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UUITP6/95, ITEPM3/95, FIAN/TD9/95
Integrability and SeibergWitten Exact Solution
[.2in] A.Gorsky ^{1}^{1}1Email address: ,
Institute for Theoretical Physics, Bern University, Sidlerstrasse 5, Bern, Switzerland
I.Krichever
^{2}^{2}2Email address:
L.D.Landau Institute for Theoretical Physics, 117940 Moscow, Russia
A.Marshakov
^{3}^{3}3Email address:
, ,
,
A.Mironov
^{4}^{4}4Email address:
, ,
Institute of Theoretical Physics, Uppsala University, Uppsala S75121, Sweden
A.Morozov
^{5}^{5}5Email address:
ITEP, Moscow, 117 259, Russia
Permanent address: ITEP, Moscow,
117 259, Russia
Permanent address: Theory Department, P. N. Lebedev Physics Institute , Leninsky prospect, 53, Moscow, 117924, Russia
and ITEP, Moscow 117259, Russia
The exact SeibergWitten (SW) description of the light sector in the SUSY YangMills theory [1] is reformulated in terms of integrable systems and appears to be a GurevichPitaevsky (GP) [2] solution to the elliptic Whitham equations. We consider this as an implication that dynamical mechanism behind the SW solution is related to integrable systems on the moduli space of instantons. We emphasize the role of the Whitham theory as a possible substitute of the renormalizationgroup approach to the construction of lowenergy effective actions.
The exact expression for the vacuumcondensate dependence of effective coupling constant in SUSY YM theory [1] provides a new basis for the search of a relevant description of vacuum structure in nonabelian theories. Especially interesting is emergence of characteristic features of integrable structures in essentially problem. In this letter we explain that the SW answer for theory is just the same as the GP solution of elliptic Whitham equations, which in its turn is a simple analog of solutions to “string equations” arising in the context of (world sheet) string theories and gravity models ^{6}^{6}6To be exact we discuss throughout this letter the first GP solution, which arises as a step decay in the KdV theory, while the second one rather corresponds to string equations.. A more detailed discussion will be presented elsewhere.
1. We begin with a survey of the relevant statements from the general theory of YM fields and from ref.[1].
The simplest dynamical characteristic of YM theory is the effective coupling constant (defined as a coefficient in front of in effective action) as a function of normalization point (roughly speaking, the IR cutoff in the integration over fast quantum fluctuations).
Within the perturbation theory for nonabelian model this function is given by Fig.1a. If there is a scalarfield condensate ^{7}^{7}7See [1] for notational details. , spontaneously breaking original gauge symmetry to , one gets instead the picture like Fig.1b. If the original symmetry is larger than there is a series of transitions at various points . In the SUSY YM theory the function is zero, and what one obtains is Fig.1c. In this model one is actually interested in the function of two variables, since there is a valley in the effective potential and the value of is a priori arbitrary ( is a dynamical variable). Because of the simple pattern in Fig.1c, the function
(1) 
can be considered as carrying a certain information about the most intriguing quantity . In other words, one can substitute the typical confinementphase problem (of evaluation of ) by the typical Higgsphase one (of evaluation of ) and the latter one definitely makes sense even beyond the perturbation theory. It is also natural to introduce the full complex coupling constant , where is the coefficient in front of the ”topological” term . Within the perturbation theory does not depend on and (see, however, [3]).
The definition of , as well as identifications like eq.(1), beyond the perturbation theory gets ambiguous. However, a qualitative description is well known in the instantongas approximation [4]. The new behaiviour, as compared to the perturbation theory, is the occurence of dependence of , which results in renormalization of the bare at to at , and deconfinement (occurence of zero of function at ) at . Both effects are described [5], [6] by the characteristic renormalizationgroup flow shown in Fig.2a. The analytic description is given by equations
(2) 
where and are some positive functions depending on a particular model.
Beyond the instantongas approximation one should represent as some parameter of the effective theory on the universal moduli space of instantons.^{8}^{8}8In general this theory can have different phases. One of them – believed to be relevant for confinement in QCD – is known in less formal terms as that of instanton fluid [7]. In the SUSY case, where perturbation theory is almost trivial (for example, the perturbative function has only oneloop contributions, [8]), one can expect that the relevant dynamical system is especially simple. One of the possible ideas is that it somehow possesses integrable properties, peculiar for dynamics on known moduli spaces (see for example [9], [10], [11] ). The results of [1], as well as their generalizations in [12], look as being consistent with this integrable dynamics.
Namely, in [1] is identified with the coordinate on the modular halfplane for the onedimensional complex tori, see Fig.2b, while is interpreted as a parameter (one of ramification points) in their elliptic representation
(3) 
i.e.
(4) 
The generalizations are described in [12] in terms of moduli of the specific subclass of hyperelliptic surfaces,
(5) 
where is any polynomial of degree . These expressions provide an explicit way to avoid the singular point (where , i.e. and – in accordance with qualitative Fig.2a; note that was restricted to be real in that picture) by analytic continuation into the complex plane. It also introduces one more singularity at another point (), while the vicinity of the last singular point () is described by the ordinary perturbation theory (thus, the three “infinitelyremote” points are not identical, and the theory actually lives on the covering of moduli space – again, as suggested by naive Fig.2a .
Most impressive, [1] implies that the Riemann surfaces themselves – not just their moduli – have some physical significance. Namely, the spectrum of excitations in the theory is identified as
(6) 
where
(7) 
and
(8) 
is a particular 1differential on the surface with the double pole and the double zero at the ramification points and respectively.^{9}^{9}9 In terms of parametrization of the spectral curve (see (8)), the integrals (7) for are just the actionintegrals in the SineGordon model over the classically allowed and forbidden domains at a given ”energy” , see Fig.3. Note that in this parametrization is identified with . It also deserves mentioning that for the elliptic solution the surface itself is isomorphic to its Jacobian, thus the periods of differentials play the role of periods of the real motion in potential (8) of the ”auxuliary” quantummechanical problem.
This poses the question of what is the reason for Riemann surfaces to appear in this theory: while and are present in it from the very beginning, the surface is something new and emerges dynamically only in the lowenergy effective theory.
2. The answer to this question is, of course, more general than the particular SW example ^{10}^{10}10The answer seems to be similar to that one from the (string theory) case where the arising (targetspace!) spectral curve might be associated with the ”scaleparameter” curve. The nonperturbative effects imply that such a surface has a nontrivial topology while the mechanism of arising the higher topologies is not yet clear..
The effective dynamics in the space of coupling constants, like and , substitutes original dynamics in the ordinary spacetime by a set of Ward identities (lowenergy theorems), which normally have the form of nonlinear differential equations for effective action (which in this context is often refered to as generalized function). When these equations belong to (generalization of) KP/Todatype hierarchy – as it often happens after appropriate choice of variables – their solutions (i.e. acceptable shapes of effective actions) are parametrized in terms of some auxiliary “spectral surfaces” also known as “target space” curves (not worldsheet) in the language of string theory.
The family of “vacua” of the original model is thus naturally associated with the family of spectral surfaces, i.e. with their moduli space. It seems that only the moduli space itself has physical meaning, not the spectral surfaces but this is not however quite true. So far we discussed the effective action (KP/Todalike function) as a function of timevariables (coupling constants , etc). However, if considered as a function of moduli (e.g. of scalar condensates ), the effective function induces a new (lowenergysector) dynamics on the space of moduli. This new dynamics implies that the moduli are no longer invariants of motion: instead they are ‘‘RGslow’’ dynamical variables of the theory^{11}^{11}11The situation is much similar to the standard renormalization group. Indeed, the RG dynamics is governed by the action of some vector field . The general approach to construction of such effective actions is known as BogolyubovWhitham averaging method (see [14], [17] for a comprehensive review and references). Though this Whitham dynamics is that of the moduli, its explicit formulation is most simple and natural in terms of connections on spectral surfaces. Thus lowenergy dynamics actually gives a lot of physical significance to the spectral surfaces themselves, and, after all, it is not such a big surprise that dynamical characteristics – of which (6) is a simplest example – are expressed in terms of them.
In the SW case one could try to be more specific – but in this letter we restrict ourselves to the following simplified scheme.
One begins with considering the fielddependent dynamics on the moduli space of instantons. One can further think that some directions in the functional space are most important for the lowenergy theory. An obvious candidate for such variable is . Effective potentials are periodic in and associated excitations are always light (unless they mix with something else which is also light – as is the case with the meson in QCD). The conjugate variable to is exactly : one of our most significant (along with “time variables”). After “Legendre transform” one can think of original dynamics of fields as of a RGlike one in the space of coupling constants. Solutions to these “RG”equations identify the valley vacuum averages with moduli of spectral surfaces. Monodromies on these surfaces are natural variables of the Todachain hierarchies, with the length of the chain equal to for the gauge group. Indeed, the simplest type of dynamics for a variable in the fundamental representation of is implied by Lagrangian
(9) 
which in the Cartan sector reduces to:
Possible higherderivative corrections to can be associated with the higher Hamiltonians of the Todachain hierarchy.
Now comes the first miracle. According to [13], the finitegap solutions to the Todachain systems are characterized exactly by hyperelliptic surfaces of the peculiar type (5).^{12}^{12}12 We remind that the data (a spectral complex curve, a point on it and a complex coordinate in the vicinity of the point) is always in onetoone correspondence with the solutions to KPhierarchy, explicit relation being given in terms of the BakerAhiezer function (the curve itself can be also described by the evolutioninvariant equation , where is the Laxoperator). Particular reductions of KP correspond to restrictions on the choice of Riemann surfaces. In particular, generic hyperelliptic surfaces correspond to solutions to KdV, while the subclass (5) describes solutions to Todachain hierarchy. The most spectacular in the last relation is that the power of polynomial in (5) is exactly the length of the chain, i.e. the size of the matrices in the fundamental representation.
The next task is to consider effective Whitham dynamics. With the “first miracle” in mind – and with the knowledge that all the Todachain systems are particular members of the KP/Todalattice family – we can just use the wellknown Whitham theory of integrable hierarchies [14], [17] ^{13}^{13}13 This context can actually be not so arrow as it seems. As often happens, different original (nonrenormalized) models produce the same kind of effective (renormalized) dynamics, and at the end of the day it can happen that integrable systems just label the classes of universality of effective actions. In other words, the concrete type of Whitham dynamics, even if derived from the study of integrable hierarchy, can have much broader significance. Moreover, the Whitham equations are themselves integrable, and – according to the previous remark – it is mostly this integrability that we refer to in the title of this letter. (these are exactly the ones that arised in the recent studies of topological theories/gravities [16], [17], [18], [19] and describe exact solutions to the string equations [20], [21].)
3. Now let us turn to the next observation. If one takes as a characteristic of effective dynamics in the vicinity of the classical solution the SW formulas (68), one immediately recognizes them as familiar objects from the theory of the Whitham equations. Namely, in (7) is exactly the generating 1differential arising in the first GurevichPitaevsky problem [2].^{14}^{14}14 This problem came from physics of fluids and concerns the decay of a step (Heavyside) function under the KdV evolution. The exact KdV dynamics,
In formal terms, the Whitham equations can be described as follows. The KP/Todatype function associated with a given spectral Riemann surface is equal to
where is a Riemann thetafunction and are meromorphic 1differentials with poles of the order at a marked point . They are fully specified by normalization relations
(10) 
and
(11) 
where is the local coordinate in the vicinity of . The moduli of the spectral surface are invariants of KP flows,
(12) 
and label the “vacua” – the (finitegap) solutions to the KP system. The effective dynamics on the space of these “vacua”, generated by the BogolyubovWhitham method, arises with respect to some a priori new “slow” Whitham times . The way the moduli depend on is defined by the Whitham equations (induced by the fast KP/Todatype equations), which for the twodimensional integrable systems were first derived in [15] in the following form
(13) 
These equations imply that
(14) 
with some ”generating” 1differential , whose periods can be interpreted as the effective ”slow” variables. Note that the selfevident relation (14) was crucially used in the constructing the exact solutions to the Whitham equation that was proposed in [15]. The equations for moduli, implied by this system, are of peculiar linear form:
(15) 
with some (in general complicated) functions , which depend on the type of ‘‘vacua’’ under consideration ^{15}^{15}15These formulas imply a special choice of basis in the moduli space, taking coordinates ( variables ) coming from commuting KPflows. The relation which defines the period matrix in terms of the N=2 superpotential [1] has also appeared in the theory of topological theories, see [18].
In the KdV case all the spectral surfaces are hyperelliptic, takes only odd values , and
(16) 
the coefficients of the polynomials being fixed by normalization conditions (10), (11) (one usualy takes and the local parameter in the vicinity of this point is ). In this case the equations (15) can be diagonalized if coordinates on the moduli space are taken to be the ramification points:
(17) 
Now an important remark is that after one swichtes on the Whitham dynamics the periods of the differential defined by (14) become the periods of the ”modulated” function (S0.Ex5). We will see below that it gives us the SW spectrum.
4. Let us be more specific in the elliptic (GP/SW) case and restrict ourselves to the first two timevariables, . The elliptic (onegap) solution to KdV is
(18) 
where is the Weierstrass function, and
The observation, that we refered to at the beginning of sect.3, is that a particular solution to eqs.(14) in the elliptic case is the same as the differential in (8).
Indeed, as we are going to demonstrate,
(20) 
where is a calculable function of Whitham times with pole only at of the order , if and all the . The reason why has this particular form (i.e. pocesses double zero at ) is simple. Normally, derivative of a meromorphic object over moduli has more poles (since after a change of the complex structure the holomorphic object becomes nonholomorphic),and moduli in the hyperelliptic parametrization are located at ramification points. In our case there is just one ramification point, , which is dependent, and, in order to cancel the pole at in (which does not occur in ), one needs to put some power of in the numerator of – once appeared in the denominator. Since is not a singlevalued function on the surface, one needs to take its square.
From (20) one derives:
and comparison with explicit expressions (S0.Ex6) implies:
In other words, this construction provides a (GP) solution to the Whitham equation
(21) 
with
(22) 
which can be expressed through elliptic integrals [2].
We see that (7) can be reinterpreted as
being the periods of ”modulated” GP elliptic solution. This implies that in generic situation (for nonelliptic surfaces and all ) the SW formula (6) should be
(23) 
Note also that
(24) 
while
(25) 
which are the frequencies in the original KP/Todatype solution (S0.Ex5). So, the periods of the ”modulated” Whitham solution give rise to the mass spectrum in the SW exact solution and its generalizations.
5. All the quantities entering the Whitham equations have the meaning of the averaged characteristics of the bare elliptic solution (3). Note also, that the further speculation of the meaning of the GP solution in the SW context of gauge theory is possible. If one relates KP times , with (the functions of) bare and then the main object under consideration – the KdV ”potential” becomes related to the correlator . This looks quite hopefull since such correlators contain the information about topological exitations in the gauge theory. Now after the averaging the ”slow” times in the Whitham system can be identified with the functions of the ”renormalized” KP times (coupling constants). Moreover the form of the GP solution suggests its interpretation as a ”decay” of the topological exitations in SW theory in the nonperturbative regime . The GP solutions have automodel form and this can be relatd to emergence of holomorphic coupling constant . We will return to this correspondence in a separate publication.
6. To conclude, we see that the central formula (6) of [1] can be interpreted as (23), i.e. in terms of periods of the central object in the theory of the Whitham hierarchy. This observation seems to be important since there exists a general beleif that lowenergy effective actions are proper objects to be reffered to as generalized functions. One should add that conceptually the Whitham method is precisely the averaging over fast fluctuations, which is necessary to produce the effective action for slow variables i.e. plays the role of the nonperturbative analog of renormalization group. We believe that this analogy deserves attention and further studies^{16}^{16}16Among related studies, one should mention [22], [23], [24] and [25]. It also deserves noting that – as usual – the study of some analogues of the YM theory can shed an additional light on the problem. The obvious relation is to topological theories and string equations. In both cases the Whitham dynamics is known to arise in description of effective actions (see [26] for discussion of parallels with the physics). will put them on a more solid ground.
We are indebted to E.Akhmedov, V.Fock, A.Gurevich, S.Kharchev, I.Polyubin, V.Rubtsov, K.Selivanov, A.Smilga and A.Zabrodin for fruitfull discussions. We aknowledge the hospitality of A.Niemi at Uppsala University where part of this work was done. A.G thank to H.Leutwyler for the warm hospitality in the Bern University where this work was finished. The work of A.G., A.Mar. and A.Mir. is partly supported by ISF grant MGK000 and grants of Russian fund of fundamental research RFFI 930203379 and RFFI 930214365. The work of A.Mar. was also supported by NFRgrant No FGF 06821305 of the Swedish Natural Science Research Council.
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