Using a generalised Noether prescription we are able to extract all the currents and their conservation laws in space dependent shift symmetric theories. Various identities among the currents in the matter sector are found that form the basis for revealing a dual picture when the full interacting theory is considered by coupling to gauge fields. The coupling is achieved in terms of vector fields by adhering to a modified minimal prescription which is also supported by an iterative Noether scheme. Further, this scheme shows that couplings can also be introduced using higher rank tensor gauge fields that have appeared in recent discussions on fractons. We reveal a connection among these descriptions (using vector or tensor fields) through certain duality maps that relate the various fields (gauge, electric and magnetic) in the two cases. A correspondence is established among the Gauss' law, Faraday's law and Ampere's law. Explicit calculations are provided for linear and quadratic shift symmetric lagrangians.
We generalize our previous lattice construction of the abelian bosonization duality in $2+1$ dimensions to the entire web of dualities as well as the $N_f=2$ self-duality, via the lattice implementation of a set of modular transformations in the theory space. The microscopic construction provides explicit operator mappings, and allows the manifestation of some hidden symmetries. It also exposes certain caveats and implicit assumptions beneath the usual application of the modular transformations to generate the web of dualities. Finally, we make brief comments on the non-relativistic limit of t…
We analyze in detail the global symmetries of various (2+1)d quantum field theories and couple them to classical background gauge fields. A proper identification of the global symmetries allows us to consider all non-trivial bundles of those background fields, thus finding more subtle observables. The global symmetries exhibit interesting 't Hooft anomalies. These allow us to constrain the IR behavior of the theories and provide powerful constraints on conjectured dualities.
We extend our exploration of nonstandard continuum quantum field theories in 2+1 dimensions to 3+1 dimensions. These theories exhibit exotic global symmetries, a peculiar spectrum of charged states, unusual gauge symmetries, and surprising dualities. Many of the systems we study have a known lattice construction. In particular, one of them is a known gapless fracton model. The novelty here is in their continuum field theory description. In this paper, we focus on models with a global $U(1)$ symmetry and in a followup paper we will study models with a global $\mathbb{Z}_N$ symmetry.
Following our earlier analyses of nonstandard continuum quantum field theories, we study here gapped systems in 3+1 dimensions, which exhibit fractonic behavior. In particular, we present three dual field theory descriptions of the low-energy physics of the X-cube model. A key aspect of our constructions is the use of discontinuous fields in the continuum field theory. Spacetime is continuous, but the fields are not.
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.
We discuss nonstandard continuum quantum field theories in 2+1 dimensions. They exhibit exotic global symmetries, a subtle spectrum of charged excitations, and dualities similar to dualities of systems in 1+1 dimensions. These continuum models represent the low-energy limits of certain known lattice systems. One key aspect of these continuum field theories is the important role played by discontinuous field configurations. In two companion papers, we will present 3+1-dimensional versions of these systems. In particular, we will discuss continuum quantum field theories of some models of fracton…
The duality symmetries of WZW and coset models are discussed. The exact underlying symmetry responsible for semiclassical duality is identified with the symmetry under affine Weyl transformations. This identification unifies the treatement of duality symmetries and shows that in the compact and unitary case they are exact symmetries of string theory to all orders in $α'$ and in the string coupling constant. Non-compact WZW models and cosets are also discussed. A toy model is analyzed suggesting that duality will not generically be a symmetry.
We study three-dimensional gauge theories based on orthogonal groups. Depending on the global form of the group these theories admit discrete $θ$-parameters, which control the weights in the sum over topologically distinct gauge bundles. We derive level-rank duality for these topological field theories. Our results may also be viewed as level-rank duality for $SO(N)_{K}$ Chern-Simons theory in the presence of background fields for discrete global symmetries. In particular, we include the required counterterms and analysis of the anomalies. We couple our theories to charged matter and determine…
There is substancial overlap with hepth-9211081. More results are presented for duality in the non-compact case. It is argued that duality persists as a symmetry also in that case.
In these proceedings for the First School on Field Theory and Gravitation (Vitória, Brasil), a brief introduction is given to superstring theory and its duality symmetries. This introduction is intended for beginning graduate students with no prior knowledge of string theory.
Target space duality (T duality), which interchanges Kaluza--Klein and winding-mode excitations of the compactified heterotic string, is realized as a symmetry of a world-sheet action. Axion-dilaton duality (S duality), a conjectured nonperturbative SL(2,Z) symmetry of the same theory, plays an analogous role for five-branes. We describe a soliton spectrum possessing both duality symmetries and argue that the theory has an infinite number of dual string descriptions.
Duality symmetries for strings moving in non-trivial spacetime backgrounds are analysed. It is shown that for backgrounds generated from WZW and coset CFT models such duality symmetries are exact to all orders in string perturbation theory. Their implications for string dynamics in non-trivial/singular spacetimes are discussed. (Talk given at the EPS 93 Conference, held at Marseille, July 22-27. To appear in the Proceedings.)
This is a general introduction to duality in field theories. The existence and breaking of global symmetries is used as a guideline to systematically prove duality between different field theories. Systems discussed include abelian and non-abelian T-duality in string theory, abelian and nonabelian bosonization, and duality for massless and massive antisymmetric tensor field theories in arbitrary number of dimensions. Open questions regarding these techniques are also discussed. (Lectures given at 33rd Karpacz Winter School `Duality: Strings and Fields' .)
Many two-dimensional classical field theories have hidden symmetries that form an infinite-dimensional algebra. For those examples that correspond to effective descriptions of compactified superstring theories, the duality group is expected to be a large discrete subgroup of the hidden symmetry group. With this motivation, we explore the hidden symmetries of principal chiral models and symmetric space models.
Spontaneous breaking of non-invertible symmetries and duality to beyond-Landau transitions
Institute for Quantum Information and Matter, California Institute of Technology（加州理工学院量子信息物质研究所）；Department of Physics, California Institute of Technology（加州理工学院物理系）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；Department of Physics, Harvard University（哈佛大学物理系）；Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder（科罗拉多大学博尔德分校物理系与量子物质理论中心）；C.N. Yang Institute for Theoretical Physics, Stony Brook University（纽约州立大学石溪分校杨振宁理论物理研究所）
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 discuss the different discrete duality symmetries in six dimensions that act within and between (i) the 10-dimensional heterotic string compactified on $T^4$, (ii) the 10-dimensional Type IIA string compactified on $K3$ and (iii) the 10-dimensional Type IIB string compactified on $K3$. In particular we show that the underlying group-theoretical structure of these discrete duality symmetries is determined by the proper cubic group ${\cal C}/\Z_2$. Our group theoretical interpretation leads to simple rules for constructing the explicit form of the different discrete Type II duality symmetries…
We discuss the duality symmetries of Type II string effective actions in nine, ten and eleven dimensions. As a by-product we give a covariant action underlying the ten--dimensional Type IIB supergravity theory. We apply duality symmetries to construct dyonic Type II string solutions in six dimensions and their reformulation as solutions of the ten--dimensional Type IIB theory in ten dimensions.
Anomalous U(1) gauge symmetries can appear both in heterotic and type I string theories. In the heterotic case we find a single anomalous U(1), while in open string theories several such symmetries can appear. Nonetheless, there is a conjectured duality symmetry that might connect these two theories. We review the properties of anomalous gauge symmetries in various string theories as well as the status of this heterotic-type I/II duality. We also comment on the possible phenomenological applications of anomalous gauge symmetries in string theory.
Recently, a duality web for three dimensional theories with Chern-Simons terms was proposed. This can be derived from a single bosonization type duality, for which various supporting arguments (but not a proof) were given. Here we explicitly derive this bosonization, in the Abelian case and for a particular regime of parameters. To do this, we use the particle-vortex duality in combination with a Buscher-like duality (both considered in the regime of low energies). As a corollary, Son's conjectured duality is derived in a somewhat singular limit of vanishing mass.
General conjectures about the SL(2,Z) modular transformation properties of N=4 super-Yang-Mills correlation functions are presented. It is shown how these modular transformation properties arise from the conjectured duality with IIB string theory on AdS_5 x S^5. We discuss in detail a prediction of the AdS duality: that N=4 field theory, in an appropriate limit, must exhibit bonus symmetries, corresponding to the enhanced symmetries of IIB string theory in its supergravity limit.
S-dualities in scale invariant N=2 supersymmetric field theories are derived by embedding those theories in asymptotically free N=2 theories with higher rank gauge groups. S-duality transformations on the coupling of the scale invariant theory follow from global symmetries acting on the Coulomb branch of the higher rank theory. Since these global symmetries are exact in the asymptotically free theory, this shows that S-duality is an exact equivalence of N=2 theories and not just a property of their supersymmetric states.
Duality symmetries for strings moving in non-trivial spacetime backgrounds are analysed. It is shown that, for backgrounds generated from compact WZW and coset models, such duality symmetries are exact to all orders in string perturbation theory. A global treatment of duality symmetries is given, by associating them to the known symmetries of affine current algebras (affine-Weyl group and external automorphisms). It is argued that self-duality symmetries of WZW and coset models generate the duality symmetries of their moduli space. Some remarks are presented, concerning the survival of such sy…
We review the parity anomaly and a duality web in 2+1 dimensions. An odd dimensional non-interacting Dirac fermion theory is not parity invariant at quantum level. We demonstrate the parity anomaly in a three dimensional non-interacting Dirac fermion theory and a one dimensional non-interacting Dirac fermion theory. These theories can generate non-gauge invariant Abelian Chern-Simons terms at a finite temperature through an effective action. The parity anomaly also leads us to study the duality web in 2+1 dimensions, in which the 2+1 dimensional duality web begins from the conjecture of a dual…
We realize the string/$(D-5)$-brane duality on the action level between the $T^{10-D}$-compactified heterotic string effective action and the $(D-5)$- brane effective action in $D$ dimensions by managing a Lagrange multiplier field. A dual dictionary is composed to be available for the translation between the elementary or solitonic solutions of the dual pair of actions. In the same way the $S$ duality is also reconstructed on the action level as a double dualization for the $T^6$-compactified heterotic string effective action.
In this paper we study U(1) gauge transformations on the space-time coordinates and on the background fields $g_{μν}$ and $ϕ$. For some special gauge functions, gauged coordinates and gauged U(1) field are equivalent to the rotated coordinates and rotated gauge field. We find gauge transformations that are symmetries of the string action. Also we obtain general $T$-duality transformations for the background fields. For special background fields this duality is equivalent to a gauge transformation.
We derive the duality symmetries relevant to moduli dependent gauge coupling constant threshold corrections, in Coxeter $ {\bf Z_N} $ orbifolds. We consider those orbifolds for which the point group leaves fixed a 2-dimensional sublattice $Λ_2$, of the six dimensional torus lattice $Λ_6$, where $Λ_6 $ cannot be decomposed as $Λ_2 \bigoplus{Λ_4}.$
We construct a web of non-supersymmetric dualities in four spacetime dimensions for theories with theta and Maxwell terms. Our construction mirrors the recipe used in (2+1)-dimensions to demonstrate a similar duality for theories with Chern-Simons terms. As in the 3-dimensional case, the web consists of boson-boson, fermion-fermion and boson-fermion dualities, any one of which can be taken as a seed for the remaining two. To be concrete, we begin by constructing the basic bosonization duality which we then use to generate the web, using an analog of the 3-dimensional massive construction given…
A way of covariantizing duality symmetric actions is proposed. As examples considered are a manifestly space-time invariant duality--symmetric action for abelian gauge fields coupled to axion-dilaton fields and gravity in D=4, and a Lorentz-invariant action for chiral bosons in D=2. The latter is shown to admit a manifestly supersymmetric generalization for describing chiral superfields in n=(1,0) D=2 superspace.
We discuss the spontaneous supersymetry breaking within the low-energy effective supergravity action of four-dimensional superstrings. In particular, we emphasize the non-universality of the soft supersymmetry breaking parameters, the $μ$-problem and the duality symmetries.
On a supersymmetric sigma model the covariantly constant forms are related to the conserved currents that are generators of a super W-algebra extending the superconformal algebra. The existence of covariantly constant forms restricts the holonomy group of the manifold. Via duality transformation we get new covariantly constant forms, thus restricting the holonomy group of the new manifold.
We study higher symmetries and anomalies of 4d $\mathfrak{so}(2n_c)$ gauge theory with $2n_f$ flavors. We find that they depend on the parity of $n_c$ and $n_f$, the global form of the gauge group, and the discrete theta angle. The contribution from the fermions plays a central role in our analysis. Furthermore, our conclusion applies to $\mathcal{N}=1$ supersymmetric cases as well, and we see that higher symmetries and anomalies match across the Intriligator-Seiberg duality between $\mathfrak{so}(2n_c)\leftrightarrow\mathfrak{so}(2n_f-2n_c+4)$.
This paper serves to elucidate the nature of toric duality dubbed in hep-th/0003085 in the construction for world volume theories of D-branes probing arbitrary toric singularities. This duality will be seen to be due to certain permutation symmetries of multiplicities in the gauged linear sigma model fields. To this symmetry we shall refer as ``multiplicity symmetry.'' We present beautiful combinatorial properties of these multiplicities and rederive all known cases of torically dual theories under this new light. We also initiate an understanding of why such multiplicity symmetry naturally le…
The evolution of a closed bosonic string is envisaged in the time-dependent background of its massless modes. A duality transformation is implemented on the spatial component of string coordinates to obtain a dual string. It is shown that the evolution equations are manifestly $O(d,d)$ invariant. The tree level string effective actions for the original and the dual string theory are shown to be equivalent.
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 derive the period structure of several one-modulus Calabi-Yau manifolds. With this knowledge we then obtain the generators of the duality group and the mirror map that defines the physical variable $t$ representing the radius of compactification. We also describe the fundamental region of $t$ and discuss its relation with automorphic functions. As a byproduct of our analysis we compute the non-perturbative corrections of Yukawa couplings.
The matrix theory description of the discrete light cone quantization of $M$ theory on a $T^{2}$ is studied. In terms of its super Yang- Mills description, we identify symmetries of the equations of motion corresponding to independent rescalings of one of the world sheet light cone coordinates, which show how the $S$ duality of Type IIB string theory is realized as a Nahm-type transformation. In the $M$ theory description this corresponds to a simple $9-11$ flip.
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.
In ${\cal N}=5, 6, 8$ supergravities there are hidden symmetries of equations of motion, described by duality groups $SU(1,5), \, SO^*(12), \, E_{7(7)}$ respectively. UV divergences and known candidate counterterms violate the deformed duality symmetry current conservation. Extra higher derivative terms in the action are required to restore duality. We study the effect of a two-vector part of the counterterm for ${\cal N}\geq 5$ supergravities using the universality of the symplectic structure of extended supergravities. We construct a compact form of a deformed action with infinite number of …
We study various duality webs involving the 3d FT[SU(N)] theory, a close relative of the T[SU(N)] quiver tail. We first map the partition functions of FT[SU(N)] and its 3d spectral dual to a pair of spectral dual q-Toda conformal blocks. Then we show how to obtain the FT[SU(N)] partition function by Higgsing a 5d linear quiver gauge theory, or equivalently from the refined topological string partition function on a certain toric Calabi-Yau three-fold. 3d spectral duality in this context descends from 5d spectral duality. Finally we discuss the 2d reduction of the 3d spectral dual pair and stud…
Following hep-th/0412336 we use the non-linear realisation of the semi-direct product of E11 and its vector representation to construct brane dynamics. The brane moves through a spacetime which arises in the non-linear realisation from the vector representation and it contains the usual embedding coordinates as well as the world volume fields. The resulting equations of motion are first order in derivatives and can be thought of as duality relations. Each brane carries the full E11 symmetry and so the Cremmer-Julia duality symmetries. We apply this theory to find the dynamics of the IIA and II…
We revisit universal features of duality in linear and nonlinear relativistic scalar and Abelian 1-form theories with single or multiple fields, which exhibit ordinary or generalized global symmetries. We show that such global symmetries can be interpreted as generalized Killing isometries on a suitable, possibly graded, target space of fields or its jet space when the theory contains higher derivatives. This is realized via a generalized sigma model perspective motivated from the fact that higher spin particles can be Nambu-Goldstone bosons of spontaneously broken generalized global symmetrie…
We study the $m$-graded quiver theories associated to CY $(m+2)$-folds and their order $(m+1)$ dualities. We investigate how monodromies give rise to mutation invariants, which in turn can be formulated as Diophantine equations characterizing the space of dual theories associated to a given geometry. We discuss these ideas in general and illustrate them in the case of orbifold theories. Interestingly, we observe that even in this simple context the corresponding Diophantine equations may admit an infinite number of seeds for $m\geq 2$, which translates into an infinite number of disconnected d…
We study various dualities in condensed matter systems. The dualities in three dimensions could be derived from a conjecture of a duality between a Dirac fermion theory and an interacting scalar field theory at the Wilson-Fisher fixed point and zero temperature in three dimensions. We show that the dualities are not affected by the non-trivial holonomy, use a mean-field method to study, and discuss the dualities at a finite temperature. Finally, we combine a bulk theory, which is an Abelian $p$-form theory with a theta term in $2p+2$ dimensions, and a boundary theory, which is a $2p+1$ dimensi…
We consider a space-time invariant duality symmetric action for a free Maxwell field and an $SL(2,{\bf R})\times SO(6,22)$ invariant effective action describing a low-energy bosonic sector of the heterotic string compactified on a six-dimensional torus. The manifest Lorentz and general coordinate invariant formulation of the models is achieved by coupling dual gauge fields to an auxiliary vector field from an axionic sector of the theory.
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.
The Homogeneous sine-Gordon (HSG) theories are integrable perturbations of $G_k/U(1)^{r_G}$ coset CFTs, where $G$ is a simple compact Lie group of rank $r_G$ and $k>1$ is an integer. Using their T-duality symmetries, we investigate the relationship between the different theories corresponding to a given coset, and between the different phases of a particular theory. Our results suggest that for $G=SU(n)$ with $n\geq5$ and $E_6$ there could be two non-equivalent HSG theories associated to the same coset, one of which has not been considered so far.
Duality between the `axion' field $a$ and the antisymmetric tensor field $B_{μν}$ is traced after a nonperturbative effect, gaugino condensation, breaks the Peccei-Quinn symmetry $a \rightarrow a +c$. Even though the PQ symmetry was at its origin, duality is nevertheless {\it not} broken by this effect. Below condensation scale, the axion simply gets a mass, but in the `stringy' version, the $ B_{μν}$ field disappears from the propagating spectrum. Its place is taken by a massive 3-index antisymmetric field $H_{μνρ}$ which is the one dual to the massive axion. This is a particular case of a ge…
We show that the non-linear realisations of all the very extended algebras G^{+++}, except the B and C series which we do not consider, contain fields corresponding to all possible duality symmetries of the on-shell degrees of freedom of these theories. This result also holds for G_2^{+++} and we argue that the non-linear realisation of this algebra accounts precisely for the form fields present in the corresponding supersymmetric theory. We also find a simple necessary condition for the roots to belong to a G^{+++} algebra.
