T-duality of string theory can be extended to the Poisson-Lie T-duality when the target space has a generalized isometry group given by a Drinfel'd double. In M-theory, T-duality is understood as a subgroup of U-duality, but the non-Abelian extension of U-duality is still a mystery. In this paper, we study membrane theory on a curved background with a generalized isometry group given by the $\mathcal{E}_n$ algebra. This provides a natural setup to study non-Abelian U-duality because the $\mathcal{E}_n$ algebra has been proposed as a U-duality extension of the Drinfel'd double. We show that the standard treatment of Abelian U-duality can be extended to the non-Abelian setup. However, a famous issue in Abelian U-duality still exists in the non-Abelian extension.
We investigate the gauge/gravity duality in the interaction between two spherical membranes in the 11-dimensional pp-wave background. On the supergravity side, we find the solution to the field equations at locations close to a spherical source membrane, and use it to obtain the light cone Lagrangian of a spherical probe membrane very close to the source, i.e., with their separation much smaller than their radii. On the gauge theory side, using the BMN matrix model, we compute the one-loop effective potential between two membrane fuzzy spheres. Perfect agreement is found between the two sides.…
We study U-duality symmetries of toroidally compactified M theory from the membrane worldvolume point of view. This is done taking the most general set of bosonic background fields into account. Upon restriction to pure moduli backgrounds, we are able to find the correct U-duality groups and moduli coset parameterizations for dimensions D>6 as symmetries of the membrane worldvolume theory. In particular, we derive the D=8 U-duality group SL(2)xSL(3). For general background fields, we concentrate on the case D=8. Though the SL(2) part of the symmetry appears to be obstructed by certain terms in…
We consider a lightcone wrapped supermembrane compactified on 2-torus in the matrix regularization. We examine the double dimensional reduction technique and deduce the free matrix string of (p,q)-string in type IIB superstring theory explicitly from the matrix-regularized wrapped supermembrane. In addition we obtain the (2+1)-dimensional super Yang-Mills action in curved background. We also examine the duality.
The problem with the U-duality transformation of membrane on $T^{n}$ is recently addressed in [arXiv:1509.02915 [hep-th]]. We will consider the U-duality transformation rule of membrane on $T^{n}\times R$. It turns out that winding modes on $T^{n}$ should be taken into account, since the duality transformation may bring the membrane configuration without winding modes into the one with winding modes. With the winding modes added, the membrane worldvolume theory in lightcone gauge is equivalent to the $n+1$ dimensional super-Yang-Mills (SYM) theory in $\tilde{T}^{n}$, which has $SL(2,Z)\times S…
We outline a formulation of membrane dynamics in D=8 which is fully covariant under the U-duality group SL(2,Z) x SL(3,Z), and encodes all interactions to fields in the eight-dimensional supergravity, which is constructed through Kaluza-Klein reduction on T^3. Among the membrane degrees of freedom is an SL(2,R) doublet of world-volume 2-form potentials, whose quantised electric fluxes determine the membrane charges, and are conjectured to provide an interpretation of the variables occurring in the minimal representation of E_{6(6)} which appears in the context of automorphic membranes. We solv…
Just as string T-duality originates from transforming field equations into Bianchi identities on the string worldsheet, so it has been suggested that M-theory U-dualities originate from transforming field equations into Bianchi identities on the membrane worldvolume. However, this encounters a problem unless the target space has dimension $D = p + 1$. We identify the problem to be the nonintegrability of the U-duality transformation assigned to the pull-back map. Just as a double geometry renders manifest the $O(D,D)$ string T-duality, here we show in the case of the M2-brane in $D = 3$ that a…
We test the background geometry of the BFSS model using a D4-brane probe. This proves a sensitive test of the geometry and we find excellent agreement with the D4-brane predictions based on the solution of a membrane corresponding to the D4-brane propagating on this background.
In this paper we study duality properties of the M(atrix) theory compactified on a circle. We establish the equivalence of this theory to the strong coupling limit of type IIB string theory compactified on a circle. In the M(atrix) theory context, our major evidence for this duality consists of identifying the BPS states of IIB strings in the spectrum and finding the remnant symmetry of SL(2,Z) and the associated tau moduli. By this IIB/M duality, a number of insights are gained into the physics of longitudinal membranes in the infinite momentum frame. We also point out an accidental affine Li…
The conjectured equivalence of the heterotic string to a $K_3$ compactified type IIA superstring is combined with the conjectured equivalence of the latter to a compactified 11-dimensional supermembrane to derive a string membrane duality in seven dimensions; the membrane is a soliton of the string theory and vice versa. A prediction of this duality is that the heterotic string is a $K_3$ compactification of the solitonic 11-dimensional fivebrane. It is verified that the worldsheet action of the D=10 heterotic string is indeed obtainable by $K_3$ compactification of the worldvolume action of t…
We show that the $E-S \sim \log S$ behaviour found for long strings rotating on $AdS_5\times S^5$ may be reproduced by membranes rotating on $AdS_4\times S^7$ and on a warped $AdS_5$ M-theory solution. We go on to obtain rotating membrane configurations with the same $E-K \sim \log K$ relation on $G_2$ holonomy backgrounds that are dual to ${\mathcal{N}}=1$ gauge theories in four dimensions. We study membrane configurations on $G_2$ holonomy backgrounds systematically, finding various other Energy-Charge relations. We end with some comments about strings rotating on warped backgrounds.
We show that the open membrane action on $T^3\times S^1/Z^2$ is equivalent to the closed membrane action on K3. The main difference between the two actions is that one generates the KK modes in the worldvolume action which is the strong coupling limit of \IIA\ while the other action generates the KK modes in a worldsheet action. Thus explaining membrane-string duality in D=7, which naturally leads to string-string duality in D=6.
We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realized as S-duality for $U(\infty)$ N=4 Super-Yang-Mills in 3+1D. We argue that Kaluza Klein states of gravitons correspond to electric fluxes, wrapped membranes become magnetic fluxes and instantonic membranes are related to Yang-Mills instantons. The T-duality transformation of gravitons into wrapped membranes is interpreted as the duality betw…
A manifestly S-dual, and `12 dimensional', IIB superstring action with an $Sl(2;\bR)$ doublet of `Born-Infeld' fields is presented. The M-theory origin of the 12th dimension is the M-2-brane tension, which can be regarded as the flux of a 3-form worldvolume field strength. The latter is required by the fact that the M-2-brane can have a boundary on an M-5-brane.
We analyse supermembrane instantons (fully wrapped supermembranes) by computing the partition function of the three-dimensional supersymmetrical U(N) matrix model under periodic boundary conditions. By mapping the model to a cohomological field theory and considering a mass-deformation of the model, we show that the partition function exactly leads to the U-duality invariant measure factor entering supermembrane instanton sums. On the other hand, a computation based on the quasi-classical assumption gives the non U-duality invariant result of the zero-mode analysis by Pioline et al. This is su…
With the deconstruction technique, the geometric information of a torus can be encoded in a sequence of orbifolds. By studying the Matrix Theory on these orbifolds as quiver mechanics, we present a formulation that (de)constructs the torus of {\em generic shape} on which Matrix Theory is ``compactified''. The continuum limit of the quiver mechanics gives rise to a $(1+2)$-dimensional SYM. A hidden (fourth) dimension, that was introduced before in the Matrix Theory literature to argue for the electric-magnetic duality, can be easily identified in our formalism. We construct membrane wrapping st…
We compare the bosonic and maximally supersymmetric membrane models. We find that in Hoppe regulated form the bosonic membrane is well approximated by massive Gaussian quantum matrix models. In contrast the similarly regulated supersymmetric membrane, which is equivalent to the BFSS model, has a gravity dual description. We sketch recent progress in checking gauge/gravity duality in this context.
We present a 1-loop toroidal membrane winding sum reproducing the conjectured $M$-theory, four-graviton, eight derivative, $R^4$ amplitude. The $U$-duality and toroidal membrane world-volume modular groups appear as a Howe dual pair in a larger, exceptional, group. A detailed analysis is carried out for $M$-theory compactified on a 3-torus, where the target-space $Sl(3,\Zint)\times Sl(2,\Zint)$ $U$-duality and $Sl(3,\Zint)$ world-volume modular groups are embedded in $E_{6(6)}(\Zint)$. Unlike previous semi-classical expansions, $U$-duality is built in manifestly and realized at the quantum lev…
The act of implementing non-Abelian duality in two dimensional sigma models results unavoidably in an additional reducible symmetry. The Batalin-Vilkovisky formalism is employed to handle this new symmetry. Valuable lessons are learnt here with respect to non-Abelian duality. We emphasise, in particular, the effects of the ghost sector corresponding to this symmetry on non-Abelian duality.
We consider a pure U(1) quantum gauge field theory on a general Riemannian compact four manifold. We compute the partition function with Abelian Wilson loop insertions. We find its duality covariance properties and derive topological selection rules. Finally, we show that, to have manifest duality, one must assume the existence of twisted topological sectors besides the standard untwisted one.
Recent development on non-Abeliann vortices and monopoles is reviewed with an emphasis on their relevance on confinement and duality. A very recent construction of non-Abelian vortices which do not dynamically Abelianize is crucial in this context.
We start by describing two of the main proposals for duality in Abelian gauge theories, namely $F$(ield strength)-duality approach and the $S$% -duality formalism. We then discuss how $F$-duality and $S$-duality can be applied to the case of linearized gravity. By emphasizing the similarities and differences between these two type of dualities we explore the possibility of combining them in just one duality formalism.
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.
We compute the partition function of five-dimension Abelian sYM including a graviphoton term on a five-torus. The result agrees with Abelian M5 brane partition function with zero characteristics on an associated six-torus.
For non-Abelian tensor gauge fields we have found an alternative form of duality transformation, which has the property that the direct and the inverse transformations coincide. This duality transformation has the desired property that the direct and the inverse transformations map Lagrangian forms into each other.
In this paper we perform some non-trivial tests for the recently obtained open membrane/D-brane metrics and `generalized' noncommutativity parameters using Dp/NS5/M5-branes which have been deformed by light-like fields. The results obtained give further evidence that these open membrane/D-brane metrics and `generalized' noncommutativity parameters are correct. Further, we use the open brane data and supergravity duals to obtain more information about non-gravitational theories with light-like noncommutativity, or `generalized' light-like noncommutativity. In particular, we investigate various …
A method for implementing non-Abelian duality on string backgrounds is given. It is shown that a direct generalisation of the familiar Abelian duality induces an extra local symmetry in the gauge invariant theory. The non-Abelian isometry group is shown to be enlarged to a non-semi-simple group. However, upon eliminating the gauge fields to obtain the dual theory the new algebra does not close. Therefore the gauge fixing procedure becomes problematic. The new method proposed here avoids these issues and leads to a dual theory in the proper sense of duality.
We find new type II backgrounds with non-relativistic symmetries via non-Abelian T-duality. First we consider geometries with Galilean symmetries in type IIA, which have been identified as non-relativistic generalizations of the ABJM background and massive IIA supergravities. We then consider the non-Abelian T-duality transformation on the backgrounds with Lifshitz symmetry constructed by Donos and Gauntlett. Using gauge/gravity duality we study aspects of the field theory dual to these backgrounds.
We construct a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold.
We construct the duality-symmetric actions for a large class of six-dimensional models describing hierarchies of non-Abelian scalar, vector and tensor fields related to each other by first-order (self-)duality equations that follow from these actions. In particular, this construction provides a Lorentz invariant action for non-Abelian self-dual tensor fields. The class of models includes the bosonic sectors of the 6d (1,0) superconformal models of interacting non-Abelian self-dual tensor, vector, and hypermultiplets.
We construct explicit canonical transformations producing non-abelian duals in principal chiral models with arbitrary group G. Some comments concerning the extension to more general $σ$-models, like WZW models, are given.
In this short article we develop recent proposals to relate Yang-Baxter sigma-models and non-abelian T-duality. We demonstrate explicitly that the holographic space-times associated to both (multi-parameter)-$β$-deformations and non-commutative deformations of ${\cal N}=4$ super Yang-Mills gauge theory including the RR fluxes can be obtained via the machinery of non-abelian T-duality in Type II supergravity.
The standard notion of the non-Abelian duality in string theory is generalized to the class of $\si$-models admitting `non-commutative conserved charges'. Such $\si$-models can be associated with every Lie bialgebra $(\cg ,\cgt)$ and they possess an isometry group iff the commutant $[\cgt,\cgt]$ is not equal to $\cgt$. Within the enlarged class of the backgrounds the non-Abelian duality {\it is} a duality transformation in the proper sense of the word. It exchanges the roles of $\cg$ and $\cgt$ and it can be interpreted as a symplectomorphism of the phase spaces of the mutually dual theories. …
Just as non-commutative gauge theories arise from quantising open strings in a large magnetic field, non-Abelian two-form gauge theories may conceivably be constructed by quantising open membranes in a large three-form magnetic background. We make some observations that arise in following this strategy, with an emphasis on the relation to the quantisation of volume-preserving diffeomorphisms (vpd). In particular, we construct consistent non-Abelian interactions of a two-form in 3+1 dimensions, based on gauge invariance under vpd.
We examine the electric-magnetic duality for a U(1) gauge theory on a general 4-manifold. The partition function for such a theory transforms as a modular form of specific weight. However, in the canonical approach, we show that S-duality, like T-duality, is generated by a canonical transformation leading to a modular invariant partition function.
We examine T-duality transformations for supersymmetric strings with target space geometry with compact abelian isometries. We consider the partition function of these models and we show that although T-duality is not a symmetry, due to an anomaly, it relates type IIA to type IIB strings. In this way we extend the corresponding result for toroidal compactification to the general case of non-trivial backgrounds with abelian isometries and for world sheets of any genera.
We analyse abelian T-duality for WZW models of simply-connected groups. We demonstrate that the dual theory is a certain orbifold of the original theory, and check that it is conformally invariant. We determine the spectrum of the dual theory, and show that it agrees with the spectrum of the original theory.
Non--Abelian duality in relation to supersymmetry is examined. When the action of the isometry group on the complex structures is non--trivial, extended supersymmetry is realized non--locally after duality, using path ordered Wilson lines. Prototype examples considered in detail are, hyper--Kahler metrics with SO(3) isometry and supersymmetric WZW models. For the latter, the natural objects in the non--local realizations of supersymmetry arising after duality are the classical non--Abelian parafermions. The canonical equivalence of WZW models and their non--Abelian duals with respect to a vect…
We briefly review the essential points of our recent work in non-Abelian T-duality. In particular, we show how non-abelian T-duals can effectively describe infinitely high spin sectors of a parent theory and how to implement the transformation in the presence of non-vanishing Ramond fields in type-II supergravity.
We examine non-Abelian T-duality on backgrounds preserving N=1 supersymmetry. We show how backgrounds with SU(3)-structure are mapped to backgrounds with SU(2)-structure. We provide the transformation rules for the corresponding pure spinors. We use these techniques to construct new flavoured solutions of type IIA supergravity.
An outline of the conformality approach to the gauge hierarchy is given including the use of non-abelian orbifolds to give unified models of the left-right type.
Electric-magnetic duality and higher dimensional analogues are obtained as symmetries in generalized coset constructions, similar to the axial-vector duality of two dimensional coset models described by Rocek and Verlinde. We also study global aspects of duality between p-forms and (d-p-2)-forms in d-manifolds. In particular, the modular duality anomaly is governed by the Euler character as in four and two dimensions. Duality transformations of Wilson line operator insertions are also considered.
A general study of non-abelian duality is presented. We first identify a possible obstruction to the conformal invariance of the dual theory for non-semisimple groups. We construct the exact non-abelian dual for any Wess-Zumino-Witten (WZW) model for any anomaly free subgroup, and the corresponding extension for coset conformal field theories. We characterize the exact non-abelian dual for $σ$-models with chiral isometries and extend the standard notion of duality to anomalous subgroups of WZW models, thus giving a way of constructing dual transformations for different groups on the left and o…
Non-Abelian duality transformations built on non-semi-simple isometry groups are analysed. We first give the conditions under which the original non-linear sigma model and its non-Abelian dual are equivalent. The existence of an invariant and non-degenerate bilinear form for the isometry Lie algebra is crucial for this equivalence. The non-Abelian dual of a conformally invariant sigma model, with non-semi-simple isometries, is then constructed and its beta functions are shown to vanish. This study resolves an apparent obstruction to the conformal invariance of sigma models obtained via non-Abe…
We investigate non-abelian gaugings of WZNW models. When the gauged group is semisimple we are able to present exact formulae for the dual conformal field theory, for all values of the level $k$. The results are then applied to non-abelian target space duality in string theory, showing that the standard formulae are quantum mechanically well defined in the low energy limit if the gauged group is semisimple.
We analyse super non-Abelian T-duality for principal chiral models, symmetric space sigma models, and semi-symmetric space sigma models for general Lie supergroups. This includes T-duality along both bosonic and fermionic directions. As an example, we perform the explicit dualisation of the OSp(1|2) principal chiral model, and, whilst the target superspace of this model is a three-dimensional supergravity background, we find that its super non-Abelian T-dual falls outside the class of such backgrounds.
Abelian Chern-Simons gauge theory is known to possess a `$S$-self-dual' action where its coupling constant $k$ is inverted {\it i.e.} $k \leftrightarrow {1 \over k}$. Here a vector non-abelian duality is found in the pure non-abelian Chern-Simons action at the classical level. The dimensional reduction of the dual Chern-Simons action to two-dimensions constitutes a dual Wess-Zumino-Witten action already given in the literature.
We examine non-abelian duality transformations in the open string case. After gauging the isometries of the target space and developing the general formalism, we study in details the duals oftarget spaces with SO(N) isometries which, for the SO(2) case, reduces to the known abelian T-duals. We apply the formalism to electrically and magnetically charged 4D black hole solutions and, as in the abelian case, dual coordinates satisfy Dirichlet conditions.
We present a convenient method for deriving the transformation of the dilaton under T-duality in the path-integral approach. Subtleties arising in performing the integral over the gauge fields are carefully analysed using Pauli-Villars regularization, thereby clarifying existing ambiguities in the literature. The formalism can not only be applied to the abelian case, but, and this for the first time, to the non-abelian case as well. Furthermore, by choosing a particular gauge, we directly obtain the target-space covariant expression for the dual geometry in the abelian case. Finally it is show…
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.
A modification of the Abelian Duality transformations is proposed guaranteeing that a (not necessarily conformally invariant) $σ$-model be quantum equivalent (at least up to two loops in perturbation theory) to its dual. This requires a somewhat non standard perturbative treatment of the {\sl dual} $σ$-model. Explicit formulae of the modified duality transformation are presented for a special class of block diagonal purely metric $σ$-models.
With the help of M-theory configuration of N=1 supersymmetric QCD, we analyze the strong coupling moduli of N=1 theory and its relation with S-duality transformation in N=2 supersymmetric QCD. As a result we confirm that two type-IIA descriptions for N=1 supersymmetric QCD, one of them being strong coupling description and another being weak, correspond to a single M-theory description for N=1 supersymmetric QCD. The existence of singlet fields is also discussed.
It has been argued by some authors that the parent action approach cannot be used in order to establish the duality between the 2+1 Abelian and non-Abelian Self-Dual (SD) and Yang-Mills-Chern-Simons (YMCS) models for all the coupling regimes. We propose here an alternative (perturbative) point of view, and show that this equivalence can be achieved with the parent action approach.
We investigate duality transformations in a class of backgrounds with non-Abelian isometries, i.e. Bianchi-type (homogeneous) cosmologies in arbitrary dimensions. Simple duality transformations for the metric and the antisymmetric tensor field, generalizing those known from the Abelian isometry (Bianchi I) case, are obtained using either a Lagrangian or a Hamiltonian approach. Applying these prescriptions to a specific conformally invariant $\s$-model, we show that no dilaton transformation leads to a new conformal background. Some possible ways out of the problem are suggested.
We explicitly construct the creation operators for the quantum field configurations associated to quantum membranes (2-branes) in BF and generalized Chern-Simons theories in a spacetime of dimension D=5. The creation operators for quantum excitations carrying topological charge are also obtained in the same theories. For the case of D=5 generalized Chern-Simons theory, we show that this operator actually creates an open string with a topological charge at its tip. It is shown that a duality structure exists in general, relating the membrane and topological excitation operators and the correspo…
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-abelian U-duality originates from the construction of exceptional Drinfel'd algebra (EDA), which extends the constriction of the classical Drinfel'd double. This symmetry is a natural extension of Poisson--Lie T-duality and is believed to be a symmetry of Type II string/M-theory or their low-energy effective theories. In this paper, we consider non-abelian U-dualities of 11- or 10-dimensional backgrounds starting with E${}_{n(n)}$ EDA with $n\leq 6$ with vanishing trombone gauging. The latter guarantees that all dual backgrounds satisfy the standard supergravity equations of motion. In par…
We present a brief review on the canonical transformation description of some duality symmetries in string and gauge theories. In particular, we consider abelian and non-abelian T-dualities in closed and open string theories as well as S-duality in abelian and non-abelian non-supersymmetric gauge theories.
For a general class of SO(4) symmetric backgrounds in type II-supergravity, we show that the action of non-Abelian T-duality can be described via consistent truncation to seven dimensional theories with seemingly massive modes. As such, any solution to these theories uplifts to both massive type IIA and IIB supergravities presenting an invertible map between the two. For supersymmetric backgrounds, we show that for spinors transforming under SO(4) the non-Abelian T-duality transformation breaks the original supersymmetry by half. We use these mappings to generate the non-Abelian T-duals of the…
