Off-shell M5 Brane, Perturbed Seiberg-Witten Theory, and Metastable Vacua
aa r X i v : . [ h e p - t h ] J a n arXiv:0801.2154 CALT-68-2668ITFA-2008-01
Off-shell M5 Brane, Perturbed Seiberg-Witten Theory,and Metastable Vacua
Joseph Marsano , Kyriakos Papadodimas , and Masaki Shigemori California Institute of Technology 452-48, Pasadena, CA 91125, USA Institute for Theoretical Physics, University of AmsterdamValckenierstraat 65, 1018 XE Amsterdam, The Netherlands marsano_at_theory.caltech.edu , kpapado_at_science.uva.nl , mshigemo_at_science.uva.nl We demonstrate that, in an appropriate limit, the off-shell M5-brane worldvolume actioneffectively captures the scalar potential of Seiberg-Witten theory perturbed by a small super-potential and, consequently, any nonsupersymmetric vacua that it describes. This happensin a similar manner to the emergence from M5’s of the scalar potential describing certaintype IIB flux configurations [1]. We then construct exact nonholomorphic M5 configurationsin the special case of SU (2) Seiberg-Witten theory deformed by a degree six superpotentialwhich correspond to the recently discovered metastable vacua of Ooguri, Ookouchi, Park [2],and Pastras [3]. These solutions take the approximate form of a holomorphic Seiberg-Wittengeometry with harmonic embedding along a transverse direction and allow us to obtain geo-metric intuition for local stability of the gauge theory vacua. As usual, dynamical processes inthe gauge theory, such as the decay of nonsupersymmetric vacua, take on a different characterin the M5 description which, due to issues of boundary conditions, typically involves runawaybehavior in MQCD. ontents R Λ Σ | ds | . . . . . . . . . . . . . . . . . . . 13 N = 2 SU (2) Gauge Theory . . . . . . . . . . . . . . . . . . . . . 25 N = 2 Curve in Parametric Representation . . . . . . . . . . . . . . . . . 316.4 Turning on a Superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.5 The Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.6 The Perturbed N = 2 Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.6.1 The OOPP Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.6.2 The Approximate Curve and Gauge Theory Quantities . . . . . . . . . 366.6.3 Explicit Connection to Seiberg-Witten Potential . . . . . . . . . . . . . 38 Concluding Remarks 44A Some Basic Results on Riemann Surfaces 45B Parametric Representation of Genus 1 Seiberg-Witten Geometry 47C Symmetry Argument for the Ansatz (6.32) q -expansions 52E Type IIA Limit of the M5 Curve 52 String theory has a rich history of providing geometric intuition for the structures that appearin supersymmetric field theories. For example,
N S /D M -theory lifts naturally give rise to the Riemann surfaces [4, 5, 6] which playsuch a crucial role on the gauge theory side [7, 8]. In this manner, the relation of geometry tothe structure and properties of supersymmetric vacua is made completely manifest.Since the recent discovery [9] that supersymmetric gauge theories often admit metastableSUSY-breaking vacua, significant effort has been devoted to the study of their stringy real-izations [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 1, 28, 29, 30, 31, 32, 33, 34, 35,36, 37, 38, 39, 40]. This is of particular interest for two reasons. First, one might hope to learnabout the role played by geometry in the structure of nonsupersymmetric vacua in gaugetheory. Second, stringy embeddings of metastable vacua can potentially provide new meansby which SUSY-breaking can be achieved in string theory. Such constructions are typicallylocal in nature and are thus well-suited to the sort of stringy model building advocated in [41].One potential pitfall is that classical brane constructions in type IIA and their M -theorylifts can be well-studied only in parameter regimes that are far from those in which thecorresponding gauge theory description is valid. As such, there is no guarantee at the outsetthat string theory knows anything about physics away from supersymmetric vacua, wherequantities are protected. Despite this fact, considerable progress has been made along thisdirection in the case of the ISS vacua of N = 1 SQCD [9] and a number of generalizations[13, 14, 15, 19, 22, 23, 29, 37, 38, 40], and it has been demonstrated that type IIA/M-theoryrealizations provide valuable intuition for the metastability of those vacua.In this paper, we shall focus instead on the stringy realization of Seiberg-Witten theorydeformed by a superpotential. Because one can tune the degree to which N = 2 SUSY2s broken, the K¨ahler potential is under some degree of control. This not only allows oneto reliably compute the scalar potential in the gauge theory, it also gives us hope that thestandard stringy realizations might allow us to engineer the full potential, in a suitable sense,along with any non-SUSY vacua that it describes.As is well-known by now, deformed Seiberg-Witten theory admits two possible low energydescriptions depending on the various scales involved in the problem. If the characteristicscale, g , of the superpotential that controls the mass of the adjoint scalar is sufficiently smallcompared to the dynamical scale, Λ, of the gauge group, then the superpotential only playsa role in the deep IR after one has moved to the effective Abelian theory on the Coulombbranch. This is the situation referred to above in which the K¨ahler potential is under tunablygood control. On the other hand, if g is much larger than Λ, we should integrate out theadjoint scalar before passing to the IR. The N = 2 SUSY is broken to N = 1 and thegauge group is Higgs’ed at a high scale, below which non-Abelian factors confine. Certainaspects of the low energy dynamics, such as chiral condensate and the value of superpotentialin supersymmetric vacua, are captured by an effective theory for the corresponding glueballsuperfields S i ∼ π tr W α i [42, 43] and corrections from the massive adjoint scalar lead togeneration of the Dijkgraaf-Vafa superpotential [44, 45, 46, 47].As pointed out by [48] and [18] in the context of the type IIB realization of this story vialarge N duality [49, 50, 51], the appearance of FI terms suggests that the glueball effectivetheory actually possesses a spontaneously broken N = 2 supersymmetry. This again givesone control over the K¨ahler potential and allows a scalar potential to be reliably computed .Quite remarkably, the IIA brane construction which realizes deformed Seiberg-Witten theoryadmits an M -theory lift that incorporates and unifies the scalar potentials of both regimes.In particular, each can be obtained from a different limit of the M .As an application of this result, we then proceed to describe stringy embeddings of themetastable vacua of deformed Seiberg-Witten theory recently discovered by Ooguri, Ookouchiand Park [2] and Pastras [3], which we will refer to as OOPP vacua. More specifically, wewill find exact minimal area M5’s which reduce in the appropriate limit to minima of the The result of [1] that the resulting potential also arises in a type IIA description provides evidence for thestructure of spontaneously broken N = 2 SUSY in the case of IIB on local Calabi-Yau in the presence of flux. That it should be possible to realize this potential is not surprising because it has been well-establishedthat the M5 realization of Seiberg-Witten theory correctly captures the K¨ahler potential and fails to reproducenonholomorphic quantities only at four derivative order and higher [52].
N S /D T -dual type IIB constructions where a single-trace superpotentialdetermines the geometry and the rank of the gauge group simply specifies the number of D D The organization of this paper is as follows. We begin in section 2 with a brief review of thetype IIA/M construction of deformed Seiberg-Witten theory. In section 3, we then review themechanism described in [1] by which this construction is able to realize not only the space ofsupersymmetric vacua, but also the full scalar potential for the light degrees of freedom in theDijkgraaf-Vafa regime. We then perform a similar analysis in the Seiberg-Witten regime insection 4 to demonstrate that the IIA/M picture captures the scalar potential there as well. Insection 5, we review the manner in which nonsupersymmetric vacua can be engineered in theSeiberg-Witten regime by choosing a suitable superpotential [2, 3]. In section 6, we specializeto the case of SU (2) and construct exact minimal-area M As we shall see, the manner in which the scalar potential arises in our setup suggests that one must relaxsome assumptions of [35] in order to see it in that context.
Let us begin with a brief review of the type IIA/M realization of SU ( N ) N = 2 supersym-metric Yang-Mills theory deformed by a superpotential for the adjoint scalar. As first shownby Witten [4], the theory without superpotential can be engineered in type IIA by startingwith two NS5-branes and N D4-branes extended along the 0123 directions. The NS5’s arefurther extended along a holomorphic direction parametrized by the combination v = x + ix (2.1)and the D4’s are suspended between them along x as depicted in figure 1(a). If we scale theNS5 separation L along x to zero, the D4 worldvolume becomes effectively four-dimensionalwith gauge coupling constant given by 8 π g Y M = Lg s √ α ′ . (2.2)The resulting configuration preserves 8 supercharges, so this worldvolume theory is nothingother than the celebrated N = 2 SU ( N ) supersymmetric Yang-Mills theory studied by Seibergand Witten [7]. Because of the protection afforded by supersymmetry, we expect that the moduli space ofvacua can be seen directly in the brane constructions, even away from the strict L → v . To describe the quantum modulispace, however, it is necessary to accurately treat the NS5/D4 intersection points. While thestrength of the string coupling in the NS5 throat makes this difficult to do directly in typeIIA, Witten pointed out [4] that one can make progress by noting that NS5’s and D4’s aretwo different manifestations of the same object, namely the M5 brane of M-theory. As aresult, the configuration of figure 1(a) should be replaced by a single M5 with four directionsalong 0123 and the remaining two, in the probe approximation, extended on a nontrivial For the discussion of supersymmetric vacua, we can be a bit careless about validity of the probe approx-imation but a detailed description of the necessary conditions, which are needed when considering nonsuper-symmetric configurations, can be found in [1]. v x NS5 Branes (a) NS5/D4 realization of the classical Seiberg-Witten moduli space.
M5 Brane v x (b) Cartoon of M5 lift which realizes thequantum Seiberg-Witten moduli space. Figure 1: Type IIA/M realization of classical and quantum moduli space of Seiberg-Wittentheory.two-dimensional surface Σ of minimal area. From the IIA point of view, this will take theform of a curved NS5-brane with flux .The general structure of the M5 curve associated to figure 1(a) can be seen by observingthat, roughly speaking, the stacks of D4-branes blow up into tubes as illustrated in figure 1(b).Consequently, a point on the classical moduli space where the D4’s form n distinct stackswill correspond to a Riemann surface Σ of genus n − . A convenientparametrization of this surface is as a double-cover of the v -plane with n cuts , as depictedin figure 2. Roughly speaking, we can think of each sheet as an NS5 and the cuts as thetubes associated to the D4’s. In this case, the full embedding is specified by providing the v -dependence of x and x , which are naturally paired into the complex combination [4] s = R − (cid:0) x + ix (cid:1) , (2.3)where R = g s √ α ′ (2.4)is the radius of the M-theory circle. We will set α ′ = 1 henceforth.The embedding relevant for figure 1(a) must be holomorphic due to the supersymmetryand is determined by imposing boundary conditions appropriate for the system at hand. In The punctures represent the points at infinity from which RR flux associated to the D4’s can flow. This relies on the fact that Σ is hyperelliptic [4, 5]. and specify thenumber of D4’s in each stack. These conditions can be conveniently summarized in terms ofconstraints on the periods of the 1-form ds around the ˆ A and ˆ B cycles of figure 2 πi I ˆ A j ds = N j and 12 πi I ˆ B j ds = − α j . (2.5)Here N j is the number of D4-branes in the j th stack and α j = − πig Y M + θ j π (2.6)with g Y M denoting the bare coupling at scale Λ . In general, one can consider nontrivialrelative θ j angles by allowing the α j to differ by integers. From this point onward, however,we shall restrict for simplicity to the case in which all α j are equivalent: α j ≡ α for all j. (2.7)To find the family of curves which satisfy v A A A B B B Figure 2: Sample parametrization of the M5curve as double-cover of v -plane for the case n = 3 with ˆ A and ˆ B cycles indicated. the constraints (2.5) for various choices of N j , Witten [4] noted that since the periodsof ds take integer values, the coordinate t ≡ Λ N e − s (2.8)must be well-defined on the curve. With thisobservation, he demonstrated that the M5lifts are described by nothing other than theSeiberg-Witten geometries t − P N ( v ) t + Λ N = 0 , (2.9)where P N ( v ) is a polynomial of degree N andΛ N = Λ N e πiα (2.10)is the dynamical scale of the gauge group. Generic choices of P N ( v ) lead to (degenerate) genus N surfaces and hence correspond to lifts of classical configurations in which all D P N ( v ), however, the curves (2.9) degenerate into surfaces oflower genus which describe the lifts of configurations with some N j > This is essentially equivalent to fixing the bare coupling constant in the Yang-Mills theory at a UV cutoffscale. It should be noted that the brane configurations under study are UV completed into MQCD and notsimply an asymptotically free gauge theory so it will only make sense to compare with the gauge theory atscales below this cutoff. Note that there are n ˆ A -cycles and n ˆ B -cycles despite the fact that the curve has genus n −
1. This allowsus to treat the curve with marked points as a degenerate Riemann surface of genus n , putting meromorphic1-forms with single poles, such as ds , on an equal footing with holomorphic 1-forms. x ∆ wv NS5 branes x Figure 3: Sample NS5/D4 realization of supersymmetric vacua in U ( N ) N = 2 supersym-metric Yang-Mills with a cubic superpotential. To similarly engineer the N = 2 theory deformed by a polynomial superpotential W n (Φ) ofdegree n + 1, we need only modify the NS5/D4 construction of figure 1(a) by curving theNS5-branes appropriately [5, 53, 54]. More specifically, we introduce the complex combination w = x + ix (2.11)and extend the NS5’s along the holomorphic curves w ( v ) = ± W ′ n ( v ). An example of such aconfiguration for the case of a cubic superpotential is depicted in figure 3. Classically, thesupersymmetric vacua correspond to configurations where the D4-branes sit at zeros of W ′ n ( v ),in accordance with our expectations from the gauge theory side.At quantum level, the supersymmetric vacua are again effectively captured by moving tothe M5 lift. The only new ingredient here is the addition of an extra nontrivial embeddingcoordinate whose v -dependence we must specify. The boundary conditions appropriate forthat coordinate are determined by the asymptotic geometry of the NS5’s, meaning that wemust impose w ( v ) ∼ ± W ′ n ( v ) (2.12)near the points at infinity. Holomorphic solutions to this constraint take the form w ( v ) = p W ′ n ( v ) − f n − ( v ) (2.13)for f n − ( v ) a generic polynomial of degree n −
1. If we further require consistency of the w ( v )and s ( v ) embeddings, we are led to the factorization formulae P N ( v ) − N = S N − n ( v ) (cid:0) W ′ n ( v ) − f n − ( v ) (cid:1) (2.14)8f n < N and similarly (cid:0) P N ( v ) − N (cid:1) H n − N ( v ) = W ′ n ( v ) − f n − ( v ) (2.15)if n > N . For given W ′ n ( v ), there are at most a finite number of choices for P N ( v ), f n − ( v ),and S N − n ( v ) ( H n − N ( v )), which satisfy (2.14) ((2.15)). This reflects the familiar fact thatadding a superpotential to Seiberg-Witten theory lifts all but a discrete set of points on themoduli space. That these conditions yield vacua in agreement with the gauge theory analysishas been well-established [55, 56] and will also fall out naturally from our general formalismto follow. We now turn to a study of physics away from the supersymmetric vacua. As discussed inthe introduction, the appropriate IR description depends strongly on the relative sizes of thecharacteristic mass g of the adjoint scalar field and the dynamical scale Λ. In this section, weconsider the case where g is sufficiently large compared to Λ. On the gauge theory side, when g is sufficiently large the gauge group is Higgs’ed and N =2 supersymmetry broken to N = 1 at a high scale. The remaining non-Abelian factorsconfine and certain aspects of the IR dynamics can be captured by the glueball superfields S i = − π tr W i α as well as the N = 1 gauge multiplets corresponding to the overall U (1)’s .The leading contribution to the glueball superpotential is given by the Veneziano-Yankielowiczterm [42] and corrections arising from the presence of the adjoint scalar can be computedby integrating it out order by order in perturbation theory [47]. This calculation receivescontributions only from planar diagrams which can in turn be computed with an auxiliaryholomorphic matrix model Z = Z D Φ exp (cid:18) − g s tr W n (Φ) (cid:19) . (3.1)In the end, one arrives at the famous Dijkgraaf-Vafa superpotential [44, 45, 46] W = α i S i + N i ∂ F ∂S i ( S j ) , (3.2) Note that the situation is quite subtle when Abelian factors remain after Higgs’ing because the stringyconstructions still contain glueballs for them [57]. We will largely avoid this technicality here and refer theinterested reader to [57], where this issue is discussed in greater detail. N i denote ranks of the confining non-Abelian factors, α i are as in (2.6) and F is theplanar contribution to the free energy of (3.1) from the saddle point where N i eigenvalues ofΦ sit at the i th critical point of W n (Φ). Note that in writing F as a function of S i , we mustmake the identification S i ∼ g s N i . This Dijkgraaf-Vafa superpotential was also derived fromanomaly arguments in [43].As usual, the matrix model computation can be reinterpreted in a geometric languagebased on the corresponding spectral curve. In the case at hand, this auxiliary Riemannsurface takes the form w = W ′ n ( v ) − f n − ( v ) (3.3)and the function F is simply its prepotential. To make this more precise, let us use thedouble-cover of the v -plane to parametrize (3.3) and specify ˆ A and ˆ B cycles as in figure 2.The quantities S i and ∂ F /∂S j are then given by suitable ˆ A and ˆ B period integrals S j = 12 πi I ˆ A j w dv ∂ F ∂S j = 12 πi I ˆ B j w dv. (3.4)As usual, ∂ j ∂ k F ≡ ˆ τ jk yields the period matrix of the Riemann surface (3.3).These geometric formulae arise quite naturally when this theory is engineered in type IIBusing D N duality relates the brane setup to a deformed Calabi-Yau with flux, whose superpotentialtakes the precise form (3.2). In this context, it has been conjectured that the IR physicscontains an underlying N = 2 structure [48, 18], which gives one control over the K¨ahlermetric. Such a possibility is suggested by the observation that the IR degrees of freedomnaturally combine into n N = 2 vector multiplets and the superpotential (3.2) is simply alinear combination of electric and magnetic Fayet-Iliopoulos parameters. If the conjecture istrue, it would imply that we have some control over the K¨ahler metric and, in particular, thatit can be identified with the imaginary part of the period matrix, Im ˆ τ ij , of (3.3). In thatcase, the scalar potential takes a relatively simple form V DV = ( α i + N k ˆ τ ki ) (Im ˆ τ ) ij (cid:0) α j + ˆ τ jℓ N ℓ (cid:1) . (3.5)In the next subsection, we will review the result of [1] that this potential is naturallyencoded in the type IIA/M realization of the gauge theory. The resulting configurationis actually related to the IIB setup described above by a T -duality [58, 59]. Because theIIA/M and IIB descriptions are only reliable in widely-separated regions of parameter space, We use the notation ˆ τ ij instead of τ ij for the period matrix of a generic hyperelliptic curve. The reasonfor this is to avoid confusion later then τ is used as the complex structure modulus for an auxiliary torus. N = 2 supersymmetry may bepresent. From the point of view of our NS5/D4 constructions, the Dijkgraaf-Vafa regime has thenatural interpretation as one where the D4’s are essentially pinned to the zeroes of W ′ n ( v ).This means that the low energy modes involve not motion of the D4’s, but rather fluctuationsin the size of the tubes into which they blow up in the M5 lift. In other words, this is a regimewhere the w ( v ) part of the curve is essentially rigid while the s ( v ) embedding can fluctuate.Such a limit is in fact quite natural because the s coordinate comes with a natural scale,namely the radius of the M-circle R = g s √ α ′ . If we take R to be small, in which case ourminimal area M5 instead has the interpretation of a curved NS5-brane with flux, we expectthat variations of the s coordinate of the embedding actually comprise the lightest excitationsof the system. To make this more precise, let us introduce a “worldsheet” coordinate, z , toparametrize the nontrivial part of the M5 and define our embedding by the functions s ( z, ¯ z ) , v ( z, ¯ z ) , w ( z, ¯ z ) . (3.6)In the probe approximation, the worldvolume theory is described by the Nambu-Goto action S ∼ R Z Σ d z p g ( s, v, w ) , (3.7)where g ( s, v, w ) is the induced metric on the “worldsheet” Σ. If we suppose that R ds ≪ dw, dv then this action can be expanded to quadratic order as S ∼ R Z Σ d z p ˜ g ( v, w ) + Z Σ | ds | + . . . , (3.8)where ˜ g ( v, w ) is the induced metric that arises from the v and w parts of the embeddingalone. At small R ds , the dominant term of this action is the first one, whose equationsof motion constrain the w and v coordinates to describe a minimal area embedding. If wefurther impose the holomorphic boundary conditions w ( v ) ∼ ± W ′ n ( v ) near infinity, we areagain led to the family of holomorphic curves w = W ′ n ( v ) − f n − ( v ) . (3.9)Note that the √ ˜ g term of (3.8) has flat directions corresponding to the complex moduli of(3.9). 11urning now to the second term of (3.7), the equations of motion for ds imply that itmust be a harmonic 1-form on (3.9). Because we fix the ˆ A and ˆ B periods of ds according to(2.5), though, this leads to a unique ds for each curve of the family (3.9). In particular, the1-form ds that we obtain exhibits explicit dependence on the complex moduli of (3.9) andhence, plugging this result back into the action, we find that R Σ | ds | serves as a potential onthe space of complex structures. It is precisely this object that will correspond to the scalarpotential (3.5). Our M5’s are noncompact, though, so computations of their area must be carefully regulated.While we can do this quite easily by introducing a cutoff Λ along v , it is important tomake sure that this procedure leads to a result that is meaningful. The issue of computingregulated areas in this context has been discussed previously by de Boer et al. [60]. Oursituation is essentially the same as theirs because, in the regime under consideration, the off-diagonal components of the induced metric ˜ g ( v, w ) are negligible. Let us proceed to reviewtheir result, focusing for now only on the evaluation of R Λ Σ | ds | .Because ds is harmonic, it can be written as the sum of a holomorphic 1-form ds H and anantiholomorphic 1-form d ¯ s A ds = ds H + d ¯ s A . (3.10)In this language, the object we want to compute is Z Λ Σ | ds | = 12 i Z Λ Σ [ ds H ∧ d ¯ s H + ds A ∧ d ¯ s A ]= 12 i Z Λ Σ [( ds H ∧ d ¯ s H − ds A ∧ d ¯ s A ) + 2 ds A ∧ d ¯ s A ]= 12 i Z Λ Σ [ ds ∧ d ¯ s + 2 ds A ∧ d ¯ s A ] . (3.11)We have chosen to write it in this manner because ds ∧ d ¯ s is the restriction to Σ of a closed2-form in the target space. For this reason, its integral over Σ should in fact be independentof the moduli. In practice, however, the regulated integral of ds ∧ d ¯ s will depend on boththe moduli and the cutoff, Λ . Consequently, to obtain a meaningful regularization, we mustchoose the cutoff Λ to vary with the moduli in such a manner that the regulated quantity12 i Z Λ Σ ds ∧ d ¯ s (3.12) Another way to see that this integral must be taken constant is to T -dualize the system into type IIB,where this integral R ds ∧ d ¯ s gets related to the integral of 3-form flux R CY G ∧ ¯ G , G = F RR + ( i/g IIB s ) H NS [1].Because this is a topological quantity, we must keep this constant as we vary the moduli.
12s indeed constant on the moduli space.With such a scheme in place, the potential R Λ Σ | ds | is simply given by Z Λ Σ | ds | ∼ i Z Λ Σ ds A ∧ d ¯ s A , (3.13)where we have simply dropped the constant term (3.12). In the work of de Boer et al. , thequantity analogous to (3.13) was cutoff-independent so this was the end of the story. In ourcase, however, this result may still exhibit a nontrivial dependence on the cutoff. Nevertheless,(3.13) provides a suitable notion of regularized area provided we also include the necessarymoduli-dependence of Λ . R Λ Σ | ds | Let us turn now to evaluation of (3.13). We begin by using the constraint (2.5) to obtain anexpression for ds in terms of the period matrix ˆ τ ij of (3.9). We write ds H and ds A as ds H = h i d ˆ ω i and ds A = ℓ i d ˆ ω i , (3.14)where the d ˆ ω i comprise a basis of n holomorphic 1-forms , satisfying12 πi I ˆ A j d ˆ ω i = δ ij . (3.15)The ˆ B -periods of d ˆ ω i yield elements of the period matrix12 πi I ˆ B j d ˆ ω i = ˆ τ ij (3.16)and the constraints (2.5) become h i − ¯ ℓ i = N i ,h i ˆ τ ij − ¯ ℓ i ¯ˆ τ ij = − α j . (3.17)This implies that h i = − i (cid:0) Im ˆ τ − (cid:1) ij (cid:0) α j + ¯ˆ τ jk N k (cid:1) , ¯ ℓ i = − i (cid:0) Im ˆ τ − (cid:1) ij (cid:0) α j + ˆ τ jk N k (cid:1) . (3.18) More precisely, the d ˆ ω i are meromorphic 1-forms with poles of degree at most 1 at ∞ on the two sheets.They correspond to holomorphic 1-forms if we consider Σ to be a degenerate Riemann surface of genus n . ds , it is easy to evaluate1 i Z Σ ds A ∧ d ¯ s A = − X j I ˆ A j ds A I ˆ B j d ¯ s A ! = 8 π ℓ i (Im ˆ τ ) ij ¯ ℓ j = 2 π ( α i + ˆ τ ik N k ) (cid:0) Im ˆ τ − (cid:1) ij (cid:0) α j + ˆ τ jℓ N ℓ (cid:1) ∼ V DV . (3.19)This is precisely the scalar potential (3.5), as promised.One potential pitfall in this calculation, though, is the fact that the period matrix ˆ τ ij involves the computation of noncompact ˆ B periods that must be regulated by Λ . Thismeans that ˆ τ ij will in general depend on Λ and hence could exhibit additional dependenceon the moduli in our regularization scheme. To see that this doesn’t happen, note that1 i Z Σ ds H ∧ d ¯ s H = − X j I ˆ A j ds H I ˆ B j d ¯ s H ! = 8 π h i (Im ˆ τ ) ij ¯ h j = 2 π (cid:0) α i + ¯ˆ τ ik N k (cid:1) (cid:0) Im ˆ τ − (cid:1) ij (cid:0) α j + ¯ˆ τ jℓ N ℓ (cid:1) = 2 π ( α i + ˆ τ ik N k ) (cid:0) Im ˆ τ − (cid:1) ij (cid:0) α j + ˆ τ jℓ N ℓ (cid:1) − N j (Im α ) j . (3.20)The regulated integral of ds ∧ d ¯ s is therefore independent of the moduli2 Z Σ ds ∧ d ¯ s = − N j (Im α ) j , (3.21)meaning that we can take our cutoff Λ to be a large, moduli-independent constant. We now turn to the main subject of interest in this paper, namely the Seiberg-Witten regime,where g is sufficiently small that the superpotential can be treated as a deformation of theIR effective description of Seiberg and Witten [7]. In what follows, we will proceed to reviewsome aspects of the gauge theory side. We will then turn to the type IIA/M description anddemonstrate that the scalar potential arises in a manner quite analogous to what we saw inthe Dijkgraaf-Vafa regime above. This will further suggest that, by analogy to [1], criticalpoints in the Seiberg-Witten regime can be associated to full M5 solutions in a suitable sensethat we shall describe. 14 . . . . . . . A A B B B v A Figure 4: Depiction of Seiberg-Witten geometry (4.1) along with the basis of A and B cyclesthat we use to analyze the Seiberg-Witten regime.. Before studying the superpotential deformation, let us first review some aspects of the Seiberg-Witten effective description [7] of the low energy physics. The classical moduli space of thetheory is parametrized by eigenvalues a i of the adjoint scalar, Φ. The quantum moduli space,on the other hand, is equivalent to the moduli space of hyperelliptic curves of the form t − P N ( v ) t + Λ N = 0 , (4.1)where Λ is the dynamical scale of the gauge group and P N ( v ) = v N − N X k =2 s k v N − k = N Y i =1 ( v − a i ) . (4.2)Here, the s k are symmetric polynomials of the a i and comprise one convenient parametrizationof the moduli space.As usual, we view (4.1) as a double-cover of the v plane with n ≤ N branch cuts. It isconventional to choose the cuts so that they encircle pairs of branch points which coalesce inthe classical limit Λ →
0. As for 1-cycles in this geometry, it will be useful in our analysis ofthe Seiberg-Witten regime to use the basis of A and B cycles depicted in figure 4 as opposedto the ˆ A and ˆ B cycles introduced previously Note in particular that we have no need tointroduce an N th A -cycle because we will consider only 1-forms with vanishing residues inwhat follows. This choice of basis differs from that used in our analysis of the Dijkgraaf-Vafa regime by the removal of one A -cycle and the replacement of noncompact ˆ B cycles with compact ones. This basis facilitates comparisonto the traditional analysis of Seiberg-Witten theory and is also more naturally adapted to the boundaryconditions we will use in section 4.2. s k nicely describe the moduli space of curves (4.1), a more convenient parametriza-tion is given by expectation values of powers of Φ u p = 1 p h tr Φ p i , (4.3)where 2 ≤ p ≤ N . These are usefully encoded in the resolvent, a meromorphic 1-form withfirst order poles on the curve (4.1) defined as T ( v ) dv = (cid:28) tr dvv − Φ (cid:29) . (4.4)An explicit expression for this 1-form can be obtained by noting that its A -periods and residueat ∞ are completely determined by the classical limit Λ →
0. In particular,12 πi I A j T ( v ) dv = N j , πi I v = ∞ T ( v ) dv = N, (4.5)where N j is the multiplicity of the j th eigenvalue of Φ in the classical limit. Because ameromorphic 1-form on (4.1) with a first order pole at ∞ is uniquely fixed by its A -periodsand residue, we arrive at the result [61] T ( v ) dv = − dtt = ds, (4.6)where s and t are related as in (2.8). From this, one can easily compute u p via u p = 12 πip I v = ∞ v p ds. (4.7)As for the eigenvalues a i themselves, they combine with their magnetic duals a D j to forma holomorphic section of an Sp (2 n, Z ) bundle over the moduli space. This section can becomputed explicitly by integrating the Seiberg-Witten 1-form dλ SW = v ds (4.8)about A and B cycles of the curve (4.1) a i = I A i dλ SW , a D,j = I B j dλ SW . (4.9)In the absence of a superpotential, the low energy dynamics are determined solely by themoduli space metric, which determines the kinetic terms for the u p g r ¯ s = ∂a i ∂u r (Im ˆ τ ) ij ∂ ¯ a j ∂ ¯ u ¯ s . (4.10)16e now return, however, to the situation at hand in which we deform the theory by a nontrivialsuperpotential of the form W n (Φ) = n X m =0 g m tr Φ m +1 m + 1 . (4.11)If the g m are all suitably small, we can treat this as a perturbation of the effective theorywhose precise form at a point u p of the moduli space is simply given by its expectation value W eff ( u p ) = h W n ( v ) i| u p = n X m =0 I v = ∞ g m v m +1 m + 1 T ( v ) dv. (4.12)This superpotential, combined with the K¨ahler metric (4.10), completely determines the IRdynamics of the deformed Seiberg-Witten theory in this regime. Because we know both, thescalar potential that controls the full vacuum structure of the theory can be written downreliably as V SW = ∂ u r W eff ( u p ) (cid:18) ∂u r ∂a i (cid:19) (cid:0) Im ˆ τ − (cid:1) ij (cid:18) ∂u s ∂a j (cid:19) ∂ u s W eff ( u q ) , (4.13)which we refer to as the Seiberg-Witten potential. It is this object that we shall see arisingfrom the type IIA/M description in the next subsection.Before moving on, though, let us note that while we have considered the SU ( N ) situationabove and will continue to focus on this example throughout the rest of this paper, it is easyto generalize to U ( N ) by allowing s = 0, leading to an additional term of degree v N − in(4.2). All of the formalism readily generalizes. From the point of view of our NS5/D4 constructions, the Seiberg-Witten regime has thenatural interpretation as one where the curvature of the NS5’s is sufficiently small that theD4’s can move along the v direction away from zeroes of W ′ n ( v ) with very little cost in energy.This suggests that configurations relevant for the Seiberg-Witten regime are those whoselightest excitations correspond to changes in the w embedding coordinate. As such, they canbe approximately described by expanding the action (3.7) for small dw . By analogy to (3.8),we find S ∼ R (cid:18)Z Λ Σ d z p ˜ g ( v, s ) + Z Λ Σ | dw | + . . . (cid:19) , (4.14)where ˜ g ( v, s ) is the induced metric that arises from the v and s parts of the embedding alone.When dw is suitably small , the dominant term of this action is the first one, whose equations The precise condition is easy to work out in explicit examples, as we shall see later.
17f motion constrain the v and s coordinates to describe a minimal area embedding Furtherimposing the conditions (2.5), we again arrive at the Seiberg-Witten geometry (2.9) t − P N ( v ) t + Λ N = 0 . (4.15)The complex structure moduli of (4.15) correspond to flat directions of the first term in (4.14).The equations of motion for dw which follow from the second term of (4.14) imply thatit must describe a harmonic 1-form on the curve (4.15). As we shall see in a moment, theboundary conditions for w lead to a unique such 1-form for each curve of the family (4.15).Plugging this back into (4.14), we find that R Λ Σ | dw | serves as a potential on the space ofcomplex structures. It is precisely this object that will correspond to the scalar potential(4.13).We now turn our attention to the computation of R Λ Σ | dw | . First, however, we mustaddress the problem of finding the appropriate dw for a given point in moduli space. Theboundary conditions that we impose are twofold. First, we require as usual that w ( v ) ∼ ± W ′ n ( v ) = n X m =0 g m v m (4.16)near v = ∞ on both sheets. Second, however, is the condition that w be single-valued. Thismeans that the integral of dw over any compact period must vanish: I A i dw = 0 for all i, I B j dw = 0 for all j. (4.17)The next step is to expand dw in a manner analogous to (3.14). Because of the polynomialbehavior (4.16) at ∞ , dw is comprised not only of holomorphic 1-forms but also meromorphicones with poles of degree 2 through n + 1. Before moving on, let us make our choice of basis 1-forms a little more precise. To start,we consider the holomorphic 1-forms on (4.15). If we let r k denote some generic set of moduli,which could be s k , u k , or some other suitable choice, a standard collection of 1-forms is givenby ( ∂P N ( v ) /∂r k ) dv p P N ( v ) − Λ N = ∂∂r k ( s dv ) . (4.18)For example, if we use the s k in this construction, the result is the standard basis of 1-forms ∂∂s k ( s dv ) = − v N − k p P N ( v ) − Λ N . (4.19) Note that the absence of logarithmic behavior at ∞ implies that dw must have vanishing residue at ∞ .It is for this reason that we did not introduce an N th A -cycle in figure 4 in order to treat the Seiberg-Wittengeometry as a degenerate Riemann surface of genus N . u k dη k = ∂∂u k ( s dv ) . (4.20)The dη k are not canonically normalized but this is easily fixed. In particular, if we define σ jk ≡ I A j dη k (4.21)then a collection of 1-forms dω i satisfying12 πi I A j dω i = δ ij (4.22)can be easily written as dω i = σ − ik dη k . (4.23)Note that we already see one piece of (4.13) appearing because σ jk = ∂∂u k I A j s dv = − ∂∂u k I A j v ds = − ∂∂u k I A j dλ SW = − ∂a j ∂u k . (4.24)Now that we have discussed our explicit basis for holomorphic 1-forms, we now turn to themeromorphic 1-forms of degrees 2 to n + 1 that can enter into dw . These are often referredto as meromorphic differentials of the second kind and in our situation generically take theform Q ( v ) dv p P N ( v ) − Λ N , (4.25)where Q ( v ) is a polynomial of degree ≥ N . A convenient choice of basis elements d Ω m , m ≥ A -periods are vanishing I A i d Ω m = 0 (4.26)and behave near v = ∞ as d Ω m = (cid:0) ± mv m − + O (cid:0) v − (cid:1)(cid:1) dv. (4.27)To see that such a collection can indeed be constructed, let us start from generic meromorphic1-forms d ˜Ω m of the form d ˜Ω m = Q N + m − ( v ) dv p P N ( v ) − Λ N . (4.28) Note that the boundary term that arises upon integrating by parts gives no contribution because it issimply the period of ds , which is a modulus-independent constant. Q N + m − ( v ) a polynomial of degree N + m −
1. The constraint (4.26) yields N − ∞ yields another m +1. In the end, this gives N + m constraints, which is equivalent to the number of coefficients in the polynomial Q N + m − ( v ).Fortunately, we will not need to know the explicit form of Q N + m − ( v ) in what follows, as onlythe leading behavior (4.27) is relevant for us.We can now finally expand dw as dw = dw H + d ¯ w A (4.29)with dw H = T m d Ω m + h i dω i dw A = R p d Ω p + ℓ i dω i . (4.30)The boundary condition (4.16) implies that T m = g m , ¯ R p = 0 , (4.31)while the vanishing of A and B periods (4.17) leads to h i = ¯ ℓ i = 14 π (cid:0) Im τ − (cid:1) ij K jm g m , (4.32)where K jm = I B j d Ω m . (4.33)Our boundary conditions have thus led to a unique choice of dw for each point on the modulispace. Now turning to the evaluation of R Λ Σ | dw | , we recall from our discussion of regular-ization in section 3.2.1 that it is sufficient to evaluate Z Λ Σ | dw | ∼ i Z Λ Σ dw A ∧ d ¯ w A . (4.34)This is a straightforward task and leads to1 i Z Λ Σ dw A ∧ d ¯ w A = − X j I A j dw A I B j d ¯ w A ! = 8 π ℓ i (Im τ ) ij ¯ ℓ j = 12 ( K im g m ) (cid:0) Im τ − (cid:1) ij ( K jp g p )= 12 ( K im g m σ ir ) (cid:18) ∂u r ∂a k (cid:19) (cid:0) Im τ − (cid:1) kℓ (cid:18) ∂u s ∂a ℓ (cid:19) (cid:0) σ sj K jp g p (cid:1) , (4.35)20here we have used the relation (4.24). To establish that R Λ Σ | dw | is proportional to (4.13),it remains only to show that σ sj K jp g p is equivalent to ∂ u s W eff ( u q ). To proceed, we relate the B -periods K im to residues involving dω i by using the identity0 = Z Σ dω i ∧ d Ω m = X j "I A j dω i I B j d Ω m − I A j d Ω m I B j dω i + X p = ∞ + , ∞ − I p (Ω m dω i )= K im + X p = ∞ + , ∞ − I p (Ω m dω i ) . (4.36)The residues appearing in this expression can be simplified even further using the asymptoticbehavior of the d Ω m (4.27) I ∞ ± (Ω m dω i ) = I ∞ v m dω i . (4.37)This means that K im = − I ∞ v m dω i (4.38)and hence − σ sj K jm g m = I ∞ g m v m σ sj dω j = I ∞ g m v m dη s = ∂∂u s I ∞ g m v m s dv = − ∂∂u s I ∞ g m v m +1 m + 1 ds = − ∂∂u s (cid:28) g m v m +1 m + 1 (cid:29) = − ∂∂u s W eff ( u q ) . (4.39)From this we see that, up to an overall constant, R Λ Σ | dw | is precisely the scalar potential(4.13) that controls the vacuum structure in the Seiberg-Witten regime. Namely, M-theorycorrectly captures physics of perturbed Seiberg-Witten theory, because for each (off-shell)configuration in gauge theory there is an M5 curve of the form (4.29), (4.30) which hasprecisely the same energy. In principle, we can be more specific about our particular cutoff Note that contributions from subleading behavior of Ω m clearly vanish because dω i also vanishes at ∞ . required to ensurethat R Λ Σ dw ∧ d ¯ w is a constant. Because the potential (4.13) exhibits no explicit dependenceon the cutoff scale Λ , though, we are precisely in the situation of [60] and this is completelyunnecessary. While we have seen that the potential (4.13) arises naturally from the M-theory framework,let us digress for a moment to discuss the implications of this result. Equations of motionthat follow from the expanded action (4.14) combined with the constraints (2.5), (4.17) andboundary conditions (4.16) imply not only that the moduli sit at critical points of the Seiberg-Witten potential (4.13) but also lead to a specific M5 embedding with holomorphic s, v andharmonic w . As such, any exact solution to the full M5 equations of motion which also satisfies(2.5), (4.17), and (4.16) must reduce to an embedding of precisely this type when w is scaledto be sufficiently small. Among other things, this provides another way of seeing that theM-theory realization correctly captures supersymmetric vacua of the Yang-Mills theory.The fact that the off-shell potential of M-theory agrees with that of gauge theory meansthat even nonsupersymmetric vacua must be captured by M-theory, not just the supersym-metric ones. However, there is some tension between this claim and the known facts aboutM5 curves. In particular, as first pointed out in [15], holomorphic boundary conditions ofthe sort (4.16) generically preclude the existence of any exact solutions other than the super-symmetric ones . In that case, what are we to make of the approximate nonsupersymmetricembeddings that follow from (4.14)?A crucial point is that in order to make sense of (4.14) and extract from it the Seiberg-Witten potential (4.13), we had to regulate it by introducing a cutoff scale Λ . As such, theregulated action does not capture the behavior of the full M5 in any sense; rather, it describeslocal fluctuations of the M5 within a finite volume region that does not extend all the wayto ∞ . Approximate solutions obtained from the regulated action can thus only hope to bereliably obtained from exact ones for which we impose the condition (4.16) on some surfacealong v ∼ Λ in the interior. Extending out toward ∞ , the exact embeddings may exhibitstarkly different behavior, even becoming wildly nonholomorphic at ∞ .This should not be completely unexpected because sorts of stringy realizations that weare studying can only admit Yang-Mills descriptions at sufficiently low energies. Taking theNS5/D4 configurations in their entirety corresponds to providing a specific UV completionto this description which differs quite significantly from that of an asymptotically free gauge We will see this later in the context of specific examples when discussing exact M5 embeddings. . Note that, while the Λ -independence of theregulated M5 action suggests that it may accurately capture the decay of a nonsupersymmetricconfiguration into a supersymmetric one, one of course has the usual runaway beyond thecutoff scale associated to the difference in boundary conditions at ∞ . In the previous sections we showed how M5-branes capture the physics of N = 2 supersym-metric gauge theories perturbed by a small superpotential. Theories of this type have beenstudied recently [2, 3] and it was found that they allow supersymmetry breaking metastablevacua (OOPP vacua). Because we have already demonstrated that the full gauge theorypotential in the Seiberg-Witten regime can be reproduced from off-shell M5-brane configura-tions when dw is sufficiently small, those OOPP vacua are guaranteed to correspond to certainnonholomorphic M5-brane configurations of the form (4.15) and (4.29) which approximatelysolve the equations of motion. After quickly reviewing the results of [2, 3], in the next sectionwe will proceed to find such explicit M5-brane curves corresponding to the OOPP metastablevacua, concentrating for simplicity on the case of SU (2) N = 2 gauge theory. If we perturb an N = 2 gauge theory by a small superpotential, then to lowest order in theperturbation, the resulting scalar potential is exactly computable. If u i are coordinates onthe Coulomb branch with K¨ahler metric g i , then the addition of the superpotential W ( u i )generates a scalar potential equal to: V ( u i ) = g i ∂ i W ∂ W , (5.1)which can also be written as (4.13) in special coordinates.It was already suggested in [9] that there might be appropriate choices of the superpotentialfor which the scalar potential (5.1) has non-supersymmetric local minima. In the same paperthe simplest case of N = 2 SU (2) perturbed by a quadratic superpotential W = g tr(Φ ) = gu for the adjoint scalar field was considered. It was shown that in this case the perturbation does23ot generate any local nonsupersymmetric minima. The only critical points of the resultingpotential are the two supersymmetric vacua and a saddle point at u = 0.It was then discovered by [2, 3] that superpotentials of higher order can indeed generatemetastable points. More specifically [2] showed that, for a generic choice of a point on theCoulomb branch of any N = 2 theory, it is possible to find a superpotential perturbationwhich generates a metastable vacuum at that point. As we explain below, this possibility isbased on the fact that the sectional curvature of the K¨ahler metric on the Coulomb branchof any N = 2 supersymmetric gauge theory is positive semi-definite. This follows directlyfrom the fact that the K¨ahler metric in (rigid) special coordinates is the imaginary part of aholomorphic function: g i = Im ∂ F ( a ) ∂a i ∂a j , (5.2)where F ( a ) is the holomorphic prepotential in terms of special coordinates a i on the Coulombbranch.Following [2], let us quickly review how one can find the appropriate superpotential per-turbation. Consider a point p on the Coulomb branch M of an N = 2 gauge theory, at whichwe want to generate the metastable vacuum. Let u i be complex coordinates on the modulispace near the point p . We introduce K¨ahler normal coordinates z i around p , defined bythe expansion: z i = u ′ i + 12 ˜Γ ijk u ′ j u ′ k + 16 ˜ g im ∂ l (˜ g nm ˜Γ njk ) u ′ j u ′ k u ′ l , (5.3)where u ′ = u − u ( p ) and ˜ means evaluation at p .In these coordinates the metric takes the form: g i ( z, z ) = ˜ g i + ˜ R ikl z k z l + O ( z ) . (5.4)We choose the superpotential: W = k i z i (5.5)and find that the scalar potential around p has the expansion: V = ˜ g i ∂ i W ∂ W = ˜ g i k i k + k i k ˜ R ikl z k z l + O ( z ) . (5.6)As shown in [2], the quadratic term is positive definite at generic points on the moduli space,therefore the vacuum at z i = 0 is naturally metastable. Because local stability depends onlyon the leading terms in the expansion (5.4), however, the superpotential (5.5) is typically The familiar Riemann normal coordinates on a general K¨ahler manifold are not holomorphic. For thisreason, it is more useful to introduce the holomorphic K¨ahler normal coordinates, which are naturally adaptedto the complex structure of the manifold [62, 63]. N = 2 SU (2) Gauge Theory
In the case of SU (2) the moduli space M can be parametrized by the gauge invariant quantity: u = 12 (cid:10) tr Φ (cid:11) , (5.7)which is a good global complex coordinate on M . According to our previous discussion, thegeneral form of the required superpotential to generate a metastable vacuum at any point p ∈ M is: W ( u ) = g ′ ( β ′ u + γ ′ u + u ) , (5.8)where the coefficients g ′ , β ′ , γ ′ depend on the choice of p and can be easily computed from theK¨ahler normal coordinate expansion around p . As written, this is a multi-trace superpotential.This is not very convenient, because in the M-theory constructions the information of thesuperpotential is introduced by starting with the tree level superpotential in a single tracerepresentation: W (Φ) = g (cid:18) β + γ + 12 tr Φ (cid:19) (5.9)and imposing the boundary conditions for the M5 brane on the w - v plane: w ( v ) ∼ ± W ′ ( v ) ∼ ± g ( βv + γv + v ) . (5.10)So we need to rewrite (5.8) in the form (5.9). Of course for general gauge group a multi-tracesuperpotential cannot always be written, using trace identities, as a sum of single traces.However this is always possible in the case of SU (2). To find the precise relationship betweenthe coefficients g ′ , β ′ , γ ′ and g, β, γ we need to use the relations for the chiral ring of N = 2 SU (2) gauge theory. Following the analysis of [43] we have: u = 14 (cid:10) tr Φ (cid:11) = u ,u = 16 (cid:10) tr Φ (cid:11) = u u Λ , (5.11)where the terms proportional to Λ are the quantum corrections of the chiral ring due toinstantons. Using (5.11) the multitrace superpotential (5.8) can be written as (5.9), In fact, the truncation is also important for supersymmetry-breaking because the theory with full su-perpotential (5.5) can realize a non-manifest N = 1 supersymmetry which is then preserved at the z i = 0vacuum [65, 64]. An Exact M5 Curve for SU(2)
In this section, we explicitly construct the M5 curve corresponding to the OOPP vacua [2, 3]in gauge theory, for the special case of SU (2) gauge group. In section 4, we worked in a regimewhere the expansion (4.14) in small dw was valid and used this to obtain a correspondencebetween M5-brane curves and gauge theory states. We could restrict ourselves to the sameregime, where the problem of finding a minimal-area M5 curve is tantamount to minimizingthe scalar potential (4.13), and obtain the M5 curve of the form (4.29) and (4.30) whichcorrespond to the OOPP vacua. However, in this section, we will endeavor to find the exact M5 curve without any approximation by attacking the honest minimal-area problem.Finding minimal-area M5 curves in such general cases is complicated but, at the sametime, illuminates the point raised in section 4.3. As pointed out in [15], if we look for M5curves with holomorphic boundary conditions (5.10) at infinity, all we can have are holomor-phic curves, corresponding to supersymmetric vacua. Therefore, if we want nonholomorphiccurves, corresponding to nonsupersymmetric vacua, we are forced to consider nonholomorphic boundary conditions. On the other hand, however, in section 4 we saw that we can realizeboth supersymmetric and nonsupersymmetric vacua using M5 curves in the Seiberg-Wittenregime with the holomorphic boundary condition such as (5.10). For things to be consistent,it must be that, when the “bending” along w becomes small, the nonholomorphicity of theM5 curve at infinity becomes very small and consequently one can make the holomorphiccondition (5.10) hold to arbitrary precision on a boundary surface at Λ in the interior .In this section, we will present exact solutions to the minimal area equations with preciselythis feature. This will allow us to accurately define a limit in which (4.14) can be trusted andto demonstrate that, in this limit, the exact solutions reduce to approximate ones which solvethe corresponding equations of motion. In particular, their moduli sit precisely at criticalpoints of the Seiberg-Witten potential (4.13).As in [1], nonholomorphic M5 embeddings are most easily described using a parametricframework. As such, we shall begin by reviewing the parametric description of the M5 con-figuration realizing pure SU (2) N = 2 gauge theory. We will then turn on a superpotentialof the sort (5.9) by suitably “bending” the M5 brane in the w and v directions. Within thissetup, we will look for minimal area nonholomorphic solutions corresponding to the OOPPvacua of the gauge theory. That one cannot simply take the boundary surface to ∞ in these solutions reflects an order of limits issuein this problem. In particular, we will see in this example that the parameter regime for which the expansion(4.14) remains valid, allowing us to reproduce the Seiberg-Witten potential (4.13), is one which requires g in(5.10) to approach zero as Λ is taken to ∞ . .1 The Minimal Area Problem First, however, let us review a few technical results that will be useful for our parametricrepresentation of M5 configurations. The mathematical problem that we have to solve isto find a minimal-area embedding of a Riemann surface Σ in the space R × S , which isparametrized by the complex coordinates w, v, s . Because of the identification s ∼ s + 2 πi , itis convenient to introduce the coordinate t = Λ e − s as in (2.8). In the case of supersymmetric M F ( v, t ) = 0 ,F ( v, w ) = 0 . (6.1)Such a representation of the surface is not very convenient when we want to consider non-holomorphic embeddings, as we have found [1] that the parametric description is often moresuitable for this purpose. For this, we consider a Riemann surface Σ with holomorphic coor-dinate z on its “worldsheet”, and describe the embedding of the M5 brane by the functions: v ( z, z ) , w ( z, z ) , s ( z, z ) . (6.2)Since the M X µ ( σ a ) embedded in an ambient space of metric G µν , thenthe induced metric on the surface is: g ab = G µν ∂ a X µ ∂ b X ν (6.3)and the area is given by the expression: A = Z Σ p det( g ) (6.4)As we know very well from the case of the bosonic string, it is very useful to pick the coor-dinates σ i in such a way that the induced metric is in conformal gauge. We use the complexvariable z for this class of coordinates. In this gauge the equations of motion from varyingthe “Nambu-Goto” action (6.4) become equivalent to two conditions: the first is that theembedding functions X µ ( z, z ) must be harmonic and the second that the Virasoro constraintmust be satisfied: G µν ∂X µ ∂z ∂X ν ∂z = 0 . (6.5)27n our case, the situation is particularly simple because the background metric G µν is flat andharmonic functions in two-dimensions are simply sums of holomorphic and antiholomorphicones. As such, our embedding functions take the form v ( z, z ) = v H ( z ) + v A ( z ) ,w ( z, z ) = w H ( z ) + w A ( z ) ,s ( z, z ) = s H ( z ) + s A ( z ) (6.6)for holomorphic v H/A , w H/A , s H/A and the constraint (6.5) becomes ∂v H ∂v A + ∂w H ∂w A + R ∂s H ∂s A = 0 . (6.7)Note that although this is a nonlinear condition, it is nevertheless a holomorphic one, aproperty which makes it considerably easier to solve. For any set of holomorphic functions v H/A , w H/A , s H/A satisfying (6.7), the embedding given by (6.6) extremizes the area (6.4).Finally, to specify the lift of a given NS5/D4 configuration, we must impose suitableconditions on our embedding functions as discussed in section 2. In particular, for s we mustfix the period integrals (2.5) about the ˆ A - and ˆ B -cycles of figure 212 πi I ˆ A j ds = N j and 12 πi I ˆ B j ds = − α j . (6.8)As for w and v , they take values in R and hence dw and dv must have vanishing periodsaround all compact cycles in the geometry. Phrased in terms of the ˆ A - and ˆ B -cycles of figure2, this condition amounts to I ˆ A j dw = 0 for all j, I ˆ B i − ˆ B j dw = 0 for all i, j (6.9)and similar equations for dv . Their asymptotic behavior at ∞ is then fixed by the superpo-tential w ∼ ± W ′ ( v ) . (6.10)In terms of the embedding functions v ( z, ¯ z ) and w ( z, ¯ z ), this condition imposes nontrivialrelations between their pole structures at the punctures. More precisely meromorphic since they can have poles at the punctures of the Riemann surface. Checking whether this extremum is truly a minimum (locally stable) is extremely difficult. To answer thisquestion one has to consider the second variation of the area functional (6.4) around the local extremum, andcheck that there are no deformations that locally decrease the area. The authors are not aware of any generalmethod of checking local stability for an arbitrary embedding. In the simplifying limits of sections 3 and 4,though, this question is easier to address and translates into the matter of local stability in the correspondingfield theory potentials. Of course, as discussed earlier in this section we will have to suitably relax this constraint later whenlooking for nonsupersymmetric solutions. A A v BB (a) Parametrization of the genus 1 M5 em-bedding as a double cover of the v -planewith ˆ A and ˆ B cycles indicated a a A B z τ A B (b) Parametrization of the genus 1 M5 em-bedding via the fundamental parallelogramon the z plane with ˆ A and ˆ B cycles indicated Figure 5: Parametrizations of the genus 1 M5 embeddingTo summarize, the problem we have to solve is to find meromorphic functions v H/A , w H/A , s H/A on a (punctured) Riemann surface Σ, satisfying the Virasoro constraint (6.7) and suchthat the embedding functions (6.6) have the monodromies (6.8), (6.9), and correct boundaryconditions (6.10).
We now specialize to the case of SU (2), where our setup is particularly simple. The M5 lift,Σ, is a genus one curve with two punctures corresponding to the points at ∞ on the twoNS5-branes. Consequently, the embedding is described by meromorphic functions on a torus.In this section, we will introduce a convenient collection of such meromorphic functions andreview their basic properties.To parametrize the M5 curve, we use the complex z -plane subject to the identifications z ∼ z + 1 ∼ z + τ. (6.11)We also have to specify the two marked points a , a which correspond to the points at ∞ ofthe NS5-branes. Notice that we can always use the conformal killing vectors of the torus to setone of the punctures to any desired point on the torus so that only the difference a = a − a has an invariant meaning. In figure 5, we depict both the v and z parametrizations of ourcurve along with the corresponding realizations of the ˆ A - and ˆ B -cycles.As we saw in the previous section, the minimal area equations imply that the embeddingcoordinates of the M z , possibly withsingularities at the punctures, with fixed monodromies around the compact cycles of the29urface. As we explain in Appendix A, it is possible to write the most general harmonic1-form f on a punctured Riemann surface in terms of the standard differentials, namely,holomorphic differentials (a.k.a. meromorphic differentials of the first kind) ω i , meromorphicdifferentials of the second kind d Ω Pn , and meromorphic differentials of the third kind d Ω P,P ′ .In the case of our genus one surface, there is only one holomorphic differential, namely ω = 2 πi dz, (6.12)whose periods on the standard compact A and B cycles are:12 πi Z ˆ A ω = 1 , πi Z ˆ B − ˆ B ω = τ. (6.13)To explicitly construct meromorphic differentials with desired poles at the punctures, we startwith the basic theta function: θ ( z ) = ∞ X n = −∞ e iπn τ +2 πinz , (6.14)and use it to define the (quasi-)elliptic functions: F ( z ) = ln θ ( z − ˜ τ ) , ˜ τ ≡
12 ( τ + 1) , (6.15) F ( n ) i = (cid:18) ∂∂z (cid:19) n F ( z − a i ) . (6.16)Because F ( z ) ∼ ln z near z = 0, the function F ( n ) i introduces an n th order pole at the point a i .For n ≥ n = 0 , F i ( z + 1) = F i ( z ) ,F i ( z + τ ) = F i ( z ) + iπ − πi ( z − a i ) ,F (1) i ( z + 1) = F (1) i ( z ) ,F (1) i ( z + τ ) = F (1) i ( z ) − πi. (6.17)Useful information about these elliptic functions, including their relation to Weierstrass func-tions, can be found in the appendices of [1].Before we close this section, let us summarize the relation between these elliptic functionsand the standard meromorphic differentials. On a genus one surface the only holomorphicdifferential is ω (6.12). One basis for meromorphic differentials of the second kind having a( n + 1)th order pole at the point a i are d Ω a i n = ( − n ( n − F ( n +1) i dz, n ≥ . (6.18)30inally the meromorphic differential of the third kind with simple poles at two points a , a and opposite residues is: d Ω a ,a = ( F (1)1 − F (1)2 + iπ ) dz. (6.19) N = 2 Curve in Parametric Representation
To see how we can use the elliptic functions to set up the parametric representation of thesurface, let us try to write the N = 2 curve which describes the Coulomb branch of the gaugetheory without superpotential in this formalism . As usual, the IIA configuration consists ofparallel NS5’s with two D4’s suspended in between, with its M-theory lift given by a singleM5 extended on the genus one holomorphic Riemann surface corresponding to the SU (2)Seiberg-Witten curve t − P ( v ) t + Λ = 0 , (6.20)where P ( v ) = v − u . (6.21)Now we want to describe a parametric representation of this curve using the elliptic func-tions introduced in the previous section. We start with the embedding v ( z ). This functionmust have first order poles at the two punctures of the torus, which correspond to the infiniteregions of the NS5-branes. We are thus led to the expression: v ( z ) = b (cid:16) F (1)1 − F (1)2 + iπ (cid:17) , (6.22)where the iπ shift has been added for convenience and b is a constant whose value will befixed later.The function v ( z ) is holomorphic and single-valued as it should be. On the other hand,we know from (2.5) that s ( z ) is multivalued, having winding number 1 around each of theˆ A -cycles of Figure 5, 12 πi I ˆ A j ds = 1 , j = 1 , . (6.23)Up to a constant shift, there is a unique holomorphic s ( z ) which satisfies these conditions s ( z ) = 2 ( F − F + iπz ) . (6.24)We now turn to the ˆ B period-constraints (2.5): I ˆ B j ds = − α, j = 1 , . (6.25) Such a parametric representation is also described in Appendix B of [35]. Without this shift, we would obtain a generic U (2) curve with nonzero u . Including this shift will set u = 0, leaving us with the SU (2) curve. (cid:18)I ˆ B − I ˆ B (cid:19) ds = 0 = ⇒ s ( z + τ ) = s ( z ) (6.26)is particularly simple and fixes the distance a between the punctures to be a ≡ a − a = τ . (6.27)The final constraint fixes b as a function of τ and the dynamical scale, Λ. The simplest wayto derive the correct relation is to demonstrate, as we do in Appendix B, that v ( z ) (6.22) and s ( z ) (6.24) satisfy the equation (6.20) under the identification b = Λ ℘ ( τ / − g , (6.28)where ℘ ( z ) is the Weierstrass ℘ -function with the half-periods ( ω , ω ) = (1 / , τ /
2) and g isone of the Weierstrass elliptic invariants.The result is a family of holomorphic embeddings parametrized by the complex structuremodulus, τ . As shown in Appendix B, we can trade τ for the more conventional modulus u via u = 3 b ℘ ( τ / . (6.29) Introduction of a superpotential is achieved by bending the M5 brane in the w directionthrough the boundary conditions w ( v ) ∼ ± W ′ ( v ) (6.30)as v → ∞ . As we saw in section 2, it is rather simple to find the supersymmetric config-urations compatible with these boundary conditions. The w bending (6.30) can be realizedby a holomorphic embedding at discrete points on the Coulomb branch which correspondprecisely to extrema of the superpotential. Moreover it is possible to find these points by thefactorization conditions (2.14),(2.15), which together with (2.9) elegantly give the form of thefunctions v, w, s for the supersymmetric embeddings.As discussed before, when we want to consider nonholomorphic minimal area embeddings,the situation is more complicated. The main difficulty is that, strictly speaking, it is notpossible to find nonsupersymmetric minimal area embeddings satisfying (6.30). To see why,let us return to the constraint (6.7) ∂v H ∂v A + ∂w H ∂w A + R ∂s H ∂s A = 0 (6.31)32nd study the pole structure of various terms that appear. Because the embedding is locallyone-to-one as v → ∞ , v H has first order poles at the marked points. The holomorphicfunction w H , on the other hand, has poles of order n , the degree of W ′ ( v ). Imposing purelyholomorphic boundary conditions w ( v ) near v → ∞ further implies that v A and w A have nopoles at all . This means that the v contribution to (6.31) contains poles of order 2 andlower while the w contribution contains poles of order n + 1 and lower. Finally, because ds has at most first order poles, the s contribution to (6.31) also contributes poles of order 2 andlower. As a result, the contribution from w cannot be cancelled by either of the others andan exact solution is not possible .To find solutions, then, it is clear that we will have to introduce antiholomorphic contri-butions to v with higher order poles and even possibly new holomorphic contributions as well.In what sense can we make a connection with the analysis of section 4, then? As discussedin section 4.3, there is no reason for us to expect that exact nonholomorphic solutions existwhich satisfy the condition (6.30) at v = ∞ . Rather, we can only expect that it holds atsmall w to arbitrary precision on a boundary surface in the interior.That this can happen is quite easy to see directly from our analysis so far. If we startwith an exact solution and scale w to be parametrically small, (6.31) suggests that the anti-holomorphic contributions to v will decrease even more rapidly, provided of course that thecorresponding holomorphic ones remain finite. In that sense, we may expect to find solutionswhich approximately approach embeddings of the form advertised in section 4.3, namely thosefor which s and v describe holomorphic Seiberg-Witten geometries with a harmonic embed-ding along w . Because the antiholomorphic terms contain poles, however, their contributioncan never be made parametrically small at ∞ . Rather, we must introduce a fixed cutoffsurface along v ∼ Λ and then scale w to zero while keeping Λ fixed. For | v | < | Λ | , suchsolutions will take the form of section 4.3. Farther toward ∞ , however, they will remainwildly nonholomorphic. As a result, if we insist on finding minimal area solutions, we willhave to relax the boundary condition (6.30) slightly. In this subsection, we will motivate a suitable ansatz and present a family of exact solutionswhich will be relevant for the SU (2) gauge theory. In the following subsection, we will then Even though this implies that any nontrivial contribution to v A or w A undergoes monodromies aroundthe nontrivial cycles of the torus, this is not a problem because it is only the combinations v H + ¯ v A and w H + ¯ w A that must be well-defined. Note that we can in principle have solutions where w is holomorphic and s, v are not. From our analysisof section 4, though, it is clear that the solutions we are looking for are not of this type. z the holomorphic coordinate on the worldsheet of the torus. In the supersym-metric case the function v ( z, z ) has first order poles around the punctures. For the reasonsmentioned in the last section, we consider a more general ansatz where the function v ( z, z )has also higher order and nonholomorphic poles. We will take the ansatz for w ( z, z ) to bedictated by the degree of the superpotential, and include higher order and nonholomorphicpoles in v ( z, z ) that are minimal for satisfying the Virasoro constraint. For s ( z, z ), we includeonly logarithmic bending. As we discuss in Appendix C, the form of the ansatz is also con-strained by requiring the symmetries that supersymmetric curves possess in the SU (2) case.From all these considerations, we are led to study the following ansatz: w = A (cid:16) F (5)1 + F (5)2 (cid:17) + A (cid:16) F (3)1 + F (3)2 (cid:17) + A (cid:18) F (1)1 + F (1)2 + 2 πτ (cid:2) ( z − A ) − ( z − A ) (cid:3) − πi (cid:19) ,v = b (cid:16) F (3)1 − F (3)2 (cid:17) + b (cid:16) F (1)1 − F (1)2 + iπ (cid:17) + c (cid:16) F (3)1 − F (3)2 (cid:17) + c (cid:16) F (1)1 − F (1)2 + iπ (cid:17) ,s = (2 + ν ) ( F − F + iπz ) + ν (cid:0) F − F − iπz (cid:1) , (6.32)where A ≡ a + a τ = τ + iτ . (6.34)Finally we have to fix the undetermined coefficients by demanding that our ansatz satisfiesthe Virasoro constraint. This leads to the solution b = 0 ,c = − πA A b τ ,c = − πA ( A + 20 A ℘ ( τ / b τ ,R τ ¯ ν ( ν + 2) = − π ¯ A (cid:2) A + 48 A (cid:0) ℘ ( τ / − g (cid:1) + 12 A ℘ ( τ / (cid:3) , τ ( A + 12 A ℘ ( τ / (cid:0) g − ℘ ( τ / (cid:1) + A [2 π + ( ℘ ( τ / − η ) τ ] . (6.35) There is also an analogous solution with c = 0 which is related to this one by complex conjugation of v . .Once we go away from the small superpotential limit or the Seiberg-Witten regime, thereis some degree of arbitrariness in turning on higher order and nonholomorphic terms in w, v, s .For example, although in the ansatz (6.32) we required the symmetries that supersymmetric SU (2) curves possess, we could have considered an ansatz which does not possess this sym-metry and might have ended up with a solution which is different from and presumably muchmore complicated than (6.35). However, the point here is to show the existence of an exactsolution which reduces to the approximate solution derived in the last section, thus justifyingthe expansion (4.14). We will see that the simple solution (6.32) does have such a property,and therefore it is a sufficient solution for our purpose.In the following section we will study the physical meaning of the various quantities whichparametrize this family. N = 2 Regime
In this section we would like to consider a limit of the solution (6.32), (6.35) such that, withinsome appropriate boundary surface, the curve takes the approximate form of a holomorphicSeiberg-Witten geometry s ( v ) with a harmonic embedding along w . Within this limit, wewill then attempt to impose boundary conditions of the sort (5.10) on that boundary surface.This will be self-consistent only provided g is small in a sense that we will make more precisebelow. In the end, this will allow us to explicitly realize OOPP vacua in the manner suggestedby the analysis of section 4. The parametric representation (6.22) and (6.24) of the SU (2) Seiberg-Witten curve givessome guidance as to what our limiting curve should look like. In particular, it suggests thatwe consider a regime in which we can simply neglect the terms proportional to c , c , and ν in (6.32) so that our solution takes the approximate form w = A (cid:16) F (5)1 + F (5)2 (cid:17) + A (cid:16) F (3)1 + F (3)2 (cid:17) + A (cid:18) F (1)1 + F (1)2 + 2 πτ (cid:2) ( z − A ) − (¯ z − ¯ A ) (cid:3) − πi (cid:19) ,v = b (cid:16) F (1)1 − F (1)2 + iπ (cid:17) ,s = 2 ( F − F + iπz ) . (6.36) Actually we only know that these are surfaces with extremal area. We have not been able to find apractical method to determine when the area is truly minimal in generic situations. In the OOPP limit, thisquestion can be answered by studying the corresponding scalar potential, which was discussed in section 5. A , A , and A tozero. A parametric separation can be achieved which results in the approximate form (6.36)because A , A , and A are linearly related to one another while c , c , and ν depend onthem in a quadratic manner. We must be a bit careful, though, because the c , c , and ν terms of (6.32) that we want to neglect nevertheless become arbitrarily large near the markedpoints, where they exhibit divergences of varying degree. This means that we can never takea limit in which our solution looks everywhere like (6.36). Rather, the best we can hope for isthat our curve approximately resembles (6.36) only after the divergences have been suitablyregulated. For this, we introduce a cutoff by removing a circle of radius ǫ about each markedpoint. Holding ǫ fixed, we can then take A , A , and A sufficiently small that our solution(6.32) approaches (6.36) to arbitrary precision inside the regulated surface. We shall hereafterrefer to this as the “OOPP limit”.Precise inequalities which yield the OOPP limit are most easily determined by studyingthe conditions for which (6.36) itself becomes an approximate solution to the equations ofmotion. More specifically, we want to scale A , A , and A in such a way that the Virasoroconstraint (6.7) holds to arbitrary precision inside the cutoff surface. It is easy to see that,for generic moduli τ , we simply need A A τ ≪ ǫ A A τ ≪ ǫ (6.37)These can in turn be translated into relations involving the physical cutoff Λ by noting that,in the OOPP limit, ǫ = b Λ . (6.38)Note that applying the conditions (6.37) to our exact solution (6.32)–(6.35) leads to suppres-sion of the c , c , and ν terms as expected . We now turn to a study of the approximate curve (6.36) to which our exact solution (6.32)reduces in the OOPP limit. More specifically, we have a family of curves characterized by fiveindependent parameters, A , A , A , b , and τ . On this family, we are now free to specify theboundary condition (5.10) w ( v ) ∼ ± g (cid:0) βv + γv + v (cid:1) . (6.39) The ǫ suppression might seem to be larger than necessary at first glance but we must keep in mind that itis not sufficient for the nonholomorphic terms in (6.32) to simply scale like a positive power of ǫ . Rather, wewant nonholomorphic corrections to the boundary conditions at Λ to become negligible and, for this, higherorder suppression of various nonholomorphic parts of the embedding can be needed.
36y studying the expansion of the elliptic functions F ( n ) i , this fixes A , A , and A in terms of g , β , and γ as A = g βb ,A = g γb + 5 βb ℘ ( τ / ,A = g b + 6 γb ℘ ( τ /
2) + βb [30 ℘ ( τ / − g ]2 . (6.40)Furthermore, recognizing the v and s embeddings as equivalent to our parametric descriptionof the Seiberg-Witten geometry, (6.22) and (6.24), we can immediately read off the relation(6.28) b = Λ ℘ ( τ / − g , (6.41)which follows from the boundary conditions that we impose along s . Already this is enough torephrase the OOPP limit (6.37) in terms of quantities which enter our boundary conditions.For example, if we suppose that τ is generic and both β Λ and γ Λ are of O (1), we see thatthe approximate solution (6.36) with boundary conditions (6.39) can only arise as the OOPPlimit of our exact solution provided g ≪ (cid:18) ΛΛ (cid:19) . (6.42)Of course, we can have more complicated inequalities for non-generic choices of parameters.In our previous analysis we simply said that g had to be “sufficiently small” so it is nice tosee a more precise condition arise .Let us now return to the one remaining unfixed parameter, namely the complex structuremodulus τ . From our analysis of section 4, we expect that this should be determined by theSeiberg-Witten potential (4.13). In the case at hand, however, we note that τ is fixed by theequations of motion of the exact solution from which our approximate curve (6.36) descends.Indeed, looking at (6.35), the first four equations can be thought of as fixing the parameters b , c , c , and ν , which are all negligible in the OOPP limit. The last equation, however,yields a nontrivial relation which must be satisfied among the four remaining parameters A , A , A , and τ of the approximate solution (6.36). Plugging in the values (6.40), this equation Note that the appearance of Λ provides yet another demonstration of the importance of regulatingthe surface before imposing (6.39). If we take Λ to be strictly infinite then our reduced curve is never anapproximate solution for any nonzero g no matter how small. b g (cid:0) g − ℘ ( τ / (cid:1) (cid:0) βb ℘ ( τ /
2) + γb (cid:1) + b g (cid:2) βb (cid:0) ℘ ( τ / − g (cid:1) + 6 γb ℘ ( τ / (cid:3) (cid:18) ℘ ( τ / − η + 2 πτ (cid:19) . (6.43)To keep notation from getting out of control, we do not plug in the result (6.41) for b .Nevertheless, we should remember that b is itself a function of τ .In the end, we have found that, in the OOPP limit, our exact solution reduces to aholomorphic Seiberg-Witten geometry with harmonic w embedding and complex structuremodulus τ determined by solving the complicated equation (6.43). From the general discussion in section 4, we expect that the equation (6.43) is nothing otherthan the condition for τ to be a critical point of the Seiberg-Witten potential associated tothe superpotential W (Φ) = g (cid:18) β Φ γ Φ (cid:19) . (6.44)We will now come full circle and demonstrate this explicitly by studying the correspondingSeiberg-Witten potential (4.13) given by V SW = 1 τ | ∂ u W eff ( u ) | (cid:12)(cid:12)(cid:12)(cid:12) ∂u ∂a (cid:12)(cid:12)(cid:12)(cid:12) , (6.45)where W eff ( u ) = g ( βu ( u ) + γu ( u ) + u ) . (6.46)Expressing the various ingredients of (6.45) in terms of τ is quite straightforward. We havealready seen that u = 3 b ℘ ( τ /
2) (6.47)and recall from (5.11) that u = u , u = u u Λ . (6.48)To determine ∂u /∂a will require a bit more work. We first compute the Seiberg-Witten1-form in our parametric formalism dλ SW = v ds = h − b (cid:16) F (2)1 + F (2)2 (cid:17) + 2 b ( ℘ ( τ / − η ) i dz (6.49)in order to evaluate a = I ˆ A dλ SW = 2 b ( ℘ ( τ / − η ) . (6.50)38e then compute ∂u ∂τ = b iπ (cid:0) ℘ ( τ / − g (cid:1) , ∂a∂τ = b πi (cid:0) ℘ ( τ / − g (cid:1) (6.51)from which it follows that ∂u ∂a = 2 b . (6.52)Putting it all together, we find in the end that ∂u ∂a ∂ u W ( u ) = gb (cid:0) γb ℘ ( τ /
2) + βb (cid:0) ℘ ( τ / − g (cid:1)(cid:1) , (6.53)which we recognize as nothing other than twice the value of A (6.40) required by our boundaryconditions! This means that V SW can simply be written as V SW ∼ τ − | A | . (6.54)A direct evaluation of the regulated integral R Λ Σ | dw | on the curve (6.36) can easily be seen toyield an identical result, as expected from our general arguments in section 4. Differentiatingthis with respect to τ , we find that critical points occur whenever the following condition issatisfied2 + 6 γb ℘ ( τ /
2) + βb (cid:0) ℘ ( τ / − g (cid:1) = 4 τ (3 ℘ ( τ / − g ) ( b γ + 6 βb ℘ ( τ / π + τ ( ℘ ( τ / − η ) . (6.55)This is precisely equivalent to the condition (6.43) imposed by requiring that the curve (6.36)arises via the OOPP limit of an exact solution of the form (6.32), (6.35). In this section, by taking a semiclassical limit of the exact curve obtained in the previoussection, we discuss the basic mechanism for the metastability of the OOPP vacua. We willsee that the metastability is the result of the balance between two forces, both of which canbe understood in a geometric manner.
Let us consider the situation where the distance between the two tubes corresponding to thetwo D4-branes is much larger than the size of the tubes. In this limit the Riemann surface ofthe M5-brane becomes a long torus with τ ≫
1. If τ ≫
1, using the q -expansion formulas(D.1) in (6.29) and (6.28), we obtain q = e iπτ = Λ u . (7.1)39herefore, τ ≫ u ≫ Λ , namely the semiclassical limit of gauge theory.In this semiclassical limit, the size of the tubes are negligible and we can think of thesystem approximately as made of two NS5-branes with D4-branes stretching between them.The NS5-branes are curved in the w direction along w = ± ∂ v W ( v ) and, at the same time,logarithmically bent in the x direction so that the distance L between two NS5-branes is afunction of v ; see Figure 6. The relation between s and v is, from eq. (4.15), L ( v ) W ’ ( v ) = w =− w W ’ ( v ) v w x NS5NS5 D4D4
Figure 6: A semiclassical picture of the configuration. The two NS5’s are curved in the w direction along w = ± W ′ ( v ) and bent in the x directions such that the distance between themis L ( v ). We depicted this situation by the two NS5’s being curved along curves w = ± W ′ ( v )(blue) in two bent w - v planes (dotted lines) which are at x = ± L ( v ) / s = log P ( v ) ± p P ( v ) − Λ Λ ! . (7.2)For | v | ≫ | Λ | , this gives s ∼ ( v/ Λ) , /v ) . (7.3)Therefore, the distance along the x = Re s direction is given by L = ∆ x = R ∆ Re s = 2 R log (cid:12)(cid:12)(cid:12) v Λ (cid:12)(cid:12)(cid:12) − R log (cid:12)(cid:12)(cid:12)(cid:12) Λ v (cid:12)(cid:12)(cid:12)(cid:12) = 2 R log (cid:12)(cid:12)(cid:12) v Λ (cid:12)(cid:12)(cid:12) . (7.4)Let us place one D4-brane at v and the other at − v . Then the D4-branes tilt in the w direction by a small angle θ ≈ | ∂ v W | L , (7.5)as one can see from Figure 7. One may wonder that the tilted D4-branes pull the NS5-branes also in the w direction, so that the NS5-branes will not lie on the curve w = W ′ ( v ) but on some “distorted” curve. However, such an effect is of higherorder in θ and ignorable in the present approximation. v and the one at − v have a Coulomb potential energy V C . In addition, there is potential energy coming from the tension of the D4-branes, whichis V T = 2 T L . Here, T is the tension of a D4-brane and the factor 2 is because we havetwo D4-branes. If the D4-branes were not tilted in the w direction, i.e. , if θ = 0, then thesystem would be supersymmetric and the two potentials would exactly cancel each other: V C + V θ =0 T = 0. However, if θ = 0, the D4-branes are longer by ∆ L = ( L/ cos θ ) − L ≈ Lθ / θ = 0. As the result, V T gets increased by∆ V T = V θT − V θ =0 T = 2 T ∆ L ≈ T θ L ≈ T | ∂ v W | L . (7.6)On the other hand, V C is not affected by θ because the Coulomb force depends only on thedistance between D4-brane endpoints. Therefore, the total potential energy when θ = 0 isgiven by V sc = V C + V θT = 4 T | ∂ v W | L . (7.7)The equation of motion derived from this potential energy is0 = ∂ v V sc ∝ ∂ v W∂ v W − ∂ v LL . (7.8)One can interpret the first term in (7.8) as a force which pushes the D4-branes towards thevalues of v for which | ∂ v W | is smaller, and the second term as a force towards the values of v for which L is larger. As can be seen from (7.4), the latter force is logarithmic and tend tomove the D4-branes toward large | v | . On the other hand, the former force is polynomial andcan be tuned by choosing the polynomial W ( v ). Therefore, it is natural to expect that, forany value of v , one can choose the superpotential W ( v ) appropriately so that the equationof motion (7.8) is satisfied for that v . This is the geometric understanding of the metastablevacua found in [2].As a quick check, let us compare the potential (7.7) obtained above with the potentialenergy in gauge theory. In the semiclassical regime where | u | ≫ | Λ | , the moduli spacemetric for u is known to be [7]: g u u ≈ π | u | log (cid:12)(cid:12) u Λ (cid:12)(cid:12) . (7.9)41herefore, the gauge theory potential is: V gt = g u ¯ u | ∂ u W | ≈ π | u | log (cid:12)(cid:12) u Λ (cid:12)(cid:12) | ∂ u W | . (7.10)If we rewrite this in terms of v = √ u , V gt = π (cid:12)(cid:12) v Λ (cid:12)(cid:12) | ∂ v W | , (7.11)which agrees with the semiclassical potential (7.7) up to a numerical factor if we use (7.4).So, the potential (7.7) is indeed correct in the semiclassical regime.One can also understand the parameter W ’ W ’ x wv (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1) θ L D4NS5 − NS5
Figure 7: D4-branes tilted along w ν that appears in the exact M5 curve bya semiclassical reasoning, although strictlyspeaking ν is vanishing in the OOPP limit( ν = O ( g )). If a D4-brane parallel to the x -axis is ending on an NS5-brane, then thelift, an M5-brane, has the following curve: x = 2 R log | v | = R (log v + log v ) , x = 2 R arg | v | = − iR (log v − log v ) . (7.12)In the NS5 language, the v -dependence of x represents the pull of the tension of the D4-branealong the x direction, while the v -dependence of x represents the flux inserted into the NS5worldvolume by the D4-brane. Instead, if the D4-brane is not parallel to the x -axis butmakes a small angle θ with it, then the tension will reduce by factor cos θ and hence (7.12) isreplaced by x = R cos θ (log v + log v ) , (7.13)whereas x is unchanged because the amount of flux inserted by a D4 is independent of theangle between the D4 and the NS5. Therefore, s = x + ix goes as s = R [(cos θ + 1) log v + (cos θ −
1) log v ] ≈ R (cid:20)(cid:18) − θ (cid:19) log v − θ v (cid:21) . (7.14)Comparing this with the expression of the exact curve (6.32), we obtain ν = − θ − | ∂ v W | L . (7.15)We will see below that this is indeed satisfied in our M5-brane curve.42 .2 Semiclassical Limit of the M5 Curve In the previous subsection, we showed that, in general, the semiclassical potential V sc (eq.(7.7)) and the gauge theory potential V gt are the same for | v | ≫ | Λ | . Here we will checkthat the M5 curve obtained in section 6 satisfies the equations of motion derived from thesepotentials.In the M5 curve (6.36), the quantities τ, ν are determined in terms of the parameters A , A , A , b , or equivalently in terms of ( g, β, γ ), b , by solving the last two equations in (6.32).In the semiclassical limit τ ≫ q = e iπτ is very small and we can use the q -expansions (D.1)to simplify those equations. After dropping all O ( q ) terms and subleading terms in g , we canwrite the resulting equations as follows: τ = 2( βv + γv + 1) π (5 βv + 3 γv + 1) = 2 π ∂ v W ( v ) v ∂ v W ( v ) (7.16a) ν = − | gv | R (5 βv + 3 γv + 1) ( βv + γv + 1) βv + γv + 1 = − (cid:18) | ∂ v W ( v ) | πg τ (cid:19) , (7.16b)where we have set v = iπb and ∂ v W = g ( βv + γv + v ). Semiclassically, D4-branes aresitting at v and − v .Also, using the q -expansion (D.1) in (6.28), we obtain q = Λ / π b . Therefore, we canexpress τ in terms of v as follows: τ = 1 π log (cid:12)(cid:12)(cid:12)(cid:12) π b Λ (cid:12)(cid:12)(cid:12)(cid:12) = 1 π log (cid:12)(cid:12)(cid:12)(cid:12) v Λ (cid:12)(cid:12)(cid:12)(cid:12) ≈ π log (cid:12)(cid:12)(cid:12) v Λ (cid:12)(cid:12)(cid:12) . (7.17)Therefore, (7.16a) can be written as ∂ v W ( v ) ∂ v W ( v ) = 2 πvτ ≈ /v log (cid:12)(cid:12) v Λ (cid:12)(cid:12) . (7.18)If we recall the expression for L , eq. (7.4), we immediately see that this is nothing but (7.8).So, indeed, the M-theory curve satisfies the semiclassical equation of motion derived from V sc or V gt .Similarly, one can show that the value of ν in the M5 curve given by (7.16b) agrees withthe one given by the semiclassical expression (7.15). These agreements confirm that thereason for the existence of (meta)stable M5-brane configurations is the geometric mechanismdiscussed in the previous subsection. For this agreement, the numerical coefficient in (7.15) is important. As mentioned in footnote 33, (7.15)was derived assuming that there is no “distortion” of the NS5-branes in the w direction. Therefore, thisagreement means that the approximation to ignore the “distortion” was consistent.
43s we show in Appendix E, we can also take the type IIA limit of the M5 curve, which isdefined as the R → x direction disappears. In order for the distance between the two D4’s toremain finite in the limit, we must take τ → ∞ as τ ∼ /R , as one can see in (E.7).Therefore, (7.16a) in this limit means that the D4’s sit at v for which ∂ v W ( v ) = w ′ ( v ) = 0.This is understood as follows: In the IIA limit, there is no log bending of NS5’s along x andthe associated force, the second term in (7.8), is not there. So, the D4’s must sit at pointswhere they can sit in equilibrium without any force (although unstable in this limit), whichhappens for w ′ ( v ) = 0. If one makes R = 0 and the quantum effects of bending are turnedon, though, these points can be stabilized if the superpotential is properly tuned. In this paper, we demonstrated that the M5 lifts of NS5/D4 configurations which realize per-turbed Seiberg-Witten theory accurately capture, in a suitable limit, the full scalar potentialof the gauge theory and hence all nonsupersymmetric vacua that it describes. Within thisregime, M5 configurations that approximately solve the equations of motion can be foundand take the simple form of a holomorphic Seiberg-Witten geometry harmonically embeddedalong a transverse direction. These represent stringy realizations of the metastable vacua de-scribed by [2,3]. Crucial to this story, however, was the fact that the scalar potential emergedonly after the M5 worldvolume action was properly regularized. As such, the approximatesolutions are not valid all the way out to ∞ , but rather only within an appropriately chosencutoff surface. We were able to make this completely explicit for the simple case of SU (2)gauge group by finding exact solutions and demonstrating the existence of the appropriate“OOPP limit.”Our SU (2) example also allowed us to obtain a geometric understanding for the gaugetheory potential, at least in the semiclassical limit, as well as the mechanism for stability.Stated in IIA language, the logarithmic bending of NS5’s by the D4’s which end on themcreates pockets where the D4’s want to remain in order to minimize their length. The tendencyto sit in these pockets, however, receives competition from the logarithmic repulsion of theD4’s stemming from their interaction through the NS5 worldvolumes. Nevertheless, one cansuitably tune the asymptotic NS5 geometry in order to achieve stability. This is precisely theOOPP mechanism [2, 3] at work.It should be noted that the story described here is quite similar to that considered in [1],44here a different limit was studied. In that regime, the bending of the NS5’s is quite large andthe lightest modes are fluctuations of the tubes into which the D4’s blow up in the M5 lift. Theasymptotic w ( v ) geometry is thus considered to be rigid with only s ( v ) dynamical . Applyinga T -duality along x to this system, one obtains a deformed Calabi-Yau geometry determinedby w ( v ) with R and NS field strengths dual to s ( v ). The limit of [1], then, corresponds on theIIB side to one in which the presence of nontrivial R and NS fluxes generates a potential forthe moduli of a rigid Calabi-Yau. This potential can be computed in IIB [18] and is preciselyequivalent to that which follows from the M5 worldvolume action along the lines of [1] andsection 3. In light of this, it is natural to ask whether the scalar potential of perturbedSeiberg-Witten theory studied in this paper also has a type IIB counterpart. The most naiveapproach, namely T -dualizing along x , leads to an awkward regime in which nonnormalizabledeformations of the geometry play a role in generating the potential for complex structuremoduli rather than the fluxes. Applying T -duality along other directions, though, can leadto more natural constructions that are currently under investigation [66]. Acknowledgments
We would like to thank J. de Boer, R. Dijkgraaf, L. Mazzucato, H. Ooguri, C. S. Park,K. Skenderis, M. Taylor and especially Y. Ookouchi for valuable discussions. We would alsolike to thank Y. Ookouchi for collaboration at the early stage of this work. The work ofJ.M. was supported in part by Department of Energy grant DE-FG03-92ER40701 and by aJohn A McCone postdoctoral fellowship. The work of K.P. was supported by Foundationof Fundamental Research on Matter (FOM). The work of M.S. was supported by an NWOSpinoza grant.
A Some Basic Results on Riemann Surfaces
In this appendix we summarize a few basic properties of Riemann surfaces, for more detailssee [67].A compact Riemann surface Σ g is a one-dimensional compact complex manifold. Itstopology is completely characterized by an integer, the genus g . The middle cohomology grouphas dimensionality dim H (Σ g ) = 2 g . The intersection form on H (Σ g , Z ) is antisymmetricand by Poincar´e duality unimodular, which means that we can pick a basis of one-cycles This is equivalent to what we called the Dijkgraaf-Vafa regime earlier in section 3, though the configura-tions of [1] contained both D4’s and D4’s so are not relevant for describing a gauge theory. i , B j with intersection: A i ∩ A j = 0 , B i ∩ B j = 0 , A i ∩ B j = δ ij . (A.1)Such a basis is unique up to a symplectic transformation in Sp (2 g, Z ),A surface Σ g of genus g has a complex structure moduli space M g of dimensionalitydim M g = 3 g − , g ≥ ω on a Riemann surface is called a holomorphic differential if in a local coordinatepatch it has the form: ω = f ( z ) dz (A.2)with f ( z ) holomorphic. We will also consider meromorphic differentials , for which we allowthe function f ( z ) to have poles at certain points on the surface. Now we present a standardbasis for holomorphic and meromorphic differentials on a general Riemann surface: Holomorphic differentials ω i : Once we pick a symplectic basis of one-cycles, there is acanonical basis of holomorphic differentials ω i , i = 1 , .., g , with the following periods: I A i ω j = δ ij , I B i ω j = ˆ τ ij . (A.3)The (symmetric) matrix ˆ τ ij is the period matrix of the surface, and depends on the complexstructure of Σ g . Meromorphic differentials of the second kind, d Ω Pn ≥ : These are characterized by apoint P on the surface where the differential has a pole of order n + 1 with n ≥
1. The arenormalized so that in local complex coordinates z where z ( P ) = 0 they have the Laurentexpansion: d Ω Pn ∼ n dzz n +1 + regular . (A.4) Meromorphic differentials of the third kind, d Ω P,P ′ : characterized by two points P, P ′ ,where the differential has first order poles with opposite residues. Around P we have: d Ω P,P ′ ∼ dzz + regular (A.5)and similarly around P ′ with the opposite sign. These are also called meromorphic differentials of the first kind. A periods of the meromorphic differentials vanish: I A i d Ω Pn = 0 . (A.6)In general, it is not possible to simultaneously set the B periods to zero. Instead we have: I B i d Ω Pn = K Pin , (A.7)where the matrix K Pin depends on the complex structure moduli of the Riemann surface andthe position of the puncture P . Harmonic functions on Riemann surfaces:
In two dimensions a function is harmonicif it satisfies ∂∂f ( z, z ) = 0. Locally it is a sum of a holomorphic and an antiholomorphicfunction. On a compact surface, a harmonic function is necessarily constant. If we allowfor poles at points { P a } we can also have nonconstant harmonic functions. They can beconstructed using the meromorphic differentials. Consider the 1-form: ω = X a X n ≥ T an d Ω P a n + X a,b T ab d Ω P a ,P b + X i h i ω i + c.c. (A.8)Given ω , locally we can always find a function f such that ω = df , and it is easy to see that f is harmonic. To make sure the f can be globally well defined on the Riemann surface, wemust make sure that the compact periods of ω vanish: I A i ω = 0 , I B i ω = 0 (A.9)and that the total residue at each of the points P a vanishes. B Parametric Representation of Genus 1 Seiberg-Witten Geome-try
In this appendix, we describe the parametric representation of the genus 1 M5 curve of section6.3 in more detail as a means of demonstrating basic techniques for manipulating the ellipticfunctions F ( n ) i . In particular, we focus on the M5 lift of an NS5/D4 configuration of thetype depicted in figure 1(a) with two parallel NS5’s extended along v and two D4-branes See [1] for a detailed description of the F ( n ) i and their properties. s ( z ) and v ( z ) defined on the fundamental parallelogram of figure 5(b). As discussedin section 6.3, s ( z ) and v ( z ) take the form v ( z ) = b (cid:16) F (1)1 − F (1)2 + iπ (cid:17) ,s ( z ) = 2 ( F − F + iπz ) . (B.1)Moreover, one of the ˆ B -period constraints for ds fixes the distance between the marked pointsin terms of τ a ≡ a − a = τ . (B.2)The other ˆ B -period constraint determines how the dynamical scale Λ is related to curveparameters. Rather than studying the ˆ B -period constraint directly, though, let us instead tryto determine the explicit relationship between v ( z ) and s ( z ) in (B.1) and read it off from theSeiberg-Witten geometry (recall that t ≡ Λ N e − s ) t − P N ( v ) t + Λ N = 0 . (B.3)It is easiest to work with elliptic functions so, to start, we study the derivative of s . Thisleads us to observe that ds = 2 (cid:16) F (1)1 − F (1)2 + iπ (cid:17) dz = 2 v dzb . (B.4)From this, it is clear that we can easily determine s ( v ) by integration once we are able towrite dvdz as a function of v . As such, we turn to G ( z ) ≡ dvdz = b (cid:16) F (2)1 − F (2)2 (cid:17) (B.5)and seek an algebraic relationship between the elliptic functions G ( z ) and v ( z ). Because G ( z )has second order poles while v ( z ) has first order ones, our first guess might be that G ( z ) isgiven by a quadratic polynomial in v ( z ). This is impossible, though, because the second orderpoles of G ( z ) have opposite signs while those of v ( z ) have the same sign. Because of this, weinstead consider the possibility that G ( z ) is equivalent to a quartic polynomial in v ( z ) G ( z ) − b (cid:0) v + C v + C (cid:1) = 0 . (B.6)Because the LHS of this equation is an elliptic function, it is completely specified, up to aconstant shift, by its pole structure. As such, we need only verify that it vanishes at the The overall factor of b can already be seen from the definitions of v ( z ) and G ( z ). The symmetry under v ↔ − v can also be verified ahead of time along the lines of appendix C. a and a . The coefficients of the poles at a and a are equivalent up topossible minus signs, though, so we only need to study G ( z ) and v ( z ) m in the vicinity of onemarked point, say a . Expanding G ( z ) , we find G ( z ) = b (cid:16) F (2)1 − F (2)2 (cid:17) ∼ b ( z − a ) − b ℘ ( τ / z − a ) + b (cid:18) g − ℘ ( τ / (cid:19) + O ( z − a ) . (B.7)On the other hand,1 b (cid:0) v + C v + C (cid:1) ∼ b ( z − a ) + C + 4 b ℘ ( τ / z − a ) + (cid:18) C b − b g C ℘ ( τ /
2) + 10 b ℘ ( τ / (cid:19) + O ( z − a ) . (B.8)From this, we see that G ( z ) = 1 b h(cid:0) v ( z ) − b ℘ ( τ / (cid:1) + b (cid:0) g − ℘ ( τ / (cid:1)i . (B.9)Returning to (B.4), this implies that ds = − v dv q ( v − b ℘ ( τ / + b ( g − ℘ ( τ / ) , (B.10)which can be integrated to yield s ( v ) = − ln (cid:16) v − u − p ( v − u ) − Λ (cid:17) + constant , (B.11)where u = 3 b ℘ ( τ /
2) Λ = b (cid:0) ℘ ( τ / − g (cid:1) . (B.12)Our choice of notation u and Λ is already quite suggestive. Indeed, it is easy to demonstratenow that, dropping the constant in (B.11), t = Λ e − s satisfies t − P ( v ) t + Λ = 0 (B.13)for P ( v ) = v − u. (B.14) The branch of the square root that we use when writing G ( z ) is correlated with how we choose to identifythe marked points a and a with the points at ∞ on the two sheets covering the v -plane. By convention, wetake a to correspond to the point at ∞ on the first sheet, hence the minus sign in (B.10). b isnow fixed in terms of τ and Λ. The only free parameter left in our solution, then, is τ . Whilethis provides a perfectly fine parametrization of the moduli space, a more conventional one isgiven by u (4.3). The relation between u and τ can be read off directly from (B.12) as u should be identified the parameter u there. We can also use (4.7), however, to compute u ( τ )directly from our parametric description (B.1) u = 14 πi I a v ds = 14 πi I a b (cid:16) F (1)1 − F (1)2 + iπ (cid:17) dz. (B.15)This can be easily evaluated by expanding F (1)1 and F (1)2 near a and results in u = 3 b ℘ ( τ / . (B.16)Consequently, u is nothing other than the parameter u in (B.12). C Symmetry Argument for the Ansatz (6.32)
In this Appendix, we motivate the ansatz (6.32) for the M5 curve describing the OOPP vacuafrom a symmetry argument.First, let us examine the symmetry of the M5 curve in the supersymmetric case. Asexplained in section 2, an M5 curve corresponding to supersymmetric vacuum of SU (2) theoryhas the following form: t − P N ( v ) t + Λ N = 0 , (C.1) w ( v ) = p W ′ n ( v ) − f n − ( v ) , (C.2)where t = Λ N e − s , and satisfies the constraint (2.14) or (2.15). Because in the SU (2) caseTr Φ k = 0 for odd k , P ( v ) = v − u and W ′ n ( v ) has only odd powers in v . By examining (2.14),(2.15), one can see that this implies that f n − ( v ) has only even powers in v . Accordingly,under v → − v , P ( v ) → P ( v ) , W ′ n ( v ) → − W ′ n ( v ) , f n − ( v ) → f n − ( v ) . (C.3)From (C.1), one obtains s ( v ) = ln P ( v ) + p P ( v ) − Λ Λ ! . (C.4)50he relations (C.2) and (C.4) define the w, s coordinates as functions on the two-sheetedcover of the complex v plane.This supersymmetric curve has the following symmetries:symmetry I) v → − v (same sheet) , w → − w, s → s ;symmetry II) v → v (different sheet) , w → − w, s → − s. (C.5)For example, if we flip v → − v remaining on the same sheet, then from (C.2)–(C.4) it is easyto see that w = p W ′ n + f n − = W ′ n + f n − W n + · · · → − W ′ n − f n − W n − · · · = − w and s → s (symmetry I). On the other hand, if move v between the first and second sheets, the squareroots in (C.2) and (C.4) flip their signs and we have w → − w , s → − s (symmetry II).Because we will write M5 curves in parametric representation on the z plane, we need toimplement these symmetries (C.5) on the z plane. In the supersymmetric case, the relationbetween v and z is given by (6.22). Using the properties of F (1) i ( z ) in (6.17), one can see that,if a = a − a = τ / v → − v (same sheet) ⇐⇒ z → a − z ∼ = 2 a − z, transformation II) v → v (different sheet) ⇐⇒ z → a + a − z. (C.6)Let us require that the exact M5 curve corresponding to the OOPP vacuum have the samesymmetries as above. Namely, under the transformations of z in (C.6), we require that s, v, w transform according to (C.5). The 1-forms ds, dv, dw have the same transformation propertyas s, v, w . The basis of 1-forms ω , d Ω a ,a , d Ω a i n ≥ , i = 1 , ω → − ω, d Ω a ,a → + d Ω a ,a , d Ω a n → ( − n d Ω a n , d Ω a n → ( − n d Ω a n , II) ω → − ω, d Ω a ,a → − d Ω a ,a , d Ω a n → ( − n d Ω a n , d Ω a n → ( − n d Ω a n . (C.7)Therefore, in order to obey (C.5), the 1-forms ds, dv, dw must have the following schematicform: dw ∼ X n =1 , , ,... ( d Ω a n + d Ω a n ) + ω + c . c .,dv ∼ X n =1 , , ,... ( d Ω a n − d Ω a n ) + c . c .,ds ∼ d Ω a ,a + c . c ., (C.8)where coefficients are omitted. In other words, dw ∼ X n =2 , , ,... ( F ( n )1 + F ( n )2 ) dz + dz + c . c .,dv ∼ X n =2 , , ,... ( F ( n )1 − F ( n )2 ) dz + c . c .,ds ∼ ( F (1)1 − F (1)2 + iπ ) dz + c . c . (C.9)51o far we have not taken into account any period constraints but, in dw , ( F (2)1 + F (2)2 ) dz mustcome with dz − dz for the period along ˆ B − ˆ B to vanish. Therefore, dw must come with( F (2)1 + F (2)2 ) dz + 4 πiτ − τ ( dz − dz ) . (C.10)In the small superpotential case studied in section 4, v ( z ) was given by (6.22) and dv ( z )had order two poles at z = a , . To consider the OOPP vacuum, we would like to take adegree six superpotential, which leads to the boundary condition (5.10). This motivates usto study dw ( z ) that has up to order six poles at z = a , and therefore contains F ( n ) i with n ≤
6. However, as was discussed below (6.31), the pole structure of v ( z ) given by (6.22) isnot enough; in order to satisfy the Virasoro constraint we need to include higher order polesin dv ( z ) than was considered in (6.22). In the present case, order four poles are sufficient.After all, we are led to the following ansatz: dw ∼ ( F (6)1 + F (6)2 ) dz + ( F (4)1 + F (4)2 ) dz + ( F (2)1 + F (2)2 ) dz + 4 πiτ − τ ( dz − dz ) ,dv ∼ ( F (4)1 − F (4)2 ) dz + ( F (2)1 − F (2)2 ) dz + c . c .,ds ∼ ( F (1)1 − F (1)2 + iπ ) dz + c . c ., (C.11)which upon integration becomes the ansatz (6.32) that we used. D q -expansions When τ ≫ q = e iπτ = e iπτ − πτ is small and the following q -expansions [1, 68] are useful: F ( z ) = ln( e πiz −
1) + ln " ∞ X k =1 (2 k + 1)( − k q k ( k +1) + 4 ∞ X k =1 q k sin ( kπz ) k (1 − q k ) ,℘ (cid:16) τ (cid:17) = − π − π ∞ X k =1 kq k q k , η = π − π ∞ X k =1 kq k − q k ,g = 4 π ∞ X k =1 k q k − q k ! , g = 8 π − ∞ X k =1 k q k − q k ! . (D.1)Note also that η , η are related by η τ − η = iπ . E Type IIA Limit of the M5 Curve
In this Appendix, we consider the type IIA limit of the M5 curve in the OOPP limit obtainedin section 6.6. The IIA limit is defined to be the R → w = ± W ′ ( v ) with D4-branes stretched in between, and that thereis no logarithmic bending of NS5-branes along the x direction. As a result, we will see thatD4-branes sit at the critical point of w ( v ), namely at v for which W ′′ ( v ) = 0.The M5 curve in the OOPP limit obtained in section 6.6 can be written as: s = 2 [ F − F + iπ ( z − A − a )] ,v = b ( F (1)1 − F (1)2 + iπ ) ,w = A ( F (5)1 + F (5)2 ) + A ( F (3)1 + F (3)2 ) + A (cid:20) ( F (1)1 + F (1)2 ) + 4 πiτ Im( z − A − a ) (cid:21) . (E.1)where the parameters A , , are related to the coefficients g, β, γ in the superpotential (5.9)and (5.10) via (6.40). Given these parameters, the modulus τ is determined by solving (6.43).Recall also that a ≡ a − a = τ / A ≡ ( a + a ) /
2. The constants in (E.1) have beenchosen for later convenience.In the type IIA limit where the distance between the tubes is much larger than the sizeof the tubes, τ ≫
1. In this limit, q = e iπτ = e iπτ − πτ is very small and, using (D.1), we canapproximate F ( z ) as F ( z ) ≈ ln( e πiz − . (E.2)Therefore, for example, we can approximate s, v in (E.1) as s ≈ (cid:20) log (cid:18) e πi ( z − a ) − e πi ( z − a ) − (cid:19) + iπ ( z − A − a ) (cid:21) ,v ≈ πib (cid:20) − e − πi ( z − a ) − − e − πi ( z − a ) + 12 (cid:21) . (E.3)We can similarly obtain an approximate form for w , but we do not display it here explicitlybecause it is too lengthy. On the other hand, applying (D.1) to (6.29), we see that, in thelarge τ limit, u ≈ − π b . Classically, the two tubes are sitting at the solution to P ( v ) = 0,namely at v = ±√ u ≈ ± iπb . Therefore, if we want to keep the distance between the twotubes to be finite, we must keep b finite.In the type IIA limit, we take R → τ and R in this limit? For this, let us look at the behavior of s and v near z = a , , where theyblow up. By examining (E.3), it is easy to see that, near z = a , , s ∼ ± (cid:20) log (cid:18) πib v (cid:19) + 12 iπa (cid:21) . (E.4) Here, we have dropped terms which are typically of order q sin(2 πz ). The modulus of this quantity is | q sin(2 πz ) | ∼ e π ( | Im z |− τ ) . As long as one stays in one fundamental region of the z -torus, this is exponentiallysmall for large τ and can be safely dropped. v = v is given by L ≡ R Re (cid:2) s ( z = a ) − s ( z = a ) (cid:3) = 4 R (cid:18) log (cid:12)(cid:12)(cid:12)(cid:12) πb v (cid:12)(cid:12)(cid:12)(cid:12) − πτ (cid:19) . (E.5)In other words, | b | = | v | π e “ πτ − LR M ” . (E.6)In the IIA limit, we send R → L and b finite. Therefore, we must send τ → ∞ as R →
0, as τ ∼ LπR . (E.7)If τ is very large, the z -torus becomes very long ver- a a a Re z Im z −τ III
Figure 8: The z -torus for large τ . Region I (Im( z − a ) ≪ z − a ) ≪
0) and region II(Im( z − a ) ≫
0, Im( z − a ) ≪ z axis) as in Figure 8. We expectthat the long, narrow regions “between” marked pointscorrespond to tubes that descend to D4-branes in the IIAlimit. More specifically, let us call the region where Im( z − a ) ≪
0, Im( z − a ) ≪ z − a ) ≫
0, Im( z − a ) ≪ s ≈ πi ( z + a − A ) ,v ≈ iπb , (region I) w ≈ πiA τ ( z + a − A ) . (E.8)So, if we move from z = a − τ (in the fundamental paral-lelogram below our fundamental parallelogram) to z = a ,then s, v, w change linearly as s = − πiτ → s = πiτ,v = iπb → v = iπb ,w = − iπA → w = iπA . (E.9)The change in x is ∆ x = R Re(∆ s ) = − πR τ = − L, (E.10)54here in the last equality we used (E.7). Therefore, region I indeed corresponds to a D4-braneat v = iπb with length L along x . Note also that this D4 is tilted along w by ∆ w = 2 πiA .Similarly, one can show that region II corresponds to a D4-brane at v = − iπb with length L along x , tilted along w by ∆ w = − πiA .Now let us turn to the regions near z = a , , which must correspond to NS5’s curved along(6.39). Near z = a , s ≈ (cid:20) log(2 πi ( z − a )) + iπτ (cid:21) , v ≈ b πi ( z − a ) ( z ≈ a ) . (E.11)If we eliminate z , v = 2 πib e − s + iπτ . (E.12)Note that (cid:12)(cid:12)(cid:12) e − s + iπτ (cid:12)(cid:12)(cid:12) = e − x R − πτ = e − R ( x + L ) . (E.13)This goes to zero in the R → x is very close to − L/
2, where the left NS5sits. If x is very close to − L/
2, within O ( R ), then we can tune x + L/ v to takeany value in C . Namely, we see that the region near z = a indeed corresponds to the leftNS5 sitting at x = − L/ v direction. How about the w direction? w goes, near z = a , as w ≈ A ( z − a ) + 2 A ( z − a ) + A z − a ( z ≈ a ) . (E.14)To see the relation between v and w , we need the expression for v up to lower powers in( z − a ) than was needed in (E.11). Near z = a ,v ≈ πib (cid:20) − e − πi ( z − a ) − (cid:21) ( z ≈ a ) . (E.15)If we eliminate ( z − a ) from (E.14) and (E.15), we obtain w = 24 A b v + 2( A + 20 π A ) b v + A + 2 π A + 16 π A b v + O (cid:18) v (cid:19) . (E.16)One can readily see that this is the expected behavior (6.39), if one uses (6.40) and that ℘ ( τ / ≈ − π / g ≈ π / τ .Similarly, one can show that the region near z = a corresponds to an NS5 at x = L/ w = − W ′ ( v ). 55o far we have been treating the parameters A , , , b , τ as if they were all arbitrary,but they are actually subject to the constraint (6.43). For large τ , this equation reduces to(7.16a): τ = 2 π W ′ ( v ) vW ′′ ( v ) . (E.17)where v = ± iπb . An important difference from the semiclassical limit considered in section7 is that we send τ → ∞ as (E.7) as we take R →
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