarity and causality at tree level. The extra massless poles which are present in the dualized model are shown to be nondynamical.
There are many physically interesting superconformal gauge theories in four dimensions. In this talk I discuss a common phenomenon in these theories: the existence of continuous families of infrared fixed points. Well-known examples include finite ${\cal N}=4$ and ${\cal N}=2$ supersymmetric theories; many finite ${\cal N}=1$ examples are known also. These theories are a subset of a much larger class, whose existence can easily be established and understood using the algebraic methods explained here. A relation between the ${\cal N}=1$ duality of Seiberg and duality in finite ${\cal N}=2$ theo…
We propose a new duality between two 3d $\mathcal{N}=8$ superconformal Chern-Simons-matter theories: the $U(3)_1 \times U(3)_{-1}$ ABJM theory and a theory consisting of the product between the $\left(SU(2)_3\times SU(2)_{-3}\right)/\mathbb{Z}_2$ BLG theory and a free ${\cal N} = 8$ theory of eight real scalars and eight Majorana fermions. As evidence supporting this duality, we show that the moduli spaces, superconformal indices, $S^3$ partition functions, and certain OPE coefficients of BPS operators in the two theories agree.
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 $N_f \le 4$ flavors, which are self-dual, and on $SU(3)$ SQCD theories, which are dual to $SU(2)^2$ quiver theories. From…
We extend the large N duality of four dimensional N=1 supersymmetric Yang-Mills theory with additional chiral fields and arbitrary superpotential recently proposed by Cachazo, Intriligator and Vafa to the case of SO/Sp gauge groups. By orientifolding the geometric transition, we investigate a large N duality between N=1, SO/Sp supersymmetric theories with arbitrary superpotential and an Abelian N=2 theory with supersymmetry broken to N=1 by electric and magnetic Fayet-Iliopoulos terms.
We determine the 2-group structure constants for all the six-dimensional little string theories (LSTs) geometrically engineered in F-theory without frozen singularities. We use this result as a consistency check for T-duality: the 2- groups of a pair of T-dual LSTs have to match. When the T-duality involves a discrete symmetry twist the 2-group used in the matching is modified. We demonstrate the matching of the 2-groups in several examples.
We study the N=1* supersymmetric gauge theories with gauge groups Sp(2n) and SO(2n+1). These theories are obtained from the corresponding N=4 supersymmetric Yang-Mills theories via a mass deformation. We show that the number of quantum vacua in the Sp(2n) theory is equal to the number of quantum vacua in the SO(2n+1) theory. This constitutes non-trivial support for S-duality between these theories. The verification of the equality of the number of quantum vacua involves a rather esoteric identity due to Ramanujan.
The Goddard, Nuyts and Olive conjecture for electric-magnetic duality in Yang-Mills theory with an arbitrary gauge group G is extended by including a non-vanishing vacuum angle $θ$. This extended S-duality conjecture includes the case when the unbroken gauge group is non-abelian and a definite prediction for the spectrum of dyons results.
Gauge theories in four dimensions can exhibit interesting low energy phenomena, such as infrared enhancements of global symmetry. We explore a class of 4d N=1 gauge theories arising from a construction that is motivated by duality walls in 5d gauge theories. Their quiver descriptions bear a resemblance to 4d theories obtained by compactifying 6d N=(1,0) superconformal field theories on a torus with fluxes, but with lower number of flavours and different number of gauge singlets and superpotentials. One of the main features of these theories is that they exhibit a flavour symmetry enhancement, …
We clarify how mirror symmetry acts on 3d theories with N=2,3 or 4 supersymmetries and non-abelian Chern-Simons terms and then construct many new examples. We identify a new duality, geometric duality, that allows us to generate large families of gauge theories, with and without Chern-Simons term, that all flow to the same conformal field theory in the infrared. In particular, we find an interesting duality of dualities: a pair of theories related via mirror symmetry can be mapped, via geometric duality, into a pair of gauge theories related by Seiberg duality. This network of dualities can be…
This thesis presents a thorough analysis of the links between the N=4 supersymmetric gauge theory in four dimensions and its three topological twisted counterparts. Special emphasis is put in deriving explicit results in terms of the vacuum structure and low-energy effective description of four-dimensional supersymmetric gauge theories. A key ingredient is the realization of Montonten-Olive duality in the twisted theories, which is discussed in detail from different viewpoints.
We review the status of duality symmetries in superstring theories. These discrete symmetries mark the striking differences between theories of pointlike objects and theories of extended objects. They prove to be very helpful in understanding non-perturbative effects in string theories. We will also briefly discuss the strange role played by open strings and their solitons in the emerging scenario.
We analyze the strong coupling behavior of the large N gauge theories in 4-dimensions with N=4 supersymmetry by making use of S-duality. We show that at large values of the coupling constant $λ=g_{YM}^2N$ the j-th non-planar amplitude $f_j(λ) (j=0,1,2 ...)$ behaves as $f_j(λ)\approx λ^j$. Implication of this behavior is discussed in connection with the supergravity theory on $AdS_5\times S^5$ suggested by the CFT/AdS correspondence. S-duality of the gauge theory corresponds to the duality between the closed and open string loop expansions in the gravity/string theory.
We extend the duality between massive and topologically massive antisymmetric tensor gauge theories in arbitrary space-time dimensions to include topological defects. We show explicitly that the condensation of these defects leads, in 4 dimensions, to confinement of electric strings in the two dual models. The dual phase, in which magnetic strings are confined is absent. The presence of the confinement phase explicitly found in the 4-dimensional case, is generalized, using duality arguments, to arbitrary space-time dimensions.
We investigate the phase structures of various N=1 supersymmetric gauge theories including even the exceptional gauge group from the viewpoint of superconvergence of the gauge field propagator. Especially we analyze in detail whether a new type of duality recently discovered by Oehme in $SU(N_c)$ gauge theory coupled to fundamental matter fields can be found in more general gauge theories with more general matter representations or not. The result is that in the cases of theories including matter fields in only the fundamental representation, Oehme's duality holds but otherwise it does not. In…
The partition functions of refined topological strings(A-models) are computed, which give rise to the circle-compactified five-dimensional supersymmetric linear quiver gauge theories in generic (not necessarily self-dual) Omega backgrounds. Based on the slicing independence conjecture of refined topological string partition functions, it is demonstrated explicitly that the duality exists between $SU(N)^{M-1}$ and $SU(M)^{N-1}$ supersymmetric linear quiver gauge theories, even in generic Omega backgrounds. It is found that the relations between string moduli and gauge moduli are deformed from t…
The relationship between the sources of physical fields and the fields themselves is investigated with regard to the coupling of topological information between them. A class of field theories that we call topological field theories is defined such that both the field and its source represent de Rham cocycles in varying dimensions over complementary subspaces and the coupling of one to the other is by way of an isomorphism of the those cohomology spaces, which we refer to as field/source duality. The deeper basis for such an isomorphism is investigated and the process is described for various …
The Renormalization group in field theories happens to resemble dynamical systems in many ways. In this paper, we discuss the unexpected connection between chaos and duality in field theories. In a sense, that various dual field theories can emerge at the end of chaotic RG trajectories, and hence strong-weak duality in quantum field theory is a direct result of the chaotic flow of the renormalization group. This suggests that various properties of field and string theories could come into existence due to chaotic RG flow. We also conjecture the existence of dual quantum field theories in the h…
