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.
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 show how the T-duality is realized for D-branes with noncommutative world-volume coordinates. We discuss D-branes wrapped on tori and the result is that the recently found noncommutative actions form a consistent collection due to the T-duality mapping between noncommutative D-branes and rotated commutative D-branes on deformed tori.
We study the generalization of S-duality to non-commutative gauge theories. For rank one theories, we obtain the leading terms of the dual theory by Legendre transforming the Lagrangian of the non-commutative theory expressed in terms of a commutative gauge field. The dual description is weakly coupled when the original theory is strongly coupled if we appropriately scale the non-commutativity parameter. However, the dual theory appears to be non-commutative in space-time when the original theory is non-commutative in space. This suggests that locality in time for non-commutative theories is a…
We generalise the electric-magnetic duality in standard Maxwell theory to its non-commutative version. Both space-space and space-time non-commutativity are necessary. The duality symmetry is then extended to a general class of non-commutative gauge theories that goes beyond non-commutative electrodynamics. As an application of this symmetry, plane wave solutions are analysed. Dispersion relations following from these solutions show that general non-commutative gauge theories other than electrodynamics admits two waves with distinct polarisations propagating at different velocities in the same…
We first perform bosonic T-duality transformation on one of the marginal TsT (T-duality, shift, T-duality)-deformed $AdS_5\times S_5$ spacetime, which corresponds to 4D $\mathcal{N}=4$ non-commutative super Yang-Mills theory (NCSYM). We then construct the solution to killing spinor equations of the resulting background, and perform the fermionic T-duality transformation. The final dual geometry becomes the usual $AdS_5\times S_5$ but with the constant NS-NS B-field depending on the non-commutative parameter. As applications, we study the gluon scattering amplitude and open string (Wilson loop)…
We investigate the complete phase diagram of the decoupled world-sheet theory of (P,Q) strings. These theories include 1+1 dimensional super Yang-Mills theory and non-commutative open string theory. We find that the system exhibits a rich fractal phase structure, including a cascade of alternating supergravity, gauge theory, and matrix string theory phases. The cascade proceeds via a series of SL(2,Z) S-duality transformations, and depends sensitively on P and Q. In particular, we find that the system may undergo multiple Hagedorn-type transitions as the temperature is varied.
We study duality transformations for two-dimensional sigma models with abelian chiral isometries and prove that generic such transformations are equivalent to integrated marginal perturbations by bilinears in the chiral currents, thus confirming a recent conjecture by Hassan and Sen formulated in the context of Wess-Zumino-Witten models. Specific duality transformations instead give rise to coset models plus free bosons.
We study issues of duality and dual equivalence in non-commutative manifolds. In particular the question of dual equivalence for the actions of the non-commutative extensions of the self-dual model (NC-SD) in 3D space-time and the Maxwell-Chern-Simons model (MCS-SD) is investigate. We show that former model {\it is not} dual equivalent the non-commutative extension of the Maxwell-Chern-Simons model, as widely believed, but a to deformed version of it that is disclosed here. Our results are not restrict to any finite order in the Seiberg-Witten expansion involving the non-commutative parameter …
We review and extend recent work on the application of the non-commutative geometric framework to an interpretation of the moduli space of vacua of certain deformations of N=4 super Yang-Mills theories. We present a simple worldsheet calculation that reproduces the field theory results and sheds some light on the dynamics of the D-brane bubbles. Different regions of moduli space are associated with D5-branes of various topologies; singularities in the moduli space are associated with topology change. T-duality on toroidal topologies maps between mirror string realizations of the field theory.
Via T-duality a theory of open strings on a D1-brane wrapped along a cycle of slanted torus is described by a U(1) gauge theory on a D2-brane in the B-field background. It is also known that there is another dual description of the D1-brane configuration by a non-commutative gauge theory on a D2-brane. Therefore, these two gauge theories on D2-branes are equivalent. Recently, the existence of a continuous set of equivalent gauge theories including these two was suggested. We give a dual D1-brane configuration for each theory in this set, and show that the relation among parameters for equivale…
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.
We use semiclassical method to study closed strings in the modified AdS_5*S^5 background with constant B-fields. The point-like closed strings and the streched closed strings rotating around the big circle of S^5 are considered. Quantization of these closed string leads to a time-dependent string spectrum, which we argue to correspond to the RG-flow of the dual noncommutative Yang Mills theory.
We provide some evidence that closed string coordinates will become non-commutative turning on H-field flux background in closed string compactifications. This is in analogy to open string non-commutativity on the world volume of D-branes with B- and F-field background. The class of 3-dimensional backgrounds we are studying are twisted tori (fibrations of a 2-torus over a circle) and the their T-dual H-field, 3-form flux backgrounds (T-folds). The spatial non-commutativity arises due to the non-trivial monodromies of the toroidal Kahler resp. complex structure moduli fields, when going around …
We study four-dimensional U(1) on a non-commutative T^2 with rational Theta. This theory has dual descriptions as ordinary SYM or as NCOS. We identify a set of massive non-interacting KK states in the SYM theory and track them through the various dualities. They appear as stretched strings in the non-commutative U(1) providing another example of the IR/UV mixing in non-commutative field theories. In the NCOS these states appear as D-strings with winding and momentum. They form an unconventional type of 1/4 BPS state with the 3-brane. To obtain a consistent picture of S-duality for compactified…
Applying a master action technique we obtain the dual of the noncommutative Maxwell-Chern-Simons theory. The equivalence between the Maxwell-Chern-Simons theory and the self-dual model in commutative space-time does not survive in the non-commutative setting. We also point out an ambiguity in the Seiberg-Witten map.
The quantum Hall system is known to have two mutually dual Chern-Simons descriptions, one associated with the hydrodynamics of the electron fluid, and another associated with the statistics. Recently, Susskind has made the claim that the hydrodynamic Chern-Simons theory should be considered to have a non-commutative gauge symmetry. The statistical Chern-Simons theory has a perturbative momentum expansion. In this paper, we study this perturbation theory and show that the effective action, although commutative at leading order, is non-commutative. This conclusion is arrived at through a careful…
We show that manifolds of fixed points, which are generated by exactly marginal operators, are common in N=1 supersymmetric gauge theory. We present a unified and simple prescription for identifying these operators, using tools similar to those employed in two-dimensional N=2 supersymmetry. In particular we rely on the work of Shifman and Vainshtein relating the $\bt$-function of the gauge coupling to the anomalous dimensions of the matter fields. Finite N=1 models, which have marginal operators at zero coupling, are easily identified using our approach. The method can also be employed to find…
We show how to construct correlators for the $CFT_1$ which is dual to non-commutative $AdS_2$ ($ncAdS_2$). We do it explicitly for the example of the massless scalar field on Euclidean $ncAdS_2$. $ncAdS_2$ is the quantization of $AdS_2$ that preserves all the isometries. It is described in terms of the unitary irreducible representations, more specifically discrete series representations, of $so(2,1)$. We write down symmetric differential representations for the discrete series, and then map them to functions on the Moyal-Weyl plane. The Moyal-Weyl plane has a large distance limit which can be…
We study one-loop corrections in scalar and gauge field theories on the non-commutative torus. For rational theta, Morita equivalence allows these theories to be reformulated in terms of ordinary theories on a commutative torus with twisted boundary conditions. UV/IR mixing does not lead to singularities, however there can be large corrections. In particular, gauge theories show tachyonic instabilities for some of the modes. We discuss their relevance to spontaneous Z_N x Z_N symmetry breaking in the Morita dual SU(N) theory due to electric flux condensation.
A particular decoupling limit of non-extremal (D1, D3) brane bound state system of type IIB string theory is known to give the gravity dual of space-space non- commutative Yang-Mills (NCYM) theory at finite temperature. We use a string probe in this background to compute the jet quenching parameter in the strongly coupled plasma of hot non-commutative Yang-Mills theory in (3+1)-dimensions from gauge/gravity duality. We give the expressions for the jet quenching parameter for both small and large non-commutativity. For small non-commutativity, we find that the value of the jet quenching paramet…
We show that noncommutative electromagnetism and Dirac-Born-Infeld (DBI) theory with scalar fields are SL(2,R) self-dual when noncommutativity is light-like and we are in the slowly varying field approximation. This follows from SL(2,R) self-duality of the commutative DBI Lagrangian and of its zero slope limit that we study in detail. We study a symmetry of noncommutative static configurations that maps space-noncommutativity into space-time (and light-like) noncommutativity. SL(2,R) duality is thus extended to space-noncommutativity. Via Seiberg-Witten map we study the nontrivial action of th…
We study an exactly marginal deformation of N=4 SUSY Yang-Mills with gauge group U(N) using field theory and string theory methods. The classical theory has a Higgs branch for rational values of the deformation parameter. We argue that the quantum theory also has an S-dual confining branch which cannot be seen classically. The low-energy effective theory on these branches is a six-dimensional non-commutative gauge theory with sixteen supercharges. Confinement of magnetic and electric charges, on the Higgs and confining branches respectively, occurs due to the formation of BPS-saturated strings…
The classical Fourier-Mukai duality establishes an equivalence of categories between the derived categories of sheaves on dual complex tori. In this article we show that this equivalence extends to an equivalence between two dual objects. Both of these are generalized deformations of the complex tori. In one case, a complex torus is deformed formally in a non-commutative direction specified by a holomorphic Poisson structure. In the other, the dual complex torus is deformed in a B-field direction to a formal gerbe. We show these two deformations are Fourier-Mukai equivalent.
We show that the non-commutative $CP^1$ model coupled with Hopf term in 3 dimensions is equivalent to an interacting spin-$s$ theory where the spin $s$ of the dual theory is related to the coefficient of the Hopf term. We use the Seiberg-Witten map in studying this non-commutative duality equivalence, keeping terms to order $θ$ and show that the spin of the dual theory do not get any $θ$ dependant corrections. The map between current correlators show that topological index of the solitons in the non-commutative $CP^1$ model is unaffected by $θ$ where as the Noether charge of the corresponding …
We investigate field theories on the non-commutative torus upon varying theta, the parameter of non-commutativity. We argue that one should think of Morita equivalence as a symmetry of algebras describing the same space rather than of theories living on different spaces (as is T-duality). Then we give arguments why physical observables depend on theta non-continuously.
We study marginal and relevant supersymmetric deformations of the N=4 super-Yang-Mills theory in four dimensions. Our primary innovation is the interpretation of the moduli spaces of vacua of these theories as non-commutative spaces. The construction of these spaces relies on the representation theory of the related quantum algebras, which are obtained from F-term constraints. These field theories are dual to superstring theories propagating on deformations of the AdS_5xS^5 geometry. We study D-branes propagating in these vacua and introduce the appropriate notion of algebraic geometry for non…
We continue the study of the nocommutative $AdS_2 / CFT_1$ correspondence. We extend our previous results obtained for a free massless scalar field to the case of a massive scalar field. Both the free and interacting cases are considered. For both cases it is confirmed that to the leading order in noncommutative corrections the 2- and 3-point correlation functions have the form that is assumed by some (yet unspecified) dual $CFT$. We also argue that there does not exist a map which connects the commutative model to its non-commutative counterpart, and therefore the conformal behaviour of the n…
We construct a nonperturbative regularization for Euclidean noncommutative supersymmetric Yang-Mills theories with four (N= (2,2)), eight (N= (4,4)) and sixteen (N= (8,8)) supercharges in two dimensions. The construction relies on orbifolds with discrete torsion, which allows noncommuting space dimensions to be generated dynamically from zero dimensional matrix model in the deconstruction limit. We also nonperturbatively prove that the twisted topological sectors of ordinary supersymmetric Yang-Mills theory are equivalent to a noncommutative field theory on the topologically trivial sector wit…
Two dimensional Non-Commutative Open String (NCOS) theory, well-defined perturbatively, may also be studied at strong coupling and for large D-string charge by making use of the Holographic duality. We analyze the zero mode dynamics of a closed string in the appropriate background geometry and map the results onto a sector of strongly coupled NCOS dynamics. We find an elaborate classical picture that shares qualitative similarities with the SL(2,R) WZW model. In the quantum problem, we compute propagators and part of the energy spectrum of the theory; the latter involves interesting variations…
Using gauge/gravity duality, we compute jet quenching parameter in confined and deconfined phases of noncommutative Sakai-Sugimoto model. In the confined phase jet quenching parameter is zero and noncommutativity does not affect it. In deconfined phase we find that the leading correction is negative i.e. it reduces the magnitude of the jet quenching parameter as compared to its value in commutative background. Moreover it is seen that the effect of leading correction is more pronounced at high temperatures
We re-examine various issues surrounding the definition of twisted quantum field theories on flat noncommutative spaces. We propose an interpretation based on nonlocal commutative field redefinitions which clarifies previously observed properties such as the formal equivalence of Green's functions in the noncommutative and commutative theories, causality, and the absence of UV/IR mixing. We use these fields to define the functional integral formulation of twisted quantum field theory. We exploit techniques from braided tensor algebra to argue that the twisted Fock space states of these free fi…
We perform canonical quantization of the dissipative Hofstadter model, which has a wide range of applications in condensed matter physics and string theory. The target space duality and the non-commutative algebra developed in string theory are discussed for the model. We show that the target space duality transformation of closed string theory, $O(2,2;R)$, removes the interaction with a uniform magnetic field. The $O(2,2;R)$ dual transformation changes the basis of oscillator operators so that the algebra of the string coordinate operators at the boundary become non-commutative. In the zero t…
Inspired by the renormalizability of the non-commutative Phi^4 model with added oscillator term, we formulate a non-commutative gauge theory, where the oscillator enters as a gauge fixing term in a BRST invariant manner. All propagators turn out to be essentially given by the Mehler kernel and the bilinear part of the action is invariant under the Langmann-Szabo duality. The model is a promising candidate for a renormalizable non-commutative U(1) gauge theory.
Here we demonstrate, firstly, the construction of dualities using the exact renormalization group approach and, secondly, that spatial non-commutativity can emerge as such a duality. This is done in a simple quantum mechanical setting that establishes an exact duality between the commutative and non-commutative quantum Hall systems with harmonic interactions. It is also demonstrated that this link can be understood as a blocking (coarse graining) transformation in time that relates commutative and non-commutative degrees of freedom.
T-duality in M(atrix) theory has been argued to be realized as Morita equivalence in Yang-Mills theory on a non-commutative torus (NCSYM). Even though the two have the same structure group, they differ in their action since Morita equivalence makes crucial use of an additional modulus on the NCSYM side, the constant Abelian magnetic background. In this paper, we reanalyze and clarify the correspondence between M(atrix) theory and NCSYM, and provide two resolutions of this puzzle. In the first of them, the standard map is kept and the extra modulus is ignored, but the anomalous transformation i…
We conjecture the existence of two new non-gravitational six-dimensional string theories, defined as the decoupling limit of NS5-branes in the background of near-critical electrical two- and three-form RR fields. These theories are space-time non-commutative Little String Theories with open branes. The theory with (2,0) supersymmetry has an open membrane in the spectrum and reduces to OM theory at low energies. The theory with (1,1) supersymmetry has an open string in the spectrum and reduces to 5+1 dimensional NCOS theory for weak NCOS coupling and low energies. The theories are shown to be T…
In this paper we discuss non-commutative and non-associative geometries that emerge in the context of non-geometric closed string backgrounds. T-duality and doubled field theory plays an important role in formulating the corresponding effective action for these kind of non-geometric string backgrounds. As we will argue, the emerging non-commutative and non-associative algebras for the closed string (dual) coordinates and (dual) momenta can be mathematically described by a twisted Poisson structure, in closed analogy to the phase space of a point particle moving in the field of a magnetic monop…
We demonstrate how a one parameter family of interacting non-commuting Hamiltonians, which are physically equivalent, can be constructed in non-commutative quantum mechanics. This construction is carried out exactly (to all orders in the non-commutative parameter) and analytically in two dimensions for a free particle and a harmonic oscillator moving in a constant magnetic field. We discuss the significance of the Seiberg-Witten map in this context. It is shown for the harmonic oscillator potential that an approximate duality, valid in the low energy sector, can be constructed between the inte…
We consider the interaction between instantons and anti-instantons in four-dimensional N=4 super-Yang-Mills theory at large N and large 't Hooft coupling as described by D-instantons via AdS/CFT duality. We give an estimate of the strength of the interaction in various regimes. We discuss also the case of Non-Commutative super Yang-Mills theory where the interaction between instantons and anti-instantons can be used as a way to probe the locality properties of the theory in the supergravity picture, without explicit reference to the definition of local operators.
We use the canonical description of T-duality as well as the formulation of T-duality in terms of chiral currents to investigate the geometric and non-geometric faces of closed string backgrounds originating from principal torus bundles with constant H-flux. Employing conformal field theory techniques, the non-commutative and non-associative structures among generalized coordinates in the so called Q-flux and R-flux backgrounds emerge by gauging the Abelian symmetries of an enlarged Rocek-Verlinde sigma-model and projecting the associated chiral currents of the enlarged theory to the T-dual co…
We give a geometric characterization of the quasi axial-vector (Kiritsis-Obers) target space duality in the spirit of the bi-algebra (Klimcik-Severa) approach. We show that the sigma-models constructed by taking quotients have non-abelian chiral currents that obey "non-commutative conservation laws" and provide the criterion for a sigma-model to have a dual using the axial-vector procedure.
We consider electric-magnetic duality(S-duality) in IIB matrix model with a D3-brane background. We propose the duality transformation by considering that of noncommutative Yang-Mills theory(NCYM) in four dimension. NCYM derived from the matrix model has a Yang-Mills coupling related to the noncommutativity of the spacetime. We argue that open strings bits as bi-local fields on the spacetime are decoupled from the bulk in NCOS decoupling limits as it is in string theory approach. We also discuss how our four dimensional spacetime relates to higher, by applying the decoupling and the commutativ…
In this paper we make an SL(2,Z)-covariant generalisation of the noncommutative theories, NCYM and NCOS on the D3-brane, and NCOS on the D5-brane in type IIB. Usually, the noncommutative theories are obtained by studying perturbative F-string theory, and the parameters governing the noncommutative theories are given by the open string data. The S-duality of NCYM and NCOS on the D3-brane has been seen by dualising the background, keeping the F-string under study fixed. We give an SL(2,Z)-covariant generalisation of the open string data relevant when one instead studies perturbative (p,q)-string…
We use the recently derived density of states for a particle confined to a spherical well in three dimensional fuzzy space to compute the thermodynamics of a gas of non-interacting fermions confined to such a well. Special emphasis is placed on non-commutative effects and in particular non-commutative corrections to the thermodynamics at low densities and temperatures are computed where the non-relativistic approximation used here is valid. Non-commutative effects at high densities are also identified, the most prominent being the existence of a minimal volume at which the gas becomes incompre…
We investigate whether the Standard Model, within the accuracy of current experimental measurements, satisfies the Hodge duality condition introduced and studied in Dabrowski, D'Andrea, Sitarz, Lett Math Phys (2018) 108:1323. We show that the neutrino and quark mass-mixing and the difference of fermion masses is necessary for this property. We demonstrate that the current data supports the new geometric feature of the Standard Model, Hodge duality, provided that all neutrinos are massive.
We discuss a new class of strong homotopy algebras constructed via inner deformations. Such deformations have a number of remarkable properties. In the simplest case, every one-parameter family of associative algebras leads to an $L_\infty$-algebra that can be used to construct a classical integrable model. Another application of this class of $L_\infty$-algebras is related with the three-dimensional bosonization duality in Chern--Simons vector models, where it implements the idea of the slightly-broken higher spin symmetry. One large class of associative algebras originates from Deformation Q…
After an introduction to some basic issues in non-commutative geometry (Gel'fand duality, spectral triples), we present a "panoramic view" of the status of our current research program on the use of categorical methods in the setting of A.Connes' non-commutative geometry: morphisms/categories of spectral triples, categorification of Gel'fand duality. We conclude with a summary of the expected applications of "categorical non-commutative geometry" to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.
Penrose limits of inhomogeneous cosmologies admitting two abelian Killing vectors and their abelian T-duals are found in general. The wave profiles of the resulting plane waves are given for particular solutions. Abelian and non-abelian T-duality are used as solution generating techniques. Furthermore, it is found that unlike in the case of abelian T-duality, non-abelian T-duality and taking the Penrose limit are not commutative procedures.
Some properties of the non-commutative versions of the sine-Gordon model (NCSG) and the corresponding massive Thirring theories (NCMT) are studied. Our method relies on the NC extension of integrable models and the master Lagrangian approach to deal with dual theories. The master Lagrangians turn out to be the NC versions of the so-called affine Toda model coupled to matter fields (NCATM) associated to the group GL(2), in which the Toda field belongs to certain representations of either $U(1){x} U(1)$ or $U(1)_{C}$ corresponding to the Lechtenfeld et al. (NCSG$_{1}$) or Grisaru-Penati (NCSG$_{…
Kuznetsov has conjectured that Pfaffian varieties should admit non-commutative crepant resolutions which satisfy his Homological Projective Duality. We prove half the cases of this conjecture, by interpreting and proving a duality of non-abelian gauged linear sigma models proposed by Hori.
We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and non-commutative combinatorics. Via the usual duality between algebra and geometry, these constructions provide categorifications for various types of affine spaces, thus our works may be regarded as a starting point towards the construction of a categorical geometry.
We study the commutative limit of the non-commutative maximally supersymmetric Yang-Mills theory in four dimensions (N=4 SYM). The commutative limits of non-commutative spaces are important in particular in the applications of non-commutative spaces for regularisation of supersymmetric theories (such as the use of non-commutative spaces as alternatives to lattices for supersymmetric gauge theories and interpretations of some matrix models as regularised supermembrane or superstring theories), which in turn can play a prominent role in the study of quantum gravity via the gauge/gravity duality.…
We derive the quiver gauge theory on the world-volume of D3-branes transverse to an L(a,b,c) singularity by computing the endomorphism algebra of a tilting object first constructed by Van den Bergh. The quiver gauge theory can be concisely specified by an embedding of a graph into a face-centered cubic lattice. In this description, planar Seiberg dualities of the gauge theory act by changing the graph embedding. We use this description of Seiberg duality to show these quiver gauge theories possess periodic Seiberg dualities whose existence was expected from the AdS/CFT correspondence.
