We bootstrap the symbol of the maximal-helicity-violating four-particle form factor for the chiral part of the stress-tensor supermultiplet in planar $\mathcal{N}=4$ super-Yang-Mills theory at two loops. When minimally normalized, this symbol involves only 34 letters and obeys the extended Steinmann relations in all partially-overlapping three-particle momentum channels. In addition, the remainder function for this form factor exhibits an antipodal self-duality: it is invariant under the combined operation of the antipodal map defined on multiple polylogarithms -- which reverses the order of the symbol letters -- and a simple kinematic map. This self-duality holds on a four-dimensional parity-preserving kinematic hypersurface. It implies the antipodal duality recently noticed between the three-particle form factor and the six-particle amplitude in this theory.
We observe that the three-gluon form factor of the chiral part of the stress-tensor multiplet in planar $\mathcal{N}=4$ super-Yang-Mills theory is dual to the six-gluon MHV amplitude on its parity-preserving surface. Up to a simple variable substitution, the map between these two quantities is given by the antipode operation defined on polylogarithms (as part of their Hopf algebra structure), which acts at symbol level by reversing the order of letters in each term. We provide evidence for this duality through seven loops.
Form Factor Alphabets and Antipodal Duality from Cluster Algebras
New Cornerstone Laboratory, Institute of Theoretical Physics, Chinese Academy of Sciences（新Cornerstone实验室，中国科学院理论物理研究所）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS & ICTP-AP（基础物理与数学学院，杭州高等研究院，UCAS & ICTP-AP）；School of Physical Sciences, University of Chinese Academy of Sciences（物理科学学院，中国科学院大学）
Antipodal Self-Duality for Generalized Fishnets from Integrability
SLAC National Accelerator Laboratory, Stanford University（斯坦福大学）；Universität Bonn（波恩大学）；Stanford University（斯坦福大学）；University of Oxford（牛津大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
Loop-tree duality allows to express virtual contributions in terms of phase-space integrals, thus leading to a direct comparison with real radiation terms. In this talk, we review the basis of the method and describe its application to regularize Feynman integrals. Performing an integrand-level combination of real and virtual terms, we obtain finite contributions that can be computed in four-dimensions. Moreover, this method provides a natural physical interpretation of infrared singularities, their origin and the way that they cancel in the complete computation.
We extend useful properties of the $H\toγγ$ unintegrated dual amplitudes from one- to two-loop level, using the Loop-Tree Duality formalism. In particular, we show that the universality of the functional form -- regardless of the nature of the internal particle -- still holds at this order. We also present an algorithmic way to renormalise two-loop amplitudes, by locally cancelling the ultraviolet singularities at integrand level, thus allowing a full four-dimensional numerical implementation of the method. Our results are compared with analytic expressions already available in the literature,…
In this paper the duality correspondence between fermion-antifermion and difermion interaction channels is established in two (2+1)-dimensional Gross-Neveu type models with a fermion number chemical potential $μ$ and a chiral chemical potential $μ_5$. The role and influence of this property on the phase structure of the models are investigated. In particular, it is shown that the chemical potential $μ_5$ promotes the appearance of dynamical chiral symmetry breaking, whereas the chemical potential $μ$ contributes to the emergence of superconductivity.
We elaborate on the color-kinematics duality for off-shell diagrams in gauge theories coupled to matter, by investigating the scattering process $gg\to ss, q\bar q, gg$, and show that the Jacobi relations for the kinematic numerators of off-shell diagrams, built with Feynman rules in axial gauge, reduce to a color-kinematics violating term due to the contributions of sub-graphs only. Such anomaly vanishes when the four particles connected by the Jacobi relation are on their mass shell with vanishing squared momenta, being either external or cut particles, where the validity of the color-kinema…
We study the Seiberg dualities with an adjoint matter for the $U(N)$ and $SU(N)$ gauge groups in three- and four-dimensions with four supercharges. The relation between three- and four-dimensional dualities is investigated. We especially derive the three-dimensional duality from four-dimensional one by the dimensional reduction including the non-perturbative effect of the $\mathbb{S}^1$-compactification. In the $U(N)$ case, we obtain the Kim-Park duality, which is known as a generalization of the Aharony duality including an adjoint matter. In the $SU(N)$ case, we obtain the duality which foll…
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.
After reviewing briefly the classical examples of duality in four dimensional field theory we present a generalisation to arbitrary dimensions and to p-form fields. Then we explain how U-duality may become part of a larger non abelian V-symmetry in superstring/supergravity theories. And finally we discuss two new results for 4d gravity theory with a cosmological constant: a new exact gravitational instanton equation and a surprizing linearized classical duality around de Sitter space.
We calculate the Hamiltonian of a compactified D4-brane, with general fluxes and moduli, and find the BPS-mass. The results are invariant under the complete U-duality SO(5,5,Z).
We transform, by means of a fiberwise duality, the partition function of QCD on a product of two two-tori, into a four-dimensional sigma-model, whose target space is the cotangent space of unitary connections on the fiber torus fiberwise.
We study heterotic vacua with four supercharges in three and four space-time dimensions and their duals obtained as M/F-theory compactified on Calabi-Yau fourfolds. We focus on their respective moduli spaces and derive the Kahler potential for heterotic vacua obtained as circle compactifications of four-dimensional N=1 heterotic theories. The Kahler potential of the dual theory is computed by compactifying 11-dimensional supergravity on Calabi-Yau fourfolds. The duality between these theories is checked for K3-fibred fourfolds and an appropriate F-theory limit is discussed.
We present supersymmetric soliton solutions of the four-dimensional heterotic string corresponding to monopoles, strings and domain walls. These solutions admit the $D=10$ interpretation of a fivebrane wrapped around $5$, $4$ or $3$ of the $6$ toroidally compactified dimensions and are arguably exact to all orders in $α'$. The solitonic string solution exhibits an $SL(2,Z)$ {\it strong/weak coupling} duality which however corresponds to an $SL(2,Z)$ {\it target space} duality of the fundamental string.
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.
We show that a necessary condition, for the partition function of four-dimensional Yang-Mills theory to satisfy a S-duality property, is that certain functional determinants, generated by the dual change of variables, cancel each other. This result holds up to non-topological boundary terms in the dual action and modulo the problem of field-strength copies for the Bianchi identity constraint.
The AGT relations reduce S-duality to the modular transformations of conformal blocks. It was recently conjectured that for the four-point conformal block the modular transform up to the non-perturbative contributions can be written in form of the ordinary Fourier transform when $β\equiv-ε_1/ε_2=1$. Here we extend this conjecture to general values of $ε_1,ε_2$. Namely, we argue that for a properly normalized four-point conformal block the S-duality is perturbatively given by the Fourier transform for arbitrary values of the deformation parameters $ε_1,ε_2$. The conjecture is based on explicit …
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…
Four-point one-loop nonsupersymmetric pure Yang-Mills amplitudes with the duality between color and kinematics manifest have been constructed in previous work. Here, we extend the discussion to fermions and scalars circulating in the loop with all external gluons. This gives another nontrivial loop-level example showing that the duality between color and kinematics holds in nonsupersymmetric gauge theory. The construction is valid in any spacetime dimension and written in terms of formal polarization vectors. We also convert these expressions into a four-dimensional form with explicit external…
Self-duality equations for Yang-Mills fields in d-dimensional Euclidean spaces consist of linear algebraic relations amongst the components of the curvature tensor which imply the Yang-Mills equations. For the extension to superspace gauge fields, the super self-duality equations are investigated, namely, systems of linear algebraic relations on the components of the supercurvature, which imply the self-duality equations on the even part of superspace. A group theory based algorithm for finding such systems is developed. Representative examples in various dimensions are provided, including the…
We examine the proposal that the dimensional reduction of the effective action of perturbative string theory on a circle, should be invariant under T-duality transformations. The T-duality transformations are the standard Buscher rules plus some higher covariant derivatives. By explicit calculations at order $α'$ for metric, dilaton and B-field, we show that the T-duality constraint can fix both the effective action and the higher derivative corrections to the Buscher rules up to an overall factor. The corrections depend on the scheme that one uses for the effective action. We have found the e…
Recently, developments in the understanding of low-energy N=1 supersymmetric gauge theory have revealed two important phenomena: the appearance of new four-dimensional superconformal field theories and a non-Abelian generalization of electric-magnetic duality at the IR fixed point. This report is a pedagogical introduction to these phenomena. After presenting some necessary background material, a detailed introduction to the low-energy non-perturbative dynamics of N=1 supersymmetric $SU(N_c)$ QCD is given. The emergence of new four-dimensional superconformal field theories and non-Abelian dual…
We investigate the connections between four-dimensional, N=2 M-theory vacua constructed as orbifolds of type II, heterotic, and type I strings. All these models have the same massless spectrum, which contains an equal number of vector multiplets and hypermultiplets, with a gauge group of the maximal rank allowed in a perturbative heterotic string construction. We find evidence for duality between two type I compactifications recently proposed and a new heterotic construction that we present here. This duality allows us to gain insight into the non-perturbative properties of these models. In pa…
We reduce the dual version of $D=10$, $N=1$ supergravity coupled to $n$ vector fields to four dimensions, and derive the $SL(2,R)\times O(6,6+n)$ transformations which leave the equations of motion invariant. For $n=0$ $SL(2,R)$ is also a symmetry of the action, but for $n>0$ only those $SL(2,R)$ transformations which act linearly on all fields leave the action invariant. The resulting four-dimensional theory is related to the bosonic part of the usual formulation of $N=4$ supergravity coupled to matter by a duality transformation.
We generalize our picture in [arXiv:0904.1744], and consider a pure abelian gauge theory on a four-manifold with nonlocal operators of every codimension arbitrarily and simultaneously inserted. We explicitly show that (i) the theory enjoys exact S-duality for certain choices of operator parameters; (ii) if there are only trivially-embedded surface operators and Wilson loop operators, or if there are only Wilson-'t Hooft loop operators, the theory enjoys a more general and exact SL(2,Z) or Γ_0(2) duality; (iii) the parameters of the loop and surface operators transform like electric-magnetic ch…
A general outlook is presented on the study of multiloop topologies appearing for the first time at four loops. A unified description and representation of this family is provided, the so-called N$^4$MLT universal topology. Based on the Loop-Tree Duality framework, we discuss the dual opening of this family and expose the relevance of a causal representation. We explore an alternative procedure for the search of causal singular configurations of selected N$^4$MLT Feynman diagrams through the application of a modified Grover's quantum algorithm.
We discuss the structure of heterotic/type II duality in four dimensions as a consequence of string-string duality in six dimensions. We emphasize the new features in four dimensions which go beyond the six dimensional vacuum structure and pertain to the way particular K3 fibers can be embedded in Calabi-Yau threefolds. Our focus is on hypersurfaces as well as complete intersections of codimension two which arise via conifold transitions.
We discuss type I -- heterotic duality in four-dimensional models obtained as a Coulomb phase of the six-dimensional U(16) orientifold model compactified on T^2 with arbitrary SU(16) Wilson lines. We show that Kahler potentials, gauge threshold corrections and the infinite tower of higher derivative F-terms agree in the limit that corresponds to weak coupling, large T^2 heterotic compactifications. On the type I side, all these quantities are completely determined by the spectrum of N=2 BPS states that originate from D=6 massless superstring modes.
Montonen and Olive's conjecture in 1977 that some theories possess a duality symmetry that interchanges the electrically charged particle with that of negatively charged t' Hooft-Polyakov monopoles which relates strong coupling to weak coupling. This will be nearly the same as Shapere et al, who showed that the equation of motion of the coupled Einstein-Maxwell-axion dilaton system, occurs in low energy in String theory, is invariant under electromagnetic duality transformation; that also interchanges the strong and weak coupling limit of the theory. These ideas have been persued in the $SO(6)…
We present several pieces of evidence for strong-weak coupling duality symmetry in the heterotic string theory, compactified on a six dimensional torus. These include symmetry of the 1) low energy effective action, 2) allowed spectrum of electric and magnetic charges in the theory, 3) allowed mass spectrum of particles saturating the Bogomol'nyi bound, and 4) Yukawa couplings between massless neutral particles and massive charged particles saturating the Bogomol'nyi bound. This duality transformation exchanges the electrically charged elementary string excitations with the magnetically charged…
We consider a twisted version of the four-dimensional N=4 supersymmetric Yang-Mills theory with gauge groups SU(2) and SO(3), and bare masses for two of its chiral multiplets, thereby breaking N=4 down to N=2. Using the wall-crossing technique introduced by Moore and Witten within the u-plane approach to twisted topological field theories, we compute the partition function and all the topological correlation functions for the case of simply-connected spin four-manifolds of simple type. By including 't Hooft fluxes, we analyse the properties of the resulting formulae under duality transformatio…
We discuss the relationship between target space modular invariance and discrete gauge symmetries in four-dimensional orbifold-like strings. First we derive the modular transformation properties of various string vertex operators of the massless string fields. Then we find that for supersymmetric compactifications the action of the duality elements, leaving invariant the multicritical points, corresponds to a combination of finite Kähler and gauge transformations. However, those finite gauge transformations are not elements of a remnant discrete gauge symmetry. We suggest that, at least in the…
We provide evidence in favor of the conjectured duality between color and kinematics for the case of nonsupersymmetric pure Yang-Mills amplitudes by constructing a form of the one-loop four-point amplitude of this theory that makes the duality manifest. Our construction is valid in any dimension. We also describe a duality-satisfying representation for the two-loop four-point amplitude with identical four-dimensional external helicities. We use these results to obtain corresponding gravity integrands for a theory containing a graviton, dilaton, and antisymmetric tensor, simply by replacing col…
A naive dimensional reduction of the $N=1, D=10$ supergravity theory that naturally arises in five-brane models is used to determine the rôle of two fields which are basic ingredients of string models: the dilaton and, among the moduli, the breathing mode. It is shown that, under the duality transformation that relates five-branes and strings, these two fields exchange the rôles of 10-dimensional dilaton and radius of the compact manifold. A description of this phenomenon in terms of the linear multiplets of the 4-dimensional supergravity is also presented.
We consider N=1 dualities in four dimensional supersymmetric gauge theories as a geometrical realization of wrapping D 6-branes around 3-cycles of Calabi-Yau threefolds in type IIA string theory. By extending the recent work of Ooguri and Vafa to the case of $SU, SO and Sp$ gauge groups with additional fields together with defining fields, we give simple geometrical descriptions of the interrelation between the electric theory and its magnetic dual in terms of the configuration of D 6-branes wrapped 3-cycles.
We identify the states in string theory which are responsible for the entropy of near-extremal rotating four-dimensional black holes in $N=8$ supergravity. For black holes far from extremality (with no rotation), the Bekenstein-Hawking entropy is exactly matched by a mysterious duality invariant extension of the formulas derived for near-extremal black holes states.
We use the method of stable degenerations to study the local geometry of Calabi-Yau fourfolds for F-theory compactifications dual to heterotic compactifications on a Calabi-Yau threefold with fivebranes wrapping holomorphic curves in the threefold. When fivebranes wrap intersecting curves, or when many fivebranes wrap the same curve, the dual fourfolds degenerate in interesting ways. We find that some of these can be usefully described in terms of degenerations of the base of the elliptic fibrations of these fourfolds. We use Witten's criterion to determine which of the fivebranes can lead to …
Under heterotic/F-theory duality it was argued that a wide class of heterotic five-branes is mapped into the geometry of an F-theory compactification manifold. In four-dimensional compactifications this identifies a five-brane wrapped on a curve in the base of an elliptically fibered Calabi-Yau threefold with a specific F-theory Calabi-Yau fourfold containing the blow-up of the five-brane curve. We argue that this duality can be reformulated by first constructing a non-Calabi-Yau heterotic threefold by blowing up the curve of the five-brane into a divisor with five-brane flux. Employing hetero…
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 …
Using generalised geometry we study the action of U-duality acting in three and four dimensions on the bosonic fields of eleven dimensional supergravity. We compare the U-duality symmetry with the T-duality symmetry of double field theory and see how the $SL(2)\otimes SL(3)$ and SL(5) U-duality groups reduce to the SO(2,2) and SO(3,3) T-duality symmetry groups of the type IIA theory. As examples we dualise M2-branes, both black and extreme. We find that uncharged black M2-branes become charged under U-duality, generalising the Harrison transformation, while extreme M2-branes will become new ex…
We study N=1 dualities in four dimensional supersymmetric gauge theories in terms of wrapping D 6-branes around 3-cycles of Calabi-Yau threefolds in type IIA string theory. We generalize the recent work of geometrical realization for the models which have the superpotential corresponding to an $A_k$ type singularity, to various models presented by Brodie and Strassler, consisting of $D_{k+2}$ superpotential of the form $W=Tr X^{k+1} + Tr XY^2$. We discuss a large number of representations for the field $Y$, but with $X$ always in the adjoint (symmetric) [antisymmetric] representation for $SU (…
We review the approach to the geometric Langlands program for algebraic curves via S-duality of an N=4 supersymmetric four dimensional gauge theory, initiated by Kapustin and Witten in 2006. We sketch some of the central further developments. Placing this four dimensional gauge theory into a six dimensional framework, as advocated by Witten, holds the promise to lead to a formulation which makes geometric Langlands duality a manifest symmetry (like coavariance in differential geometry). Furthermore, it leads to an approach toward geometric Langlands duality for algebraic surfaces, reproducing …
Using a six-orientifold, fourbranes and four fivebranes in type IIA string theory we construct $\mathcal{N}$=1 supersymmetric gauge theories in four dimensions with product group $SU(M)\times SO(N)$ or $SU(M)\times Sp(2N)$, a bifundamental flavor and quarks. We obtain the Seiberg dual for these theories and rederive it via branes. To obtain the complete dual group via branes we have to add semi-infinite fourbranes. We propose that the theory derived from branes has a meson deformation switched on. This deformation implies higgsing in the dual theory. The addition of the semi-infinite fourbrane…
We show how the supersymmetric KdV equation can be obtained from the self-duality condition on Yang-Mills fields in four dimension associated with the graded Lie algebra OSp(2/1). We also obtain the hierarchy of Susy KdV equations from such a condition. We formulate the Susy KdV hierarchy as a vanishing curvature condition associated with the U(1) group and show how an Abelian self-duality condition in four dimension can also lead to these equations.
We study, as hypersurfaces in toric varieties, elliptic Calabi-Yau fourfolds for F-theory compactifications dual to E8xE8 heterotic strings compactified to four dimensions on elliptic Calabi-Yau threefolds with some choice of vector bundle. We describe how to read off the vector bundle data for the heterotic compactification from the toric data of the fourfold. This map allows us to construct, for example, Calabi-Yau fourfolds corresponding to three generation models with unbroken GUT groups. We also find that the geometry of the Calabi-Yau fourfold restricts the heterotic vector bundle data i…
We consider configurations of rotated NS-branes leading to a family of four-dimensional N=1 super-QCD theories, interpolating between four-dimensional analogues of the Hanany-Witten vacua, and the Elitzur-Giveon-Kutasov configuration for N=1 duality. The rotation angle is the N=2 breaking parameter, the mass of the adjoint scalar in the N=2 vector multiplet. We add some comments on the relevance of these configurations as possible stringy proofs of N=1 duality.
Construction of integrable field theories in space with a boundary is extended to fermionic models. We obtain general forms of boundary interactions consistent with integrability of the massive Thirring model and study the duality equivalence of the MT model and the sine-Gordon model with boundary terms. We find a variety of integrable boundary interactions in the $O(3)$ Gross-Neveu model from the boundary supersymmetric sine-Gordon theory by using boson-fermion duality.
Some recent results on the applications of duality (and related) transformations to general four-dimensional, spherically symmetric, asymptotically flat and time-independent string configurations are summarized. Two classes of results have been obtained. First, these transformations are used to generate the general such solution to the lowest-order field equations in the alpha' expansion. Second, the action and implications of duality (based on time-translation) on the general configuration is determined. It is found to interchange two pairs of the six parameters which label these configuratio…
We overview some attempts to find S-duality analogues of non-supersymmetric Yang-Mills theory, in the context of gravity theories. The case of MacDowell-Mansouri gauge theory of gravity is discussed. Three-dimensional dimensional reductions from the topological gravitational sector in four dimensions, enable to recuperate the 2+1 Chern-Simons gravity and the corresponding S-dual theory, from the notion of self-duality in the four-dimensional theory.
From the underlying non-linear realisation we compute the complete E11 invariant equations of motion in eleven dimensions, at the linearised level, up to and including level four in the fields. Thus we include the metric, the three and six forms, the dual graviton and three fields at level four. The fields are linked by a set of duality equations, which are first order in derivatives and transform into each other under the E11 symmetries. From these duality relations we deduce second order equations of motion, including those for the usual supergravity fields. As a result the on-shell degrees …
We compute the partition function of four-dimensional abelian gauge theory on a general four-torus T4 with flat metric using Dirac quantization. In addition to an SL(4, Z) symmetry, it possesses SL(2,Z) symmetry that is electromagnetic S-duality. We show explicitly how this SL(2, Z) S-duality of the 4d abelian gauge theory has its origin in symmetries of the 6d (2,0) tensor theory, by computing the partition function of a single fivebrane compactified on T2 x T4, which has SL(2,Z) x SL(4,Z) symmetry. If we identify the couplings of the abelian gauge theory τ= {θ\over 2π} + i{4π\over e^2} with …
We investigate the electromagnetic duality properties of an abelian gauge theory on a compact oriented four-manifold by analysing the behaviour of a generalised partition function under modular transformations of the dimensionless coupling constants. The true partition function is invariant under the full modular group but the generalised partition function exhibits more complicated behaviour depending on topological properties of the four-manifold concerned. It is already known that there may be "modular weights" which are linear combinations of the Euler number and Hirzebruch signature of th…
It has recently pointed out that a four-dimensional analog of Chern-Simons theory provides an elegant framework for understanding integrable models with spectral parameters. The goal of this short note is to better understand the relation of this theory to the more standard three-dimensional Chern-Simons theory. We point out that two Chern-Simons theories, in four dimensions and three dimensions, are related by a novel T-duality in field theory. We then discuss this T-duality in string theory. Our T-duality prescription applies to a more general class of topological quantum field theories, pro…
