The Yang-Mills String as the A-Model on the Twistor Space of the Complex Two-Dimensional Projective Space with Fluxes and Wilson Loops: the Beta Function
Marco Bochicchio
备注：
36 pages, latex; more extended version, introduced almost complex structure with vanishing first Chern class, change in the Lagrangian submanifold, beta function unchanged
We argue that the string theory dual to a certain sector of the four-dimensional Yang-Mills theory at large-N is the A-model wrapping N-Lagrangian D-branes on the twistor space of the complex two-dimensional projective space, with certain fluxes and Wilson loops, by finding that the target-space beta function of this A-model coincides with the large-N beta function of the Yang-Mills theory
This article reviews the recent progress in twistor approaches to Wilson loops, amplitudes and their duality for N=4 super Yang-Mills. Wilson loops and amplitudes are derived from first principles using the twistor action for maximally supersymmetric Yang-Mills theory. We start by deriving the MHV rules for gauge theory amplitudes from the twistor action in an axial gauge in twistor space, and show that this gives rise to the original momentum space version given by Cachazo, Svrcek and Witten. We then go on to obtain from these the construction of the momentum twistor space loop integrand usin…
We consider the kinematics of the locally BPS super-Wilson loop in $\mathcal{N}=4$ super-Yang-Mills with scalar coupling from a twistorial point of view. We find that the kinematics can be described either as supersymmetrized pure spinors or as a point in $G_{2\vert 2}(4\vert 4) \times G_{2\vert 2}(4\vert 4)$. In this description of the kinematics the scalar--scalar correlation function appearing in the one-loop evaluation of the super-Wilson loop can be neatly written as a sum of four super-determinants.
We give an explicit realization of the 4d local operator / 2d conformal block correspondence of Costello and Paquette in the case of gauge theories. This is accomplished by lifting the 4d local operators to non-local operators in twistor space using a holomorphic generalization of the Wilson line. This procedure automatically constructs the 2d conformal blocks corresponding to the local operator. We interpret this lifting as effectively integrating out the 2d degrees of freedom living on the defect. We present some 2d chiral CFT representations of the defect algebra whose correlators reproduce…
We propose that the complete planar S-matrix of N=4 super Yang-Mills - including all N^kMHV partial amplitudes to all loops - is equivalent to the correlation function of a supersymmetric Wilson loop in twistor space. Remarkably, the entire classical S-matrix arises from evaluating the correlation function in the self-dual sector, while the expansion of the correlation function in powers of the Yang-Mills coupling constant provides the loop expansion of the amplitudes. We support our proposal with explicit computations of the n particle NMHV and NNMHV trees, the integrands of the 1-loop MHV an…
This thesis is concerned with the study of scattering amplitudes in four-dimensional conformal field theories, more particularly the N=4 super-Yang-Mills theory. We study this theory first at tree level by using twistor space techniques and review the twistor string models that were proposed to describe it. Then, we turn to the issue of iteration relations and all loop ansatze for scattering amplitudes. We review the unitarity method for computing scattering amplitudes and discuss the Wilson loop--scattering amplitude duality that was inspired by the strong-coupling prescription of Alday and M…
We study a complex analogue of a Wilson Loop, defined over a complex curve, in non-Abelian holomorphic Chern-Simons theory. We obtain a version of the Makeenko-Migdal loop equation describing how the expectation value of these Wilson Loops varies as one moves around in a holomorphic family of curves. We use this to prove (at the level of the integrand) the duality between the twistor Wilson Loop and the all-loop planar S-matrix of N=4 super Yang-Mills by showing that, for a particular family of curves corresponding to piecewise null polygons in space-time, the loop equation reduce to the all-l…
The S-matrix for planar N = 4 super Yang-Mills theory can be computed as the correlation function for a holomorphic polygonal Wilson loop in twistor space. In an axial gauge, this leads to the construction of the all-loop integrand via MHV diagrams in twistor space. We show that at MHV, this formulation leads directly to expressions for loop integrands in d log form; i.e., the integrand is a product of exterior derivatives of logarithms of rational functions. For higher MHV degree, it is in d log form multiplied by delta functions. The parameters appearing in the d log form arise geometrically…
This is a review of recent developments in the study of perturbative gauge theory and gravity using action functionals on twistor space. It is intended to provide a user-friendly introduction to twistor actions, geared towards researchers or graduate students interested in learning something about the utility, prospects, and shortcomings of this approach. For those already familiar with the twistor approach, it should provide a condensed overview of the literature as well as several novel results of potential interest. This work is based primarily upon the author's D.Phil. thesis. We first con…
We consider the ratio of the correlation function of n+1 local operators over the correlator of the first n of these operators in planar N=4 super-Yang-Mills theory, and consider the limit where the first n operators become pairwise null separated. By studying the problem in twistor space, we prove that this is equivalent to the correlator of a n-cusp null polygonal Wilson loop with the remaining operator in general position, normalized by the expectation value of the Wilson loop itself, as recently conjectured by Alday, Buchbinder and Tseytlin. Twistor methods also provide a BCFW-like recursi…
The amplituhedron provides a beautiful description of perturbative superamplitude integrands in N=4 SYM in terms of purely geometric objects, generalisations of polytopes. On the other hand the Wilson loop in supertwistor space also gives an explicit description of these superamplitudes as a sum of planar Feynman diagrams. Each Feynman diagram can be naturally associated with a geometrical object in the same space as the amplituhedron (although not uniquely). This suggests that these geometric images of the Feynman diagrams give a tessellation of the amplituhedron. This turns out to be the cas…
We find a new family of AdS_4 vacua in IIA string theory. The internal space is topologically either the complex projective space CP^3 or the "flag manifold" SU(3)/(U(1)xU(1)), but the metric is in general neither Einstein nor Kaehler. All known moduli are stabilized by fluxes, without using quantum effects or orientifold planes. The analysis is completely ten--dimensional and does not rely on assumptions about Kaluza--Klein reduction.
Recently a reformulation of the MHV diagram method in N=4 supersymmetric Yang-Mills theory in momentum twistor space was presented and was shown to be equivalent to the perturbative expansion of the expectation value of a supersymmetric Wilson loop in momentum twistor space. In this note we present related explicit Feynman rules in dual momentum space, which should have the interpretation of Wilson loop diagrams in dual momentum space. We show that these novel rules are completely equivalent to ordinary spacetime MHV rules and can be naturally viewed as their graph dual representation.
We prove that in the limit when its insertion points become pairwise null-separated, the ratio of certain n-point correlation functions in N=4 SYM is equal to a supersymmetric Wilson loop on twistor space, acting in the adjoint representation. In the planar limit, each of these objects reduces to the square of the complete n-particle planar superamplitude. Our proof is at the level of the integrand.
We construct a U(1) bundle over N(1,1), usually considered as an SO(3) bundle over CP^2, and show that type IIB supergravity can be consistently compactified over it. With the five form flux turned on, there is a solution for which the metric becomes Einstein. We also turn on 3-form fluxes and show that there is a one parameter family of solutions. In particular, there is a limiting solution of large 3-form fluxes for which two U(1) fiber directions of the metric shrink to zero size. We also discuss compactifications over N(1,1) to AdS_3. All solutions turn out to be non-supersymmetric.
In this paper we consider tree-level gauge invariant off-shell amplitudes (Wilson line form factors) in $\mathcal{N}=4$ SYM. For the off-shell amplitudes with one leg off-shell we present a conjecture for their Grassmannian integral representation in spinor helicity, twistor and momentum twistor parameterizations. The presented conjecture is successfully checked against BCFW results for MHV$_n$, NMHV$_4$ and NMHV$_5$ off-shell amplitudes. We have also verified that our Grassmannian integral representation correctly reproduces soft (on-shell) limit for the off-shell gluon momentum. It is shown …
We propose a new diagrammatic formulation of the all-loop scattering amplitudes/Wilson loops in planar N=4 SYM, dubbed the "momentum-twistor diagrams". These are on-shell-diagrams obtained by gluing trivalent black and white vertices defined in momentum twistor space, which, in the reduced diagram case, are known to be related to diagrams in the original twistor space. The new diagrams are manifestly Yangian invariant, and they naturally represent factorization and forward-limit contributions in the all-loop BCFW recursion relations in momentum twistor space, in a fashion that is completely di…
We use momentum twistors to evaluate planar loop integrals. Infrared divergences are regulated by the recently proposed AdS-inspired mass regulator. We show that two-loop amplitudes in N=4 super Yang-Mills can be expanded in terms of basis integrals having twistor numerators. We argue that these integrals are considerably simpler compared to the ones conventionally used. Our case in point is the two-loop six-point MHV amplitude. We present analytical results for the remainder function in a kinematical limit, and find agreement with a recent Wilson loop computation. We also provide two-loop evi…
We study the flavor structure of 4D effective theories, which are derived from extra dimensional theories with magnetic fluxes and non-Abelian Wilson lines. We study zero-mode wavefunctions and compute Yukawa couplings as well as four-point couplings. In our models, we also discuss non-Abelian discrete flavor symmetries such as $D_4$, $Δ(27)$ and $Δ(54)$.
We incorporate all gauge-invariant local composite operators into the twistor-space formulation of N=4 SYM theory, detailing and expanding on ideas we presented recently in arXiv:1603.04471. The vertices for these operators contain infinitely many terms and we show how they can be constructed by taking suitable derivatives of a light-like Wilson loop in twistor space and shrinking it down to a point. In particular, these vertices directly yield the tree-level MHV super form factors of all composite operators in N=4 SYM theory.
We present a compact analytic formula for the two-loop six-particle MHV remainder function (equivalently, the two-loop light-like hexagon Wilson loop) in N = 4 supersymmetric Yang-Mills theory in terms of the classical polylogarithm functions Li_k with cross-ratios of momentum twistor invariants as their arguments. In deriving our result we rely on results from the theory of motives.
In this note we continue the exploration of the polytope picture for scattering amplitudes, where amplitudes are associated with the volumes of polytopes in generalized momentum-twistor spaces. After a quick warm-up example illustrating the essential ideas with the elementary geometry of polygons in CP^2, we interpret the 1-loop MHV integrand as the volume of a polytope in CP^3x CP^3, which can be thought of as the space obtained by taking the geometric dual of the Wilson loop in each CP^3 of the product. We then review the polytope picture for the NMHV tree amplitude and give it a more direct…
In a certain (non-commutative) version of large-N SU(N) Yang-Mills theory there are special Wilson loops, called twistor Wilson loops for geometrical reasons, whose v.e.v. is independent on the parameter that occurs in their operator definition. There is a semigroup that acts on the parameter by rescaling and on the functional measure, resolved into anti-selfdual orbits by a non-supersymmetric version of the Nicolai map, by contracting the support of the measure. As a consequence the twistor Wilson loops are localized on the fixed points of the semigroup of contractions. This localization is a…
In this paper we study correlation functions of circular Wilson loops in higher dimensional representations with chiral primary operators of N=4 super Yang-Mills theory. This is done using the recently established relation between higher rank Wilson loops in gauge theory and D-branes with electric fluxes in supergravity. We verify our results with a matrix model computation, finding perfect agreement in both the symmetric and the antisymmetric case.
By exploiting in large-N YM the change of variables from the gauge connection to the ASD part of its curvature by a non-SUSY version of the Nicolai map, we show that certain twistor Wilson loops supported on a Lagrangian submanifold of twistor space are localized on lattices of surface operators of Z(N) holonomy that form translational invariant sectors labelled by the magnetic charge k=1,2,...,N-1 at a point. The localization is obtained reducing the loop equation in the ASD variables in the holomorphic gauge, regularized by analytic continuation to Minkowski space-time, to a critical equatio…
We revisit the calculation of one-loop corrections to the partition functions of a string ending on a circular or straight line in type IIB string theory on $AdS_3\times S^3 \times T^4$ supported by a mixture of 3-form fluxes depending on a parameter $q$ (with $q=0$ being R-R and $q=1$ pure NS-NS cases). Correcting earlier results in the literature, we find that the partition function depends on $q$ only through rescaled tension $T= T_0/(1-q^2)$.
Motivated by the recent proposal for the S-matrix in $AdS_3\times S^3$ with mixed three form fluxes, we study classical folded string spinning in $AdS_3$ with both Ramond and Neveu-Schwarz three form fluxes. We solve the equations of motion of these strings and obtain their dispersion relation to the leading order in the Neveu-Schwarz flux $b$. We show that dispersion relation for the spinning strings with large spin ${\cal S}$ acquires a term given by $-\frac{\sqrtλ}{2π} b^2\log^2 {\cal S}$ in addition to the usual $\frac{\sqrtλ}π \log {\cal S}$ term where $\sqrtλ$ is proportional to the squa…
We compute analytically the two-loop contribution to the correlation function of the Lagrangian with a four-sided light-like (or null) Wilson loop in N=4 super Yang-Mills. As a non-trivial test of our result, we reproduce the three-loop value of the cusp anomalous dimension upon integration over the insertion point of the Lagrangian. The method we used involved calculating a dual scattering amplitude. Moreover, we give a simple representation of the loop integrand of the latter in twistor variables.
We discuss how to implement, in lattice gauge theories, external charges which are not commensurate with an elementary gauge coupling. It is shown that an arbitrary, real power of a standard Wilson loop (or Polyakov line) can be defined and consistently computed in lattice formulation of non-abelian, two dimensional gauge theories. However, such an observable can excite quantum states with integer fluxes only. Since the non-integer fluxes are not in the spectrum of the theory they cannot be created, no matter which observable is chosen. Also the continuum limit of above averages does not exist…
We study the supersymmetric Wilson loops in the four-dimensional N=4 super Yang-Mills theory in the context of AdS/CFT correspondence. In the gauge theory side, it is known that the expectation value of the Wilson loops of circular shape with winding number k is calculable by using a Gaussian matrix model. On the other hand, in the gravity side, it has been conjectured that the expectation value of the Wilson loop is given by the classical value of the action for a probe D3-brane with k electric fluxes. Given such correspondence, we pursue the interpretation of the matrix model eigenvalue dens…
In the holographic framework, a half BPS Wilson loop in ${\cal N}=4$ supersymmetric Yang-Mills in the fundamental, symmetric or antisymmetric representation of SU(N), is best described by a fundamental string, a D3 brane or a D5 brane with fluxes in their worldvolumes, respectively. We derive the spectrum of excitations of such D3 brane in $AdS_5\times S^5$ explicitly, considering its action in both the bosonic and the fermionic sectors, and demonstrate that it is organized according to short multiplets of the supergroup $OSp(4^*|4)$. We also show that the modes of the fundamental string form …
The half-supersymmetric Wilson loop in $\mathcal N=4$ SYM is arguably the central non-local operator in the AdS/CFT correspondence. On the field theory side, the vacuum expectation values of Wilson loops in arbitrary representations of $SU(N)$ are captured to all orders in perturbation theory by a Gaussian matrix model. Of prominent interest are the $k$-symmetric and $k$-antisymmetric representations, whose gravitational description is given in terms of D3- and D5-branes, respectively, with fluxes in their world volumes. At leading order in $N$ and $λ$ the agreement in both cases is exact. In …
A systematic treatment is given of the Dirac quantisation condition for electromagnetic fluxes through two-cycles on a four-manifold space-time which can be very complicated topologically, provided only that it is connected, compact, oriented and smooth. This is sufficient for the quantised Maxwell theory on it to satisfy electromagnetic duality properties. The results depend upon whether the complex wave function needed for the argument is scalar or spinorial in nature. An essential step is the derivation of a "quantum Stokes' theorem" for the integral of the gauge potential around a closed l…
We present a general set-up for inflation in string theory where the inflaton field corresponds to Wilson lines in compact space in the presence of magnetic fluxes. T-dualities and limits on the value of the magnetic fluxes relate this system to the standard D-brane inflation scenarios, such as brane-antibrane inflation, D3/D7 brane inflation and different configurations of branes at angles. This can then be seen as a generalised approach to inflation from open string modes. Inflation ends when the Wilson lines achieve a critical value and an open string mode becomes tachyonic. Then hybrid-lik…
We propose new backgrounds of extra dimensions to lead to four-dimensional chiral models with three generations of matter fermions, that is $T^2/Z_N$ twisted orbifolds with magnetic fluxes. We consider gauge theory on six-dimensional space-time, which contains the $T^2/Z_N$ orbifold with magnetic flux, Scherk-Schwarz phases and Wilson line phases. We classify all the possible Scherk-Schwarz and Wilson line phases on $T^2/Z_N$ orbifolds with magnetic fluxes. The behavior of zero modes is studied. We derive the number of zero modes for each eigenvalue of the $Z_N$ twist, showing explicitly examp…
We study 10D super Yang-Mills theory with the gauge groups $E_6$, $E_7$ and $E_8$. We consider the torus/orbifold compacfitication with magnetic fluxes and Wilson lines. They lead to 4D interesting models with three families of quarks and leptons, whose profiles in extra dimensions are quasi-localized because of magnetic fluxes.
We explicitly construct the fully back-reacted half-BPS solutions in Type IIB supergravity which are dual to Wilson loops with 16 supersymmetries in $\mathcal{N}=4$ super Yang-Mills. In a first part, we use the methods of a companion paper to derive the exact general solution of the half-BPS equations on the space $AdS_2 \times S^2 \times S^4 \times Σ$, with isometry group $SO(2,1)\times SO(3) \times SO(5)$ in terms of two locally harmonic functions on a Riemann surface $Σ$ with boundary. These solutions, generally, have varying dilaton and axion, and non-vanishing 3-form fluxes. In a second p…
We study aspects of gauge theory on tori which are a consequences of Morita equivalence. In particular we study the behavior of gauge theory on noncommutative tori for arbitrarily close rational values of Theta. For such values of Theta, there are Morita equivalent descriptions in terms of Yang-Mills theories on commutative tori with very different magnetic fluxes and rank. In order for the correlators of open Wilson lines to depend smoothly on Theta, the correlators of closed Wilson lines in the commutative Yang-Mills theory must satisfy strong constraints. If exactly satisfied, these constra…
An exactly soluble non-perturbative model of the pure gauge QCD is derived as a weak coupling limit of the lattice theory in plaquette formulation. The model represents QCD as a theory of the weakly interacting field strength fluxes. The area law behavior of the Wilson loop average is a direct result of this representation: the total flux through macroscopic loop is the additive (due to the weakness of the interaction) function of the elementary fluxes. The compactness of the gauge group is shown to be the factor which prevents the elementary fluxes contributions from cancellation. There is no…
Morse-Smale-Floer homology associates the critical points of the action functional of a classical field theory over a manifold to its homology. We associate to the intersection homology of certain Lagrangian submanifolds of R^4 the critical points of a quantum effective action of large-N SU(N) YM. For this purpose we construct in YM a trivial Topological Field Theory defined by twistor Wilson loops whose v.e.v. is 1 in the large-N limit for any shape of the loops supported on certain punctured Lagrangian submanifolds. We derive a new holomorphic loop equation for the twistor Wilson loops, that…
We calculate the energy of a Yang-Mills vortex as function of its magnetic flux or, else, of the Wilson loop surrounding the vortex center. The calculation is performed in the 1-loop approximation. A parallel with a potential as function of the Polyakov line at nonzero temperatures is drawn. We find that quantized Z(2) vortices are dynamically preferred though vortices with arbitrary fluxes cannot be ruled out.
In this paper we present our results concerning the dependence of Wilson loop expectation values on the size of the lattice and the rank of the SU(N) gauge group. This allows to test the claims about volume independence in the large N limit, and the crucial dependence on boundary conditions. Our highly precise results provide strong support for the validity of the twisted reduction mechanism and the TEK model, provided the fluxes are chosen within the appropriate domain.
We analyze large gauge invariance in combined nonabelian and thermal QFT and their physical consequences for D=3 effective actions. After briefly reviewing the structure of bundles and large gauge transformations that arise in non-simply connected 3-manifolds and gauge groups, we discuss their connections to Chern-Simons terms and Wilson-Polyakov loops. We then provide an invariant characterization of the ``abelian'' fluxes encountered in explicit computations of finite temperature effective actions. In particular we relate, and provide explicit realizations of, these fluxes to a topological i…
We have computed in [arXiv:1107.4320 (hep-th)] the glueballs spectrum in a certain sector of the large-N YM theory by solving by a change of variables the holomorphic loop equation for cusped twistor Wilson loops supported on certain Lagrangian submanifolds and by evaluating the correlators of surface operators supported on these Lagrangian submanifolds. We have shown that the correlators of composite surface operators of length L reproduce in the large-L limit the leading logarithms of perturbation theory of the corresponding glueballs propagators, including the correct anomalous dimensions. …
We recently introduced a new representation for loop quantum gravity, which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay out the classical framework underlying this new formulation. The central objects in our construction are the so-called integrated fluxes, which are defined as the integral of the electric field variable over surfaces of codimension one, and related in turn to Wilson surface operators. These integrated flux observables will play an important role in the coarse graining of states in loop quantum gra…
We propose that inflation is driven by a (complex) neutral Higgs of the MSSM extension of the SM, in a chaotic-like inflation setting. The SUSY breaking soft term masses are of order $10^{12}-10^{13}$ GeV, which is identified with the inflaton mass scale and is just enough to stabilise the SM Higgs potential. The fine-tuned SM Higgs has then a mass around 126 GeV, in agreement with LHC results. We point out that the required large field excursions of chaotic inflation may be realised in string theory with the (complex) inflaton/Higgs identified with a continuous Wilson line or D-brane position…
