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\title{Amplitudes in N=4 SYM from the
Quantum geometry of the Momentum Space}

\classification{04.65.+e, 11.25.-w}
\keywords{quantum geometry,super Yang-Mills}
\author{A. S. Gorsky}{address={ITEP, Moscow, Russia, B. Cheryomushkinskaya 25}}

%\date{}
\maketitle

\begin{abstract}
We  discuss  loop MHV
amplitudes in the $N=4$ SYM theory in terms of the
effective gravity in the momentum space with the IR regulator
branes as degrees of freedom.
Rapidities of  external particles
yield the moduli space of  complex structures
providing the playground for the Kodaira-Spencer(KS) type gravity.
We suggest the fermionic
representation for the loop MHV amplitudes in the $N=4$ SYM theory assuming the
identification of
KS fermions with the IR regulator  branes in the B model.
The two-easy mass box diagram is treated as the four fermion correlator
on the spectral curve and it plays the role of
a building block in the whole picture. The
BDS anzatz has the interpretation as  semiclassical limit
of a fermionic correlator.
It is argued that   fermionic representation implies
integrability on the moduli spaces
which fixes the dependence of the amplitudes on the cross-ratios
of the external momenta.
\end{abstract}
\section{Introduction}

The $N=4$ SYM theory provides a possibility to recognize  some
features of the theories with less amount of SUSY. While
$N=4$ SYM  is far from the QCD-like theories in the infrared because of
the lack of confinement it shares  common features in  UV region
where  physics in asymptotically free theories is described
within a perturbation theory. That is considering the perturbative
expansion in $N=4$ SYM  coupling constant which does not run we
could try to clarify some generic properties  of the perturbative
expansion in the gauge theories.

It is of the prime importance  to discover any hidden symmetries at the
high energies or equivalently hidden integrable structures providing
the nontrivial conservation laws restricting the form of the
scattering amplitudes. In the four-dimensional setup  the
integrability behind the amplitudes is known  only at the Regge
limit when the SL(2,C) spin chain gets materialized
\cite{lipatov1,fadkor}(see \cite{bbgk} for review).




The simplest objects  at  generic kinematics are the MHV
amplitudes which are the perfect starting point for any discussion
since at the planar limit they can be described in terms of the
single kinematical function. Even at the tree level MHV amplitudes
\cite{parke} enjoy some remarkable properties. They are localized on
the complex curves in the twistor space \cite{witten}  and can be described as the correlators of chiral
bosons on the genus zero Riemann surface \cite{nair}. It turns out
that the generating function for the tree MHV amplitudes is just the
particular solution to the self-duality equation in YM theory \cite{bardeen,rs}.
It substitutes the naive superposition of the plane waves of the
same chirality in a nonlinear theory.  Moreover this solution
provides the symplectic transformation \cite{gr}(see also
\cite{mansfield}) of the YM theory in the light-cone gauge
formulation  into the so-called tree MHV Lagrangian formulated in
\cite{mhv} which to some extend is the analogue of the t'Hooft
effective vertex generated by instantons. However this approach
becomes less clear when going to higher loops. Indeed, the attempt
to formulate the one-loop MHV amplitudes in a twistor-like manner  was not
successful \cite{csw} and certainly calls for additional insights on the problem.









More recently Bern, Dixon and Smirnov (BDS) have formulated the  conjecture
\cite{bds} that all-loop MHV amplitudes get exponentiated and
factorize into  IR divergent and  finite parts. Moreover it
was conjectured that the finite part of all-loop amplitude involves
only all-loop cusp anomalous dimension $\Gamma_{cusp}(\alpha)$ and finite part of one-loop amplitude.
Inspired by this conjecture Alday and Maldacena have calculated the
amplitude at  strong coupling regime via  minimal surfaces in
$AdS$-type geometry with the proper boundary conditions \cite{am}.
They have found  unexpected relation between the MHV amplitudes in planar
limit of $N=4$ SYM theory and Wilson polygons in the momentum space.



The Wilson polygon-amplitude duality  refreshes the
problem but deserves for the explanation itself. It was originally
formulated at  strong coupling when the Wilson loop is calculated
in terms of minimal surface in the $AdS_5$ geometry upon a kind of
T-duality transform.  Later it was
shown that duality holds true at the perturbative regime as well
\cite{drum1} which puts it on  more firm ground. The important point was the formulation of the
anomalous Ward identities for the special conformal transformations
with respect to the dual conformal group.
It fixes the kinematical dependence of the amplitudes up to five
external legs \cite{drum1}. However  Ward identities tell
nothing about the functional form of the amplitudes  starting
from six external legs. Recently the dual superconformal
group was identified as the symmetry of the worldsheet theory
of the superstring in $AdS_5\times S^5$ geometry \cite{bermal, beisert}.







Finally it was recognized that   BDS anzatz fails  at weak coupling
at two loop level for six external legs \cite{six1,six2} and at strong
coupling  for infinitely large number of external legs.
Moreover the BDS anzatz seems not fit well with the Regge limit
\cite{regge}. On the other hand at two loop level the duality between
Wilson polygon and MHV amplitude survives.


There  are a lot of pressing questions to be answered. Just  mention
a few;
\begin{itemize}


\item{Is there some geometrical picture behind the BDS anzatz which would
suggest the way of its necessary generalization?}
\item{Is there the generalization of the dual conformal Ward identity
which would fix the functional form of the one-loop amplitude for
any number of external legs?}
\item{Is there the fermionic representation for the loop amplitudes
which would imply the hidden integrability?}
\item{What is the
origin of the Wilson polygon - amplitude duality ?}
\item{Is there clear geometrical picture behind the
reggeization of the gluon?}

\end{itemize}


To some extend we shall try to generalize the geometrical  picture for the tree
amplitudes suggested in \cite{witten}. At the tree level in \cite{witten}
the Euclidean D1 "instanton" branes with the attached open strings have been
considered in the twistor space. The D1 brane is localized
at the point in the Minkowski space in agreement with the
locality of the vertex generating tree MHV amplitude in the MHV formalism. To describe
the loop amplitudes we shall adopt a little bit different picture
and consider $C^4$ manifold in the B model as a "twistor-like" manifold for the complexified
Minkowski space. The   D3 branes
substitute "D1 instantons"  and are embedded in $C^4$.
The somewhat similar objects
were also introduced as the IR regulator branes in the
Alday-Maldacena calculation. Indeed, it was shown \cite{am} that dilaton field
gets changed upon the T-duality in the RG radial coordinate which means
that D-instanton is added to the background. After the Fourier transform
along flat four-dimensions D-instanton gets transformed into the D3 brane
we shall work with.
The Wilson polygon which corresponds
to the boundary of the string worldsheet and is presumably dual to the amplitude is
located just on these IR regulator branes.
Contrary to the  previous considerations the positions
of the regulator branes are not free but determined dynamically in terms of the
cross-ratios of the external momenta.



The physics of the scattering at the loop level can be treated from the different perspectives.
>From the point of view of the KS gravity on the moduli space we are calculating the correlator
of the fermions or the
fermionic currents  which can be identified with the
tau-function of the 2d integrable system. The second viewpoint
concerns  the 4d gauge theory on the regulator
branes whose number  is fixed by the number of external
particles. Finally one could consider the worldsheet viewpoint
where the regulator branes provide the proper boundary conditions
for the string.
These viewpoints are complimentary and allow to check the self-consistency
of our approach.

Within the KS perspective we shall discuss the
fermionic representation behind the loop MHV amplitudes which would
generalize the Nair's fermionic representation for the tree
amplitudes. The fermionic picture is a heart of the integrability
which admits the representation in terms of the chiral fermions on
the Riemann surface in the external gauge field. The gauge field
on the Riemann surface represents the "point of Grassmanian" or in physical terms the
particular Bogolyubov transformation between the fermionic vacua.
This approach was summarized in \cite{vafa}.
It was argued that fermions in the KS gravity correspond to mirror
of Lagrangian branes  in the A model. These branes are also
refereed to as Kontsevich or
noncompact branes and their positions on the Riemann surface yield
the  "times" in the corresponding integrable systems. Note  that in
the framework of the topological strings in A-model we discuss the
Kahler geometry while in B-model the complex geometry is captured by
the KS \cite{kodaira} theory.

















The fermion
one-point function corresponds to the Baker-Akhiezer
function in the integrable system framework and to the single regulator
brane insertion at some point on the moduli space. Since generically we are interested in the quantum integrable system
the Riemann surface gets quantized and yields the corresponding Baxter equation
\cite{sklyanin}. The
semiclassical solutions to the Baxter equation  which are the generating functions
for the Lagrangian sub-manifolds in the particular integrable system  play
important role
in the analysis. They serve as the building blocks for the
correlators in the $N=4$ YM theory and can be considered as
the "semiclassical D3 brane wave
function" or  as the
effective action in the 4d gauge theory on the brane worldvolume.
>From the moduli space  viewpoint the solution to
the Baxter equation provides the generating function of the
Lagrangian sub-manifold.
The natural integrable system on the moduli space can be identified with the  3-KP system however similar to the N=2 SYM one could expect the
pair of integrable system - 2D field theory and finite dimensional one.
The natural finite dimensional integrable system which is responsible for the hidden
symmetries at the generic kinematics is conjectured to be related
to the Faddeev-Volkov model \cite{volkov} and the corresponding statistical
model \cite{bms} based on the discrete quantum conformal
transformations.

Since  we are trying to sum  the perturbation series
the YM coupling constant is expected to be involved into
some algebraic structure behind the all-loop answer. It is this
hidden symmetry which provides the choice of the particular solution
to the Yang-Baxter equation. The Faddeev-Volkov solution to
the Yang-Baxter implies that we are actually trying to relate the
YM coupling constant with the parameter $q$ of $U_{q}(SL(2,R))$.
The proper identification turns out to be nontrivial problem since
in particular
it has to respect the S-duality group in N=4 theory. It will
be argued that the BDS anzatz corresponds to the limit
$q\rightarrow 1$ while the Regge limit seems to be related
to the opposite "strong coupling regime" of the quantum group.





The consideration of the four-dimensional
theories on the regulator brane worldvolume is useful as well. The
theory is in the Coulomb phase and the position of the regulator brane on
the particular Riemann surface corresponds to the coordinate on the Coulomb moduli space.
Since all regulator D3 branes are at different positions on the moduli space the theory generically has the gauge group $U(1)^k$ where k is related
to the number of the external gluons .
The effective action of each U(1) gauge theory plays the role
of the wave function of the two-dimensional fermions in KS gravity.
A little bit surprisingly one has to consider not the real part
of the effective action in the external field but the imaginary
one involving dilogarithm. This is natural from the
Euclidean viewpoint while in the Minkowski space we actually
consider the probability of the pair production.





It is important to discuss separately the special Regge kinematical
region were the hidden symmetries of the amplitudes where found
for the first time. The hidden symmetries were captured at one loop
by the SL(2,C) spin chains \cite{lipatov1,fadkor}. It was shown in \cite{gkk}
that the N-reggeon dynamics belongs to the same universality class
as conformal N=2 SQCD with $N_f=2N$ at the strong coupling orbifold point.
We shall argue that the brane geometry in the  reggeon case is similar
to the one in SQCD which provides  the qualitative
explanation of the same universality class for both theories.
The new object  is the open
string stretched between two regulator branes and is the
analogue of the massive vector bosons in the conventional N=2 SYM
theory. Here we shall tempt to interpret these open strings
as the "reggeons". The masses of these effective degrees of freedom
correspond to the differences of the positions of the
regulator branes on the Riemann surface.





























\section{The loop results for the MHV amplitudes}

Let us remind the main results concerning the loop MHV amplitudes.
The  MHV gluon
amplitudes  involve two gluons of the negative chiralities and
the rest of gluons have  positive chiralities. Consider the ratio of
all-loop and tree answers. The following form of the all-loop amplitudes has been
suggested in \cite{bds}

\beq log(\frac{M_{all=loop}}{M_{tree}})= (F_{div}+
\Gamma_{cusp}(\lambda) M_{one-loop}) \label{af} \eeq which involves
only the all-loop answer for the cusp anomaly $\Gamma_{cusp}$ and
one-loop MHV amplitude. The IR divergent part $F_{div}$ gets
factorized in the all-loop answer. The cusp anomaly measures  UV
behavior of the contour with cusp \cite{polyakov}. Recently the
closed integral equation has been found for the cusp anomalous
dimension in $N=4$ SYM theory \cite{bes} which correctly reproduces
the weak and strong coupling expansions.

The finite part of the one-loop MHV which presumably defines the all-loop
answer  can be written in terms of the finite part of the
so-called two-mass easy box function $F^{2em}$ \cite{easy}


\beq M_{one-loop,finite}= \sum_{p,q} F^{2em,f}(p,q,P,Q)\eeq This
function can be expressed in terms of the dilogarithms only

\beq F^{2em,f}(p,q,P,Q)= Li_2(1-aP^2) + Li_2(1-aQ^2)
- Li_2(1-a(q+P)^2) - Li_2(1-a (p+P)^2) \eeq
where

\beq a=\frac{P^2 + Q^2 -(q+P)^2 -(p+P)^2}{ P^2 Q^2 -(q+P)^2(q+P)^2}
\eeq
and $p+q+P+Q=0$. One more expression for the function $F^{2em,f}$
which will be useful later can be written in terms of the variables
$x_{i,k}=p_i-p_k$ in terms of  the sums \cite{drum1}

\beq \sum_i \sum_r Li_2 ( 1-
\frac{x_{i,i+r}^2x_{i=1,i+r+1}^2}{x_{i,i+r+1}^2x_{i-1,i+r}^2})\eeq
where
\beq
x_i=p_{i+1}-p_i \eeq
Since all external momenta are on the mass shell the arguments of
dilogarithms are expressed in terms of the cross-ratios of the
scalar products of the momenta only.






The BDS anzatz (\ref {af}) has been checked at weak and strong
coupling regimes. At strong coupling  analyzed in the
stringy setup \cite{am} one considers first the T-duality
transformation on the worldsheet   which effectively interchanges UV
and IR regions in the $AdS_5$ geometry. Then the calculation of the
amplitude reduces to the calculation of the minimal surface in the
dual AdS space bounded by the polygon formed by the external
on-shell gluon momenta. For the four external legs the answer fits
with the BDS anzatz for all-loop amplitude.

It was conjectured in \cite{am} that any MHV  N-leg amplitude
follows from the vacuum expectation value of the Wilson loop of the
special form \beq \frac{M_{all-loop}}{M_{tree}}=<W(p_1,p_2,...,p_N)>
\label{am} \eeq where the closed Wilson loop polygon has light-like
momenta at the edges and vertexes at $x_i$. Its closeness is provided
by the total momentum conservation.

At weak coupling to check this polygon-amplitude duality one
considers the expansion of the Wilson polygon in the YM coupling
treating Wilson loop as one in the coordinate space. The perfect
matching of Wilson loop and amplitudes has been found for one- and
two loop answers up to six external legs \cite{six1,six2}. Moreover it
was demonstrated that the anomalous Ward identities for the special
conformal transformations of the form

\beq K^{\nu}W(x_1,\dots x_N)= \sum_{i=1}^{n}(2x_i^{\nu}x_i \partial_i -x_i^2 \partial_i^{\nu}) W(x_1,\dots x_N)          =
\frac{1}{2}\Gamma_{cusp}\sum_{i=1}^n ln \frac{x_{i,i+2}^2}{x_{i-1,i+1}^2} x_{i,i+1}^{\nu}
\eeq
fix the answer up to four
external legs \cite{drum07}.
The BDS anzatz has to be modified for generic amplitude while
the Wilson polygon-MHV amplitude duality has the chance to be
all-loop exact.
\section{Finite part of N=4 SYM MHV amplitudes and momentum space geometry}

\subsection{Fermionic picture}
Let us now consider the four-dimensional case and formulate our
proposal for finite part of the MHV loop amplitudes.
Remind that the tree amplitudes were described in terms of the
D1  string instanton embedded into the twistor manifold \cite{witten}.
The instanton is localized at point in the Minkowski space and open strings
representing gluons are attached to  it. To describe the loop
amplitude we shall substitute D1 brane by the IR regulator  branes
embedded into the proper manifold. The gluons are attached
to the regulator branes whose embedding coordinates
are considered as dynamical degrees of freedom.
Contrary to tree case  regulator branes are localized at the sub-manifold of the
complexified Minkowski space.


%\subsection{The brane picture}
The starting
point is the representation of the N=4 theory via geometrical
engineering \cite{vafageom} as the IIA superstring compactified on the
three-dimensional Calabi-Yau manifold which was identified as  the
$K3\times T^2$ geometry in the singular limit. One has to
consider the singular limit of K3 manifold when it develops $A_{N-1}$
singularity, where N becomes the rank of the gauge group, and upon
blowing up procedure it can be represented as $ALE_N$ geometry. On
the other hand the Kahler class of the $T^2$ can be identified with
the coupling constant \beq Area(T^2)=1/g_{YM}^2 \eeq At weak
coupling  the torus is large and can be approximated by the complex
plane.  That is the geometry can be roughly approximated by $C^3$
upon the particular blow-ups.

As we have seen the one-loop answer for the MHV amplitude
determining  the BDS form of the amplitude involves the sum
of the dilogarithms depending on the cross-ratios of the
$x_i$ variables. Below we shall try to explain how such
functions  with cross-ratio arguments emerge naturally both in A-model
and B-model frameworks. As is well-known the A-model captures
the information about the Kahler moduli while the B-model
about the complex moduli and we shall see where these moduli comes from.
The brane description of the scattering amplitude involves
the set of the Lagrangian branes in the A-model and the
corresponding B-model branes. It is these branes which provide the
corresponding moduli spaces.

























Let us interpret the BDS anzatz in
terms of the correlator of the noncompact Euclidean D3 branes
embedded into the four dimensional complex space. Consider
3d complex manifold  which is mirror to the  topological
vertex \cite{vertex}. This manifold  classically is
described by the equation in the  $C^4$ with coordinates $x,y,u,v$

\beq xy=e^{u} +e^{v}+1 \eeq
At the discriminant locus it defines  the Riemann surface
\beq H(v,u)= e^{u}+e^{v}+1=0
\label{surface}
\eeq
of genus zero with three different asymptotic regions.
We shall argue that the loop MHV amplitudes can be identified with the
fermionic correlators on the Riemann surface (\ref{surface}). Fermions on the surface
(\ref{surface}) represent the degrees of freedom in the KS gravity
that is the IR regulator D3 branes imbedded into
$C^4$ geometry.

There are two D3 branes  defined by the equations

\beq
x=0 \quad H(v,u)=0 \eeq
and
\beq
y=0 \quad H(v,u)=0 \eeq
which  intersect along the Riemann surface.
The intersecting branes provide
the natural fermionic degrees of freedom on the intersection surface
from the
open strings stretched between these branes.
The fermions are in  external field amounted from the worldvolume
gauge connection on the intersecting branes. In addition to  two  branes
intersecting along the Riemann surface we introduce the set of Kontsevich -like
branes classically localized at the points $(v_i,u_i)$ at the Riemann surface.
The number of such branes is fixed by the number of the external gluons
and the coordinates of these branes on the surface are defined by some particular
cross-ratios.
At quantum level D3 branes are extended along the Lagrangian submanifold
in the $(u,v)$ space. The cross-ratios are the natural coordinates
on the moduli space of the punctured spheres that is the $(u,v)$ space
is related to the $T^{*}M_{0,4}$.







At the next step the
Riemann surface gets quantized and the branes-fermions should obey
the equation of the quantum Riemann surface that is
Baxter equation which provides
the wave functions depending on the  separated variables.
The Baxter equation in our problem reads as
\beq (e^{\hbar \partial_v} +e^{v} +1)Q(v)=0 \eeq
Its solution  turns out to
be the  quantum dilogarithm \cite{vafa}. Note that the solution to the Baxter
equation in our case  can not be presented in the polynomial form that is we have infinite number of the Bethe roots.






To get the MHV all-loop amplitude in the BDS form we take the
semiclassical limit of the fermionic correlator on this surface.
Indeed using the semiclassical limit for the quantum dilogarithm
we can represent the four-point fermionic correlator as

\beq <\bar{\Psi}(z_1)\bar{\Psi}(z_2) \Psi(z_3)\Psi(z_4)>\propto exp(\hbar^{-1}(
Li_2(z_3)+ Li_2(z_4)-Li_2(z_1)-Li_2(z_2))\eeq
This expression exactly coincides   with the expression for the
contribution of the single 2-easy mass box diagram
hence upon  the identification of the Planck constant

\beq \hbar^{-1}=\Gamma_{cusp}(\lambda) \eeq
we reproduce BDS anzatz for the finite part of the amplitude. Indeed the one-loop
answer for the MHV amplitude can be expressed purely in terms of the sum
of 2-mass easy box diagrams with different grouping of the gluon momenta
and therefore in terms of the fermionic correlators.


Since the regulator brane ( D1 "instanton") yielding the tree amplitude is localized in
the complexified Minkowski space $M^c$ \cite{witten} one could ask about similar localization of regulator branes
responsible for the higher loop calculations. To this aim recall that
$M^c$ is equivalent
to the Grassmanian $Gr(2,4)$. On the other hand the factor
of the Grassmanian by the maximal torus action is related to the
compactified moduli space \cite{kapranov}
\beq
Gr(2,4)// T =\bar{M}_{0,4}
\eeq
This representation allows us to represent the complexified
Minkowski space
itself as the fancy divisor of the $M_{0,4}$ \cite{altmann}.
We suggest that this realization implies the localization
of the regulator branes on the submanifold of
$T^{*}(M^c//T)$. It is natural to identify
this manifold with the Riemann surface where the KS degrees
of freedom live.

Let us present the qualitative argument  concerning
the corresponding A-model picture.
In the A-model we introduce  the set of Lagrangian branes
with topology  $S^1\times R^2$. The emergence of the dilogarithm
as the wave function of the Lagrangian brane has been discovered
in the $C^3$ geometry in \cite{saulina}.
The brane/asntibrane  can be considered as the insertion of the
fermion/antifermion \cite{saulina}  in the fermionic representation of the
topological vertex picture \cite{vertex}.






\subsection{The regulator  brane worldvolume theory}

Since fermions in KS framework are identified as the D3 regulator
branes the natural question
concerns their four-dimensional worldvolume theory.
The theory on the regulator branes share many features
with N=2 and N=1 SYM low-energy sectors. The number of the regulator branes
is fixed by the number of the external gluons  so naively one could expect
a kind of $SU(K)$ gauge theory. The worldsheet theory on the regulator branes
enjoys the complex scalar
corresponding to  the complex coordinate $z$ of the brane on the Riemann surface (\ref{surface}).
This is similar to the situation when the vev of the scalar field corresponds
to the position of the D4 branes on the $a$-plane in the IIA realization
of the N=2 SYM theory \cite{wittenm}.

Since the different regulator branes are at the different points
on the Riemann surface  we can speak about the Coulomb 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 D3 branes on the points of the moduli space $M_{0,4}$. Similar to the N=1
SYM theory when branes are localized at positions corresponding to
the discrete vacua  the D3 regulator branes are localized at some
points parameterized by the cross-ratios. These
points correspond to the local rapidities in the framework of
integrability and simultaneously have to
correspond to the minima of the effective superpotentials $W_{eff}(z_i)$
in the regulator worldvolume theory.

Since we attributed dilogarithms to the regulator brane wave functions  it is necessary
to explain where they come from in the worldvolume theory. The qualitative arguments
looks as follows. In the worldvolume theory there are massive excitations corresponding
to the open strings stretched between two regulator branes.  They are analogue of
the massive W-bosons in N=1 SYM theory on the Coulomb branch. In our case the masses
of these particles are related to the cross-ratios. To recover the dilog let us remind
that usually in the external field the effective action develops the imaginary part
corresponding to the pair creation. The probability of the pair creation on the
external field is described by the classical trajectory in the Euclidean space
and in the leading approximation reads as
\beq
Im S_{eff}\propto e^{-\frac{m^2}{eE}}
\eeq
for a particle of the mass m in the external field E. Upon taking into account the multiple
wrapping and the quadratic fluctuations one gets for the scalar particle Schwinger pair
production
\beq
Im S_{eff}\propto \sum_{n} \frac{1}{n^2}e^{-\frac{nm^2}{eE}}
\eeq
that is dilog plays the role of the decay probability. Hence one can
say that we are considering the Euclidean version of the regulator
worldvolume theory and the amplitude from this viewpoint is described
via bounce type configuration corresponding to the creation of the
pairs of the effective massive degrees of freedom. Note that the real part
of the effective action corresponds to the summation over the loops
of the same degrees of freedom in the loops.


..













In the A-model one can similarly consider the worldvolume
theory on the D2 Lagrangian regulator branes. In this case the
corresponding dilog functions emerge upon summation over
the disc instantons with boundaries located at the corresponding
Lagrangian branes which provide the effective superpotential
in the worldvolume theory
\beq
W_{eff}\propto \sum_{n} \frac{d_n}{n^2} e^{-nA}
\eeq
where A -is the corresponding area of the target disc.
Note that in
the A model   D2 branes wrapped around the
ideal tetrahedrons whose Kahler classes are defined by the cross-ratios
provide the masses of the same effective "W-bosons" as in B-model.






%\subsection{$\hbar^{-1} = \Gamma_{cusp}(\alpha)$ ?}
Let us comment on the identification of the
Planck constant providing the quantization of the KS gravity
as the inverse cusp anomalous dimension inspired
by the BDS anzatz. At the first glance it looks completely groundless
however  the argument supporting this
identification goes as follows. The emergence of the cusp anomaly
in the exponent means from the worldsheet viewpoint that it plays the role of the effective
string tension or equivalently the inverse Planck constant.
Such effective tension emerges if one considers the string whose boundary
is extended along the light-like contours.
It was shown \cite{alday3} that in the limit suggested in \cite{bgk2}
the string worldsheet action can be identified with
$O(6)$ sigma model  and the energy of the ground state in $O(6)$ model
is proportional to the length of the string multiplied
by the $\Gamma_{cusp}(\alpha)$. That is indeed $\Gamma_{cusp}(\alpha)$
plays the role of the effective tension of the string in this special
kinematics. Since in our
case  the boundary of the string worldsheet lies on the Wilson polygon
the effective tension involving the
cusp anomalous dimension is natural.



However certainly this point is far
from being clarified. For instance in the Ward identity for the
special conformal transformation  $\Gamma_{cusp}$ enters as
the multiplier in the anomalous contribution. This claim has been
explicitly checked at the first loops in the gauge theory calculations
and the arguments that it holds true at all orders have been presented.
This means  that  in the anomalous Ward identity it plays the role
of the Planck constant not the inverse one. To match both arguments
we could suggest that in the Ward identity we are considering
the S-dual formulation and therefore the D1 string worldsheet action
instead of the F1 one in $O(6)$ sigma model. This would imply that
the Wilson polygon equivalent to the MHV amplitude could be
considered as the boundary of the D1 string as well.


In more general setup it is highly desirable to realize the meaning
of the relation of such type  in the first quantized language. Since the cusp anomalous
dimension is just the renormalization factor for the self-crossing of the
worldline it is very interesting to understand if such self-crossing
is involved into the quantization issue. In particular in the Ising model
the effect of the self-crossing is captured  by the topological term
and in the description of the topological string on $C^3$ somewhat
similar $\theta$  term in six dimensions plays the role of the quantization
parameter indeed \cite{foam}. In the gauge theory language such
objects are related to the renormalization of the double-trace operators
couplings.


























\section{Integrability behind the scattering amplitudes }
\subsection{General remarks}
In this Section  we shall
discuss the hidden integrability behind the scattering amplitudes and
present the arguments that similarly to the integrability pattern
behind effective actions in N=2 SYM theory
two integrable systems are involved. The degrees of freedom of both
integrable systems are related to the coordinates of the regulator branes.
One of these systems which we identify as the Whitham-like  3-KP one plays the
role of RG flows in the regulator brane worldsheet theory or equivalently
the motion of the regulator brane along the "radial" RG-coordinate.
The second integrable system generalizing the Hitchin-like or spin chain models
involves the effective interactions between the regulator branes. We shall
give arguments that this system is based on the Faddeev-Volkov
solution to the Yang-Baxter equation
for the infinite-dimensional representations of the noncompact $SL(2,R)$ group.

Recall  how  two integrable systems are involved into the description of the
low energy effective actions of N=2 SYM theories. The first finite dimensional system is
of the Hitchin or spin chain type and its complex Liouville tori
are identified with the Seiberg-Witten curves.
This spectral curve emerges in the gauge theory
upon the summation over the
infinite number of instantons \cite{nekrasov}.


Following \cite{duality} one can
canonically define the dual integrable system whose phase space is built on the
integrals of the motion of the first one. In the simplest
case of $SU(2)$ theory
the phase space for the dual system has the symplectic structure \cite{losev}
\beq
\omega= da\wedge da_{D}
\eeq
where the variables $(a,a_{D})$ are the standard variables in N=2 SYM framework
\cite{sw}. The prepotential $\cal{F}$  can be identified with the generating function
of the Lagrangian sub-manifold in the dual system with the $a,a_{D}$ phase space
\beq
H(a(u), \frac{\partial \cal{F}}{\partial a})) =u
\eeq
and obeys the Hamilton-Jacobi equation
\beq
 \frac{\partial \cal{F}}{\partial log \Lambda}=H
\eeq
In the brane setup the prepotential defines the semiclassical "wave function"
of the D4 brane $\Psi(a)\propto exp(\hbar^{-1} {\cal{F}}(a))$
in the IIA brane picture where perturbatively the argument
of the wave function can be identified with  coordinate
of the D4 brane on the NS5 brane. The total  perturbative
prepotential in $SU(N_c)$ can be considered as a sum of the exponential
factors in the product of the wave functions of  $N_c$
D4 branes. In the A-model side these wave functions can be considered
in the Kahler gravity framework and the arguments of the wave function
have to be treated as the Kahler classes of the blow-upped spheres.


The integrals of motion provide the moduli space of the complex structures
in the Calabi-Yau geometry in the B model hence
we are precisely in the KS framework. In this B-model
formulation we consider the argument of the brane wave function
as the coordinate on the moduli space of the complex structures.
The dual Whitham-type integrable system naturally
defines the $\tau$-function of the 2d Toda theory
formulated in terms of the
chiral fermions on the Riemann surface with two marked points.




















\subsection{3-KP system}

Let us turn to the integrable structure relevant for the scattering
amplitudes at generic kinematics and
first  identify the degrees of freedom
and evolution "times". As we have described
above   the fermionic degrees of freedom correspond to the
noncompact branes  localized on the Riemann surface. The
two-dimensional field theory corresponds to the reduction of the KS
theory on the two-dimensional surface.
The fields on the surface are in the
external abelian connection of the Berry type which tells how the B- branes
transform under the change of the complex structure fixed by the momenta
of external particles.



The form of the Riemann surface $H(u,v)=0$
dictates that there are three infinities and therefore we are dealing with the
particular solution to 3-KP  integrable system.
To describe the integrable system it is convenient
to introduce the chiral fermions
with the following mode expansion
\beq
\psi(x_i)= \sum_{n}\psi^{i}_{n+1/2}x_i^{-n-1},\qquad
\psi^{*}(x_i)= \sum_{n}\psi^{*i}_{n+1/2}x_i^{-n-1}\eeq
around the i-th infinity, $i=1,2,3$ and the
commutation relations
\beq
\{\psi_{n}^{i},\psi_{m}^{*j}\}=\delta^{ij} \delta_{n+m,o}
\eeq
Defining the vacuum state by relations
\beq
\psi_n|0>=0, \quad \psi_n^{*}|0>=0, \quad n>0
\eeq
 the generic state $|V>$ can be presented in the form
 \beq
 |V>=exp(\sum_{i,j} \sum_{n,m} a_{nm}^{ij}\psi^{i}_{-n-1/2}\psi^{*i}_{-m-1/2})|0>
 \eeq
 where the point of Grassmanian representing the topological vertex
 was derived in \cite{vafa}.




Hence we can define the classical $\tau$ function of the 3-KP system we are working with

\beq \tau(T_k)= <t|\Psi(z_1)....\Psi(z_k)|V_{tv}> \eeq

It is this tau-function of the 3-KP system that plays the role of the generating function
for the  MHV amplitudes. In fact the semiclassical limit of the tau-function
is of the most interest when we consider the classical Riemann surface before
any quantization. In the semiclassical approximation we can safely consider
the differential
\beq
dS=vdu
\eeq
which yields the semiclassical brane wave function
\beq
\Psi_{qs}\propto exp(- \hbar^{-1} \int ^{x} v(u)du)
\eeq
involving the dilogs. The tau-function obeys the
3-KP equation and there are the additional
$W_{1+\infty}$ Ward identity written in terms of the fermions
\beq
\oint_{u}\psi^{*}(u)e^{nu}\psi(u) +
(-1)^n \oint_{v}\psi^{*}(v)e^{nv}\psi(v) +
\oint_{s}\psi^{*}(s)e^{ns}\psi(s)=0
\eeq
where the sum over three asymptotic regions is considered.



The quantization of the system can be done most effectively
in terms of the Baxter equation.
The  Baxter equation implies that the regulator branes
are localized on the surface. Hence the whole set of the
equations determining amplitudes involve the dual
conformal transformations on the regulator worldvolume
and the set of Ward identities  for the coordinate
of regulator brane in the transverse moduli space. It is these
Ward identities which fix the dependence of the amplitude
on the conformal invariants for large number of external legs.

The precise higher Hamiltonians from $W_{1+\infty}$ responsible for the
higher conservation laws in the scattering amplitude problem can
be written as the fermionic bilinears \cite{vafa}.
Generically as was discussed in \cite{vafa} one
has some unbroken part of $W_{\infty}$ which
annulate the $\tau$-function corresponding to the
topological vertex and therefore
the scattering amplitude in the form of BDS anzatz.


\subsection{On the Faddeev-Volkov model}
Let us turn now to the description of the second integrable system
representing the particular solitonic sector of the infinite-dimensional
integrable system.
We shall conjecture that the integrable system at the generic kinematics
is the generalization of the SL(2,C) spin chain relevant for the
Regge limit of the amplitudes.

The finite-dimensional integrable systems can be usually defined in terms of the R-matrix.
The Faddeev-Volkov model is defined via the Drinfeld
solution  to the Yang-Baxter equation which provides
the  universal R-matrix acting on $U_{q}(SL(2,R))\otimes U_{q}(SL(2,R))$.
The corresponding statistical model describes the discrete quantum Liouville theory \cite{bms} with
the following partition function
\beq
Z=\int \prod_{ij}W_{p-q}(S_i-S_j)\prod_{kl}\bar{W}_{p-q}(S_k-S_l)\prod_{i} dS_i
\eeq
where the Boltzmann weights depend only on the differences of the spins $S_k$
at the neighbor cites and rapidity variables
at the ends of the edge. The first product is over the horizontal
edges(i,j) while the second product is over the vertical edges (k,l). The integral
is over all internal spin degrees of freedom. In the fundamental R-matrix
the cross-ratios of the relative rapidities of the particles play the
role of the local inhomogeneities in the lattice model and Boltzmann weights
are defined as \cite{bms}
\beq
W_{\theta}(s)= F(\theta)^{-1}e^{2\eta \theta s}\frac{\Psi(s+ic_b \theta /\pi)}
{\Psi(s-ic_b \theta /\pi)}
\eeq
where spin $s$ and local rapidity variables $\theta$ are combined together in  the argument of the function
\beq\Psi_{b}(z) = exp(\frac{1}{4}\int \frac{e^{-2izx}dx}{x sinh(bx)sinh(b^{-1} x)})\eeq
$ c_b= 1/2(b + b^{-1})$ and $F(\theta)$ is normalization factor. The relative importance
of the spin variables and the local inhomogeneities depends on the
value of the YM coupling constant and the kinematical region.



Semiclassically when $b\rightarrow 0$ the spin variables are frozen  and the Boltzmann weight behaves as
\beq
W_{\theta}(\rho/2\pi b))=exp(-\frac{A(\theta|\rho)}{2\pi b^2} + ...)
\eeq
where
\beq A(\theta|\rho)=iLi_2(-e^{\rho-i\theta}) -i Li_2(-e^{\rho+i\theta})
\eeq
The extremization of the semiclassical action yields
the Bethe Anzatz type equations connecting the dynamical spin variables
with the local rapidities
\beq
\prod_{i} \frac{e^{\rho_i}+e^{\rho_j +\theta_{ij}}}{e^{\rho_j}+e^{\rho_i +\theta_{ij}}}=1
\eeq

The Regge limit is described in terms of the
SL(2,C) spin chains when the number of sites
in the chain corresponds to the
number of reggeons. The possible limit which could
yield such spin chain from the Faddeev-Volkov model
or statistical model \cite{bms}
looks as follows. In the model \cite{bms} the statistical weights
depend on the sum of the local rapidities  and the spin variables.
It is clear that one can not expect the quasiclassical limit
of the quantum dilogarithm to be relevant since the reggeization
of the gluon happens upon   the nontrivial resummation of
the perturbation series.



Fortunately there is the limit \cite{bms} corresponding to the strong
coupling region in the Liouville theory when
the quantum dilogarithms  reduce to the ratio of Gamma
functions depending on the $SL(2,R)$ spin variables
\beq
\Psi_{c_b\rightarrow 0}(s+\eta x)\propto \frac{\Gamma(1-s+ix/2)}{\Gamma(1-s-ix/2)}
\eeq
where $|b|=1$ .
The leading argument depends on the difference
of two infinite-dimensional representations in the neighbor
sites and the expression coincides with the
fundamental R-matrix involved into the
$SL(2,R)$ spin chains.  That is in this particular limit we
get the statistical weights
or R-matrixes depending only on the SL(2,R) spins similar
to the BFKL-type Hamiltonian \cite{bfkl} while  the local rapidity  yields
the "time" variable $log s$. Note that clearly this
suggestive argument need for further clarification.









\section{Conclusion}
We have suggested the relation  between the loop
MHV amplitudes and the KS gravity in the momentum space
which allows us to recover the relevant integrability pattern.
The key idea is that the scattering of the particles induces
the back-reaction on the geometry of the momentum space through
the nontrivial dynamics on the emerging moduli space.
That is one can  say that the tree
amplitude is dressed by the gravitational degrees of freedom  which can be
treated within the Kahler gravity  in the A type geometry or
KS gravity in the type B model. They  are
identified with the coordinates of Lagrangian branes in the A model or the
corresponding noncompact
branes in the B model. On the field theory side the
four-fermion correlator on the moduli space  is identified with the two-mass
easy box amplitude which is the basic block in the whole answer.


The BDS anzatz corresponds to the
semiclassical limit in the KS gravity and $\Gamma_{cusp}$ has
to be identified with the inverse Planck constant in KS gravity.
There are
several natural generalizations of the BDS anzatz. First one could imagine that
the quantization parameter can be generalized to more complicated
function than cusp anomalous dimension respecting the S-duality of N=4 theory. The next
evident point concerns the  full quantum
theory in the KS framework which effectively substitutes the
dilogarithm function in the BDS anzatz by the quantum dilogarithm.
However these modifications do not produce higher polylogaritms
which are known to appear in higher loop calculations of the
amplitudes and Wilson polygons. The most natural way to get
higher polylogarithms in our picture is to consider the
nontrivial Feynman diagrams in the two-dimensional KS theory
probably involving loops. Indeed increasing the number
of vertexes in the KS tree diagrams we increase the
trancendentality of the answer. We expect that all mentioned
generalizations are necessary to get the correct all-loop
answer.







We have identified the most natural integrable structure
behind the scattering amplitudes which are considered as
a kind of the "wave functions" in the particular model.
The KS gravity in our case naturally involves
the 3-KP hierarchy and the role of the "time" variables
are played by the combination of the conformal cross-ratios.
The second finite-dimensional integrable system is conjectured
to be related to the Faddeev-Volkov model however this point
deserves for further investigation.
The integrability is responsible for the
conservation laws in addition to the dual superconformal
symmetry.
The relevant  Ward identities
correspond to the area preserving symplectomorphysms of the spectral
curve.


The additional IR regulator branes
added into the picture
are responsible for the blow up of the internal
momentum space in the manner dictated by
the scattering process. The blow up of the internal geometry
physically corresponds to the IR regularization of the field theory and
the anomaly in the transformations
in the momentum space tells that the
regularization does not decouple completely. This a
little bit surprising picture implies that  we  have to take into
account the dynamics of the regulator degrees of freedom as well.
Naively they are treated semiclassically but generically the fermions representing the
regulator branes obey the quantum  Baxter equation.








One of the most inspiring findings  is the
appearance of the hidden  "new massive degree of freedom". They correspond
on the A model side to the D2 brane wrapped around the 2-cycle created by the
scattering states or the open string stretched between
two IR regulator branes in the B model. It is somewhat similar  to the W-boson state
however  its mass is fixed by the kinematical invariants
of the scattering particles.
In the Regge limit  we anticipate its important role in the Reggeon field theory.





In is evident that the results of this paper are qualitative
in many respects  and represent only
part of the whole picture. In particular the clear understanding
of the amplitudes of the gluon scattering with  generic chiralities
is absent and our proposal for the improvement
of the BDS anzatz deserves for the further evidences. Nevertheless we
believe that the dual picture we have  suggested is the useful
step towards the clarification of the scattering geometry
responsible for the summation of the perturbative series in YM theory.

\begin{theacknowledgments}
I would like to thank the organizers of Landau Memorial Conference for providing
the nice atmosphere. The work  was supported in part by grants
INTAS-1000008-7865 and PICS- 07-0292165.
\end{theacknowledgments}


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\bibitem{regge}
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\bibitem{bfkl}
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  %%CITATION = NUPHA,B175,365;%%









\end{thebibliography}


\end{document}