Quantization of charges and fluxes in warped Stenzel geometry
aa r X i v : . [ h e p - t h ] A p r MAD-TH-11-02
Quantization of charges and fluxes in warped Stenzel geometry
Akikazu Hashimoto a and Peter Ouyang ba Department of Physics, University of Wisconsin, Madison, WI 53706, USA b Department of Physics, Purdue University, West Lafayette, IN 47907, USA
Abstract
We examine the quantization of fluxes for the warped Stiefel cone and Stenzel geome-tries and their orbifolds, and distinguish the roles of three related notions of charge:Page, Maxwell, and brane. The orbifolds admit discrete torsion, and we describe theassociated quantum numbers which are consistent with the geometry in its large radiusand small radius limits from both the type IIA and the M-theory perspectives. Thediscrete torsion, measured by a Page charge, is related to the number of fractionalbranes. We relate the shifts in the Page charges under large gauge transformations tothe Hanany-Witten brane creation effect.
Introduction
Recently, a holographic duality for superconformal Chern-Simons-Matter theories in 2+1dimensions with N = 6 and N = 8 supersymmetry was proposed [1, 2]. These field theorieshave U ( N ) k × U ( N + l ) − k product gauge symmetry (where the subscripts refer to the Chern-Simons level associated with each gauge group) and bifundamental matter fields. In thelarge N limit, the field theory has a dual gravity description in terms of M-theory as N M2-branes on the orbifold C /Z k (where the orbifold acts by rotating each of the complexplanes by an angle 2 π/k simultaneously) and l fractional branes. The supergravity solutioncorresponding to this brane system is AdS × S /Z k , and the quantum number l is encodedin the discrete torsion of the H ( Z ) = Z k homology group of S /Z k .The M-theory background can also be described in type IIA supergravity by dimension-ally reducing along the Hopf-fiber of S /Z k . In this description, the geometry has the form AdS × CP and is the effective description when 1 ≪ N ≪ k . Homologically, CP is verydifferent from S /Z k , particularly in that CP has no discrete torsion cycles, but it doespossess integral homology. Even for this simple example, the relationship of the spectrum ofcharges and fluxes in the M-theory and the type IIA descriptions is subtle.One way to gain some perspective on the physical meaning of the defining data of thegravity side of these correspondences is to realize the superconformal field theory as eitherthe UV or IR fixed point of a holographic renormalization group flow. For example, asuperconformal Chern-Simons theory can arise as the IR fixed point of an RG flow from aChern-Simons-Yang-Mills-Matter theory [3, 4]. Several related realizations have also beenconsidered [5]. These renormalization group flows are dual to transverse geometries whichdiffer from R , and many of these constructions have the interesting property that the dualgeometry admits a normalizable 4-form. In M-theory, this allows one to introduce a nontrivial4-form flux. The freedom of tuning the 4-form flux has a specific interpretation in termsof tuning the parameters of the dual field theory, and in some examples, one can exploredynamical features such as phase transitions in the low energy effective field theory from thegeometry of the supergravity dual [4, 5].In this article, we investigate the duality of N = 2 Chern-Simons quiver theories dual to AdS × V , /Z k where V , is a homogeneous Sasaki-Einstein seven-manifold. This dualitywas originally considered by Martelli and Sparks in [6]. On the field theory side, it generalizesthe model of ABJM by adding a chiral multiplet in the adjoint representation to each factorof the U ( N ) k × U ( N + l ) − k gauge group. The gravity dual description can be deformed inthe IR, giving rise to a geometry known as the warped Stenzel metric. At the moment, littleis known about the field theory interpretation of this IR deformation. In order to facilitate1n its interpretation, it is useful to enumerate the the discrete and continuous parametersassociated with this system. This is related to the problem of quantizing charges and fluxesin the gravity dual.In type IIA (and IIB) supergravity, there is a well-known subtlety in imposing chargequantization, which arises in the example studied in this paper. The V , /Z k geometry,reduced to IIA along the U (1) isometry along which the Z k acts, is a space M which hasthe same homology structure as CP [6]; in particular there is a nontrivial 4-cycle. Now, onemight want to quantize the four-form flux through this cycle, but the natural gauge-invariantfour-form ˜ F = dA + H ∧ A (1.1)is not closed, and therefore its integral through the 4-cycle is not conserved and cannot bequantized. A similar issue arises in the flux of ∗ ˜ F through M . These apparent difficul-ties have also appeared in earlier examples considered in [4, 5] and their resolution is wellunderstood. The four-form flux satisfies a modified Bianchi identity, d ˜ F = − H ∧ F (1.2)so to define a conserved charge we should not integrate ˜ F but a modified flux which ischosen to be closed: Q P age = 1(2 πl s ) g s Z ( − ˜ F + B ∧ F ) . (1.3)This new charge, known as the Page charge, is one of the three subtle notion of chargesidentified by Marolf [7]. The three charges being referred here are the Maxwell charge, branecharge, and the Page charge, and they can take distinct values in gauge theories involvingChern-Simons terms as is the case for type IIA supergravity. Each of these charges respectssome, but not all, of the properties commonly associated with charges in simpler contexts:gauge invariance, conservation, localization, and integer quantization. Page charge turnsout to respect conservation, localization, and integer quantization, but fails to be invariantwith respect to large gauge transformations which shift the period of B . This ambiguity isprecisely what is required to interpret the Hanany-Witten brane creation effects in the braneconstruction of these models is intimately connected to the duality of the field theory.In this article, we will analyze the quantization of fluxes in AdS × V , /Z k geometry andits Stenzel deformation from the type IIA perspective, and relate the gauge ambiguity toHanany-Witten brane creation effects along the lines of [4, 5]. In [6], it was argued that theStenzel deformation is incompatible with the presence of discrete torsion which gives rise toa non-vanishing value of l in U ( N ) k × U ( N + l ) − k . On the contrary, we find that some valuesof l are allowed, and explain the source of this apparent discrepancy. We will also examinethe compatibility of the IIA and the M-theory perspectives.2 Stiefel, Stenzel, and the N = 2 Chern-Simons-Quiver theory
In this section, we briefly review the construction of N = 2 Chern-Simons-Quiver theory, itsgravity dual, and its Stenzel deformation. We closely follow the presentation of [6]. The starting point is a non-compact Calabi-Yau 4-fold z n + z + z + z + z = 0 (2.1)where we take n = 2. This geometry is a cone whose base is a Sasaki-Einstein seven manifold V , , also known as the Stiefel manifold. Had one taken n = 1, the geometry of the Calabi-Yau 4-fold would have been R which is formally a cone over S . For n >
2, the geometryis not a cone over a Sasaki-Einstein manifold [6].When M2 branes are placed at the tip of the cone, we obtain a warped geometry
AdS × V , . The base Y = V , has a torsion 3-cycle H ( Y , Z ) = Z .The Z k orbifold is taken on the U (1) b isometry which rotates( z , z + z , z + z , z − z , z − z ) (2.2)with weights (0 , , , − , − Y /Z k , this changes the torsion group from Z to H ( Y /Z k , Z ) = Z k , so 1(2 πl p ) Z Σ C = l − k k (2.3)for Σ which generates H ( Y /Z k ). Here we have shifted l by k compared to what is writtenin (3.26) of [6]. Both l and k are integers so this shift is a matter of convention in describingthe supergravity background.When reduced to IIA along the U (1) b direction parametrized by γ , the Sasaki-Einsteinspace Y /Z k decomposes into ds ( Y /Z k ) = ds ( M ) + wk ( dγ + kP ) (2.4)and the IIA string frame metric becomes ds = √ w R k (cid:18) ds (AdS ) + ds ( M ) (cid:19) (2.5)with F = kg s l s Ω , Ω = dP . (2.6)3ince C = A + B ∧ dγ (2.7)with this dimensional reduction, B turns out to have the period1(2 πl s ) Z B = l k − . (2.8) The field theory dual is conjectured in [6] to arise from the low energy limit of a networkof D3-branes, an NS5-brane and a (1 , k ) 5-brane in type IIB on R , × S × R × C . TheD3-branes wind along R , × S . The NS5 is extended along R , , one of the R in R andalong the curve w = − iw where C is parametrized by ( w , w ). The (1 , k ) 5-brane isextended along R , , a line at an angle relative to the NS5-brane in R , and along w = iw in C . There may also be fractional D3-branes stretching between the NS5 and the (1 , k )5-brane at ( w , w ) = (0 , S keeping the other 5-brane fixed. Ifthere were N integer and l fractional branes to start with, moving the 5-brane once aroundthe circle will give rise to a shift N → N + ll → l + 2 k . (2.9) In this subsection, we will briefly review the IR deformation of the Stiefel cone. As analgebraic curve, it amounts to deforming (2.1) to z + z + z + z + z = ǫ . (2.10)The tip of the cone is blown up by an S parametrized by X i =0 (Re z i ) = ǫ , Im z i = 0 (2.11)and the full geometry can be viewed as the cotangent bundle over S . This geometry is alsoknown as the Stenzel geometry [8] and admits an explicit metric [9]. In the notation adoptedin [10], the metric takes the form 4 s = c ( dr + ν ) + a X i =1 σ i + b X i =1 ˜ σ i (2.12)where a ( r ) = 3 − / λ ǫ / (2 + cosh 2 r ) / cosh r,b ( r ) = 3 − / λ ǫ / (2 + cosh 2 r ) / sinh r tanh rc ( r ) = 3 / λ ǫ / (2 + cosh 2 r ) − / cosh r (2.13)and ν , σ i , and ˜ σ i are left-invariant one-forms of the coset SO (5) /SO (3) (for which a niceexplicit basis appears in [10].)At r = 0, the geometry collapses to an S . At large r , the geometry asymptotes to a coneover V , . Formally, this geometry is similar to the deformed B space [11] which collapsesto an S near the tip, and asymptotes to cone over a squashed 7-sphere, but there are afew important differences. One is the fact the Z k orbifold along the U (1) b of the Stenzelgeometry has fixed points at antipodal points of S at r = 0. We will comment on otherdifferences below.One important feature of the Stenzel geometry is that it admits a self-dual 4-form whichcan be written, explicitly, as G = m ( ǫ coth r (cid:20) a cν ∧ σ ∧ σ ∧ σ + 12 b cdr ∧ ˜ σ ∧ ˜ σ ∧ ˜ σ (cid:21) + 12 ǫ coth r (cid:20) a bcǫ ijk dr ∧ σ i ∧ σ j ∧ ˜ σ k + ab cǫ ijk ν ∧ σ i ∧ ˜ σ j ∧ ˜ σ k (cid:21)) . (2.14)Because the four-form is self-dual, and the background geometry is Calabi-Yau, one canturn on this flux in eleven-dimensional supergravity without breaking supersymmetry [12].Moreover, it gives rise to a solution where the background geometry is unmodified exceptfor the presence of a warp factor H , as in the standard warped product ansatz ds = H − / ( − dt + dx + dx ) + H / ds F = dt ∧ dx ∧ dx ∧ d ˜ H − + mG . (2.15)The warp factor itself can be determined by solving the four-form field equation, d ∗ G = 12 G ∧ G , (2.16)where in general there can be additional source terms due to the presence of explicit M2-branes. 5
Quantization of fluxes in Stiefel cones and Stenzel space
Let us now consider the quantization of fluxes in the warped Stiefel cones and Stenzel ge-ometries in order to identify the discrete parameters characterizing the background. Thereare two guiding principles which we follow in carrying out the quantization. One is thatquantized fluxes should be invariant under deformation of Gaussian surfaces unless the dis-crete unit of charge crosses the surface. The other is for the quantization condition to beinvariant under string dualities.
We begin by considering the quantization of fluxes for the Stenzel geometry in the IIAdescription. While the IIA description of the Stenzel geometry is singular near the core, onestill expects Gauss law considerations to lead to a consistent picture far away from the coreregion, where the geometry looks essentially like the warped Stiefel cone.The relevant fluxes to consider then are the flux of ˜ F through the generator of H ( M , Z )and the flux of ∗ ˜ F through the six cycle M . As we mentioned earlier, however, these fluxesdepend on the radius r at which we identify the base M for the background in consideration.The resolution to these apparent difficulties is the realization that one is dealing witha situation where the Maxwell, brane, and Page charge are distinct from one another, andthat care is required in applying quantization conditions on the appropriate charge.Let us recall the specific definition of three charges. In type IIA supergravity, the fourform ˜ F = dA + H ∧ A is gauge invariant and well defined but is not closed and doesnot respect Gauss’ law. One can nonetheless compute the period of ˜ F on the generator of H ( M , Z ) in the r → ∞ limit. This defines the Maxwell charge. In contrast, the period ofPage flux − ( ˜ F + B ∧ F ) on H ( M , Z ) is independent of r , although it is ambiguous withrespect to large gauge transformation of B . This quantity defines the Page charge. Finally,the amount of charge carried by a brane source through its Wess-Zumino couplings definesthe brane charge. Brane charge includes the contribution of lower-brane charges from thepull-back of the NSNS 2-form in the Wess-Zumino coupling. Therefore, if the backgroundcontains a non-uniform NSNS 2-form B , the brane charge is not conserved with respect tochanges in the position of the branes. Some of these subtleties appeared originally in [13].The triplet of charges exists for the other forms, e.g. the six form F = ∗ ˜ F and aresummarized in appendix B of [4]. For the flux of F = ∗ F , is is also useful to introduce the6otion of bulk charge Q bulk which is the total charges carried by the bulk fields Q bulk = Z Y G ∧ G . (3.1)Then, the bulk charge can be understood as being related to the brane and Maxwell chargesvia Q Maxwell = Q brane + Q bulk . (3.2)To correctly quantize the supergravity solution, one should impose the discreteness conditionon the Page charges, and not on Maxwell, brane, or bulk charges. To illustrate the integrality of Page charges and the non-integrality of the other charges, letus first carryout the quantization procedure for the Stiefel cone.First, consider the flux of ˜ F . The Stiefel geometry has vanishing fourth Betti number,so there is no G to consider in M-theory, and after dimensional reduction, the IIA flux ˜ F also vanishes. We are not done yet, however, because we still have to consider the Page flux(1.3), which contains a term B ∧ F , and F is nonvanishing in the dimensional reductionof the orbifolded Stiefel cone. Requiring the Page flux to be integer quantized imposes thequantization condition Z B = − l k + 12 (3.3)which we inferred independently from M-theory considerations earlier in section 2.1.Next, we consider the quantization of flux of D2 charge through M . We are interestedin determining the Maxwell charge when the Page charge is set to N . One finds Q Maxwell = N − l ( l − k )2 k (3.4)which can essentially be viewed as the sum of a contribution from N M2-branes and acontribution from the discrete torsion, along the lines of [14]. The Maxwell charge Q Maxwell has several notable features. First, it is not necessarily integer valued. Second, it is invariantunder N → N + l, l → l + 2 k . (3.5)This is consistent with the property of Maxwell charge that it is conserved under continuousdeformations corresponding to moving one of the 5-branes around the S in the type IIBbrane construction. Finally, Q Maxwell can go to zero or negative for some range of ( N, l, k ).This suggests that the entropy of the superconformal field theory is going to zero or negative,signaling a phase transition. The condition that Q Maxwell is positive is also related to the7ondition necessary for supersymmetry to be unbroken as was highlighted in related contextsin [4, 5]. Let us now extend our analysis of flux quantization to the case where the Stiefel cone isdeformed into the Stenzel geometry, as described in Section 2.3. To keep a general set ofcharges under consideration, we will study the case where the Stenzel manifold has beenquotiented by a Z k orbifold action.The most important feature of the geometry in the deep IR is its singularity structureafter the orbifold has been taken. At the tip of the deformed orbifolded cone, the geometryhas the local structure ( R × S ) /Z k , and in particular it has two fixed points which we canthink of as the north and south poles of the S /Z k . At each of the fixed points, the localgeometry is R /Z k [6]. This geometric feature has a nice implication. The supersymmetryof the deformed Stenzel cone is consistent with adding some mobile M2-branes, and we arefree to move some number of them to either of the orbifold fixed points. Then the theory onthe M2-branes in the deep IR should simply be two copies of the ABJM theory.At any finite excitation energy the theory should feel the effects of curvature and theself-dual four-form flux in the background which break the supersymmetry from N = 6 to N = 2. However, for issues such as charge quantization, we should be able to work in theextreme low energy limit and use our intuition from the ABJM case. In particular one mightexpect that it is possible to turn on discrete torsion at each singularity, and we will see thatthis is correct, although the torsion will be subject to some global constraints.First we will consider the type IIA reduction of this geometry. This geometry develops adilaton and curvature singularity near the tip. However, because the geometry asymptotesto the Stiefel cone away from the tip, and because quantization of Page fluxes in type IIAdescription appropriately respects Gauss law/localization of charge sources, we are able topartially constrain the discrete parameters of the supergravity ansatz. We will then continueto consider the geometry and fluxes near the core region from the M-theory perspective, andidentify additional constraints which further restrict the parameters of the ansatz.The Stenzel manifold admits the self-dual four form flux (2.14) which can be derivedfrom a three-form potential C [10] C = mβ + α Ω ∧ dγ (3.6) β = acǫ cosh r " (2 a + b )˜ σ ∧ ˜ σ ∧ ˜ σ + a ǫ ijk σ i ∧ σ j ∧ ˜ σ k , (3.7)8here Ω and γ are as defined in [6]. Here we have added an exact term proportional to α ,which does not affect the gauge invariant four-form flux. This exact term is present in the AdS × V , /Z k system with discrete torsion [6] which is the UV limit of the Stenzel solution.In quantizing the flux of the type IIA Page flux through the four cycle of M , we imposethe condition Z S G + nk Z ˜ S /Z k C = (2 π ) nα = − (2 πl p ) ( l − k ) , n = 2 (3.8)which constrains α . Note that in the asymptotically conical limit, the torsion is Z k -valued,and so l takes integer values in the range 0 ≤ l ≤ k − G through S /Z k is independently quantized to be integral.This implies Z S /Z k G = 8 π √ k m = (2 πl p ) q (3.9)for integer q . This constraint has no counterpart in the Stiefel cone as neither the S cyclenor the self-dual 4-form exist in that case.Now let us consider the quantization conditions that arise from considering M-theorynear the orbifold fixed points; we will show that the expected charges at the singularities arecompatible with the IIA calculations. At the north pole of S /Z k , the pull-back of G onthe R /Z k fiber was computed in [6]:1(2 πl p ) Z R /Z k G = q ≡ ˜ M M (3.10)where M and ˜ M are the variables used in [6]. This means that the integral of C (includingboth the nontrivial flux and the discrete torsion contribution) on S /Z k at the north pole is1(2 πl p ) Z ˜ S /Z k C = − l k + 12 − q . (3.11)Suppose that at the north pole we impose the condition that the system is described bycharges as in the ABJ case with l N units of discrete torsion (including a shift by 1/2 a unitas discussed in [4].) This is compatible with (3.11) provided that − l N k + 12 = − l k + 12 − q l N = l + kq . (3.13) For the interested reader, γ is the angular coordinate which is quotiented by the orbifold action, and Ω is proportional to the geometric flux associated with γ .
9t the south pole, the computation is very similar, except that the flux quantum q appears with a minus sign: l S = l − kq C between the north and the south pole is just the totalflux q , while the discrete torsion contribution must be the same at the north and south polesbecause the torsion has no associated flux. How should we interpret the formulas (3.13) and (3.14)? The first thing to note is that l N and l S are equal mod k , so if they had described decoupled systems we would have said thatthey were equivalent up to a large gauge transformation. However, they are not decoupled,and there is no large gauge transformation that sets them equal to each other. Instead, thepicture that has emerged is that l N and l S locally appear to describe the same torsion, butglobally there is a topologically nontrivial twist relating them, and the winding number ofthe twist is just the number of units of G flux in S /Z k .The second thing to note is that in the local ABJ models at the north and south poles, l N and l S should themselves be integers, or in other words l − kq must be even. This meansthat for a given q and k , l must take either only even or only odd values. In the undeformedtheory, l described a Z k -valued discrete torsion, but we see that our local considerationsat the tip of the Stenzel geometry remove half of the possible values of l , and the discretetorsion in the deformed case is Z k -valued. This phenomenon is reminiscent of the deformedconifold; the “singular” conifold admits a Z -valued discrete torsion which is not present inthe deformed conifold [16].We can now examine the quantization of the six form flux though M in IIA or the7-form flux through V , in M-theory which measures the charge of D2/M2 branes in thisbackground.One way to approach this issue is to first examine the brane charges present in this setup.Before adding any explicit 2-branes, there are 2-brane charges arising from the discretetorsion at the Z k fixed points at north and south poles [14]. These should have the sameform as what was computed in [4], so we find Q torsion = − l N k + 12 ! + − l S k + 12 ! = − l ( l − k )4 k − kq . (3.15)If, in addition, we were to introduce N In the coordinates of [10, 15], the U (1) b quotient as defined in [6] is imposed on the angular coordinate φ . With this choice of U (1) action, the poles of the S /Z k are located at ( τ = 0 , α = π/ , ψ = 0 , π ). In thevicinity of the poles, one can check that the one-forms ˜ σ i differ by a sign, ˜ σ i ( N ) = − ˜ σ i ( S ), so the three-form β in (3.7) also changes by a sign from the north pole to the south pole. N to the brane charge Q brane = N − l ( l − k )4 k − kq . (3.16)Since Maxwell charge is the sum of brane charge, and since the bulk charge is given by Q bulk = 1(2 πl p ) Z M G ∧ G = 2 m vol( V , )(2 πl p ) = kq Q Maxwell = N − l ( l − k )4 k . (3.18)It also follows that the Page charge Q P age = N .This result is gratifying for several reasons. First, this result reflects the accounting ofall identifiable charge sources in an otherwise consistent and smooth M-theory backgroundaside from the orbifold fixed point. The final answer is the same as what we inferred for theundeformed Stiefel cone (3.4). It then follows that the gauge invariant Maxwell charge isinvariant under the shifts N → N + l, l → l + 2 k (3.19)which arises naturally from several perspectives mentioned earlier.The only additional constraint imposed by the Stenzel deformation is the restriction onthe parity of l so that l is congruent to kq mod 2. This is far milder than what was foundin [6]. In this article, we reviewed the quantization of fluxes in warped Stiefel cone and its Stenzeldeformation which is conjectured to be the holographic dual of N = 2 Chern-Simons mattertheory in 2+1 dimensions. We described the subtle difference between several different yetrelated notions of charges, and recovered a structure compatible with the pattern of Hanany-Witten brane creation effects and duality cascades.There are a number of interesting features which one can infer from the structure ofthe gravity solution. Q brane is a measure of the number of degrees of freedom in the deepinfrared of this system. When Q brane is zero or negative, we expect the system to breaksupersymmetry and flow to a different universality class of vacuum as was the case for manyrelated system [4, 5]. It would be very interesting to better understand the nature of theeffective low energy physics when the system is in this new phase. This question can beaddressed in the simple context of k = 1 where there are no Z k orbifold fixed points, and by11a) (b) (c)Figure 1: Potential V ( ψ ) for p anti D3-brane blowing up to an NS5-brane wrapping an S of fixed latitude in ψ in S at the tip of the Klebanov-Strassler solution. (a), (b), and (c)corresponds to p/M = 0 . p/M = 0 .
08, and p/M = − .
03, respectively. These figuresoriginally appeared in figure 2 of [17].taking q to be even, we can even set l = 0 and disregard the contribution from the discretetorsion.One way to probe the fate of pushing the system which is slightly perturbed into thisnew phase is to start with a background with q large but Q brane = 0 (which can easily bearranged for q even and k = 1). Consider now adding p ≪ q anti M2-brane as a probe. Thissetup is very similar to adding anti D3-brane in warped deformed conifolds [17] which hasreceived a lot of attention (and controversy) as a possible prototype as a gravity dual of ametastable vacua [18–20]. For the Stenzel manifold, the effective action of the brane probeundergoing a KPV-like transition [17] works essentially in the same way as is illustrated infigure 1. However, from the point of view of the bound Q brane >
0, one expects the stablesupersymmetric minima not to exist when p anti M2-branes are introduced.Tentatively, we interpret these facts as follows. The computation of the potential V ( ψ )neglected the backreaction of the anti-branes, and when the number of anti-branes is para-metrically small ( p ≪ q ) this probe approximation is valid. In particular, the existence ofthe non-BPS local minimum in 1.(a) is a robust prediction in this limit. However, when thestate in the metastable false vacuum illustrated in figure 1.(a) tunnels to the putative “true”vacuum, the amount of charge carried by the probe grows to q − p which is not parametri-cally small compared to q . The backreaction due to this charge can be significant, and sothe computation of the tunneling potential is (at least) not obviously self-consistent. It istempting to speculate that the supersymmetric vacuum might actually be spurious and thatthe non-BPS local minimum is the global minimum which characterizes the dynamics in the Q brane < p/q .Similar considerations apply to the BPS domain wall one constructs for p < S at the tip of the Stenzel manifold. A12-brane wrapped on a 4-cycle is effectively a string, and in 2+1 dimensions, a string formsa domain wall. It would be very interesting to understand the nature of vacua separatedby these domain walls. Since M5-brane wrapped on S with q units of flux must have q additional M2-branes ending on it to cancel the anomaly, some quantum numbers of thevacuum must shift to reflect this. Nonetheless, one expects the Maxwell charges Q Maxwell and Q Maxwell to be invariant as one crosses the domain wall, as these charges are conserved.Making complete sense of these expectations requires taking the full back reaction of theM5-brane and the q anomalous M2-branes into account. Unfortunately, q M2-branes cannot be treated reliably as a probe, making systematic analysis of these issues a challenge.Let us also mention that similar issues of stable/metastable non-BPS vacua, domainwall, and low energy effective field theories can be discussed in the closely related B systembuilding on the analysis of [5] and [21]. Quantization of charges and the enumeration ofbrane, Maxwell, and Page charges for this system was carried out in [5]. Here, however, weencounter one additional puzzle. It was argued in [21] that the 4-form flux through S atthe tip of the B cone is half integral as a result of the shift originally due to Witten [22].This would appear to require half integer units of M2-branes to end on the domain wallmade by wrapping the M5 on the S . Of course, the number of M2’s ending on an M5 isconstrained to be an integer. Perhaps this is indicating that odd number of M5-branes areforbidden from wrapping the S . Alternatively, this paradox is another manifestation of notsystematically taking the back reaction of the domain wall into account.Finally, let us emphasize that for the time being, the concrete field theory interpretationof the Stenzel deformation and the quantum number q is not known. The gravity dualsuggests that the parameter q is important for both the IR and the UV physics. At largeradius, q is related to the total number of units of M2-brane charge generated by the cascade,which in turn affects the UV gauge symmetry. Near the tip of the Stenzel geometry, the G flux is nonvanishing so q should also appear in the data of the IR field theory. Of course, q can only be nonzero when the geometry is deformed. Martelli and Sparks conjecturedthat this deformation was related to turning on a particular mass term on the field theoryside. One can indeed see that the null geodesic can travel from boundary at infinity tothe core in finite field theory time, and so the spectrum of glueball-like states will exhibita discrete structre whose scale is set by the deformation. If this conjecture is correct, itwould suggest that the field theory confines because of a mass deformation (reminiscent ofthe N = 1 ∗ theory in d = 4 [23] and the mass deformed ABJM theory [24–27]) rather thanas a dynamical effect, as is the case in the Klebanov-Strassler system [28]. It should be veryinteresting to understand this theory better. 13 cknowledgements We would like to thank Ofer Aharony and Shinji Hirano for collaboration on related issuesand for discussions at the early stage of this work. We also thank Igor Klebanov, Don Marolf,Dario Martelli, and James Sparks for useful comments and discussions. The work of AH issupported in part by the DOE grant DE-FG02-95ER40896 and PO is supported in part byDOE grant DE-FG02-91ER40681.
A Charge quantization and duality transformations
One of the subtle features arising in quantizing the supergravity background in this articleis the fact that some fluxes lifts or dualizes to a quantity encoded not by the period of a fieldstrength, but by a quantity like the discrete torsion which is the period of a potential field.In this appendix, we illustrate several examples, in a simpler context, giving rise to similarsubtleties.
A.1 Charge quantization for duals of
T N × S The KK-monopole, also known as the Taub-NUT space, is a well known Ricci-flat gravita-tional background. The metric for
T N × S has a simple form (cid:18) R r (cid:19) ( dr + r ( dθ + sin θdφ )) + R (cid:16) R r (cid:17) (cid:18) dψ + 12 cos θdφ (cid:19) + R dη . (A.1)We take φ , ψ , and η to have period 2 π , and 0 ≤ θ ≤ π . R and R are the radius of S parametrized by ψ and η , respectively.Being Ricci flat, this metric can easily be embedded in M-theory. Reducing to IIA along η will give rise to a KK5-brane in type IIA string theory. Reducing to IIA along ψ will giverise to a D6 brane extended along η .Consider a general linear transformation on the coordinates ψ and ηη = aη ′ + bψ ′ , ψ = cη ′ + dψ ′ . (A.2)This will modify the last two terms of (A.1) to R R ( c R + a R V ) (cid:18) ( ad − bc ) dψ ′ − a θdφ (cid:19) + a R + c R V ! ( dη ′ + A ) (A.3)14ith A = cR a ( c R + a R V ) + 1( ad − bc ) ba ! (cid:18) ( ad − bc ) dψ ′ + a θdφ (cid:19) − ad − bc ) b θdφ . (A.4)At the level of classical supergravity, this is a solution generating transformation, butnot all of the solutions obtained in this fashion are sensible backgrounds of string theory.Rather, there is a certain discrete subset of these solutions which is consistent with chargequantization. A.2 Twisted Z p orbifold One example of such a discrete subset is to take a bc d = q/p /p (A.5)and impose the periodicity η ′ = η ′ + 2 π and ψ ′ = ψ ′ + 2 π . This can be viewed as a twisted Z p orbifold of T N × S as outlined in (3.4) of [29]. When reducing to IIA, one finds a RR1-form potential A = qp dψ ′ + q θdφ ! − q θdφ = qp dψ ′ . (A.6)Upon further T-dualizing this background along the ψ ′ direction, we obtain a supergravitysolution for a ( p, q ) 5-brane smeared along the ψ ′ direction. This can easily be seen fromthe RR 2-form potential A = − q θdφ ∧ d ˜ ψ ′ (A.7)and the NSNS 2-form potential B = − p θdφ ∧ d ˜ ψ ′ (A.8)which one finds from the duality transformation. The 3-form field strength is closed andnaturally encodes the flux through S × S parametrized by θ , φ , and ˜ ψ ′ for arbitrary r .The issue stems from attempting to understand the q units of D6 charge from the IIAdescription prior to taking the T-duality along the ψ ′ direction. One expects the D6 chargeto be encoded by the flux of the RR 2 form field strength F = dA (A.9) The p and q are switched from what is in [29] so that p counts the number of NS5-brane and q countsthe number of D5-brane in the dual IIB description. vanishes for the background (A.6) under consideration. For this example, the hintfor where the quantum number of the D5 charge is encoded in the IIA description is staringat our face in equation (A.6). It is the fractionally valued vector potential arising as aresult of the non-trivial twist, q , in the Z p . This may be thought of as the simplest exampleillustrating the point that charge is sometimes encoded in the period of a potential, i.e. aWilson line, rather than the field strength. A.3 SL (2 , Z ) dual of T N × S Let us now consider a different example, where we take the SL (2 , Z ) subset of the generallinear transformation (A.2). In this case, we have ad − bc = 1, simplifying (A.4) to A = cR a ( c R + a R V ) + ba ! (cid:18) dψ ′ + a θdφ (cid:19) − b θdφ . (A.10)Once again, T-dualizing on ψ ′ will give rise to a RR 2-form A = − b θdφ ∧ d ˜ ψ ′ . (A.11)In fact, if we take a = p and b = q , the IIB 5-brane charges are identical to the example in theprevious section although the background differs in the asymptotic value of the axiodilaton.The puzzle, once again, is the status of the D6 charge in the IIA frame. This time,the RR 2-form field strength does not vanish, so one might try to define a D6-brane chargeby integrating F over a suitable 2-cycle. However, no such nonsingular 2-cycle exists. Forexample, integrating on the natural S parametrized by θ, φ would give a charge that dependson the radius r . This apparent failure of the Gauss law can be traced to this S not actuallybeing a well-defined 2-cycle.Notice that in the ordinary IIA reduction of (A.1) on the circle parametrized by ψ , theprocedure of integrating F on the S at fixed radius is the correct one for counting thenumber of D6 branes. On the other hand, for the IIA reduction on η (or in M-theory) theintegrality of the D6 brane charge follows from demanding that the KK5 metric does nothave a singularity. In a generic duality frame, such as the one given by reduction on ψ ′ ,neither condition is correct.Instead, one might try to define a modified flux quantization condition that mixes theflux and geometry in such a way as to obtain a conserved charge. This can be done byconsidering the combination Q D = 12 π (cid:18)Z S F + aad − bc Z S A (cid:19) = bad − bc (A.12)16here S is the 2-sphere parametrized by ( θ, φ ), and S is the circle parametrized by ψ ′ .Note that this reduces to the same prescription as in the previous subsection if F happensto vanish. In fact, one could have also considered applying the prescription of reading offthe period of A precisely at the radius r = 0 where F would have vanished.Formally, this procedure is equivalent to considering Q = 12 π Z S ( F + f ∧ A ) (A.13)discussed briefly in appendix A of [4] based on the language of [30–33]. Although theprocedure of computing the period of this “modified” flux will give the correct answer, it issomewhat unsettling that the procedure is not generally covariant. The aim of this articleis to identify the origin of this charge from discrete data of the potential. A.4 Quantization of gravity duals of field theories in 2+1 dimensions
Let us now examine the quantization of fluxes in gravity duals of various 2+1 dimensionalfield theories and compare their features with the example of the previous section.We will begin by reviewing the case of ABJM and ABJ for which the gravity theory is M-theory on
AdS × S /Z k [1, 2]. The transverse S /Z k has a torsion 3-cycle on which one canwrap l M5-branes in the range 0 ≤ l < k . These M5-branes will not source M-theory 4-formflux. Instead, they give rise to discrete torsion parametrized by a flat C in the background,supported on the S /Z k . This is quite similar to the case of the one form potential (A.6) inthe case of twisted Z p orbifold of T N × S we described in section A.2 and A.3. There aretwo refinements to this story.One is that the quantization condition for the discrete torsion has an anomalous shiftdue to the Freed-Witten anomaly, and reads k Z S /Z k C = l − k . (A.14)The presence of Freed-Witten anomaly was inferred in the IIA description of this backgroundin [4]. At the moment, it is not clear how one understands this shift strictly in the M-theoryperspective, but since it is required in the IIA reduction, we will adopt it in the M-theorylift as well. One can simply view this as an overall shift in the charge lattice. As we will seebelow, this shift turns out to be consistent with a rather non-trivial consistency test.The second refinement concerns the relation between the M2 charge and the radius ofanti de Sitter geometry. In the absence of discrete torsion, the radius of anti de Sitterspace is directly proportional to the number of M2 branes giving rise to the near horizon17 dS × S geometry. In the presence of discrete torsion, however, the relation receives acorrection. This issue was investigated originally in [14] which left out the contribution fromthe Freed-Witten anomaly. Taking the Freed-Witten anomaly into account [4], one findsthat R = (2 πl p ) Q k (A.15)where Q = N − l ( l − k )2 k + Q curv (A.16)with Q curv = − (cid:18) k − k (cid:19) (A.17)is the contribution from the C ∧ R correction to the M-theory action.In the IIA reduction along the Hopf fiber of S /Z k , the Q can also be written Q = N + k ! + l − k ! b + 12 kb + Q curv , b = − lk + 12 (A.18)where b is the pull-back of B on S level surface of R /Z k reduced on S . This expres-sion makes the interpretation of Q as including the contribution from the B -field in theWess-Zumino term for k D6-brane and l − k/ k/ Q curv . The k/ Q / . This is a rather non-trivial test for theconsistency of the details of the shifts in curvature due to discrete torsion, including thedetailed form of the effects of Freed-Witten anomaly. They in fact confirm specifically thepresence of a shift in the D2 charge by k/ − k/
24 = k/
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