supersymmetric Yang-Mills theory from quantum geometry of momentum space

Amplitudes in the N ¼ 4 supersymmetric Yang-Mills theory from quantum geometryof momentum space

A. Gorsky

Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow, 117259, Russia(Received 2 July 2009; published 3 December 2009)

We discuss multiloop maximally helicity violating amplitudes in the N ¼ 4 supersymmetric Yang-

Mills theory in terms of effective gravity in the momentum space with IR regulator branes as degrees of

freedom. Kinematical invariants of external particles yield the moduli spaces of complex or Kahler

structures which are the playgrounds for the Kodaira-Spencer or Kahler type gravity. We suggest

fermionic representation of the loop maximally helicity violating amplitudes in the N ¼ 4 supersym-

metric Yang-Mills theory assuming the identification of the IR regulator branes with Kodaira-Spencer

fermions in the B model and Lagrangian branes in the A model. The two-easy mass box diagram is related

to the correlator of fermionic currents on the spectral curve in the B model or hyperbolic volume in the A

model and it plays the role of a building block in the whole picture. The Bern-Dixon-Smirnov–like ansatz

has the interpretation as the semiclassical limit of a fermionic correlator. It is argued that fermionic

representation implies a kind of integrability on the moduli spaces. We conjecture the interpretation of the

reggeon degrees of freedom in terms of the open strings stretched between the IR regulator branes.

DOI: 10.1103/PhysRevD.80.125002 PACS numbers: 11.15.Bt, 11.25.Tq

I. INTRODUCTION

The N ¼ 4 supersymmetric Yang-Mills (SYM) theoryprovides a possibility to recognize some features of thetheories with less amount of supersymmetry. While N ¼ 4SYM is far from the QCD-like theories in the infraredbecause of the lack of confinement it shares commonfeatures in the UV region where physics in asymptoticallyfree theories is described within a perturbation theory. Thatis, considering the perturbative expansion in the N ¼ 4SYM coupling constant which does not run, we could try toclarify some generic properties of the perturbative expan-sion in the gauge theories.

It is of prime importance to discover any hidden sym-metries at high energies or equivalently hidden integrablestructures providing the nontrivial conservation laws whichrestrict the form of the scattering amplitudes. In the four-dimensional setup the integrability behind the amplitudesis known only at the Regge limit when the SLð2; CÞ spinchain gets materialized [1,2] (see [3] for a review).

The simplest objects at generic kinematics are the maxi-mally helicity violating (MHV) amplitudes which are theperfect starting point for any discussion since at the planarlimit they can be described in terms of the single kinemati-cal function. Even at the tree level MHV amplitudes [4]enjoy some remarkable properties. They are localized onthe complex curves in the twistor space [5] (see [6] for areview) and can be described as the correlators of chiralbosons on the genus zero Riemann surface [7]. It turns outthat the generating function for the tree MHVamplitudes isjust the particular solution to the self-duality equation inYM theory [8,9]. It substitutes the naive superposition ofthe plane waves of the same chirality in a nonlinear theory.Moreover this solution provides the symplectic transfor-

mation [10] (see also [11]) of the YM theory in the light-cone gauge into the so-called tree MHV Lagrangian for-mulated in [12] which to some extent is the analogue of thet’Hooft effective vertex generated by instantons. However,this approach becomes less clear when going to higherloops. Indeed, the attempt to formulate the one-loopMHVamplitudes in a twistor-like manner was not success-ful enough [13] and certainly calls for additional insightson the problem.One more line of development based on a first quantized

picture for the loop calculations was initiated in [14]. It wasshown that the three-point amplitude written in theSchwinger parametrization implies the identification ofthe Schwinger proper ‘‘time’’ parameter with the radialcoordinate in the AdS geometry providing some rationalefor the appearance of the AdS space. The similar interpre-tation holds true for the calculation of the one-loop effec-tive actions in the different backgrounds [15]. Howeverstarting from a one-loop four-point amplitude the situationbecomes more subtle because of the emerging modulispace. It was suggested in [16] that the Feynman diagramcan be presented in terms of the skeleton graph parame-trized by the set of Schwinger parameters. This set ofparameters can be mapped for a planar limit of then-point amplitude into the manifold M0;n � Rnþ where

M0;n is the moduli space of the n-punctured sphere. The

mapping of Schwinger parameters into the coordinates onthe moduli space is quite nontrivial and, for instance, doesnot respect the special conformal transformations [17]. Thegluing of the segments of the skeleton diagram is subtle butsome arguments based on the operator product expansionsupporting this picture were presented [18]. Hence wecould expect that loop amplitudes in the SYM theory canbe expressed as the correlators of some vertex operators on

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the moduli space of the complex structures which is theframework of the B model.

Another important starting point for the multiloop gen-eralizations was provided by the geometrical picture foundin [19]. It was argued that the one-loop amplitudes can beidentified with the hyperbolic volume of the ideal tetrahe-dron in the space of the Feynman parameters. The corre-sponding Kahler moduli are fixed by the kinematicalinvariants. The hyperbolic volumes of three-dimensionalmanifolds are the natural playground for the A model andthe SLð2; CÞ gravity provides the natural generalizations ofthe one-loop answer. It is possible to consider the partitionfunction of SLð2; CÞ gravity in terms of the proper gluingof the three-dimensional manifold from the ideal tetrahe-drons [20].

More recently Bern, Dixon and Smirnov (BDS) haveformulated the conjecture [21] that all-loop MHV ampli-tudes get exponentiated and factorized into IR divergentand finite parts. Moreover it was conjectured that the finitepart of the all-loop amplitude involves only the all-loopcusp anomalous dimension �cuspð�Þ and finite part of the

one-loop amplitude. Inspired by this conjecture Alday andMaldacena have calculated the amplitude at a strong cou-pling regime via minimal surfaces in AdS-type geometrywith the proper boundary conditions [22]. They have foundan unexpected relation between the MHVamplitudes in theplanar limit of N ¼ 4 SYM theory and Wilson polygons inthe ‘‘momentum space.’’

The Wilson polygon-amplitude duality refreshes theproblem but deserves an explanation. It was originallyformulated at strong coupling when the Wilson loop iscalculated in terms of the minimal surface in the AdS5geometry upon a kind of T-duality transform. Later it wasshown that duality holds true at the perturbative regime aswell [23,24] which puts it on firmer ground. Recently theexplicit derivation of the one-loop duality has been pre-sented [25]. The important anomalous Ward identity forthe special conformal transformations with respect to thedual conformal group has been derived. It fixes the kine-matical dependence of the amplitudes up to five externallegs [26,27]. However Ward identities tell nothing aboutthe functional form of the amplitudes starting from sixexternal legs. Recently the dual superconformal groupwas identified as the symmetry of the world sheet theoryof the superstring in AdS5 � S5 geometry [28,29].

Finally it was recognized that the BDS ansatz fails atweak coupling at the two loop level for six external legs[30,31] and at strong coupling [32,33] for an infinitelylarge number of external legs. Moreover the BDS ansatzseems not to fit well with the Regge limit [34] (see, how-ever, [35]). On the other hand at the two loop level theduality between the Wilson polygon and MHV amplitudesurvives. The current status of the whole problem has beenreviewed in [36].

There are a lot of pressing questions to be answered.This just mentions a few:

(i) Is there some geometrical picture behind the BDS-like ansatz which suggests the way to generalize it?

(ii) Is there a generalization of the dual conformal Wardidentity which would fix the functional form of theone-loop amplitude for any number of externallegs?

(iii) Is there a fermionic representation for the loopamplitudes which would imply the hiddenintegrability?

(iv) Is there a clear geometrical picture behind theReggeization of the gluon?

In what follows we shall suggest the answers to some ofthese questions and make a couple of conjectures.To some extent we shall try to generalize the geometrical

picture for the tree amplitudes suggested in [5]. At the treelevel in [5] the Euclidean D1 ‘‘instanton’’ branes with theattached open strings have been considered in the twistorspace. The D1 brane is localized at the point in theMinkowski space in agreement with the locality of thevertex generating tree MHVamplitude in the MHV formal-ism. To describe the loop amplitudes we shall adopt aslightly different picture and consider a C4 manifold inthe B model. The B branes substitute ‘‘D1 instantons’’ andare embedded in C4. The somewhat similar objects werealso introduced as the IR regulator branes in the Alday-Maldacena calculation. Indeed the dilaton field getschanged upon the T duality in the renormalization group(RG) radial coordinate which means that theD instanton isadded to the background. After the Fourier transformsalong flat four-dimensions the D instanton gets trans-formed into the D3 brane we shall work with. TheWilson polygon which corresponds to the boundary ofthe string world sheet and is presumably dual to the am-plitude is located just on these IR regulator branes.Contrary to the previous considerations the positions ofthe regulator branes will not be free but determined dy-namically in terms of the cross-ratios of the externalmomenta.The emerging moduli space of IR regulator branes plays

the central role in the picture. Contrary to the tree level theKodaira-Spencer (KS) degrees of freedom in the B modeldefined on the moduli space do not decouple and providethe phase space for the corresponding integrable system.The essential point in our approach concerns the quantiza-tion of the emerging moduli spaces and the identification ofthe corresponding Planck constant with some function ofthe YM coupling constant. Therefore to some extent wecould tell that the loop MHV amplitudes emerge upon akind of gravitational dressing of the tree ones within theKodaira-Spencer type gravity in the ‘‘momentum’’ ortwistor space.The physics of the scattering at the loop level can be

treated from the different perspectives. From the point ofview of the KS gravity on the moduli space we are calcu-lating the correlator of the fermions or the fermionic

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currents which can be identified with the tau function of the2d integrable system. The second viewpoint concerns aconsideration of the gauge theory on the IR regulatorbranes whose number is fixed by the number of the externalparticles. Finally, one could consider the world sheetviewpoint where the regulator branes provide the properboundary conditions for the string. These viewpoints arecomplementary and allow one to check the self-consistency of the approach.

Within the KS perspective we shall discuss the fermionicrepresentation behind the loop MHV amplitudes whichgeneralizes Nair’s fermionic representation for the treeamplitudes. The fermionic picture is at the heart of theintegrability which admits the representation in terms ofthe chiral fermions on the Riemann surface in the externalgauge field. The gauge field on the Riemann surface rep-resents the ‘‘point of Grassmanian’’ or in physical terms,the particular Bogolyubov transformation between the fer-mionic vacua. Such an emerging fermionic picture isknown to be quite generic in the set of examples whichinvolve minimal string models [37], c ¼ 1 string [38], andthe crystal melting problem [39]. This approach was sum-marized in [40] where it was argued that fermions in theKS gravity correspond to mirror the Lagrangian branes inthe A model. These B branes are also refereed to asKontsevich or noncompact branes and their positions onthe Riemann surface yield the ‘‘times’’ in the correspond-ing integrable systems. Note that in the framework of thetopological strings in the A model we discuss the Kahlergeometry while in the B model the complex geometry iscaptured by the Kodaira-Spencer [41] theory.

The natural question concerns the role of the Riemannsurface in the B model where the KS fermions live on. Inseveral examples it collects the information about theinfinite set of the anomalous Ward identities in the theory[40] and encodes the unbroken part of the W1 symmetry.More qualitatively it means that if we introduce the IRregularization of the theory corresponding to the infiniteblowup of C3 in the A model the ‘‘anomaly’’ survives uponthe removed regularization. Note some analogy with thedescription of the Seiberg-Witten solution to the low-energy N ¼ 2 SYM theory [42]. In that case we have firstto perform the summation over the pointlike instantons[43] which amounts to the particular blowups and desin-gularizes the target space theory geometry providing thenontrivial Riemann surface. The physical correlators afterall are calculated in terms of the fermions on this emergingRiemann surface.

The fermion one-point function corresponds to theBaker-Akhiezer function in the integrability frameworkand to the single regulator brane insertion at some pointon the moduli space. Since generally we are interested inthe quantum integrable system the Riemann surface getsquantized and yields the corresponding Baxter equation[44]. The semiclassical solutions to the Baxter equation

which are the generating functions for the Lagrangiansubmanifolds in the particular integrable system play im-portant role in the analysis. They serve as the buildingblocks for the correlators in the N ¼ 4 YM theory andcan be considered as the ‘‘semiclassical B brane wavefunction’’ or as the effective action in the gauge theoryon the brane worldvolume. From the moduli space view-point the solution to the Baxter equation provides thegenerating function of the Lagrangian submanifold. Thenatural integrable system on the moduli space can beidentified with the 3–Kadomtsev-Petviashvili (KP) system,however, similar to the N ¼ 2 SYM one could expect thepair of integrable systems—the 2D field theory and thefinite-dimensional one. The natural finite-dimensional in-tegrable system which is responsible for the hidden sym-metries at the generic kinematics is conjectured to berelated to the Faddeev-Volkov model [45] and the corre-sponding statistical model [46] based on the discrete quan-tum conformal transformations.Since we are trying to sum the perturbation series the

YM coupling constant is expected to be involved into somealgebraic structure behind the all-loop answers. It is thishidden symmetry which provides the choice of the particu-lar solution to the Yang-Baxter equation. The Faddeev-Volkov solution to the Yang-Baxter implies that we areactually trying to relate the YM coupling constant with theparameter q of UqðSLð2; RÞÞ. The proper identification

turns out to be a nontrivial problem since, in particular, ithas to respect the S-duality group in the N ¼ 4 theory. Itwill be argued that the BDS ansatz corresponds to the limitq ! 1 while the Regge limit seems to be related to theopposite ‘‘strong coupling regime’’ of the quantum group.The consideration of the gauge theories on the regulator

brane worldvolume is useful as well. The theory can bethought of as in the Coulomb phase and the position of theregulator brane in the particular complex plane corre-sponds to the coordinate on the Coulomb moduli space.Since all regulator branes are at different positions on themoduli space the theory generally has the gauge groupUð1Þk where k is related to the number of the externalgluons, however, there are possible enhancements to thenon-Abelian factors at some kinematical regions. The ef-fective action of each Uð1Þ gauge theory on the regulatorbrane plays the role of the wave function of the two-dimensional fermion or B brane in KS gravity on the Bmodel side. Similarly the worldvolume theory on theLagrangian regulator D2 branes can be considered on theA-model side. In this case we shall consider the twistedsuperpotentials in the worldvolume theory. The minimiza-tion of the effective superpotential amounts to the selectionof the positions of the Lagrangian branes at the basemanifold.The Riemann surface involved has the interpretation in

the regulator brane worldvolume theory as well. To thisaim, note that the coordinate on the moduli space plays the

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role of the complex scalar in the B brane worldvolumetheory. One can consider the change of variables in thetheory corresponding to the reparametrization of this scalarfield. Such transformation is familiar in the N ¼ 1 SYMtheory as the generalized Konishi transformations whichare anomalous. One can collect all Konishi transformationsor Virasoro constraints in the Dijkgraaf-Vafa approach [47]into the single generating equation which yields the par-ticular Riemann surface [48,49]. It can be considered as ananalogue of the chiral ring in the worldvolume theory onthe regulator branes.

It is important to discuss separately the special Reggekinematical region where the hidden symmetries of theamplitudes were found for the first time. The hidden sym-metries were captured at one loop by the SLð2; CÞ spinchains [1,2]. It was shown in [50] that the N-Reggeondynamics belong to the same universality class as confor-mal N ¼ 2 supersymmetric quantum chromodynamicswith Nf ¼ 2N at the strong coupling orbifold point. We

shall argue that the brane geometry in the Reggeon case issimilar to the one in supersymmetric quantum chromody-namics which provides the qualitative explanation of thesame universality class for both theories. The new object isthe open string stretched between two regulator branes andis the analogue of the massive vector bosons and mono-poles in the conventional N ¼ 2 SYM theory. Here weshall attempt to interpret these open strings as the‘‘Reggeons.’’ The ‘‘masses’’ of these effective degrees offreedom correspond to the differences of the positions ofthe regulator branes on the proper Riemann surface andtherefore depend on the kinematical invariants. We shallcomment on the possible link of this picture with theeffective Reggeon field theory [51].

The paper is organized as follows. In Sec. II we recall themain features concerning the loop MHV amplitudes. InSec. III we briefly consider the example of the c ¼ 1noncritical string which provides some information onthe calculation of the amplitudes from the target spaceperspective. In Sec. IV we review the relevant propertiesof the quantum dilogarithm. Section V is devoted to theformulation of our conjecture for the N ¼ 4 MHV ampli-tudes. In Sec. VI we make conjectures on the pair of theunderlying integrable systems. Some arguments concern-ing the Regge limit of the amplitudes are present inSec. VII. In the last section we collect the main points ofour proposal and fix the open questions to be answered.

II. THE LOOP RESULTS FOR THE MHVAMPLITUDES

Let us recall the main results concerning the loop MHVamplitudes. The MHV gluon amplitudes involve two glu-ons of the negative chiralities and the rest of gluons havepositive chiralities. Consider the ratio of all-loop and treeanswers. The following form of the all-loop amplitudes hasbeen suggested in [21]:

log

�Mall¼loop

Mtree

�¼ ðFdiv þ �cuspð�ÞMone-loopÞ; (1)

which involves only the all-loop answer for the cuspanomaly �cusp and one-loop MHV amplitude. The IR di-

vergent part Fdiv gets factorized in the all-loop answer. Thecusp anomaly measures UV behavior of the contour withthe cusp [52]. Recently the closed integral equation hasbeen found for the cusp anomalous dimension in N ¼ 4SYM theory [53] which correctly reproduces the weak andstrong coupling expansions.The finite part of the one-loop MHV which presumably

defines the all-loop answer can be written in terms of thefinite part of the so-called two-mass easy box functionF2em [24]

Mone-loop;finite ¼Xp;q

F2em;fðp; q; P;QÞ: (2)

This function can be expressed in terms of the dilogarithmsonly

F2em;fðp; q; P;QÞ ¼ Li2ð1� aP2Þ þ Li2ð1� aQ2Þ� Li2ð1� aðqþ PÞ2Þ� Li2ð1� aðpþ PÞ2Þ; (3)

where

a ¼ P2 þQ2 � ðqþ PÞ2 � ðpþ PÞ2P2Q2 � ðqþ PÞ2ðqþ PÞ2 ; (4)

and pþ qþ PþQ ¼ 0. One more expression for thefunction F2em;f can be written in terms of the variablesxi;k ¼ pi � pk as the sums [23]

Xi

Xr

Li2

�1� x2i;iþrx

2i¼1;iþrþ1

x2i;iþrþ1x2i�1;iþr

�; (5)

where

xi ¼ piþ1 � pi: (6)

Since all external momenta are on the mass shell thearguments of dilogarithms are expressed in terms of thecross-ratios of the scalar products of the momenta only.Since we shall aim to get the geometrical interpretation

of the BDS ansatz we first need the clear geometry behindone loop. It is provided by the observation in [19] that thebox diagram with all the external off-shell particles justcalculates the volume of the three-dimensional ideal hyper-bolic tetrahedron in the space of the Feynman parameters(see the Appendix). The Kahler modulus z of the tetrahe-dron is fixed by the kinematical invariants and in the two-mass easy box it reduces to the conformal ratios. Theappearance of the hyperbolic volume implies that thetopological string approach or Chern-Simons (CS) withthe SLð2; CÞ group are relevant [20]. Indeed we can con-sider the ideal tetrahedron as the knot complement andcalculate its volume via the Chern-Simons theory action


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with the complex group. To some extent the exponentiationof the one-loop answer in the BDS manner corresponds tothe calculation of the classical partition function in theSLð2; CÞ CS action which involves the expðVolðzÞÞ factor.Let us emphasize that the volume is finite if we consideroff-shell particles only and IR divergence of the amplitudecorresponds to the divergence of the volume. There hasbeen some development of this interpretation of the higherloops which can be found in [54].

The topological string picture usually can be representedboth in the A-model and B-model sides. The hyperbolicvolume calculation evidently corresponds to the Kahlergravity in the A-side so one could ask about the one-loopgeometry in the B-model language. It can be uncoveredindeed if we recall that dilogarithm is the natural object onthe moduli space Mð0;nÞ [55] where it provides the proper

canonical transformations. The natural arguments of dilogsas functions on the moduli space are just the conformalratios appearing in the one-loop integrals. We shall try tointerpret the BDS-like structure in the B-model languageas the result of the quantization of the Teichmuller spacewhen the one-loop dilogs are substituted by the quantumdilogs. This viewpoint will be useful when we shall searchfor the proper ‘‘degrees of freedom’’ involved into higherloops calculations. They will be conjectured to be IRregulator branes or equivalent fermions.

III. THE c ¼ 1 EXAMPLE

The useful examplewhich shares some essential featureswith our problem is provided by the c ¼ 1 model. Thenoncritical c ¼ 1 model describes the string in the onedimensional compact target space and because of the non-criticality the target geometry becomes two dimensionaldue to the Liouville direction. The only physical modes aremassless tachyons generally gravitationally dressed by theLiouville modes.

The theory enjoys two natural types of branes whichprovide the boundary conditions for the strings; so-calledZZ and FZZT branes. The compact ZZ branes correspondto the unstable D0 branes localized in the Liouville direc-tion. On the other hand the noncompact branes correspondto the stable D1 branes extended along the Liouville coor-dinate until the Liouville wall at the cosmological constantscale � [38].

The explicit target space description is more appropriatefor our purposes. This approach is natural in the topologi-cal string setup and was developed in [38]. The crucialpoint is the existence of the so-called chiral ring in thetheory. In the c ¼ 1 case it collects the information aboutthe set of certain anomalous relations in the theory. Themost useful description of the chiral ring involves theRiemann surface supplied with some meromorphic differ-ential. For c ¼ 1 theory it reads as

x2 � y2 ¼ const; (7)

where x, y 2 C. In terms of the Riemann surface a com-pact ZZ brane corresponds to the tunneling state in theinverted oscillator or equivalently to the closed cycle on thesurface pinched at the degeneration point. On the otherhand, the noncompact FZZT brane corresponds to the openpath on the surface. The corresponding semiclassical wavefunction

�FZZT / exp

�iZ

ydx

�(8)

transforms nontrivially when going to different coordinatepaths on the surface.Another useful language is provided by the matrix

model approach. The random matrix model is usually builton the set of an infinite number of ZZ branes whosecoordinates correspond to the eigenvalues of the matrixof the infinite size triangulating the string world sheet. Onthe other hand one can consider the matrix model of theKontsevich type on theN noncompact FZZT branes. In thiscase one deals with the N � N matrix model with thesource term Tr�X and the eigenvalues of the matrix �encode the positions of the FZZT branes. The appropriateobject in the matrix model is its resolvent

WðzÞ ¼�Tr

1

z�M

�(9)

obeying the loop equation which in the semiclassical limitcoincides with the equation of the spectral curve.All languages yield the same important feature of the

c ¼ 1 model—its hidden integrability. It turns out that taufunction of the Toda hierarchy serves as the generatingfunction for the tachyonic amplitudes [56]. This objectscan be naturally described in terms of the chiral boson�ðzÞon the spectral curve or as the corresponding fermion

�ðzÞ ¼ expðg�1s �ðzÞÞ: (10)

More precisely one considers the following matrix ele-ment in the theory of the chiral boson or its fermionizedversion

�ðt; AÞ ¼ htj expðcAc Þj0i ¼ htjVi; (11)

where jt> ¼ expðPktk�kÞ is generally the coherent stateof the chiral boson with modes �k on our Riemann surface.The matrix A encodes the scattering of fermions off theLiouville wall which essentially provides the whole answerfor the tachyonic amplitudes [56]. The integrability enc-odes the infinite number of theWard identities in the theoryfollowed from the symplectic invariance of the Riemannsurface which is the complex Liouville torus for the com-plex Hamiltonian system. Some part of the symmetries isspontaneously broken yielding the corresponding Wardidentities. Some of them are unbroken yielding the equa-tion of the quantum spectral curve [40]. The set of Wardidentities can be formulated in terms of the fermionic


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bilinears on the surface and provides the exact answers forthe correlators in the theory.

The FZZTor noncompact branes parametrize the modulispace of the complex structures in the target geometry andcan be naturally treated within the KS gravity in the targetspace. The positions of the noncompact FZZT branes ziyield the following times in the Toda integrable system:

Tk ¼ 1

k

XNi¼1

z�ki : (12)

The wave function of the FZZT brane itself can beconsidered as the Baker-Akhiezer function in the inte-grable systems. Quantum mechanically the Riemann sur-face gets quantized yielding the Baxter equation for theeigenvalue of the Baxter operator. The solution to theBaxter equation corresponds to the wave function of thesingle separated variable. The brane interpretation of theseparated variables has been suggested in [57]. The corre-lator of the Kontsevich branes inserted at points zi has thefollowing structure:

h0j�ðziÞ . . . �ðznÞjVi ¼Yi;j

ðzi � zjÞePi

�ðziÞ; (13)

where one can clearly distinguish the classical and quan-tum components of the answer. Another point to be men-tioned is the identification of the quantization parameter. Inthe commutation relation on the quantized Riemann sur-face

½x; y� ¼ gs; (14)

the Planck constant is just the string constant or the grav-iphoton field.

To summarize, the c ¼ 1 model provides the examplewhen the nonperturbative structure of the theory is stored isthe Riemann surface in the B model which collects theinformation about the chiral ring. This Riemann surfacehas to be considered as the energy level of some complexHamiltonian and the set of canonical transformations of thephase space amounts to the set of Ward identities in theinitial model which fix the scattering amplitudes. Thedeformations of the complex structures which are dynami-cal degrees of freedom in the Kodaira-Spencer theory areparametrized by the fermions on the Riemann surfacewhich upon quantization yield the Baxter equation of thecorresponding integrable system.

IV. QUANTUM DILOGARITHM

The one-loop answer is expressed in terms of the dilo-garitms, hence in this section we shall briefly review somerelevant properties of the quantum dilogarithm defined asthe following integral:

�bðzÞ ¼ exp

�1

4

Z e�2izxdx

x sinhðbxÞ sinhðb�1xÞ�: (15)

The integration contour is chosen in such way that theintegral reduced to the infinite sum via the residue calcu-lation

�bðzÞ ¼ exp

�Xn

e�nx

n½n��: (16)

It obeys the functional equations

�bðzÞ�bð�zÞ ¼ ei�z2�i�ð1þ2c2qÞ=6; (17)

and

�bðz� b�1=2Þ ¼ ð1þ e2�zb�1Þ�bðzþ b�1=2Þ; (18)

where cb ¼ i2 ðbþ b�1Þ, as well as the unitarity condition

�� bðzÞ ¼ �bð�zÞ�1: (19)

The quantum dilogarithm can be represented as the ratioof two q exponentials

�bðzÞ ¼ ðe2�ðzþcbÞb; q2Þ1ðe2�ðz�cbÞb�1

; ~q2Þ1; (20)

where q ¼ ei�b2, ~q ¼ e�i�b�2

and

ðx; qÞ1 ¼ Yn

ð1� qnxÞ: (21)

The dilogarithm enjoys the duality

�bðzÞ ¼ �b�1ðzÞ; (22)

which is essential when it is involved in the gluing of theconformal blocks in the Liouville model [58,59]. Note thatthe central charge in the corresponding Liouville theoryreads as

cliouv ¼ 1þ 6ðbþ b�1Þ2: (23)

The quantum dilogarithm is natural object from theviewpoint of the quantum torus algebra defined by therelation

U V ¼ qV U; (24)

where U ¼ expðixÞ and V ¼ expðipÞ. It is assumed that thevariables x, p obey the canonical phase space commutationrelations. In terms of the quantum torus the quantumdilogarithm is defined via the relation

�ðVÞ�ðUÞ ¼ �ðUÞ�ð�U VÞ�ðVÞ: (25)

This property represents the so-called quantum pentagonrelation [60] and reduces to the classical Rogers identityfor Li2ðzÞ in the semiclassical limit.Quantum mechanically the dilogarithm defines the op-

erator with the kernel

Kðx; zÞ ¼ �qðzÞeðð�zxÞ=2�qÞ; (26)

which serves as the generating function of the followingcanonical Backlund type transformations:


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U ! ðð1þ qUÞV; V ! U�1; (27)

which belongs to the outer automorphisms of the algebra offunctions on the quantum torus. The important property ofthis transformation is the operator version of the pentagonrelation

K 5 ¼ 1; (28)

which states that automorphism is of the fifth order.Dilogarithm plays an essential role in the symplectic

treatment of the Teichmuller space. Recall that theTeichmuller space TðSÞ is the space of the complex struc-tures on the Riemann surface S modulo trivial diffeomor-phisms homotopy equivalent to the identity while themoduli space MðSÞ is obtained via the factorization ofthe Teichmuller by the action of the mapping class group.The quantum dilogarithm plays the role of the quantumgenerating function for the particular element of the map-ping class group—flip transformation. The flip transforma-tions are responsible for the generic transition mapsbetween coordinate systems corresponding to the differenttriangulations. The quantization of the Teichmuller spacewas developed in [55,61] (see [58] for a review).

There are different ways to introduce the coordinates onthe Teichmuller space of the punctured spheres we areinterested in. Of particular interest are the coordinatesrelated to the geodesic lengths and the correspondingclassical Poisson structure can be written in a simple wayin terms of triangulations. These shear coordinates can beintroduced in terms of the cross ratio of the four points onthe real line connected by the geodesic circles in the upperhalf-plane

eðzÞ ¼ ðx2 � x1Þðx4 � x3Þðx3 � x2Þðx4 � x1Þ : (29)

The natural arguments of the dilogarithms defined on themoduli space are just cross ratios which we meet in theanswers for the MHVamplitude. In the semiclassical limitwe have

�bðxÞ ! exp

�1

bLi2ðexÞ

�: (30)

In the context of integrability dilogarithms appear as theingredients of the fundamental R matrix involved into thedescription of the Liouville and sin-Gordon theories in thediscrete space-time. In terms of the discretized version ofthe Kac-Moody currents with the commutation relationsimilar to the quantum torus

!n!nþ1 ¼ q2!nþ1!n; (31)

!n!m ¼ !m!n; jn�mj � 2: (32)

The periodicity condition for the current variables is as-sumed !n ¼ !nþN . In terms of these dynamical variablesone can define the solution to the Yang-Baxter equation

depending on the spectral parameter � [62]

Rð�;!ÞÞ ¼ �bð!Þ�bð!�1Þ�bð�!Þ�bð�!�1Þ : (33)

The product of the R matrices over the lattice cites yieldsthe evolution operator for the massive model on the space-time lattice.The finite-dimensional system discussed in the B-model

framework should carry some information on the S dualityof the N ¼ 4 gauge theory. The modular parameter of thegauge theory can be identified with the quantization pa-rameter of the integrable system. However to get the fullmodular symmetry the modular double [63] has to beincluded into the game. It unifies two quantum tori withthe S-dual moduli. It turns out that the modular doubleplays the crucial role in the quantum integrable systemsproviding the self-consistency of the local and nonlocalintegrals of motion.That is, the integrable system has to be supplied with the

following symmetry:UqðSLð2; RÞ �U~qðSLð2; RÞ. The cor-responding R matrix acting of the modular double reads as

R ¼ eð�=2Þðp3þp2Þ�ðp1þp4Þ�ðp13Þ�ðp34Þ�ðp23Þ�ðp24Þ;(34)

pik ¼ pi � I þ I � pk; (35)

and the variables pi, i ¼ 1 . . . 4 obey the commutationrelations

½pk; pkþ1� ¼ �2�iI: (36)

The integrable system with such symmetry was found in[45]. It was argued in [46] that using the R matrix for themodular double one can define the positive weights whichprovide the unitary model.Such a type of R matrix emerges naturally within the

discrete quantum Liouville theory related to the discreteconformal transformations [45,46,61]. It was argued thatthe structure of the modular double is necessary for theself-consistent description of the Liouville theory at thestrong coupling region 1< c< 25. The integrability of themodel to some extent is equivalent to its very quantumexistence [46]. The link with the dilogarithms involved inthe description of the Teichmuller space goes as follows.The universal Teichmuller space is known to be identifiedwith the coadjoint Virasoro orbit. On the other hand theLiouville action plays the role of the geometrical action onthis orbit [64], therefore it is no surprise that the triangu-lation of the moduli space is related with the discretequantum Liouville theory.In the physical setup the quantum dilogarithm corre-

sponds to the probability of the charged pair creation inthe constant external field in four-dimensional scalar QEDtheory. It is assumed that both electric (E) and magnetic(H) fields are switched then the one-loop effective actionreads as


125002-7

Lone-loop ¼ 1

16�2

Z dt

te�m2t

��

e2ac

sinhðectÞ sinhðeatÞ �1

t2� e2

6ða2 � c2Þ

�;

(37)

where a2 � c2 ¼ E2 � H2, e is the coupling constant, andac ¼ EH. The last two terms provide the proper substrac-tion to get the finite answer. The ‘‘strong coupling limit’’jbj ! 1 of the quantum dilogarithm corresponds to thealmost self-dual external field while the semiclassical limitcorresponds to the strong deviation from the self-dualregime.

V. FINITE PART OF THE N ¼ 4 SYM MHVAMPLITUDES AND MOMENTUM SPACE’’

GEOMETRY

A. The brane picture

Let us now formulate our proposal for the finite part ofthe MHV loop amplitudes. Recall that the tree amplitudeswere described in terms of the D1 string instanton em-bedded into the twistor manifold [5]. The instanton islocalized at a point in the Minkowski space and openstrings representing gluons are attached to it. To describethe loop amplitude we shall substitute D1 brane by the IRregulator brane embedded into the proper manifold. Thegluons are attached to the regulator branes whose embed-ding coordinates are considered as dynamical degrees offreedom. Contrary to the tree case regulator branes arelocalized at the submanifold of the complexifiedMinkowski space. The loop amplitudes can be consideredfrom the different perspectives: in terms of the KS gravityon the particular Riemann surface, within the worldvolumetheory on the regulator branes, and in the theory on thestring world sheet. Let us emphasize that the embedding ofthe IR regulator branes nontrivially depends on the externalmomenta.

The starting point is the representation of the N ¼ 4theory via geometrical engineering [65] as the IIA super-string compactified on the three-dimensional Calabi-Yaumanifold which was identified as the K3� T2 geometry inthe singular limit. One has to consider the singular limit ofthe K3 manifold when it develops AN�1 singularity, whereN becomes the rank of the gauge group, and upon blowingup procedure it can be represented as ALEN geometry. Onthe other hand the Kahler class of the T2 can be identifiedwith the coupling constant

Area ðT2Þ ¼ 1=g2YM: (38)

At weak coupling the torus is large and can be approxi-mated by the complex plane. That is, the geometry can beroughly approximated by C3 upon the particular blowups.

As we have seen in the one-loop answer for the MHVamplitude, determining the BDS form of the amplitude

involves the sum of the dilogarithms depending on thecross ratios of the xi variables. Below we shall try toexplain how such functions with cross-ratio argumentsemerge naturally both in A-model and B-model frame-works. As is well known the A model about the Kahlermoduli while the B model captures the information aboutthe complex moduli and we shall see where these modulicome from. The brane description of the scattering ampli-tude involves the set of the Lagrangian branes in the A-model and the corresponding B-model branes. It is thesebranes which provide the corresponding moduli spaces.

1. B-model

First, we shall discuss the B-model approach. There aretwo natural B-model setups. The first one follows from thetopological S duality [66] and corresponds to the samemanifold with the S-dual moduli. Since in our case thearea of the torus is defined in terms of the YM couplingconstant we end up with the small dual torus in the S-dualmodel at the weak coupling. This topological B-model inthe S-dual geometry can be described in terms of the non-commutative Uð1Þ gauge theory in D ¼ 6 which is natu-rally defined on the D5 brane worldvolume. Anotherviewpoint is provided by the mirror symmetry whichmaps the A-model to the B-model on the different mani-fold. It is convenient to consider the dual mirror geometryupon the infinite blowup of C3.Let us interpret the BDS ansatz in terms of the correlator

of the noncompact Euclidean B branes embedded into thefour-dimensional complex space. Consider the 3d complexmanifold which mirrors the topological vertex [67]. Thismanifold, classically, is described by the equation in the C4

with coordinates x, y, u, v,

xy ¼ eu þ ev þ 1: (39)

At the discriminant locus it defines the Riemann surface

H ðv; uÞ ¼ eu þ ev þ 1 ¼ 0; (40)

of genus zero with three different asymptotic regions.This Riemann surface emerges from the infinite blowups

of the origin of the toric fibration of C3 upon the mirrortransform and provides the part of the IR regularization ofthe theory. We shall try to argue that the loop MHVamplitudes can be identified with the fermionic correlatorson the Riemann surface (40). Fermions on the surface (40)represent the degrees of freedom in the KS gravity. Theyare identified with the IR regulator branes imbedded intoC4 geometry.There are two B branes defined by the equations

x ¼ 0; Hðv; uÞ ¼ 0; (41)

and

y ¼ 0; Hðv; uÞ ¼ 0; (42)

which intersect along the Riemann surface. The intersect-


125002-8

ing branes provides the natural fermionic degrees of free-dom on the intersection surface [68] from the open stringsstretched between them. The fermions are in an externalfield amounted from the worldvolume gauge connectionson the intersecting branes. This gauge field represents thepoint of the Grassmanian which can be read off from thetopological vertex. In addition to two branes intersectingalong the Riemann surface we introduce the set ofKontsevich-like branes classically localized at the pointsðvi; uiÞ at the Riemann surface. The number of such branesis fixed by the number of the external gluons and thecoordinates of these branes on the surface are defined bysome particular cross ratios. The cross ratios are the naturalcoordinates on the moduli space of the punctured spheres;that is, the ðu; vÞ space is related to the T�M0;4. Hence we

are in the framework of the KS gravity and the fermions onthe Riemann surface represent the KS gravity degree offreedom.

The Riemann surface gets quantized and the branes-fermions should obey the equation of the quantumRiemann surface that is Baxter equation which providesthe wave functions depending on the separated variables[57]. The Baxter equation in our problem reads as

ðe@@v þ ev þ 1ÞQðvÞ ¼ 0: (43)

Its solution corresponds to the vacuum expectation value offermionic bilinear J,

QðvÞ ¼ h0jJðvÞjVi; (44)

and turns out to be the quantum dilogarithm [40]. Note thatthe solution to the Baxter equation in our case can not bepresented in the polynomial form; that is, we have aninfinite number of the Bethe roots.

To get the MHVall-loop amplitude in the BDS form wetake the semiclassical limit of the fermionic correlator onthis surface. Indeed using the semiclassical limit for thequantum dilogarithm we can represent the four-point fer-mionic current correlator as

h �Jðz1Þ �Jðz2ÞJðz3ÞJðz4Þi / expð@�1ðLi2ðz3Þ þ Li2ðz4Þ� Li2ðz1Þ � Li2ðz2ÞÞ: (45)

This expression exactly coincides with the expression forthe finite contribution of the single two-easy mass boxdiagram, hence upon the identification of the Planck con-stant

@�1 ¼ �cuspð�Þ; (46)

we reproduce the BDS ansatz for the finite part of theamplitude. Indeed the one-loop answer for the MHV am-plitude can be expressed purely in terms of the sum of two-mass easy box diagrams with a different grouping of thegluon momenta and therefore in terms of the fermioniccorrelators.

Since the regulator brane (D1 instanton) yielding thetree amplitude is localized in the complexified Minkowskispace Mc [5] one could ask about a similar localization ofthe regulator branes responsible for the higher loop calcu-lations. To this aim recall that the complexified MinkowskispaceMc is equivalent to the Grassmanian Grð2; 4Þ. On theother hand the factor of the Grassmanian by the maximaltorus action is related to the compactified moduli space[69]

Gr ð2; 4Þ==T ¼ �M0;4: (47)

This representation allows one to represent the complexi-fied Minkowski space itself as the fancy divisor of theM0;4

[70]. We suggest that this realization implies the localiza-tion of the regulator branes on the submanifold ofT�ðMc==TÞ. It is natural to identify this manifold withthe Riemann surface where the KS degrees of freedom live.

2. A-model

Let us present the qualitative arguments concerning thecorresponding A-model picture. In the A model we intro-duce the set of Lagrangian branes with topology S1 � R2.They can be also thought of as D6 branes if we add theconventional R3;1 piece of geometry. The emergence of the

dilogarithm as the wave function of the Lagrangian branehas been discovered in the C3 geometry in [71]. The brane/antibrane can be considered as the insertion of the fermion/antifermion [71] in the fermionic representation of thetopological vertex picture [67].In this case we get the Kahler gravity as the target space

description of our geometry. The Lagrangian of the Kahlergravity was conjectured to reduce to the SLð2; CÞ CStheory on the Lagrangian branes which describes the quan-tum geometry of the hyperbolic space. Since we are tryingto identify the amplitude as the wave function of theLagrangian branes, its argument in the proper polarizationshould be Kahler modulus of the ideal tetrahedron. This isindeed consistent with the loop calculations since the boxdiagram yields the hyperbolic volume in the space of theFeynman parameters [19].The intersecting Lagrangian branes naturally provide

the set of knots. The knot complements are the naturalhyperbolic manifolds and we can consider the triangulationof the three-dimensional cusped hyperbolic space by theideal tetrahedron (see, for instance, [20]). It turns out thatquantum dilogarithm with the proper argument can beattributed to each tetrahedron and the main problem turnsout to be the gluing of the whole manifold from severalideal tetrahedra. The gluing conditions have the form of theBethe ansatz-type equations if one attributes to each tetra-hedron a kind of S matrix [20]. That is in the A-modelpicture the all-loop answer deserves the accurate gluing ofthe submanifolds in the hyperbolic spaces.


125002-9

B. The regulator brane worldvolume theory

Since fermions in KS framework are identified as theregulator B branes the natural question concerns their four-dimensional worldvolume theory. The theory on the regu-lator branes share many features with N ¼ 2 and N ¼ 1SYM low-energy sectors. The number of the regulatorbranes is fixed by the number of the external gluons sonaively one could expect a kind of SUðKÞ gauge theory.The world sheet theory on the regulator branes enjoys thecomplex scalar corresponding to the complex coordinate zof the brane on the Riemann surface (40). This is similar tothe situation when the vacuum expectation value of thescalar field corresponds to the position of the D4 branes onthe u plane in the IIA realization of the N ¼ 2 SYM theory[72].

Since the different regulator branes are at the differentpoints on the Riemann surface we can speak about theCoulomb branch of the regulator worldvolume theory.However their positions on the Riemann surface are fixed,that is, we could say about the localization of the B branesat the points of the moduli spaceMð0;4Þ. Similar to the N ¼1 SYM theory, when branes are localized at positionscorresponding to the discrete vacua the regulator branesare localized at some points parametrized by the crossratios. These points correspond to the local rapidities inthe framework of integrability and simultaneously have tocorrespond to the minima of the effective superpotentialsWeffðziÞ in the regulator worldvolume theory.

Since we identify dilogaritms as the regulator branewave functions it is necessary to explain where theycome from in the worldvolume theory. The qualitativearguments looks as follows. In the worldvolume theorythere are massive excitations corresponding to the openstrings stretched between two regulator branes. They are ananalogue of the massive W bosons in the N ¼ 1 SYMtheory on the Coulomb branch. In our case the masses ofthese ‘‘particles’’ are related to the cross ratios. To recoverthe dilogaritms let us recall that in the external field theeffective action usually develops the imaginary part corre-sponding to the pair creation. The probability of the paircreation in the external field is described by the classicaltrajectory in the Euclidean space and in the leading ap-proximation reads as

ImSeff / e�ðm2Þ=eE; (48)

for a particle of the mass m in the external field E. Upontaking into account the multiple wrapping and the qua-dratic fluctuations one gets for the scalar particleSchwinger pair production

ImSeff /Xn

1

n2e�ðnm2Þ=eE; (49)

that is, the dilog plays the role of the decay probability.Hence one can say that we are considering the Euclideanversion of the regulator worldvolume theory and the am-

plitude from this viewpoint is described via a bounce typeconfiguration corresponding to the creation of the pairs ofthe effective massive degrees of freedom. Note that the realpart of the effective action corresponds to the summationover the loop contributions of the same degrees of freedom.We have a N ¼ 1 type theory on the regulator branes

with the finite number of vacua and the complex scalarfield whose vacuum expectation value corresponds to thecoordinates of the B branes on the Riemann surface. Hencewe can discuss the role of the symmetries implied by thearea preserving diffeomorphisms of the Riemann surface.From the regulator worldvolume theory this transformationis the symmetry of the target space analogous to the trans-formations of the scalar field in N ¼ 1 gauge theory withthe adjoint scalar. In the N ¼ 1 SYM case the generalizedKonishi anomaly responsible for these transformations[48,49],

� ! fð�Þ; (50)

captures the unbroken part of W1.In the A model one can similarly consider the worldvo-

lume theory on the D2 (or D6) Lagrangian regulatorbranes. In this case the corresponding dilogarithm func-tions emerge upon the summation over the disc instantonswith boundaries located at the corresponding Lagrangianbranes

Weff /Xn

dnn2

e�nA; (51)

where Ais the corresponding area of the target disc. Theeffective twisted superpotentials in the three-dimensionalworldvolume theory with one compact dimension werediscussed in [73]. Roughly speaking the wave function ofD2 brane has the form

�ðzÞ / eWtwistedðzÞ: (52)

Note that in the A model D2 branes can be considered aswrapped around the ideal tetrahedrons whose Kahlermodulus are defined by the cross ratios providing themasses of the same effective ‘‘W-bosons’’ as in the Bmodel. The issue of the gluing of the tetrahedra getsreformulated in terms of the minimization of the totaltwisted superpotential which is equivalent to the solutionto the Bethe ansatz equations in the XXZ spin chain model[73]. The solution to the Bethe ansatz equations fixes thepositions of the Lagrangian branes. To fit this argumentwith the complex moduli remark that Bethe ansatz equa-tions for the XXZ model are related to the solution to theclassical equations of motion in the discrete Liouvillemodel [74].

C. @�1 ¼ �cuspð�Þ?Let us comment on the identification of the Planck

constant for the quantization of the KS gravity as theinverse cusp anomalous dimension inspired by the BDS


125002-10

ansatz. At the first glance it looks completely groundlesshowever the argument supporting this identification goesas follows. The emergence of the cusp anomaly in theexponent means that it plays the role of the effective stringtension or equivalently the inverse Planck constant. Sucheffective tension emerges if one considers the string whoseboundary is extended along the lightlike contours. It wasshown [75] that in the limit suggested in [76] the stringworld sheet action can be identified with the Oð6Þ sigmamodel and the energy of the ground state in theOð6Þmodelis proportional to the length of the string multiplied by the�cuspð�Þ. That is, the �cuspð�Þ plays the role of the effectivetension of the string in this special kinematic. Since in ourcase the boundary of the string world sheet lies on theWilson polygon the effective tension involving the cuspanomalous dimension is natural.

However, certainly this point is far from being clarified.For instance in the Ward identity for the special conformaltransformation �cusp enters as the multiplier in the anoma-

lous contribution. This claim has been explicitly checked atthe first loops in the gauge theory calculations and thearguments that it holds true at all orders have been pre-sented. The anomalous Ward identity reads as [26]

K�Wðx1; . . . xNÞ ¼Xni¼1

ð2x�i xi@i � x2i @�i ÞWðx1; . . . xNÞ

¼ 1

2�cuspð�Þ

Xni¼1

lnx2i;iþ2

x2i�1;iþ1

x�i;iþ1; (53)

and it has been proved at the strong coupling as well [77].In this equation �cusp plays the role of the Planck con-

stant, not the inverse one. To match both arguments wecould suggest that in the Ward identity we are consideringthe S-dual formulation and therefore the D1 string worldsheet action instead of the F1 one in theOð6Þ sigma model.This would imply that theWilson polygon equivalent to theMHVamplitude could be considered as the boundary of theD1 string as well. Similarly, the �cusp enters the loop

equation for the Wilson loops with cusps [78]

�ChWðCÞi ¼ Xcusps

�cuspð�; �iÞhWðCÞi; (54)

where �C is the Laplace operator in the loop space and thesummation goes over all cusps along the contour C. Weassume that there are no self-intersections. In this case the�cuspð�Þ has the natural interpretation as the inverse Planckconstant since the loop equation has the form of theSchrodinger equation.

In more general setup it is highly desirable to realize themeaning of the relation of such a type in the first quantizedlanguage. Since the cusp anomalous dimension is just therenormalization factor for the self-crossing of the world-line it is very interesting to understand if such self-crossingis involved into the quantization issue. In particular, in theIsing model the effect of the self-crossing is captured by

the topological term and in the description of the topologi-cal string on C3 a somewhat similar � term in six dimen-sions plays the role of the quantization parameter indeed[79]. In the gauge theory language such objects are relatedto the renormalization of the double-trace operator’scouplings.Note that generally the relation between the YM cou-

pling and string coupling involves the BNS field

1

g2YM¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVol2

T2 þ B2NS

qgs

: (55)

One could try to speculate that the self-crossing could besensitive to the BNS field. Anyway it is clear that preciseidentification of the relation between the quantization pa-rameter in the KS gravity and the YM coupling is one ofthe necessary steps in improving the BDS ansatz.

VI. INTEGRABILITY BEHIND THE SCATTERINGAMPLITUDES

A. General remarks

In this section we shall discuss the hidden integrabilitybehind the scattering amplitudes and present the argumentsthat similar to the integrability pattern behind effectiveactions in N ¼ 2 SYM theory (see [80] for the review)two integrable systems are involved. The degrees of free-dom of both integrable systems are related to the coordi-nates of the regulator branes. One of these systems whichwe identify as the Whitham-like 3-KP one plays the role ofRG flows in the regulator brane worldsheet theory orequivalently the motion of the regulator brane along the‘‘radial’’ RG-coordinate. The second integrable systemgeneralizing the Hitchin-like or spin chain models involvesthe effective interactions between the regulator branes. Weshall give arguments that this system is based on theFaddeev-Volkov solution to the Yang-Baxter equation forthe infinite-dimensional representations of the noncompactSLð2; RÞ group.Recall how two integrable systems are involved in the

description of the low-energy effective actions of the N ¼2 SYM theories. The first finite-dimensional system is ofthe Hitchin or spin chain type and its complex Liouvilletori are identified with the Seiberg-Witten curves. Thisspectral curve emerges in the gauge theory upon the sum-mation over the infinite number of instantons [43].Following [81] one can canonically define the dual

integrable system whose phase space is built on the inte-grals of the motion of the first one. In the simplest case ofSUð2Þ theory the phase space for the dual system has thesymplectic structure [82]

! ¼ da ^ daD; (56)

where the variables a, aD are the standard variables in theN ¼ 2 SYM framework [42]. The prepotential F can beidentified with the generating function of the Lagrangian


125002-11

submanifold in the dual system with the a, aD phase space

H

�aðuÞ; @F

@a

�¼ u; (57)

and obeys the Hamilton-Jacobi equation

@F@ log�

¼ H: (58)

In the brane setup the prepotential defines the semiclassicalwave function of the D4 brane �ðaÞ / expð@�1F ðaÞÞ inthe IIA brane picture where perturbatively the argument ofthe wave function can be identified with coordinate of theD4 brane on the NS5 brane. The total perturbative prepo-tential in SUðNcÞ can be considered as a sum of theexponential factors in the product of the wave functionsof Nc D4 branes. At the A-model side these wave functionscan be considered in the Kahler gravity framework and thearguments of the wave function have to be treated as theKahler classes of the blown-up spheres.

The integrals of motion provide the moduli space of thecomplex structures in the Calabi-Yau geometry in the Bmodel, hence we are precisely in the KS framework. In thisB-model formulation we consider the argument of thebrane wave function as the coordinate on the moduli spaceof the complex structures. The dual Whitham-type inte-grable system naturally defines the � function of the 2dToda theory formulated in terms of the chiral fermions onthe Riemann surface with two marked points. Upon per-turbing the N ¼ 2 theory down to N ¼ 1 the moduli spacedisappears and the number of vacua becomes finite. In theintegrability framework this is treated in the followingmanner. The Hamiltonian of the first finite-dimensionalsystem turns out to coincide with the superpotential ofthe N ¼ 1 system [83]. That is, the vacua of the gaugetheory at the classical level correspond to the equilibriumpoints in the Hamiltonian system W 0 ¼ 0.

B. 3-KP system

Let us turn to the integrable structure relevant for thescattering amplitudes at generic kinematics and first iden-tify the degrees of freedom and evolution times. As wehave described above the fermionic degrees of freedomcorrespond to the noncompact branes localized on theRiemann surface. The two-dimensional field theory corre-sponds to the reduction of the Kodaira-Spencer theory onthe two-dimensional surface. The coordinate on theRiemann surface is related with the coordinate on themoduli space M0;4. The Kodaira- Spencer theory is de-

scribed by the two-dimensional Lagrangian

LKS ¼Z �

@� �@�þ 1

�! �@�þ �

!ð@�Þ2 �@�

�; (59)

where � is the basic scalar in the KS theory, ! is one formon the surface, and � is the topological string couplingconstant. It was argued recently [68] that the cubic inter-

action term in the KS Lagrangian can be formulated as thescreening operator in the two-dimensional conformal the-ory. The fields on the surface are in the external Abelianconnection of the Berry type which tells how the B branestransform under the change of the complex structure fixedby the momenta of external particles.As we have mentioned in the c ¼ 1 example there are

two possible sets of times, ‘‘compact’’ and ‘‘noncompact’’ones. The compact ones correspond to the variation of thecomplex structure at infinities and are responsible for theinsertion of the vertex operators of the ‘‘tachyonic’’ de-grees of freedom while the noncompact ones correspond tothe insertions of the noncompact B branes at the particularvalues of the cross ratios. The gluon vertex operators in thisframework correspond to the tachyonic vertex operators inthe c ¼ 1 model. The set of Kontsevich times determinedby the positions of the B branes is defined by (12) where ziare the corresponding cross ratios.The form of the Riemann surface Hðu; vÞ ¼ 0 dictates

that there are three infinities and therefore we are dealingwith the particular solution to the 3-KP integrable system.To describe the integrable system it is convenient to in-troduce the chiral fermions with the following mode ex-pansion:

c ðxiÞ ¼Xn

c inþ1=2x

�n�1i ; c �ðxiÞ ¼

Xn

c �inþ1=2x

�n�1i ;

(60)

around the i-th infinity, i ¼ 1, 2, 3, and the commutationrelations

fc in; c

�jm g ¼ ijnþm;o: (61)

Defining the vacuum state by relations

c nj0i ¼ 0; c �nj0i ¼ 0; n > 0; (62)

the generic state jVi can be presented in the form

jVi ¼ exp

�Xi;j

Xn;m

aijnmc i�n�1=2c

�i�m�1=2

�j0i; (63)

where the point of the Grassmanian representing the topo-logical vertex was derived in [79]. For instance the diago-nal coefficients read as

aiinm ¼ ð�1Þn qmðmþ1Þ�nðnþ1Þ

½mþ nþ 1�½m�½n� : (64)

The tau function of the 3-KP system plays the role of thegenerating function for the MHV amplitudes. In the semi-classical approximation we can safely consider the differ-ential on the classical Riemann surface

dS ¼ vdu; (65)

which yields the semiclassical brane wave function

�qs / exp

��@

�1Z x

vðuÞdu�

(66)


125002-12

involving the dilogaritms. The tau-function obeys the 3-KPequation and there is the additional W1þ1 Ward identitywritten in terms of the fermions

Iuc � ðuÞenuc ðuÞ þ ð�1Þn

Ivc � ðvÞenvc ðvÞ

þIsc � ðsÞensc ðsÞ ¼ 0; (67)

where the sum over three asymptotic regions is considered.The quantization of the system can be done most effec-

tively in terms of the Baxter equation. The Baxter equationimplies that the regulator branes are localized on thesurface. Hence the whole set of the equations determiningamplitudes involves the dual conformal transformations onthe regulator worldvolume and the set of Ward identitiesfor the coordinate of the regulator brane in the transversemoduli space. It is these Ward identities which fix thedependence of the amplitude on the conformal invariantsfor alarge number of external legs.

The precise form of the higher Hamiltonians from theW1þ1 responsible for the higher conservation laws in thescattering amplitude problem can be written as the fermi-onic bilinears [40]. Generally as was discussed in [40] onehas some unbroken part of W1 which annulates the �function corresponding to the topological vertex and there-fore the scattering amplitude in the form of BDS-likeansatz.

C. On the Faddeev-Volkov model

Let us turn now to the description of the second inte-grable system representing the particular solitonic sector ofthe infinite-dimensional integrable system. We shall con-jecture that the integrable system at the generic kinematicsis the generalization of the SLð2; CÞ spin chain relevant forthe Regge limit of the amplitudes.

The finite-dimensional integrable systems can usuallybe defined in terms of the R matrix. The Faddeev-Volkovmodel is defined via the Drinfeld solution to the Yang-Baxter equation which provides the universal R matrixacting on UqðSLð2; RÞÞ �U~qðSLð2; RÞÞ. The correspond-

ing statistical model describes the discrete quantumLiouville theory [46] with the following partition function:

Z ¼Z Y

ij

Wp�qðSi � SjÞYkl

�Wp�qðSk � SlÞYi

dSi; (68)

where the Boltzmann weights depend only on the differ-ences of the spins Sk at the neighbor sites and rapidityvariables at the ends of the edge. The first product is overthe horizontal edges ði; jÞ while the second product is overthe vertical edges ðk; lÞ. The integral is over all internal spindegrees of freedom. In the fundamental R matrix the crossratios of the relative rapidities of the particles play the roleof the local inhomogeneities in the lattice model andBoltzmann weights are defined as [46]

W�ðsÞ ¼ Fð�Þ�1e2�s�ðsþ icb��Þ�ðs� icb��Þ ; (69)

where spin and local rapidity variables are combined to-gether in the argument of the function � and Fð�Þ is somenormalization factor. The relative importance of the spinvariables and the local inhomogeneities depends on thevalue of the YM coupling constant and the kinematicalregion.Semiclassically when b ! 0 the spin variables are fro-

zen and the Boltzmann weight behaves as

W�ð�=2�bÞÞ ¼ exp

��Að�j�Þ

2�b2þ . . .

�; (70)

where

Að�j�Þ ¼ iLi2ð�e��i�Þ � iLi2ð�e�þi�Þ: (71)

The extremization of the semiclassical action yields theBethe ansatz type equations connecting the dynamical spinvariables with the local rapidities

Yi

e�i þ e�jþ�ij

e�j þ e�iþ�ij¼ 1: (72)

Let us try to compare the brane geometry behind twointegrable systems behind the low-energy N ¼ 2 SYM andin the N ¼ 4 scattering problem. In the N ¼ 2 case in theIIA picture we have Nc D4 branes stretched between twoNS5 branes and coordinates of D4 branes on the NS5 branecorrespond to the vacuum expectation values of the scalars.The second set of degrees of freedom is provided by thelow-dimensional branes on D4 branes with attached stringsconnecting D4 branes. The first ‘‘fast’’ integrable systemdescribes the dynamics of the strings while the second‘‘slow’’ integrable system describes the dynamics of D4branes on the moduli space of the vacua. Very similarly inthe scattering geometry the D4 branes are substituted bythe B-model branes localized on the moduli spaceM0;4 and

their dynamics is described by the Whitham-type 3-KPhierarchy while the second integrable system with N de-grees of freedom corresponds to the dynamics of the openstrings attached to the B branes.In order to complete this section let us comment on how

the interplay between ‘‘soft’’ and ‘‘regulator’’ degrees offreedom is captured by the integrable dynamics. To thisaim recall the description of the Korteweg–De Vries (KdV)hierarchy in terms of the Liouville field. The KdVequationcan be considered as the rotator on the coadjoint orbit ofthe Virasoro algebra. The coadjoint orbit is the symplecticmanifold and the geometrical action on the Virasoro orbitis the Liouville action [64]. It can be derived upon integra-tion of the chiral fermion in the external gravitational field.On the other hand the KdV Hamiltonians are provided bythe integration of the ‘‘heavy’’ nonrelativistic degree offreedom in the same gravitational field that is log detðd2 þTÞ, where T is the two-dimensional energy stress tensor. In


125002-13

the Lax representation we consider the isospectral evolu-tion of the Baker-Akhiezer function which is the eigen-function of the Schrodinger operator and can be attributedto the heavy degree of freedom. This is similar to our casesince the wave functions of the noncompact branes can beconsidered as the Baker-Akhiezer functions of the inte-grable system.

VII. COMMENTS ON THE REGGE LIMIT

In this section we shall discuss some features whichhopefully could help in the explanation of theReggeization of the amplitude. The interpretation of theReggeon in the dual picture was discussed in [50] where itsidentification as the singleton representation in AdS3 wassuggested and the universality class of the multi-Reggeonsystem was clarified. The dual picture behind the pomeronstate was discussed in [84]. We shall conjecture on theinterpretation of the Reggeon degrees of freedom in the KSgravity framework.

Recall that the Reggeon field VðxÞ enters the effectiveLagrangian being coupled to the semi-infinite lightlikeWilson line [51] playing the role of the source

Lint ¼ � 1

g@þP exp

��g

2

Z xþ

�1Aþdx�

�@2V�

� 1

g@�P exp

��g

2

Z x�

�1A�dxþ

�@2Vþ; (73)

where xþ, x� are the light-cone coordinates and A is theconventional gluon field. That is, according to the gauge/string duality it is natural to lift the Reggeon field to thefield in the bulk. Hence the correlator of the lightlikeWilson lines could be derived by differentiation of thebulk Reggeon action with respect to the boundary valuesof the Reggeon field. It is this line of reasoning that wasimplied in [50] when interpreting the Reggeon as thesingleton in the bulk action.

The Reggeon field does not transform under the localgauge transformation, however, it carries the global colorcharge and therefore interacts with the conventional gluon.It is this interaction amounts to the Balitskii-Fadin-Kuraev-Lipatov Hamiltonian governing the t ¼ logs evolution ofthe pomeron state [85] and the corresponding multi-Reggeon BKP generalization [86]. Note that the situationis reminiscent of the standard interplay between the localand global symmetries in the brane picture. In the color Nc

branes worldvolume theory the gauge symmetry on theflavor NF branes is seen as the global flavor symmetry. Theopen strings connecting the color and flavor branes carrythe global flavor number.

We could conjecture that the Reggeized gluon can beidentified with the open string stretched between two regu-lator branes. That is, it can be considered as the massivevector ‘‘gauge’’ particle for the ‘‘flavor’’ gauge group onthe set of the IR regulator branes. Such a Reggeon field,

indeed, plays the role of the source on the regulator braneworldvolume and therefore the Wilson polygon on theregulator brane worldvolume presumably can be derivedupon differentiating the classical action in the bulk over theboundary values of the Reggeon field.The Reggeized gluon emerges upon the resummation of

the perturbative series, hence, one could try to identify theparticular limit in the KS framework which could providethe multi-Reggeon dynamics of the all-loop amplitude inthe generic kinematics. To this aim it is useful to comparethe integrable structures at the generic kinematics we dis-cussed above and the one responsible for the Regge limit[1,2].The Regge limit is described in terms of the SLð2; CÞ

spin chains when the number of sites in the chain corre-sponds to the number of Reggeons. The possible limitwhich could yield such a spin chain from the Faddeev-Volkov model or statistical model [46] looks like thefollowing. In the model [46] the statistical weights dependon the sum of the local rapidities and the spin variables. It isclear that one can not expect the semiclassical limit of thequantum dilogarithm to be relevant since the Reggeizationof the gluon happens upon the nontrivial resummation ofthe perturbation series.Fortunately there is the limit [46] corresponding to the

strong coupling region in the Liouville theory when thequantum dilogarithms reduce to the ratio of gamma func-tions depending on the SLð2; RÞ spin variables

�cb!0ðsþ xÞ / �ð1� sþ ix=2Þ�ð1� s� ix=2Þ ; (74)

where jbj ¼ 1 and x is the rescaled local rapidity. Theleading argument depends on the difference of twoinfinite-dimensional representations in the neighbor sitesand the expression coincides with the fundamental R ma-trix involved into the SLð2; RÞ spin chains. That is, in thisparticular limit we get the statistical weights or R matricesdepending only on the SLð2; RÞ spins similar to theBalitskii-Fadin-Kuraev-Lipatov–type Hamiltonian whilethe local rapidity yields the time variable logs. Note thatclearly this suggestive argument needs for furtherclarification.Another possible limit which can be compared with is

the semiclassical limit of the multi-Reggeon system whichis described in terms of the higher genus Riemann surfaceof the type

y2 ¼ P2NðxÞ � 4x2N; (75)

where N is the number of Reggeons and PN is the N-thorder polynomial depending on the higher integrals ofmotion. It was shown [50] that the Reggeon system be-longs to the same universality class as the N ¼ 2 SYMwith Nf ¼ 2Nc at the strong coupling orbifold point. In

that case the brane geometry behind the low-energy effec-


125002-14

tive action is known [5] and the theory is realized on theM5 brane with worldvolume ðR3;1;�Þ where the surface �lies in the internal space.

In the scattering geometry it is known [50] that thespectral curve of the integrable spin chain is embeddedinto the complexified ðx; pÞ space where x is the coordinatein the conventional Minkowski space. On the other handwe have discussed the geometry in the internal space ðu; vÞwhich can be roughly thought of as the T�Mð0;4Þ. The twoviewpoints can be matched if we use the realization of theT�Mð0;4Þ as the T�ðMc==TÞ; hence, we indeed can try to

treat the spectral surface as the submanifold in the com-plexified phase space.

Some comments on the role of the SLð2; CÞ group is inorder. It is just the group of the Lorentz rotations which actboth in the coordinate and the momentum space. In the Amodel the set of Lagrangian branes yields the knots andthere is the natural action of SLð2; CÞ on the knot comple-ment. That is, the SLð2; CÞ holonomies around the bound-ary torus yields the degrees of freedom in the spin chainmodel. In the B-model side one can try to relate the groupwith the SLð2; RÞ structure which has been consideredwithin the lifting of KS theory on the Riemann surface tothe three-dimensional CS theory. The derivation of the spinchain system from the set of Wilson lines in the CS theorywas discussed a long time ago [87] and the similar deriva-tion could be expected here as well.

VIII. DISCUSSION

In this paper we have suggested the relation between themultiloop MHV amplitudes and effective gravity on themoduli spaces provided by the kinematical invariants ofthe scattering particles. This viewpoint allowed us to sug-gest that the relevant integrability pattern and amplitudeswere treated as the fermionic current correlators on themoduli spaces. The key idea is that the scattering processinduces the back-reaction on the geometry of the momen-tum space through the nontrivial dynamics on the emergingmoduli space. That is, one can say that the tree amplitude isdressed by the effective gravitational degrees of freedomwhich can be treated within the Kahler gravity in the A-type geometry or KS gravity in the B-model–type. Theyare identified with the coordinates of Lagrangian branes inthe A model or the corresponding noncompact branes inthe B model. These branes serve as the effective IR regu-lators in the theory. On the field theory side the correlatorof the four fermion currents on the moduli space is iden-tified with the two-mass easy box amplitude which is thebasic block in the whole answer. Within the conventionalcalculation of the Feynman diagrams the relevant modulispaces are parametrized by the Schwinger or Feynmanparameters.

The BDS ansatz corresponds to the semiclassical limit inthe effective gravity and �cusp has to be identified with the

effective inverse Planck constant in KS gravity. The ansatz

has to be modified and our proposal suggests severalnatural directions of its generalizations. First, one couldimagine that the quantization parameter can be generalizedto a more complicated function than the cusp anomalousdimension which would respect the S duality of the N ¼ 4theory. The next evident point concerns the full quantumtheory in the gravity framework which effectively substi-tutes the dilogarithm function in the BDS ansatz by thequantum dilogarithm. However, these modifications do notproduce proper higher polylogarithms which are known toappear in higher loop calculations of the amplitudes andWilson polygons. The most natural way to get desiredhigher polylogarithms in our picture is to take into accountthe nontrivial Feynman diagrams in the two-dimensionalKS theory, which would probably involve loops. Indeed, inincreasing the number of vertices in the KS tree diagramswe increase the transcendentality of the answer. We expectthat all mentioned generalizations are necessary to be takeninto account to get the correct all-loop answer.We have identified the most natural integrable structure

behind the scattering amplitudes which are considered asan example of the wave functions in the particular model.The KS gravity in our case naturally involves the 3-KPhierarchy and the roles of the time variables are played bythe combination of the conformal cross ratios expressed interms of the external momenta. The second finite-dimensional integrable system is conjectured to be relatedto the Faddeev-Volkov model, however, this point deservesfurther investigation. The integrability is responsible forthe conservation laws in addition to the dual superconfor-mal symmetry. The relevant Ward identities correspond tothe area preserving the symplectomorphisms of the spec-tral curve similar the considerations discussed previouslyin the c ¼ 1 model.The additional IR regulator branes added into the picture

are responsible for the blowup of the internal momentumspace in the manner dictated by the scattering process. Theblowup of the internal geometry physically corresponds tothe IR regularization of the field theory and the anomaly inthe transformations in the momentum space tells that theregularization does not decouple completely. This some-what surprising picture implies that we have to take intoaccount the dynamics of the regulator degrees of freedomas well. It is highly desirable to develop the microscopicderivation of these IR branes. One can imagine that suchbranes emerge upon the peculiar summation of the non-commutative instantons in the effective Abelian targetspace description of our simplified C3 geometry.Naively, IR regulators are treated semiclassically but

generally the fermion currents representing the regulatorbranes obey the quantum Baxter equation. It is clear thatthe discrete Liouville model plays the important role in thewhole picture providing the particular gravitational dress-ing of the operators involved. We expect that these discreteLiouville modes correspond to the remnant of the repar-


125002-15

ametrization of the Wilson polygons coming from thecusps. The regulator branes to some extent play the rolesimilar to the Liouville walls in the c ¼ 1 model.

We expect that our treatment of the scalar box functionimplies that the hidden structure behind the gauge boxdiagrams holds for the non-MHV case as well. In particularwe expect the non-vanishingnonvanishing all-plus ampli-tude in the usual YM theory which is anticipated to be ofthe anomalous nature since long corresponds to the purelyanomalous part of the algebra of the symplectomorphysmsof the spectral curve.

One of the most inspiring findings of the paper is theappearance of the hidden ‘‘new massive degree of free-dom.’’ They correspond on the A-model side to the D2brane wrapped around the 2-cycle created by the scatteringstates or the open string stretched between two IR regulatorbranes in the B-model. It is somewhat similar to the W-boson or monopole states in the Seiberg-Witten descriptionof the low-energy effective action of N ¼ 2 theory how-ever its ‘‘mass’’ is fixed by the kinematical invariants of thescattering particles. It would be very interesting to developthis reasoning further and determine the correspondingwalls of marginal stability in the space of the kinematicalinvariants. In the Regge limit we anticipate its importantrole in the Reggeon field theory. We plan to elaborate thisissue further elsewhere.

It is evident that our proposal requires clarifications inmany respects. In particular the clear understanding of theamplitudes of the gluon scattering with generic chiralitiesis absent yet and our conjecture for the improvement of theBDS ansatz requires further evidence. Nevertheless webelieve that the dual representation of amplitudes in termsof the dynamical systems on the moduli space of theregulator branes we have suggested is the useful steptoward the derivation of the dual geometry responsiblefor the summation of the perturbative series in the SYMtheory.

ACKNOWLEDGMENTS

I would like to thank J. Drummond, E. Gorsky, D. Gross,S. Gukov, G. Korchemsky, A. Levin, V. Mikhailov,N. Nekrasov, A. Rosly, M. Shifman, E. Sokatchev,A. Tseytlin, A. Vainshtein, M. Voloshin, V. Zakharov,and A. Zhiboedov for useful discussions. Parts of thiswork have been done during the programs ‘‘ StrongFields, Integrability and Strings’’ at INI, Cambridge;‘‘From strings to things’’ at INT, Seattle; and ‘‘Non-Perturbative Methods in Strongly Coupled Gauge theo-ries’’ at GGI, Florence. I would like to thank their organ-izers for hospitality and support. I would like to thank alsoIHES, SUBATECH, FTPI at the University of Minnesotaand Universite Paris XI for their kind support while theparts of the work were done. The work was supported inpart by Grants No. RFBR 09-02-00308 and PICS-07-0292165.

APPENDIX

To discuss the multiloop calculations it is useful toutilize the geometrical picture behind the one-loop calcu-lations which we shall review in the following [19]. Thereexists the explicit map of the box diagram to the hyperbolicvolume of the particular simplex build from the kinemati-cal invariants of the external momenta. Introduce theFeynman parametrization of the internal generally massivepropagators with the parameters �i. If one considers theone-loopN-point function with the external momenta pi inD space-time dimensions it can be brought into the usualform

JðD;p1; . . .pNÞ /Z 1

0. . .

Z 1

0

Yd�i

�X�i � 1

�

��X

�2i m

2i þ

Xj<l

�i�jmjmlCil

�D=2�N

;

(A1)

where

Cjl ¼m2

i þm2l � k2jl

2mjmi

; kij ¼ pi � pj; (A2)

and mi is the mass in the i-th propagator.It is possible [19] to organize for the generic one-loop

diagram the N-dimensional simplex defined as follows.First introduce the N mass vectors miai, where ai are theunit vectors. The length of the side connecting the i-th and

j-th mass vectors isffiffiffiffiffiffikij

p; that is, one can define the

momentum side of the simplex. Therefore the

N-dimensional simplex involves NðNþ1Þ2 sides including N

mass sides as well as NðN�1Þ2 momentum sides. At each

vertex N sides meet and at all vertices but one there are onemass side and (N � 1) momentum sides. The volume ofsuch a N-dimensional simplex is given as follows:

VðNÞ ¼ ðQmiÞffiffiffiffiffiffiffiffiffiffidetC

pN!

: (A3)

There are (N þ 1) hypersurfaces of dimension (N � 1),one of which contains only momentum sides and can berelated with the massless N-point function.Upon the change of variables, the loop integral gets

transformed into the following form:

JðD;p1; . . .pNÞ /Y

m�1i

Z 1

0. . .

Z 1

0

Yd�i

� ð�TC�� 1Þ�X �i

mi

�N�D

: (A4)

It is useful to introduce the content of the N-dimensional

solid angle �ðNÞ subtended by the hypersurfaces at the

mass meeting point. It turns out that �ðNÞ coincides withthe content of the (N � 1) dimensional simplex in the


125002-16

hyperbolic space whose sides are equal to the hyperbolicangles �ij defined at small masses as follows:

Cij ¼ cosh�ij: (A5)

Then the integral for the case D ¼ N acquires the follow-ing form:

JðN; p1; . . .pNÞ ¼ i1�2N �N=2�ðN=2Þ�ðNÞ

N!VðNÞ ; (A6)

hence the calculation of the Feynman integral is nothingbut the calculation of the hyperbolic volume in the properspace. The case N � D can be treated similarly with somemodification [19].

To avoid IR divergence it is useful to start with the boxdiagram with all off-shell particles; that is, D ¼ N sim-plices in the hyperbolic space:

Jð4; p1; p2; p3; p4Þ ¼ 2i�2�ð4Þ

m1m2m3m4

ffiffiffiffiffiffiffiffiffiffidetC

p : (A7)

Since all internal propagators are massless, in our case weget the ideal hyperbolic tetrahedron whose vertices are atinfinity. In the massless limit we get

ðm2i m

22m

23m

24 detCÞmi!0 ¼ 1

16�ðk212k234; k213k224; k214k223Þ;(A8)

where the Kallen function �ðx; y; zÞ is defined as

�ðx; y; zÞ ¼ x2 þ y2 þ z2 � 2xy� 2yz� 2zx; (A9)

andffiffiffiffiffiffiffiffi��

pis just the area of the triangle with sides

ffiffiffiffiffiffiffiffiffiffiffiffiffik212k

234

q,


224

q,


223

q. The hyperbolic volume of the ideal

tetrahedron under consideration reads as

2i�ð4Þ ¼ Cl2ðc 12Þ þ Cl2ðc 13Þ þ Cl2ðc 23Þ; (A10)

where the dihedral angles are defined via the kinematicalinvariants

� cosc 12 ¼ k213k224 þ k214k

223 � k212k

234ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k213k223k

214k

243

q ; (A11)

� cosc 13 ¼ k214k223 þ k212k

243 � k213k


k214k223k

212k

243

q ; (A12)

� cosc 14 ¼ k212k234 þ k213k

224 � k214k


k213k224k

212k

243

q ; (A13)

and c 12 ¼ c 34, c 13 ¼ c 24, c 14 ¼ c 32. The func-tions involved are defined as

Cl2ðxÞ ¼ Im

�Li2ðeixÞ

�¼ �

Z x

0dy lnj2 siny=2j: (A14)

In the case of the two-mass easy box diagram defining theone-loop MHV amplitude, the additional simplification ofthe kinematical invariants happens since two external par-ticles are on the mass shell. In this case the arguments ofthe Li2 function degenerate to the conformal ratios of fourpoints.

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supersymmetric Yang-Mills theory from quantum geometry of momentum space