We present a geometric derivation of all-multiplicity (single-trace) tree-level Einstein Yang-Mills (EYM) amplitudes ${\cal A}(n;r)$ involving $n$ gluons and $r$ gravitons by a bilinear of two twisted differential forms on the moduli space of Riemann spheres with $m\!:=\!n\!+\!r$ punctures. The differential forms are gained by studying the underlying superstring disk amplitude and proposing an embedding of the disk onto the sphere. This map can be interpreted as a geometrical map from the open superstring to a heterotic or ambitwistor string structure. Then, the twisted intersection number of the two $m$-forms, which is obtained by integrating over the moduli space of Riemann sphere with $m$ punctures, reproduces in the infinite inverse string tension limit $α'\!\rightarrow\! \infty$ the corresponding CHY formula of the EYM amplitude. To bolster our findings we study the disk amplitude of open and closed strings using the Grassmann description of the underlying superstring amplitude, map it to a closed string amplitude and consider the $α'\!\rightarrow\! \infty$ limit. Finally, we present an all-multiplicity decomposition formula of any EYM amplitude ${\cal A}(n;r)$ as linear combination over $(m\!-\!3)!$ pure $m$ gluon subamplitudes.
Solutions to the twisted Yang-Baxter equation, arising from intertwiners for cyclic representations of $U_q(\widehat{sl}_n)$ are described via two coupled the lattice current algebras.
We formulate N=2 twisted super Yang-Mills theory with a gauged central charge by superconnection formalism in two dimensions. We obtain off-shell invariant supermultiplets and actions with and without constraints, which is in contrast with the off-shell invariant D=N=4 super Yang-Mills formulation with unavoidable constraints.
We propose a twisted D=N=2 superspace formalism. The relation between the twisted super charges including the BRST charge, vector and pseudo scalar super charges and the N=2 spinor super charges is established. We claim that this relation is essentially related with the Dirac-Kähler fermion mechanism. We show that a fermionic bilinear form of twisted N=2 chiral and anti-chiral superfields is equivalent to the quantized version of BF theory with the Landau type gauge fixing while a bosonic bilinear form leads to the N=2 Wess-Zumino action. We then construct a Yang-Mills action described by the …
Herein, we consider a topologically twisted version of maximally supersymmetric Yang-Mills theory in five dimensions which was introduced by Witten in 2011. We consider this theory on a five manifold of the form M_4 x I for M_4 an oriented Riemannian four manifold. The complete and unique action of the theory in bulk is written down and is shown to be invariant under two scalar supersymmetries.
We study the twisted N=4 super Yang-Mills theories in the Omega-background with the constant R-symmetry Wilson line gauge field. Based on the classification of topological twists of N=4 supersymmetry (the half, the Vafa-Witten and the Marcus twists), we construct the deformed off-shell supersymmetry associated with the scalar supercharges for these twists. We find that the Omega-deformed action is written in the exact form with respect to the scalar supercharges as in the undeformed case.
We propose a lattice formulation of three dimensional super Yang-Mills model with a twisted N=4 supersymmetry. The extended supersymmetry algebra of all the eight supercharges is fully and exactly realized on the lattice with a modified "Leibniz rule". The formulation we employ here is a three dimensional extension of manifestly gauge covariant method which was developed in our previous proposal of Dirac-Kaehler twisted N=2 super Yang-Mills on two dimensional lattice. The twisted N=4 supersymmetry algebra is geometrically realized on a three dimensional lattice with link supercharges and the u…
Employing a twisted superspace with eight supercharges, we describe an off-shell formulation of N=4 D=3 twisted super Yang-Mills in the continuum spacetime which underlies the recent proposal of N=4 D=3 twisted super Yang-Mills on a lattice (arXiv:0707.3533[hep-lat]). By a dimensional reduction from the N=2 D=4, we explore the two possible topological twists of N=4 D=3 and then show that the lattice formulation given in arXiv:0707.3533[hep-lat] is essentially categorized as the B-type. We also show that, amongst the two inequivalent twists of N=4 D=3, only the B-type SYM can be realized on the…
The generalization of the the Yang-Baxter relation is proposed. In this generalization the spectral parameters of the particles change after the scattering. The corresponding algebraic structures are discussed. The corresponding action of the symmetric group is studied. The most interesting examples of such action are connected with factorization of matrix polynomials and matrix $theta$-functions. The generalization of the star-triangle relation is also proposed.
We construct spectral parameter dependent R-matrices for the quantized enveloping algebras of twisted affine Lie algebras. These give new solutions to the spectral parameter dependent quantum Yang-Baxter equation.
We investigate mass deformation of twisted superalgebra of U(N) super Yang-Mills (SYM) theories in several models and in several dimensions, motivated by the method formulated in [1]. We show that there are several ways to perform the deformation, if a model possesses four scalar supercharges except for two dimensional B-model. We also evaluate classical vacuum solutions of the potentials of scalar fields for each model. We then find that it is always possible to find fuzzy sphere solution in the theories.
A lattice formulation of a three dimensional super Yang-Mills model with a twisted N=4 supersymmetry is proposed. The extended supersymmetry algebra of all eight supercharges is fully and exactly realized on the lattice with a modified "Leibniz rule". The formulation we employ here is a three dimensional extension of the manifestly gauge covariant method which was developed in our previous proposal of Dirac-Kähler twisted N=2 super Yang-Mills on a two dimensional lattice. The twisted N=4 supersymmetry algebra is geometrically realized on a three dimensional lattice with link supercharges and t…
Integrable spin chains in twisted maximally supersymmetric Yang-Mills theory
Instituto Galego de Física de Altas Enerxías (IGFAE) and Departamento de Física de Partículas, Universidade de Santiago de Compostela（加利西亚高能物理研究所和圣地亚哥德孔波斯特拉大学粒子物理系）；Institut für Physik, Humboldt-Universität zu Berlin（柏林洪堡大学物理研究所）
We investigate the large-$N$ asymptotics of the topologically twisted index of $\mathcal N=4$ SU($N$) Super-Yang-Mills (SYM) theory on $T^2\times S^2$ and provide its holographic interpretation based on the black hole Farey tail. In the field theory side, we use the Bethe-Ansatz (BA) formula, which gives the twisted index of $\mathcal N=4$ SYM theory as a discrete sum over Bethe vacua, to compute the large-$N$ asymptotics of the twisted index. In a dual $\mathcal N=2$ gauged STU model, we construct a family of 5d extremal solutions uplifted from the 3d black hole Farey tail, and compute the re…
It is shown that the dimensional reduction of the N_T=2, D=3 Blau-Thompson model to D=2, i.e., the novel topological twist of N=8, D=2 super Yang-Mills theory, provides an example of a Hodge-type cohomological theory. In that theory the generators of the topological shift, co-shift and gauge symmetry, together with a discrete duality operation, are completely analogous to the de Rham cohomology operators and the Hodge *-operation.
A necessary ingredient for our previous results on the form of the long distance effective action of the twisted lattice N=4 super Yang-Mills theory is the existence of a real space renormalization group which preserves the lattice structure, both the symmetries and the geometric interpretation of the fields. In this brief article we provide an explicit example of such a blocking scheme and illustrate its practicality in the context of a small scale Monte Carlo renormalization group calculation. We also discuss the implications of this result, and the possible ways in which to use it in order …
We report on our ongoing computation of the perturbative running of the Yang-Mills coupling using gradient flow techniques. In particular, we use the gradient flow method with twisted boundary conditions to perform a perturbative expansion of the expectation value of the Yang-Mills energy density up to fourth order in the coupling at finite flow time. We regularise the resulting integrals using dimensional regularisation, and reproduce the universal coefficient of the 1/ε term in the relation between bare and renormalised couplings. The computation of the finite part leading to a determination…
We investigate the one-loop spectral problem of $γ$-twisted, planar $\mathcal{N}$=4 Super Yang-Mills theory in the double-scaling limit of infinite, imaginary twist angle and vanishing Yang-Mills coupling constant. This non-unitary model has recently been argued to be a simpler version of full-fledged planar $\mathcal{N}$=4 SYM, while preserving the latter model's conformality and integrability. We are able to derive for a number of sectors one-loop Bethe equations that allow finding anomalous dimensions for various subsets of diagonalizable operators. However, the non-unitarity of these defor…
Twisted compactification of the 6d N=(2,0) theories on a punctured Riemann surface give a large class of 4d N=1 and N=2 gauge theories, called class S. We argue that nonperturbative dynamics of class S theories are described by 5d maximal Super-Yang-Mills (SYM) twisted on the Riemann surface. In a sense, twisted 5d SYM might be regarded as a "Lagrangian" for class S theories on R^{1,2} times S^1. First, we show that twisted 5d SYM gives generalized Hitchin's equations which was proposed recently. Then, we discuss how to identify chiral operators with quantities in twisted 5d SYM. Mesons, or ho…
We characterize the correspondence between the twisted $N=2$ super-Yang-Mills theory and the Baulieu-Singer topological theory quantized in the self-dual Landau gauges. While the first is based on an on-shell supersymmetry, the second is based on an off-shell Becchi-Rouet-Stora-Tyutin symmetry. Because of the equivariant cohomology, the twisted $N=2$ in the ultraviolet regime and Baulieu-Singer theories share the same observables, the Donaldson invariants for 4-manifolds. The triviality of the Gribov copies in the Baulieu-Singer theory in these gauges shows that working in the instanton moduli…
We extend the localization calculation of the 3D Chern-Simons partition function over Seifert manifolds to an analogous calculation in five dimensions. We construct a twisted version of N=1 supersymmetric Yang-Mills theory defined on a circle bundle over a four dimensional symplectic manifold. The notion of contact geometry plays a crucial role in the construction and we suggest a generalization of the instanton equations to five dimensional contact manifolds. Our main result is a calculation of the full perturbative partition function on a five sphere for the twisted supersymmetric Yang-Mills…
The homogeneous Yang-Baxter deformation is part of a larger web of integrable deformations and dualities that recently have been studied with motivations in integrable $σ$-models, solution-generating techniques in supergravity and Double Field Theory, and possible generalisations of the AdS/CFT correspondence. The $σ$-models obtained by the homogeneous Yang-Baxter deformation with periodic boundary conditions on the worldsheet are on-shell equivalent to undeformed models, yet with twisted boundary conditions. While this has been known for some time, the expression provided so far for the twist…
In this letter, we find the first dynamically stable non-singular solution spherically symmetric SU(2) Einstein-Yang-Mills equation. This solutions is regular at r=0 and asymptotically flat. Since the Yang-Mills field strength decay exponentially, the Einstein-Yang-Mills particle perhaps can be used to be a candidate for dark matter.
We study a discretization of ${\cal N}=2$ super Yang-Mills theory which possesses a single exact supersymmetry at non-zero lattice spacing. This supersymmetry arises after a reformulation of the theory in terms of {\it twisted} fields. In this paper we derive the action of the other twisted supersymmetries on the component fields and study, using Monte Carlo simulation, a series of corresponding Ward identities. Our results for SU(2) and SU(3) support a restoration of these additional supersymmetries without fine tuning in the infinite volume continuum limit. Additionally we present evidence s…
We examine the topologically twisted index of $\mathcal N=4$ super-Yang-Mills with gauge group $SU(N)$ on $T^2\times S^2$, and demonstrate that it receives contributions from multiple sectors corresponding to the freely acting orbifolds $T^2/\mathbb Z_m\times\mathbb Z_n$ where $N=mn$. After summing over these sectors, the index can be expressed as the elliptic genus of a two-dimensional $\mathcal N=(0,2)$ theory resulting from Kaluza-Klein reduction on $S^2$. This provides an alternate path to the 'high-temperature' limit of the index, and confirms the connection to the right-moving central ch…
We propose a lattice action for two dimensional super Yang-Mills theory with a twisted N=2 supersymmetry. The extended supersymmetry is fully and exactly realized on the lattice. The method employed is quite general and its extension to the N=4 supersymmetry in four dimensions is briefly presented. The lattice has a new type of ``fermionic'' links, where odd Grassmann variables, supercharges and fermionic connections sit. The Leibniz rule is preserved on the lattice, although in a modified ``shifted'' form that takes into account the link nature of both derivatives and supercharges. Superfield…
We discuss the asymptotic form of the static axially symmetric, globally regular and black hole solutions, obtained recently in Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theory.
Recently it has been shown how a topologically twisted version of ${\cal N}=4$ super Yang-Mills may be discretized in such a way as to preserve one scalar supersymmetry at nonzero lattice spacing. The remaining fifteen supersymmetries are broken by terms of ${\cal O}(a)$ where $a$ is the lattice spacing. One would like to know whether these remaining supersymmetries are regained in the continuum limit $a \to 0$ and, if not, how much tuning of the couplings in the lattice action is required. In this paper we derive the form of these additional twisted supersymmetries by combining a set of discr…
We define supersymmetric Yang-Mills theory on an arbitrary two-dimensional lattice (polygon decomposition) with preserving one supercharge. When a smooth Riemann surface $Σ_g$ with genus $g$ emerges as an appropriate continuum limit of the generic lattice, the discretized theory becomes topologically twisted $\mathcal{N}=(2,2)$ supersymmetric Yang-Mills theory on $Σ_g$. If we adopt the usual square lattice as a special case of the discretization, our formulation is identical with Sugino's lattice model. Although the tuning of parameters is generally required while taking the continuum limit, t…
The symmetries of the topological Yang-Mills theory are studied in the Hamiltonian formalism and the generators of the twisted N=2 superPoincaré algebra are explicitly constructed. Noting that the twisted Lorentz generators do not generate the Lorentz symmetry of the theory, we relate the two by extracting from the latter the twisted version of the internal SU(2) generator. The hermiticity properties of the various generators are also considered throughout, and the boost generators are found to be non-hermitian. We then recover the BRST cohomology condition on physical states from representati…
$\mathcal{N}=1$ SUSY Yang-Mills theory is an appealing theoretical framework that has been studied in the literature using different methods, including standard lattice simulations. Among these, the volume-reduced twisted Eguchi-Kawai model, endowed with one adjoint Majorana fermion, could play an important role in studying its large-$N$ limit via the Curci-Veneziano prescription. In this talk, we present our results on the analysis of the scale of the theory, performed via different methods based on purely gluonic observables as well as (quenched) fundamental mesons in the chiral limit. These…
We study the low-energy limit of a compactification of N=4 U(n) super Yang-Mills theory on $S^1$ with boundary conditions modified by an S-duality and R-symmetry twist. This theory has N=6 supersymmetry in 2+1D. We analyze the $T^2$ compactification of this 2+1D theory by identifying a dual weakly coupled type-IIA background. The Hilbert space of normalizable ground states is finite-dimensional and appears to exhibit a rich structure of sectors. We identify most of them with Hilbert spaces of Chern-Simons theory (with appropriate gauge groups and levels). We also discuss a realization of a rel…
On a five dimensional simply connected Sasaki-Einstein manifold, one can construct Yang-Mills theories coupled to matter with at least two supersymmetries. The partition function of these theories localises on the contact instantons, however the contact instanton equations are not elliptic. It turns out that these equations can be embedded into the Haydys-Witten equations (which are elliptic) in the same way the 4D anti-self-dual instanton equations are embedded in the Vafa-Witten equations. We show that under some favourable circumstances, the latter equations will reduce to the former by pro…
The phenomena of emergent fuzzy geometry and noncommutative gauge theory from Yang-Mills matrix models is briefly reviewed. In particular, the eigenvalues distributions of Yang-Mills matrix models in lower dimensions in the commuting (matrix or Yang-Mills) phase of these models are discussed.
A generalization of the two-dimensional Yang-Mills and generalized Yang-Mills theory is introduced in which the building B-F theory is nonlocal in the auxiliary field. The classical and quantum properties of this nonlocal generalization are investigated and it is shown that for large gauge groups, there exist a simple correspondence between the properties a nonlocal theory and its corresponding local theory.
I articulate and discuss a geometrical interpretation of Yang-Mills theory. Analogies and disanalogies between Yang-Mills theory and general relativity are also considered.
The N=1,d=10 superYang-Mills action is constructed in a twisted form, using SU(5)-invariant decomposition of spinors in 10 dimensions. The action and its off-shell closed twisted scalar supersymmetry operator Q derive from a Chern-Simons term. The action can be decomposed as the sum of a term in the cohomology of Q and of a term that is Q-exact. The first term is a fermionic Chern-Simons term for a twisted component of the Majorana-Weyl gluino and it is related to the second one by a twisted vector supersymmetry with 5 parameters. The cohomology of Q and some topological observables are define…
Using an infinite number of fields, we construct actions for D=4 self-dual Yang-Mills with manifest Lorentz invariance and for D=10 super-Yang-Mills with manifest super-Poincaré invariance. These actions are generalizations of the covariant action for the D=2 chiral boson which was first studied by McClain, Wu, Yu and Wotzasek.
We discuss the integrability structure of deformed, four-dimensional N=4 super Yang-Mills theories using Yangians. We employ a recent procedure by Beisert and Roiban that generalizes the beta deformation of Lunin and Maldacena to produce N=1 superconformal gauge theories, which have the superalgebra SU(2,2|1)xU(1)xU(1). The deformed theories, including those with the more general twist, were shown to have retained their integrable structure. Here we examine the Yangian algebra of these deformed theories. In a five field subsector, we compute the two cases of SU(2)xU(1)xU(1)xU(1) and SU(2|1)xU(…
The partition function of the topological twisted super Yang-Mills field theory on the boundary can be expanded as Jones polynomial, which can be computed as expectation values of Wilson loop operators. We show that the zero of the Jones polynomial is Lee-Yang type. Moreover, Lee-Yang phase transition is also discussed in the Jones polynomial of torus knot and the topological twisted super Yang-Mills field theory.
We determine all inhomogeneous Yang-Mills algebras and super Yang-Mills algebras which are Koszul. Following a recent proposal, a non-homogeneous algebra is said to be Koszul if the homogeneous part is Koszul and if the PBW property holds. In this paper, the homogeneous parts are the Yang-Mills algebra and the super Yang-Mills algebra.
We will discuss an integrable structure for weakly coupled superconformal Yang-Mills theories, describe certain equivalences for the Yangian algebra, and fill a technical gap in our previous study of this subject.
We review some recent results related to the notion of volume independence in SU(N) Yang-Mills theories. The topic is discussed in the context of gauge theories living on a d-dimensional torus with twisted boundary conditions. After a brief introduction reviewing the formalism for introducing gauge fields on a torus, we discuss how volume independence arises in perturbation theory. We show how, for appropriately chosen twist tensors, perturbative results to all orders in the 't Hooft coupling depend on a specific combination of the rank of the gauge group (N) and the periods of the torus (l) g…
The planar Yang-Mills theory in three spatial dimensions is examined in a particular representation which explicitly embodies factorization. The effective planar Yang-Mills theory Hamiltonian is constructed in this representation.
We discuss a non-minimal Einstein-Yang-Mills-Higgs model with uniaxial anisotropy in the group space associated with the Higgs field. We apply this theory to the problem of propagation of color and color-acoustic waves in the gravitational background related to the non-minimal regular Wu-Yang monopole.
Recent developments in the determination of the planar S-matrix of N=4 Super Yang-Mills are closely related to its Yangian symmetry. Here we provide evidence for a yet unobserved additional symmetry: the Yangian level-one helicity operator.
This is a next paper from a sequel devoted to algebraic aspects of Yang-Mills theory. We undertake a study of deformation theory of Yang-Mills algebra YM - a ``universal solution'' of Yang-Mills equation. We compute (cyclic) (co)homology of YM.
We formulate the ten-dimensional super-Yang-Mills theory in a twisted superspace with 8+1 supercharges. Its constraints do not imply the equations of motion and we solve them. As a preliminary step for a complete formulation in a twisted superspace, we give a superspace path-integral formulation of the N=2, d=4 super-Yang-Mills theory without matter. The action is the sum of a Chern--Simons term depending on a super-connection plus a BF-like term. The integration over the superfield B implements the twisted superspace constraints on the super-gauge field, and the Chern-Simons action reduces to…
The BCFW recursion relation in $\mathcal{N}=4$ super-Yang-Mills theory is solved using Yang-Baxter $R$-operators in the NMHV sector. Explicit expressions for $\mathcal{R}$-invariants are obtained in terms of the chains of $R$-operators acting on an appropriate basic state.
The large-N behavior of Yang-Mills and generalized Yang-Mills theories in the double-scaling limit is investigated. By the double-scaling limit, it is meant that the area of the manifold on which the theory is defined, is itself a function of N. It is shown that phase transitions of different orders occur, depending on the functional dependence of the area on N. The finite-size scalings of the system are also investigated. Specifically, the dependence of the dominant representation on A, for large but finite N is determined.
We present new relations for scattering amplitudes of color ordered gluons and gravitons in Einstein-Yang-Mills theory. Tree-level amplitudes of arbitrary multiplicities and polarizations involving up to three gravitons and up to two color traces are reduced to partial amplitudes of pure Yang-Mills theory. In fact, the double-trace identities apply to Einstein-Yang-Mills extended by a dilaton and a B-field. Our results generalize recent work of Stieberger and Taylor for the single graviton case with a single color trace. As the derivation is made in the dimension-agnostic Cachazo-He-Yuan forma…
We analyze in an effective Lagrangian framework the connection between Super QCD (Super Yang Mills) and QCD (Yang Mills) by highlighting the crucial role that the zero modes play in the process of decoupling gluinos and squarks.
We show that the solutions of SU(2) Yang-Mills-dilation and Einstein-Yang-Mills-dilaton theories described in a sequence of papers by Kleihaus and Kunz are not regular in the gauge field part.
The existence of a "bonus" U(1) level-one Yangian symmetry of N=4 super-Yang-Mills has recently been proposed. We provide evidence for this proposal by constructing the BRST-invariant nonlocal charge in the pure spinor sigma model corresponding to this bonus level-one symmetry. We also construct analogous charges for bonus U(1) symmetries at all odd levels of the Yangian.
