aa r X i v : . [ h e p - t h ] O c t WIS/15/08-JUL-DPP
Fractional M2-branes
Ofer Aharony a , Oren Bergman b,c and Daniel Louis Jafferis da Department of Particle PhysicsThe Weizmann Institute of Science, Rehovot 76100, Israel
[email protected] b School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA [email protected] c Department of Physics, Technion, Haifa 32000, Israel [email protected] d Department of Physics, Rutgers University, Piscataway, NJ 08855, USA [email protected]
We consider two generalizations of the N = 6 superconformal Chern-Simons-matter theories with gauge group U ( N ) × U ( N ). The first generalization isto N = 6 superconformal U ( M ) × U ( N ) theories, and the second to N = 5superconformal O (2 M ) × U Sp (2 N ) and O (2 M + 1) × U Sp (2 N ) theories. Thesetheories are conjectured to describe M2-branes probing C / Z k in the unitarycase, and C / b D k in the orthogonal/symplectic case, together with a discreteflux, which can be interpreted as | M − N | fractional M2-branes localized at theorbifold singularity. The classical theories with these gauge groups have beenconstructed before; in this paper we focus on some quantum aspects of thesetheories, and on a detailed description of their M theory and type IIA stringtheory duals.July 2008 . Introduction and summary of results Motivated by the work of Bagger and Lambert [1] (see also [2-4]), there has recentlybeen great interest in studying three dimensional superconformal field theories and theirrelation to the low-energy theory on M2-branes. The attempts to directly generalize theBagger-Lambert construction of an N = 8 superconformal theory with SO (4) gauge groupto the low-energy theory on N M2-branes have not yet been successful, and seem tolead either to non-unitary theories or to non-conformal theories. In [5] we suggested analternative route to studying M2-branes. We presented a U ( N ) k × U ( N ) − k Chern-Simonsmatter theory with explicit N = 6 superconformal symmetry . Considering the braneconstruction of this theory led us to conjecture that it was equivalent to the low-energytheory of N M2-branes on a C / Z k singularity, and we presented various pieces of evidencefor this conjecture. In the special cases of k = 1 and k = 2 this conjecture implies thatthe theories should have an enhanced N = 8 supersymmetry, whose generators cannot bedescribed locally in terms of the original fields of the theory. In particular, for k = 1 weconjectured that the U ( N ) × U ( N ) − theory is equivalent to the low-energy theory on N M2-branes in flat space.In this paper we generalize the construction of [5] in two directions, both of whichbreak the parity symmetry of the original construction. Both of these generalizations havealready been considered as classical field theories in [8], but we add more details aboutquantum aspects of these theories, and we also discuss in detail their gravitational duals.The first generalization we consider is to U ( M ) k × U ( N ) − k Chern-Simons-matter theories,with the same matter content and interactions as in [5], but with M = N . From thepoint of view of M2-branes probing a C / Z k singularity, these theories arise (for M > N )when we have ( M − N ) fractional M2-branes sitting at the singularity, together with N M2-branes that are free to move around. On the field theory side, classically this generalizationis straightforward, and still leads to N = 6 superconformal field theories. However, wewill argue (both directly in the field theory, and by using a brane construction) thatquantum mechanically these theories only exist as unitary superconformal field theorieswhen | M − N | ≤ | k | . For these cases, we argue that the gravitational dual is the same AdS × S / Z k background described in [5], just with an additional “torsion flux”, whichtakes values in H ( S / Z k , Z ) = Z k . This flux may be thought of as an M theory analog oforbifold discrete torsion, corresponding to a discrete holonomy of the M theory 3-form fieldon a torsion 3-cycle in S / Z k . This is a good description of the gravitational dual when N ≫ k . For k ≪ N ≪ k , on the other hand, the appropriate description is in termsof type IIA string theory on AdS × CP , with a discrete holonomy of the NSNS 2-form These theories are special cases of N = 4 superconformal Chern-Simons theories constructedin [6,7]. CP ⊂ CP . In this description the NSNS 2-form holonomy is quantizeddynamically.Our second generalization involves an orientifold in the original brane construction,which changes the gauge symmetry to O ( M ) × U Sp (2 N ), and reduces the charged mattercontent to two bi-fundamental chiral multiplets. The resulting gauge theories have N = 5superconformal symmetry, and again there are bounds on | M − N | in order for thequantum theory to exist (which we will describe in more detail below). We conjecturethat these theories are dual to the low-energy theory on M2-branes probing a C / b D k singularity, where b D k is the binary dihedral group with 4 k elements, and (generically) withsome discrete flux, taking values in H ( S / b D k , Z ) = Z k , at the singularity. As before, onecan also think of this discrete flux as associated with fractional M2-branes sitting at thesingularity. The gravitational dual of these theories may thus be described (for N ≫ k )by M theory on AdS × S / b D k , possibly with discrete torsion. When k ≪ N ≪ k thisbackground reduces to an orientifold of type IIA string theory on AdS × CP , again witha discrete holonomy of the NSNS 2-form.Our study provides additional examples of AdS /CF T duals with extended super-symmetry, which should hopefully be useful for understanding better the properties of thisduality. In particular, we extend the duality to theories which are not parity-invariant,and are thus closer in spirit to the Chern-Simons-matter theories encountered in condensedmatter applications. It would be interesting to analyze these theories in the ’t Hooft limit(large N and k with fixed N/k ) and to see if integrability (on the field theory or stringtheory sides of the duality [9,10]) may be used to learn about various properties of thesetheories. Some of our results also have applications to more general three dimensional the-ories. For instance, our claim (based on [11]) that the U ( M ) k N = 3 supersymmetric pureChern-Simons theories with M > k do not exist implies that almost all of the classicalsupersymmetric vacua of the mass-deformed U ( N ) × U ( N ) theory found in [12] do notsurvive in the quantum theory (when k = 1), so more vacua need to be found in order tomatch with expectations .The rest of the paper is organized as follows. In section 2 we consider the N = 6 U ( M ) × U ( N ) superconformal Chern-Simons-matter theories, their realization in termsof brane configurations in type IIB string theory, and their gravity dual descriptions. Insection 3 we consider the N = 5 O ( M ) × U Sp (2 N ) superconformal theories, their type IIBbrane realizations, and their gravity dual descriptions. The moduli space of these theoriesis analyzed in an appendix. We thank M. Van Raamsdonk for discussions on this issue. . U ( M ) × U ( N ) theories It is straightforward to generalize the field theory construction of [5] to supersymmetric U ( M ) k × U ( N ) − k Chern-Simons matter theories. As in the case of N = M , we can takethe N = 3 supersymmetric Chern-Simons-matter theory with this gauge group and withtwo bi-fundamental hypermultiplets, and argue that because of the special form of itssuperpotential, it in fact has an enhanced N = 6 supersymmetry (these theories were alsorecently discussed in [8]). Many aspects of the analysis of these theories are very similarto the case of N = M which was discussed in [5]. For instance, the moduli space of thesetheories turns out to be exactly the same as that of the U ( L ) k × U ( L ) − k theories with L = min( N, M ), and the spectrum of non-baryonic chiral primary operators is also thesame as the one in that theory. The main difference is that the effective field theory on themoduli space contains an extra U ( M − N ) k Chern-Simons (CS) theory (when
M > N ),with no massless charged fields. In this section we will describe the gravitational dual ofthe U ( M ) × U ( N ) theories, and some of the differences between these theories and the N = M theories. Let us assume to begin with that
M > N , and consider the superconformal theories U ( N + l ) k × U ( N ) − k with l ≥ k > U ( N ) k × U ( N + l ) − k theory. We will see later that there is also an equivalence relating a theoryof the first type to a theory of the second type (with l → k − l ). As in the equal rankcase, the more general superconformal theory can be obtained (at least classically) as theIR limit of an N = 3 supersymmetric gauge theory, arising as the low-energy limit of atype IIB brane construction. Recall that the brane construction of the U ( N ) k × U ( N ) − k theories involves [5] N D3-branes winding around a circle, and intersecting an NS5-braneand a (1 , k )5-brane (a bound state of an NS5-brane with k D5-branes) at specific angles(described in [5]).To get the U ( N + l ) k × U ( N ) − k theory we add to this l D3-branes which are suspendedbetween the NS5-brane and the (1 , k )5-brane on one side of the circle (the side in whichthe 5-brane intersection numbers lead to a Chern-Simons level of + k , see figure 1). Thisconstruction makes it clear that the classical moduli space is identical to the moduli spaceof the U ( N ) k × U ( N ) − k theory. It corresponds to the motion of the N free D3-branes (aswell as to the Wilson line and dual gauge field on the D3-branes). The l suspended D3-branes are locked into position by the two 5-branes, so there is no moduli space associatedwith them.There is some additional “braneology” associated with these configurations, which is3 (1,k) D3 N D3 N+l
NS5
Figure 1:
The brane construction of the U ( N + l ) k × U ( N ) − k theory. reflected in interesting properties of the corresponding field theories . First, let us try to goonto the classical moduli space, by separating the N branes which wrap all the way aroundthe circle from the other branes. This leaves only l “fractional branes” stretched betweenthe two five-branes. For l > k , it was argued in [11] that this configuration of “fractionalbranes” breaks supersymmetry. In the brane picture this follows from the “s-rule” [13]forbidding more than one D3-brane from ending on a specific NS5-brane D5-brane pair,which was generalized to this case in [11]. From the field theory point of view, at such apoint on the moduli space the low-energy field theory includes a pure N = 3 U ( l ) k YM-CStheory living on the fractional D3-branes. A simple generalization of the Witten indexcomputation of [14] implies that supersymmetry is unbroken for l ≤ k . However, for l > k , this computation suggests that supersymmetry is broken [11]. This implies that theclassical moduli space of this U ( N + l ) k × U ( N ) − k brane configuration is at least partlylifted when l > k .The second interesting bit of brane physics has to do with the brane creation effect[13]. When the NS5-brane crosses the (1 , k )5-brane, k D3-branes are created between them,corresponding to one D3-brane for each D5-brane component of the (1 , k )5-brane. If thereare l suspended D3-branes to begin with, there will be ( k − l ) suspended D3-branes afterthe crossing. Let us start with the brane configuration for the N = 3 gauge theory with U ( N + l ) k × U ( N ) − k where 0 ≤ l ≤ k , and move one of the 5-branes around the circle inthe direction that initially shortens the suspended 3-branes. The final configuration thendescribes the N = 3 gauge theory with U ( N ) k × U ( N + k − l ) − k , see figure 2. We thusexpect these two gauge theories to flow to the same superconformal theory. Below we willargue that these two superconformal theories are indeed equivalent. We thank A. Hashimoto for useful discussions on these brane configurations. Naively, moving the 5-brane around the circle multiple times, or in the other direction, N(N+l)
D3 NS5 (1,k) N D3 (N+k−l) NS5
Figure 2:
Branes are created when we move a 5-brane around the circle.
Let us now consider the N = 6 superconformal U ( N + l ) k × U ( N ) − k Chern-Simons-matter theory that is obtained in the IR limit of the above gauge theory. We will use thedescription of these theories in [5], in which they are written as special cases of N = 3supersymmetric Chern-Simons-matter theories that happen to have enhanced supersym-metry. We will now argue that the theories with l > k do not exist as unitary conformalfield theories, and that the above theory with l ≤ k is equivalent to the theory with U ( N ) k × U ( N + k − l ) − k .In Chern-Simons-matter theories with this much supersymmetry the metric on themoduli space cannot be corrected, so we can go to a generic point on the moduli space,where at low energy we get N copies of a U (1) k × U (1) − k theory (as in the U ( N ) k × U ( N ) − k case), together with a pure U ( l ) k N = 3 supersymmetric Chern-Simons theory, with nomassless charged matter fields. Naively, this supersymmetric Chern-Simons theory is thesame as the bosonic Chern-Simons theory at level k , since the other fields in the N = 3vector multiplet just have Gaussian actions in the absence of any matter fields, but in factwe should be more careful. Recall that the Chern-Simons level in [5] was defined such thatit was equal to the level of the N = 3 Yang-Mills-Chern-Simons theory which flows to theChern-Simons-matter theory in the IR. In this theory the level is not renormalized [15], butthis comes from the cancelation of two contributions. There is a shift of k → k − l · sign( k )coming from integrating out the gauginos, and another opposite shift from the one-loopcontributions of the gauge field itself (these shifts are only in the SU ( l ) level, not inthe U (1) level, but this will not affect our arguments). Thus, if we think of this theoryin bosonic terms, we can say that it includes a bosonic SU ( l ) k − l · sign( k ) Chern-Simonstheory. However this only makes sense if | k | ≥ l , since otherwise the one-loop shift in should produce more equivalences. However, all of these additional configurations lead either toanti-D3-branes or to U ( N ′ + l ′ ) k × U ( N ′ ) − k theories with l ′ > k . k → k + l · sign( k )) would havethe wrong sign and would not bring us back to the level k theory that we want. Thissuggests that the N = 3 supersymmetric U ( l ) k pure Chern-Simons theory does not existas a unitary theory for l > k (at least, it does not seem to be equivalent to any standardbosonic Chern-Simons theory). We are therefore led to conjecture that the superconformal U ( N + l ) k × U ( N ) − k theories with l > k do not exist as unitary theories. Note that thesetheories are never weakly coupled (recall that a U ( N ) k theory is weakly coupled only when | k | ≫ N ). This conjecture is consistent with what we found in the corresponding N = 3 U ( N + l ) k × U ( N ) − k Yang-Mills-Chern-Simons theories, where we showed that for l > k the moduli space was partly lifted, so these theories cannot flow to the U ( N + l ) × U ( N )Chern-Simons-matter theories (whose moduli space does not receive quantum corrections).For the superconformal theories with l ≤ k the brane picture suggested that thereshould be an equivalence between pairs of theories of the form U ( N + l ) k × U ( N ) − k = U ( N ) k × U ( N + k − l ) − k . (2 . . As (weak) evidence for this conjectured equivalence, note thaton the moduli space one obtains the U ( l ) k supersymmetric CS theory for the theory onthe left-hand side of (2.1), and the U ( k − l ) − k theory for the theory on the right-hand side.The corresponding effective bosonic theories therefore include pure SU ( l ) k − l CS and pure SU ( k − l ) − l CS, respectively. In the presence of a boundary, we obtain the correspondingWZW theories, SU ( l ) k − l and SU ( k − l ) l , which are precisely related by level-rank duality.For the special case l = k we get an equivalence between the U ( N + k ) k × U ( N ) − k theoryand the original l = 0 theory with U ( N ) k × U ( N ) − k . Note that in this case on the modulispace of the former theory we seem to obtain an extra U ( k ) k pure Chern-Simons theory,but as argued above, at least the SU ( k ) part of this is equivalent to a bosonic SU ( k ) theory, which is an empty theory . We therefore end up with a total of k inequivalenttheories for any given minimal rank N (which sets the dimension of the moduli space tobe 8 N ). Below we will see that this is consistent with the dual gravitational picture. Notethat, as expected, for the case k = 1 corresponding to M2-branes in flat space we do notfind any new theories associated with “fractional branes”. In particular, these theories seem to have a different number of degrees of freedom, but thisnumber can be significantly modified at strong coupling. There is also a remaining U (1) k pure CS theory. We will see below that in their dualdescription these two theories differ by a shift of a bulk theta angle by 2 π . The extra U (1) k partmay be related to the fact that the boundary theory changes by a U (1) Chern-Simons theorywhen the theta angle in the bulk is shifted by 2 π [16]. .2. M theory lift Lifting the type IIB brane configuration for the U ( N + l ) k × U ( N ) − k theory to Mtheory as in [5], we obtain N M2-branes moving in C / Z k , together with l “fractional M2-branes”, which correspond to M5-branes wrapped on a vanishing 3-cycle at the orbifoldpoint. This is a pure torsion cycle since H ( S / Z k , Z ) = Z k . In other words, k wrappedM5-branes are equivalent to none, and there are k inequivalent configurations. This can bethought of as an M theory analog of orbifold discrete torsion [17], classified by H (Γ , U (1)),where Γ = Z k . This discrete torsion may be realized by a discrete holonomy of the 3-form potential on the above 3-cycle, exp( i R − cycle C ) ∈ Z k . By Poincar´e duality we canalso associate the fractional branes to a pure torsion flux of the 4-form field strength in H ( S / Z k , Z ) = Z k . We therefore conclude that the U ( N + l ) k × U ( N ) − k superconformaltheory with 0 ≤ l ≤ k − N M2-branes on C / Z k with l units of discrete torsion.Our analysis of the discrete torsion generalizes the known result in the case of k = 2, wherethe two variants of C / Z are the OM + and OM − orbifold planes [18,19].Note that the theory of two M2-branes on the C / Z orbifold with discrete torsionwas also claimed [20,21] to be described by the N = 8 Bagger-Lambert theory, with gaugegroup SU (2) × SU (2), at level k = 2. Here we claim that the same theory may be describedusing the N = 6 U (3) × U (2) − Chern-Simons-matter theory, which has a hidden N = 8superconformal symmetry (as well as a hidden parity symmetry).The number of discrete torsion variants is consistent with the restriction l ≤ k , to-gether with the equivalence of the l = 0 and l = k theories, found above for the supercon-formal field theories. In fact the more general equivalence between the pairs of theoriesin (2.1) can be understood as a parity symmetry in M theory, that reflects one of thecoordinates along the M2-branes and at the same time takes C → − C . The former hasthe effect of changing the sign of the Chern-Simons term, and therefore the level k → − k ,and the latter has the effect of changing the torsion class [ G ] → k − [ G ], and therefore l → k − l in the gauge group that has level ( − k ). Taking the near-horizon limit of the brane configuration described above as in [5], wefind that our field theories are dual to M theory on
AdS × S / Z k , with N units of 4-formflux in AdS , and l units of discrete torsion: ds = R ds AdS + R ds S / Z k ,G ∼ N ǫ , Z S / Z k ⊂ S / Z k C = 2 π lk , (2 . ǫ is the unit volume form on AdS and R = (2 π kN ) / l p . For N ≫ k the theoryhas a good approximation in terms of eleven dimensional supergravity on this background.We can describe S / Z k in terms of 4 complex coordinates z i with a constraint P | z i | = 1and an identification z i ∼ e πi/k z i . In this language, the 3-cycle S / Z k can be described(for instance) as {| z | + | z | = 1 , z = z = 0 } .We can also describe the S / Z k as an S fibred over CP , in which case its metric isexpressed as ds S / Z k = 1 k ( dϕ + kω ) + ds CP , (2 . ϕ ∼ ϕ + 2 π , and dω = J where J is the K¨ahler form on CP . The 3-form whichhas the above holonomy can then be expressed locally as C ∝ lk J ∧ dϕ . (2 . Recall that for M = N , when N / ≪ k ≪ N , the M theory circle becomes small(in eleven dimensional Planck units). A better description of the background dual tothe U ( N ) k × U ( N ) − k theory is obtained [5] by reducing to type IIA string theory on AdS × CP , with N units of ˜ F flux on AdS and k units of F flux on the CP CP ; we can write this background (setting the string scale to one) as ds string = R k ( 14 ds AdS + ds CP ) ,e φ = R k ∼ N / k / = 1 N (cid:18) Nk (cid:19) / , ˜ F = 38 R ˆ ǫ ,F = kdω = kJ, (2 . R str = R /k = 2 / π p N/k .To describe the U ( N + l ) k × U ( N ) − k theories we need to add to this background thereduction of the discrete torsion in M theory. The 3-form in (2.4) has one leg on the Mtheory circle, so it reduces to the NSNS 2-form B , which attains a non-trivial holonomyon the CP CP b ≡ π Z CP ⊂ CP B = lk . (2 . b is periodic with period one. Equation (2.6)seems like a surprising result at first, since the holonomy of B is a continuous variable,8 .e. there is no discrete torsion in CP . Let us therefore explain how it arises directly inthe type IIA description of the fractional branes.The M5-branes wrapped on S / Z k reduce to D4-branes wrapped on the CP CP . Note that the low-energy theory on l such D4-branes contains a U ( l ) Chern-Simons term at level k , as we expect, due to the F ∧ A ∧ dA term in the D4-brane effectiveaction (where A is the D4-brane worldvolume gauge field). From the point of view of thetheory on AdS , the D4-branes are domain walls separating the space into two separateregions. Of course, these domain walls (which we think of as filling space and sitting at afixed radial position) are not static, but rather they feel a force driving them towards thehorizon of AdS ; this corresponds to the fact that there is no moduli space for moving thefractional branes around. Nevertheless, let us assume for a moment that we fix the branesat some fixed radial position. The D4-branes are a source for the RR 5-form field C ; thissource means that as we go from one side of the D4-branes to the other, the flux of theRR 4-form field strength ˜ F , integrated over CP ⊂ CP (which is the dual cycle to theone the branes are wrapped on), jumps by l units.Naively this suggests that the type IIA dual of the U ( N + l ) × U ( N ) theories shouldbe described by a configuration with l units of 4-form flux on CP ⊂ CP . However,such a solution does not seem to exist, and we wish to claim that the correct backgroundis actually the one described above. The point is that we also have k units of F fluxon CP , and in the presence of this flux the equation of motion of ˜ F is modified to d ˜ F = − F ∧ H . From the point of view of the effective field theory on
AdS , thismeans that f ∝ R CP ˜ F (normalized so that it is quantized to be an integer) is notconserved, but rather its equation of motion is given by d ( f + kb ) = 0. This means thatthe conserved flux which jumps by l units as we cross the D4-branes is not f but rather f + kb , and another way to realize such a jump is by having b jump by l/k . This isprecisely what we obtained above. Thus, we conjecture that the type IIA string theorydual of the U ( N + l ) k × U ( N ) − k theory is the original AdS × CP background, togetherwith a B field holonomy b = l/k . Clearly this is a solution of the classical equationsof motion of type IIA supergravity (preserving the same supersymmetries as the originalbackground), since b does not enter the equations.Note that naively b is not quantized, but the arguments above suggest that ( f + kb )is quantized, implying that when f = 0, b must be quantized in units of 1 /k . Theperiodicity b ∼ b + 1 then implies that there are k inequivalent theories, in agreementwith both M theory and the field theory arguments. As a consistency check, note thatthe equivalence of b → b + 1 naively comes from the presence of NS5-branes wrapped on CP ⊂ CP , which from the AdS point of view look like axionic strings such that as we Recall that the gauge-invariant 4-form field strength in type IIA string theory is given by˜ F = dC − C ∧ H .
9o around them, b → b +1. However, in the presence of k units of F flux, such a wrappedNS5-brane must have k D4-branes wrapped on CP ending on it. This is consistent withthe fact that inserting k fractional D4-branes is the same as taking b → b + 1, since suchD4-branes can annihilate to nothing by creating bubbles of wrapped NS5-branes whichthey can end on, and these bubbles can then shrink to nothing, leaving behind only thechange b → b + 1.Furthermore, the type IIA version of the parity symmetry is given by the reflection ofa coordinate in AdS together with b → − b . Combining this with b → b + 1 we findthat the theory with b = l/k should be related to the theory with b = ( k − l ) /k by aparity transformation, implying that the U ( N + l ) k × U ( N ) − k theory should be equivalentto the U ( N ) k × U ( N + k − l ) − k theory. This is also in complete agreement with the fieldtheory and M theory points of view described above. Our derivation of the M theory / string theory duals was of course far from rigorous,so we should now make various tests to see if our conjecture is sensible or not. First, notethat the moduli space for a single M2-brane in our M theory background is still given by C / Z k , in agreement with the moduli space we expect. Similarly, the spectrum of lightfields in our backgrounds is not affected by the holonomy of the C field or of the B field,in agreement with our claim that the spectrum of (non-baryonic) chiral primary operatorsin these theories is independent of l .What effect does this holonomy have at all, given that it does not modify the classicalequations of motion? It does change the spectrum of baryonic operators. Let us analyzethis explicitly using the type IIA description. Recall that the background above allows forD4-branes wrapped on CP ⊂ CP , which were identified in [5] with di-baryon operators of the form ( C I ) N , where C I are bi-fundamentals of U ( N ) × U ( N ). The worldvolumeaction of the D4-branes contains a coupling B ∧ F ∧ A , implying that when b = l/k ,we must have l strings ending on the D4-brane. This is exactly what we expect for thedi-baryon operators in the U ( N + l ) × U ( N ) theory, since if we take an operator like ( C I ) N ,we can contract the U ( N ) indices to a singlet of SU ( N ) with an epsilon symbol, but if wethen contract the indices of SU ( N + l ) by an epsilon symbol we are left with an object inthe l ’th anti-symmetric product of fundamentals of SU ( N + l ). This exactly matches thenumber of strings that must end on this object to form a consistent state. (Of course, the Note that these operators are not gauge-invariant under one of the U (1) gauge groups;nevertheless they are present in the bulk. Presumably one way to think of these operators islike electrons in QED, as defined together with a Wilson line that carries away the U (1) chargeto infinity. C I ) N + l similarly transforms in the l ’th anti-symmetric product of fundamentalsof SU ( N ).) Thus, the spectrum of di-baryons exactly matches what we expect.However, we have been a bit too fast in the analysis above; what really appears in thesource term on the D4-branes used above is F ∧ ( ˆ F − B ) ∧ A , where ˆ F is the gauge fieldstrength on the D4-branes, and usually we say that in the presence of a B field, ˆ F − B isquantized rather than ˆ F , so one might suspect that we must turn on some ˆ F gauge field tocancel the above effect of the B field. We claim that in our background this is not correct,and we are still allowed to have configurations with ˆ F = 0 that lead to the di-baryonsdescribed above. To see this, let us analyze the effect of the B field in a different way,by looking at the four dimensional effective action on AdS . As we discussed, the B fielddoes not affect the equations of motion, but it does affect the action. The ten dimensionalaction includes a term B ∧ ˜ F ∧ ˜ F . So, when we reduce to four dimensions, and look atthe gauge field coming from R CP ˜ F , the b component of the B field behaves as a thetaangle for this gauge field. We know that when we shift the theta angle, the Witten effect[22] implies that the electric charges of particles are shifted by the theta angle times theirmagnetic charges. In the case of the specific gauge field we are discussing, the electricallycharged particles are D2-branes wrapped on CP , while the magnetically charged particlesare D4-branes wrapped on CP . So, we expect that turning on the B field will changethe wrapped D2-brane charge of a wrapped D4-brane by an amount proportional to the B field. And indeed, this is exactly what happens, because of the B ∧ C coupling onthe worldvolume of the D4-branes. Note that again this coupling is really of the form( ˆ F − B ) ∧ C , so that we would not get this effect if ( ˆ F − B ) were quantized; butthe space-time argument above implies that we should see this effect, so we conclude thatD4-brane configurations with ˆ F = 0 are still allowed , and reproduce for us the expecteddi-baryon spectrum discussed above. It would be interesting to reproduce this result by adirect study of the quantization of ˆ F in our background. Our conjectured duality implies that at strong coupling (large λ = N/k ), all the U ( M ) k × U ( N ) − k theories with M = N, N +1 , · · · , N + k are very similar to each other, sincethey only differ by the B field, which only affects the D-brane spectrum and worldsheetinstanton effects (which are suppressed by exp( −√ λ )). In particular, it seems that allthese theories should be identical (at large N ) to all orders in perturbation theory in1 / √ λ , though not beyond this perturbation theory. For instance, the entropy of thesetheories should be the same to all orders in 1 / √ λ . Of course, it is hard to check this claim,since we do not know how to perform computations in these theories at strong coupling. This is not necessarily true for other wrapped D-branes. U ( N ) k × U ( N ) − k theoriescan easily be generalized to the U ( M ) k × U ( N ) − k theories. In particular, it would beinteresting to examine the spin chain representation of the spectrum of operators in thesetheories in the ’t Hooft limit [9], and to see if ( M − N ) can be related to a B field on the“worldsheet” of the spin chain.
3. Orientifold theories
A class of N = 5 superconformal Chern-Simons-matter theories with gauge groups O ( M ) × U Sp (2 N ) was recently constructed in [8]. We will review how one can obtain thesetheories as orientifold projections of the N = 6 superconformal theories, or alternativelyas the IR limits of gauge theories living on D3-branes in a type IIB orientifold-braneconfiguration . We will argue that these theories describe M2-branes probing the orbifold C / b D k , where b D k is the binary dihedral group with 4 k elements, together with discretetorsion (or “fractional M2-branes”). The dual description (which is useful for N, M ≫ k )is then either M theory on AdS × S / b D k , or type IIA string theory on AdS × CP / Z ,where the Z is an orientifold projection that we will describe below. Let us begin with the type IIB brane configurations. The orientifold theories areobtained by including an orientifold 3-plane wrapped on the circle, in addition to the D3-branes and the two 5-branes that we had in the previous section. Adding the orientifoldsdoes not break any additional supersymmetry. We will take the D3-branes to be “whole3-branes” in the sense that they have images in the covering space of the orientifold (theyare not identified with their own image). The 5-branes on the other hand intersect theorientifold plane, and are their own images. They are therefore “half branes” in a sense.The orientifold 3-plane comes in four varieties, which are denoted O3 − , O3 + , f O3 − and f O3 + . When we have N D3-branes sitting on top of the orientifold 3-plane, theselead to the gauge groups O (2 N ), U Sp (2 N ), O (2 N + 1) and U Sp (2 N ), respectively. TheO3-planes carry fractional D3-brane charges given by ( − /
4) for O3 − and (+1 /
4) for O3 + , f O3 − and f O3 + . In particular, the f O3 − plane can be thought of as an O3 − plane with ahalf D3-brane stuck on top of it. The two U Sp (2 N ) theories corresponding to O3 + and f O3 + are perturbatively identical, but differ in their non-perturbative (dyon) spectrum.The four types of orientifolds are related by the SL (2 , Z ) duality of type IIB string theory[24]. The O3-plane variants also correspond to a choice of discrete torsion for the NSNS A different orientifold of these theories, which breaks supersymmetry, and the correspondingbrane construction, were recently discussed in [23]. H ( RP , e Z ) = Z [24]. In particular,NSNS torsion differentiates the ( − ) and (+) variants, and RR torsion differentiates thetilde and no-tilde variants. A single 5-brane that intersects an orientifold 3-plane in twospatial dimensions will therefore change it from one type on one side to a different typeon the other side [25,19] . In particular, crossing an NS5-brane changes O3 − to O3 + and f O3 − to f O3 + , and crossing a D5-brane changes O3 − to f O3 − and O3 + to f O3 + . D3 N D3 N NS5 + (N+l) D3 + O3 − O3 ~~ NS5 + (N+l) D3O3O3 − + ++ ++ −− (1,2k) (1,2k) I,II III,IV
Figure 3:
The brane constructions of the orientifold theories.
In our circle configuration we therefore get an orthogonal gauge group on one intervalbetween the two 5-branes, and a symplectic gauge group on the other interval. For con-sistency of the configuration the D5-brane charge of the second 5-brane must be even, sowe will take it to be (1 , k ). Otherwise, the discrete torsion in the brane configuration isnot single-valued. The O ( M ) gauge field then has a Chern-Simons term at level ± k , andthe U Sp (2 N ) gauge field has a Chern-Simons term at level ∓ k . We therefore find fourclasses of theories (see figure 3)
I O (2 N + 2 l I ) k × U Sp (2 N ) − k ,II U Sp (2 N + 2 l II ) k × O (2 N ) − k ,III O (2 N + 2 l III + 1) k × U Sp (2 N ) − k ,IV U Sp (2 N + 2 l IV ) k × O (2 N + 1) − k , (3 . k → − k . We willuse a shorthand notation to denote each of these theories, for example the theory in class I at level k with l fractional branes will be denoted as I k ( l ). The matter content can be This gives another explanation of why we must start with 2 k units of D5-brane charge. Thelevel of the U Sp (2 N ) Chern-Simons term must be an integer. U (2( N + l )) × U (2 N )bi-fundamental fields C I ( I = 1 , , ,
4) : C Ia ¯ b → − M IJ C ∗ Ja ¯ c J ¯ c ¯ b , (3 . J is the invariant anti-symmetric matrix of the U Sp theory, and M IJ is a matrixacting on the four complex scalars (and their superpartners) as iσ ⊗ × . This projectsout half of the two bi-fundamental hypermultiplets that we started from, leaving a singlebi-fundamental hypermultiplet, or equivalently two bi-fundamental chiral superfields. Forexample, choosing a specific ordering of the C I , we can take the identification on thechiral superfields A i and B i described in [5] to take the form A ≡ B T J , A ≡ B T J . Wetherefore obtain an N = 3 YM-CS theory with two chiral bi-fundamental superfields A and A , and with a product of an orthogonal and symplectic gauge group in one of theclasses shown in (3.1), or their parity partners. The superpotential is the projection of theoriginal superpotential W ∝ tr( A B A B − A B A B ) using the identifications above,which gives W ∝ tr( A J A T A J A T − A J A T A J A T ) . (3 . O3+ − O3 − + O3+ − NS5 NS5’ O3 − + O3+ − + − + − O3 NS5NS5’ D3
Figure 4:
A D3-brane is created when two NS5-branes cross on an O3 ∓ -plane. As in the case without the orientifold, we expect the “s-rule” and the brane creationeffect to lead to a restriction on the number of inequivalent superconformal theories. How-ever one has to be slightly careful here since both the brane creation effect, and the “s-rule”which can be derived from it, are shifted in the presence of an O3-plane relative to thecase without the orientifold. The basic reason for the shift is an additional brane creationeffect when two purely NS5-branes, that intersect an O3-plane, cross. In particular, if theorientifold between the NS5-branes is an O3 − -plane a single D3-brane is created, and if itis an O3 + -plane a single D3-brane is annihilated (see figure 4). If the orientifold betweenthe NS5-branes is an f O3 − or f O3 + -plane there is no brane creation. The creation (or an-nihilation), or lack thereof, can be understood from the requirement that the jump in the143-brane charge across each NS5-brane is preserved when the NS5-branes cross. Thetotal number of D3-branes created by the crossing NS5-brane and (1 , k )5-brane thereforedepends on the type of O3-plane between them (on the side that initially shrinks). Inthe O3 − case k + 1 D3-branes are created, in the O3 + case k − f O3 − and f O3 + cases k D3-branes are created. This also implies that the restriction on l isshifted to l ≤ k + 1 in the first case (class I), l ≤ k − l ≤ k in the last two cases (classes III and IV). D3 N D3 N + O3 ~ + − + O3 ~ + − D3 N − + − +D3 N D3 N D3 N (1,2k) (1,2k) NS5 + − ~ + + − ~ +O3 (N+k−l) NS5O3 ++ O3 − + (N+k−l 1) NS5 + (N+l) D3O3O3 − + ++ − NS5NS5 55 (1,2k) + (N+l) D3 − ~ + + (N+l) D3 − ~ +O3 + O3 ~ + − + O3 ~ + − (1,2k) D3D3
Figure 5:
The brane description of the parity-dualities.
We are now ready to examine the effect of moving one of the 5-branes around thecircle. As before, we find that only one motion gives rise to another supersymmetricconfiguration, and therefore to another N = 3 supersymmetric gauge theory, namely theone where the 5-brane winds the circle once in the direction that initially shrinks thesuspended D3-branes. The initial configuration has l suspended D3-branes in one of the One can derive this effect from a linking number analogous to the NS5-D5 case [13]. Thecreation of branes by NS5-branes crossing on an orientifold plane was originally proposed intype IIA brane configurations for four-dimensional N = 1 gauge theories with orthogonal andsymplectic gauge groups in [25], in order to reproduce Seiberg duality for these theories [26]. Inthat case the jump in the D4-brane charge across an NS5-brane leads to an asymptotic bendingof the NS5-brane, which must be preserved in finite (zero) energy brane motions. k → − k ,and k − l D3-branes, with the possible ± I k ( l ) = I − k ( k − l + 1) ,II k ( l ) = II − k ( k − l − ,III k ( l ) = III − k ( k − l ) ,IV k ( l ) = IV − k ( k − l ) . (3 . ≤ l I ≤ k + 1 , ≤ l II ≤ k − , ≤ l III,IV ≤ k . (3 . I k ( k + 1) = II k (0) ,II k ( k −
1) = I k (0) ,III k ( k ) = IV k (0) ,IV k ( k ) = III k (0) . (3 . k inequivalent theories (for each value of N and k ).As in the case with the unitary gauge groups, the two theories related by a parity-duality are never weakly coupled at the same time, making it difficult to test this duality.However we will provide more evidence for these dualities, as well as for the maximalvalues of l , by analyzing the superconformal theories directly, and we will also show thatthis picture is consistent with the gravitational dual descriptions. O ( M ) × U Sp (2 N ) superconformal theories The three-dimensional orthogonal/symplectic N = 3 gauge theories living on thebrane configurations described above flow in the IR to superconformal Chern-Simons-matter theories. Alternatively we can also get these superconformal theories by gauging adiscrete symmetry of the N = 6 superconformal theories with U ( ˜ M ) k × U ( ˜ N ) − k whichincludes the usual orientifold action on the gauge fields, and the action on the matterfields in (3.2). This action breaks the baryon number symmetry U (1) b completely, whilethe subgroup of the R-symmetry SU (4) R left unbroken by M IJ is U Sp (4) ∼ SO (5). The16ew O ( M ) × U Sp (2 N ) gauge theory must therefore have N = 5 supersymmetry, and thiswas verified (classically) in [8] .Let us now determine the moduli space of these theories. A complete analysis of themoduli space of these theories is given in the appendix. Here we will motivate the resultby a shortcut, starting with the moduli space of the parent N = 6 superconformal theory.The latter is given by ( C / Z k ) ˜ N / S ˜ N , where the Z k is generated by the transformation C I → e πi/k C I . It is easy to see that the symmetry group of (3.2) is Z . This symmetrydoes not commute with the Z k symmetry acting on the moduli space of the originaltheory. In fact, the square of the Z generator acts trivially on the gauge fields, and on thematter fields it gives C I → − C I , which corresponds precisely to the generator of the Z k symmetry raised to the k ’th power. This is precisely the definition of the binary dihedralgroup b D k (we are using the convention that b D k is the binary dihedral group with 4 k elements). The simplest one is b D = Z , and the higher k groups are non-Abelian. Thus,we expect the moduli space after the projection to be ( C / b D k ) N /S N , and we verify in theappendix that this is indeed the case.Based on this moduli space, it is natural to conjecture that our theories are relatedto the theory of N M2-branes probing the orbifold C / b D k in M theory. This orbifoldpreserves precisely N = 5 supersymmetry [28], which is consistent with this conjecture.This conjecture implies that the dual M theory background is AdS × S / b D k . We alsoexpect a regime of the parameters N and k in which there is a weakly curved type IIAstring theory description. We will discuss both gravitational duals below.But first, let us return to the implications of the “s-rule” and brane creation effect,namely the parity-duality relations (3.4), and the maximal values of l (3.5). We will usesimilar arguments to the ones we used for the U ( N + l ) × U ( N ) theories as further evidencefor these quantum properties of the orientifold theories.We start by considering a generic point on the moduli space. This leaves at lowenergies a pure N = 3 supersymmetric Chern-Simons theory with a gauge group thatdepends on l . We can then integrate out the gauginos to get an effective bosonic Chern-Simons theory with a shifted level, where the shift is given by the dual Coxeter numberof the gauge group, k → k − h · sign( k ) [14] . The dual Coxeter numbers of the groups O (2 l ), O (2 l + 1) and U Sp (2 l ) are given, respectively, by h = 2 l −
2, 2 l − l + 1. We Note that one could also consider N = 5 supersymmetric theories with the gauge group SO ( M ) × U Sp (2 N ), but we do not obtain these theories from our brane construction or from theprojection of the unitary theories, and we will not consider them in this paper. For the specialcase of SO (2) × U Sp (2 N ) these theories have an enhanced N = 6 supersymmetry [8,27]. In [14] the shift was by − h · sign( k ) for the N = 1 theory with one gaugino. In the N = 3theory there are four gauginos, three of which contribute with the same sign, and the fourth withthe opposite sign. l , as in the previous section, by requiring thatthe level in the effective bosonic CS theory should have the same sign as the original level,so that the one-loop shift of the level in this theory will bring us back to the original level k . This gives 0 ≤ l I ≤ k + 1, 0 ≤ l II ≤ k − ≤ l III ≤ k , for the class I, II and IIItheories, respectively, precisely as predicted above.The class IV theories are a bit more subtle, because on their moduli space we obtain a U Sp (2 l IV ) × O (1) gauge theory; the O (1) theory does not have any gauge fields, but it leadsto massless matter fields in the fundamental of U Sp (2 l IV ). Thus, we cannot map thesetheories directly to bosonic CS theories. We believe that these superconformal theoriesexist precisely for 0 ≤ l IV ≤ k , and it would be interesting to verify this directly.Our main direct evidence for the parity-dualities in (3.4) comes (as in the unitarycase) from level-rank duality between the two effective bosonic Chern-Simons theories onthe moduli space. For example, the I k ( l ) theory gives at a generic point on the modulispace an N = 3 O (2 l ) k Chern-Simons theory, which is equivalent to a bosonic O (2 l ) k − l +2 Chern-Simons theory, while the I − k ( k − l +1) theory on the moduli space includes an N = 3 O (2 k − l + 2) − k theory, which is equivalent to a bosonic O (2 k − l + 2) − l theory. Thetwo WZW theories that we obtain from these in the presence of boundaries are relatedby level-rank duality, which supports the first of the four conjectured dualities in (3.4). Asimilar test of the II k ( l ) and III k ( l ) dualities confirms their consistency as well. For theclass IV theories it is again more difficult to analyze this directly because of the presenceof extra massless fields, so we do not have direct field theory arguments for the duality inthis case. The configuration of D3-branes, O3-planes, and fivebranes described above can beT-dualized and lifted to M theory as in [5], resulting in M2-branes probing a singularityin a toric hyperK¨ahler 8-manifold, which preserves 3 /
16 of the supersymmetry [29]. Thisgeometry has a T formed from the T-dualized circle and the M theory circle, fibred overa six dimensional base. The IR limit of the field theory on the D3-branes that we derivedcorresponds to taking the size of the wrapped circle small in IIB, and hence large in the T-dual picture. Further, the low energy limit of the D2-branes brings us to strong coupling, sothe superconformal field theory describes the limit of large T in the hyperK¨ahler geometry.In [5], the hyperK¨ahler geometry resulting from the theory with one NS5-brane andone (1 , k ) 5-brane was described explicitly, and its IR limit was shown to be a C / Z k singularity. The hyperK¨ahler geometries were studied in [29], and have a general formrelated to a product of two Taub-NUT spaces, ds = U ij d~x i · d~x j + U ij ( dϕ i + A i )( dϕ j + A j ) ,A i = d~x j · ~ω ji = dx ja ω aji , ∂ x ja ω bki − ∂ x kb ω aji = ǫ abc ∂ x jc U ki , (3 . U ij is a two by two symmetric matrix of harmonic functions and U ij is its inverse.In our case there will be an additional Z quotient due to the lift of the orientifold.Suppose we first consider the T-dual and lift of the O3-plane without any 5-branes.Then, in M theory the geometry would simply be a Z quotient , acting on the coordinatesabove as ~x i → − ~x i , and ϕ i → − ϕ i , where ~x i describe a flat R , and ϕ i describes the fiber T . Re-introducing the fivebranes, the torus fiber will shrink along 3-planes in the base.However, far from these centers of the Taub-NUT spaces, the geometry is not alteredsignificantly, and one sees that the Z action must be the same as above.The particular geometry obtained from an NS5-brane and a (1 , k ) 5-brane is describedby U = + (cid:18) h
00 0 (cid:19) + (cid:18) h kh kh k h (cid:19) , h = 12 | ~x + 2 k~x | , h = 12 | ~x | , (3 . C / Z k , as explained in [5]. Note that near theorigin of the hyperK¨ahler 8-manifold, one obtains C / Z k using the fact that the centerof each Taub-NUT geometry looks just like a flat R space, naturally written in polarcoordinates with a Hopf-fibred angular S .More explicitly, the Hopf fibration defines a map f : C → S which can be writtenas f ( z , z ) = (cid:0) z z ∗ ) , z z ∗ ) , | z | − | z | (cid:1) , where the S is regarded as the unitsphere in R , the base of the Taub-NUT. The orientifold Z then acts as the antipodalmap on the S , which can be seen to lift to the action z → iz ∗ , z → − iz ∗ on C , notingthat the overall phase of z , z is exactly the Hopf circle, which is also inverted, ϕ → − ϕ ,by the lift of the orientifold. This Z also acts simultaneously on the other C in the eightdimensional geometry.Therefore we conclude that the transverse geometry to the M2-branes includes a C / b D k singularity (which will control the IR field theory). The Z k subgroup of b D k is theorbifold obtained from the 5-branes, acting as ( z , z , z , z ) → e πik ( z , z , z , z ), while theother generator acts by ( z , z , z , z ) → ( iz ∗ , − iz ∗ , iz ∗ , − iz ∗ ) . (3 . Z transformation on the C / Z k orbifold, but of course it gives a Z inthe C covering space.It is not hard to check that this is identical to the C / b D k singularity mentioned in[28], in different coordinates. One sees that the b D k sits inside an SU (2) subgroup of the SO (8) rotation group of R , and thus its commutant is the SO (5) R R-symmetry of this N = 5 orbifold. More precisely, we obtain four such quotients on four different points on the T , but we willfocus here on the vicinity of one of these four orbifold points.
19n the special case of k = 1, the orbifold is simply C / Z , which is unique upto coordinate redefinitions. Therefore the O × U Sp theories actually preserve N = 6supersymmetry for k = 1, and they should be quantum mechanically dual to the N = 6 Chern-Simons-matter theories with unitary gauge groups (with the same min-imal rank) at level k = 4. In particular, the four O × U Sp theories in this caseare O (2 N ) × U Sp (2 N ) − , O (2 N + 2) × U Sp (2 N ) − , O (2 N + 1) × U Sp (2 N ) − , and U Sp (2 N ) × O (2 N + 1) − . The first two should then presumably be equivalent to the U ( N ) × U ( N ) − and U ( N + 2) × U ( N ) − theories (though we cannot say which towhich), and the last two to U ( N + 1) × U ( N ) − and U ( N + 3) × U ( N ) − . Of course,these theories are strongly coupled, so it is difficult to check these identifications, but onecan compare their spectrum of chiral operators, their moduli spaces, and so on. As asimple consistency check, note that the parity-dualities (3.4), (3.6) imply that the first two O × U Sp theories mentioned above map to themselves by a parity transformation. Thisis obviously true also for the U ( N ) × U ( N ) − theory, and it turns out to also be truefor the U ( N + 2) × U ( N ) − theory, using the equivalences (2.1). Similarly, our last two O × U Sp theories map to each other under parity, and using (2.1) we can check that thesame is true for the U ( N + 1) × U ( N ) − and U ( N + 3) × U ( N ) − theories. From the M theory lift above, it is easy to construct the M theory background dualto our orientifolded field theories (3.1). This background is simply
AdS × S / b D k , with N units of 4-form flux in AdS : ds = R ds AdS + R ds S / ˆ D k ,G ∼ N ǫ , (3 . ǫ is the unit volume form on AdS . This is obviously consistent with the modulispace of these field theories which we described above. The spectrum of chiral operatorsin this background was recently computed in [30]. The eleven dimensional supergravityapproximation is valid for k ≪ N / , as before.It is natural to describe the S / b D k as a Z quotient of the S fibred over CP thatwe had in the S / Z k case. This is equivalent to a twisted circle bundle over the non-orientable 6-manifold CP / Z (where by twisted we mean that the curvature of the circlebundle lives in the Z -twisted cohomology).The geometry described above has a Z k torsion 3-cycle; generally, the discrete torsionin M theory when we divide by a discrete group with | Γ | elements is Z | Γ | [31]. One way todescribe this cycle explicitly is as the twisted S fibration over an RP in the base. Thiscycle is precisely half the torsion cycle in S / Z k , given by the S fibred over a CP . In20 theory, one can thus turn on a discrete torsion G flux, or in other words, a flat C fieldwith holonomy in Z k ⊂ U (1) on that torsion 3-cycle. Explicitly, C ∝ l k J ∧ dϕ, (3 . S / b D k , with l = 0 , · · · , k −
1; this is invariant under the identifications describedabove. Thus, we have 4 k different field theories, just as expected from our discussionabove. Upon reduction to type IIA string theory, we will see precisely how this discretetorsion matches with the 4 k orientifold field theory duals found earlier. When N ≪ k the M theory circle becomes small, and it is more appropriate todescribe the gravitational dual using type IIA string theory (which is weakly curved for N ≫ k ). The reduction to type IIA is very similar to the one discussed in the previoussection, (2.5) (with 2 k replacing k everywhere). The only difference is the additional Z identification which we have to take into account. This identification acts on the naturalprojective coordinates of CP by (3.9). It is easy to check that this maps J → − J , so thatthe orientation of CP (and, thus, of the full type IIA background) is reversed. Thus, ourtype IIA background is an orientifold of AdS × CP (the orientifold has no fixed pointsso this is a smooth manifold). In M theory, the Z acts on the CP as above, togetherwith an inversion of the coordinate on the M theory circle (while not acting on the 3-formfield). Thus, in type IIA string theory, beyond its geometric action described above, theorientifold inverts the RR 1-form C and the NSNS 2-form B , while leaving invariant theRR 3-form C . Note that since F → − F and J → − J , this is consistent with the RR2-form flux F = 2 kJ (which is invariant under the Z ). One can think of this as k unitsof RR 2-form flux on the cycle CP / Z that we obtain by performing the identification(3.9) on the CP ⊂ CP . Note that this identification maps the CP (defined, say, by z = z = 0) to itself, so that we obtain a smaller 2-cycle, but it does not map the CP cycle (defined, say, by z = 0) to itself. Thus, the minimal 4-cycle in the orientifoldedbackground is still CP .Both U (1) gauge fields which are present on AdS × CP before the orientifold, areprojected out by the orientifold (one of them originally became massive by swallowing anaxion [5], but this axion is also projected out by the orientifold). The projection on the SU (4) gauge fields leaves an SO (5) subgroup invariant, so that the gauge group on AdS agrees with the global symmetry group of our field theories.Next, let us analyze the reduction of discrete torsion in this picture. The fluxes f and b described in the previous section are invariant under the Z . As in the previous sectionwe can define f ∝ R CP ˜ F (normalized to be an integer in the absence of 2-form flux),21ut we can now define a smaller 2-form holonomy ˜ b = π R CP / Z B (which is periodicwith period one). The same arguments as in the previous section imply that the quantizedfour-form flux is f + 4 k ˜ b (since we have 2 k units of F flux on CP , and ˜ b = b ). Thus,we have 4 k possibilities for the NSNS holonomy (when ˜ F = 0), given by ˜ b = l/ k with l = 0 , · · · , k − I k ( l ) theories are related byadding fractional branes (each of which shifts ˜ b by k ) makes it natural to identify the I k ( l ) theory with the type IIA background with ˜ b = l k . However, the fractional branepicture only tells us the differences between the values of ˜ b between the different theories,and not their absolute values, so it allows for an identification of the form ˜ b = l + c k with some (half)-integer c . The III k ( l ) theory is then related to this by adding another“half of a fractional brane” (in the brane construction), so it is natural to identify it withthe background with ˜ b = l +2 c +14 k . Using the equivalence relations (3.6), we can thendetermine the identifications of the class II and class IV theories. Our full identificationsare thus summarized by: I k ( l ) ↔ ˜ b = l + c k ,II k ( l ) ↔ ˜ b = l + c + k + 12 k ,III k ( l ) ↔ ˜ b = l + c + k ,IV k ( l ) ↔ ˜ b = l + c + k + k . (3 . c we have a one-to-one map between our space of theories and the spaceof type IIA backgrounds. In order to determine c , we require consistency with the parity-duality identifications of (3.4). Recall that the parity transformation of type IIA takes˜ b → − ˜ b = 1 − ˜ b , and we conjectured that it acts on our theory space as (3.4). All ofthese identifications turn out to be consistent for precisely two values of c : c = − k +12 or c = k − (mod 2 k ). Note that the NSNS holonomy encodes in the language of our originaltype IIB brane construction both the number of fractional branes and the NSNS and RRtorsion fluxes, as a result of the chain of dualities that we performed. The type IIA orientifold may also admit RR torsion corresponding to a discrete holonomyof the RR 3-form C on a torsion 3-cycle in CP / Z ( H ( CP / Z , Z ) = Z ). Since we have noroom for an extra quantum number associated with this, we expect that this torsion should becorrelated with the parity of 4 k ˜ b , that distinguishes the class I and II theories from the class IIIand IV theories (and is related to the RR torsion in our brane construction). This correlationmay be seen in M theory by noting that the torsion 3-cycle lifts to the same torsion cycle in Mtheory as CP / Z . It would be interesting to derive this correlation directly in type IIA string
22t is not clear how to determine which of the two possibilities for c is correct. Note thatfor both possibilities, the I k (0) theory does not map to ˜ b = 0 as one might have naivelyexpected; presumably the orientifold itself induces some discrete torsion in our background,even in the absence of any fractional branes. This is consistent with the fact that the I k (0)theory is not parity-invariant (for k > c . Since we have no massless U (1) fields,the di-baryon operators (coming from D4-branes wrapped on CP ) are not independentof the “meson”-type operators, in agreement with our gauge theory expectations. Thus,these operators do not seem to give any clues about the correct identifications.It would be interesting to analyze the mass deformations of these theories (analogousto [12]) and to try to obtain a correct count of the number of vacua in these theories. Itwould also be interesting to see if analyzing the operators in this theory in the ’t Hooftlarge N limit leads to interesting integrable spin chains, either on the field theory or onthe string theory side. Of course, at the leading (planar) order the theories of this sectionare identical to those of the previous section. The leading difference between them comesfrom non-orientable diagrams at the next order in 1 /N . Acknowledgments
We would like to thank Juan Maldacena for collaboration on the first part of thispaper, and for many useful discussions and comments. We would like to thank JohnSchwarz for comments on an early version of the paper. We would also like to thankDavide Gaiotto, Sergei Gukov, Dario Martelli, Alfred Shapere, Yuji Tachikawa, Mark VanRaamsdonk, and Edward Witten for useful discussions. The work of OA was supported inpart by the Israel-U.S. Binational Science Foundation, by a center of excellence supportedby the Israel Science Foundation (grant number 1468/06), by a grant (DIP H52) of theGerman Israel Project Cooperation, by the European network MRTN-CT-2004-512194,and by Minerva. OA would like to thank the “Monsoon workshop on string theory” atMumbai, India, for hospitality during the course of this work. OB gratefully acknowledgessupport from the Institute for Advanced Study. The work of OB was supported in partby the Israel Science Foundation under grant no. 568/05. The work of DJ was supportedin part by DOE grant DE-FG02-96ER40949. DJ would like to thank the Simons Centerfor Geometry and Physics at Stony Brook and the 2008 Simons Workshop in Mathematicsand Physics for kind hospitality during the course of this work. theory by a careful analysis of the flux quantization conditions in our background. ppendix A. Moduli spaces of the orientifold field theories In this appendix we compute the classical moduli spaces of the O (2 M ) k × U Sp (2 N ) − k and O (2 M + 1) k × U Sp (2 N ) − k field theories discussed above; these are expected to beexact quantum mechanically due to the considerable amount of supersymmetry. Recall thatthe moduli space of the U ( N ) × U ( M ) Chern-Simons-matter theory consists of diagonalmatrices [5], so one might expect the same here. However, the orientifold projection (3.2)does not preserve the diagonal form due to the presence of J . This is reflected in the factthat it is impossible to use the projected O × U Sp gauge symmetry to diagonalize a genericmatrix in the bi-fundamental, in contrast to the unitary group case.We will choose J to be made of 2 × − iσ ). Using thegauge symmetry, it is then possible to bring the vacuum expectation value of one of thecomplex bi-fundamental matter fields, A , into a 2 × A must also have this form. Within each block we thenjust need to determine the moduli space of the O (2) × U Sp (2) theory. In the first part ofthis appendix we show that the moduli space of this theory is C / b D k , on which the gaugegroup is generically broken to a U (1), in agreement with our interpretation of this theoryas describing a single M2-brane in that geometry; moreover, that is the result expected byprojecting the moduli space of the U (2 N ) × U (2 N ) theory found in [5].Therefore, the moduli space of the O ( M ) k × U Sp (2 N ) − k theories for M ≥ N is givenby N copies of the O (2) × U Sp (2) moduli space, quotiented by the unbroken permutationsymmetry S N ⊂ O (2 N ) × U Sp (2 N ), namely it is equal to Sym N (cid:16) C / b D k (cid:17) . The low-energy effective theory on this moduli space includes a residual pure N = 3 Chern-Simonstheory O ( M − N ) k .For the U Sp (2 M ) k × O (2 N ) − k theories with M ≥ N the moduli space is similarlygiven by Sym N (cid:16) C / b D k (cid:17) , where now the low-energy effective action on the moduli spaceincludes an N = 3 supersymmetric U Sp (2 M − N ) k pure Chern-Simons theory. Thesituation for the U Sp (2 M ) k × O (2 N + 1) − k theories is similar, except that the low-energyeffective theory on the moduli space now includes an N = 3 supersymmetric Chern-Simons-matter theory with gauge group U Sp (2 M − N ) k × O (1) − k , which has massless matter inaddition to the N = 3 Chern-Simons terms. In the second part of this appendix we showthat there are no additional moduli associated to this residual theory, so it does not affectthe moduli space. A.1. The moduli space of the O (2) k × U Sp (2) − k theory We begin by gauge fixing the gauge fields to zero, so that the moduli space is givenby the zero locus of the bosonic potential for the two bi-fundamental scalar fields A , A (which we write as 2 × | ∂W | (with W given by (3.3)), and from the σ -termsin the supersymmetric kinetic terms. These terms are proportional to X a =1 tr( | σ O A a − A a σ USp | ) , (A.1)where the scalar fields σ in the vector multiplet are equal (up to a constant that we willignore) to the moment maps, σ O = X a =1 ( A a A † a − A ∗ a A Ta ) , σ USp = X a =1 ( A † a A a + J A Ta A ∗ a J ) , (A.2)after integrating out the auxiliary fields in the vector multiplets. Their form can either bederived directly, or by using the projection from the unitary case.It is easy to see that both the F-term equations A T A J A T = A T A J A T , A T A J A T = A T A J A T , (A.3)and the equations coming from (A.1) are satisfied when [ J, A ] = [ J, A ] = 0, since in thatcase the matrices take the form A = x + yJ, A = z + wJ, (A.4)so that the matrices (and their conjugates and transposes) all commute with each other.The scalar fields in the vector multiplets are then equal to σ O (2) = 2( yx ∗ − xy ∗ + wz ∗ − zw ∗ ) J = σ USp (2) , (A.5)so it is easy to check that (A.1) vanishes. Therefore, the matrices (A.4) are on the modulispace for any complex numbers x, y, z, w .Generically, on the moduli space, the gauge symmetry is broken to a diagonal U (1),acting by an O (2) transformation on the left, and by the inverse transformation (which iscontained in U Sp (2)) on the right. The unbroken gauge field is thus A + = A O (2) + A ,where we define A USp (2) = P j =1 σ j A j . The other three generators of the gauge groupbecome massive. However, the off-diagonal combination A − = A O (2) − A acts on themoduli space, that is, it preserves the form (A.4). This rotation acts on the componentsas (cid:18) xy (cid:19) → (cid:18) cos φ − sin φ sin φ cos φ (cid:19) (cid:18) xy (cid:19) , (A.6)and similarly for z and w . It is convenient to define new coordinates on the moduli space u = Re( x ) + i Re( y ), u = Im( x ) + i Im( y ), u = Re( z ) + i Re( w ), u = Im( z ) + i Im( w ), sothat this broken U (1) acts by multiplying all the u I ( I = 1 , , ,
4) by e iφ .25ext, we need to take into account the effect of constant gauge transformations (whichleave the gauge fields vanishing) acting on the moduli space. Since effectively we have a U (1) × U (1) theory, this analysis is precisely the same as in [5]. The only difference isthat the level of the U (1) Chern-Simons terms is now 2 k instead of k as in [5]; this is clearfor the O (2) k piece, and one can show that it is also true for the U (1) in U Sp (2) − k byconsidering the embedding of U (1) in U Sp (2). Thus, by a similar analysis to [5] (eitherby considering which gauge transformations leave the CS terms invariant, or by dualizingthe gauge field A + to a scalar), we find that the U (1) action leads to a Z k identification,where the Z k acts as u I → exp( πin/k ) u I (for n ∈ Z ).This takes into account the connected part of the gauge group. However, O (2) has anadditional disconnected component (involving matrices with determinant ( − Z identification of the moduli space. In particular, we can considera gauge transformation A → (cid:18) (cid:19) A (cid:18) i − i (cid:19) = (cid:18) − iy − ixix − iy (cid:19) , (A.7)(and similarly for A ). This preserves the form (A.4), and acts on the coordinates definedabove as ( u , u , u , u ) → ( iu ∗ , − iu ∗ , iu ∗ , − iu ∗ ). This is exactly the orientifold action wediscussed in section 3, which squares to u I → − u I which is an element of the Z k thatwe already identified. Thus, after imposing all the identifications, the moduli space isprecisely C / b D k . A.2. The moduli space of the residual
U Sp (2 l ) k × O (1) − k theories In these theories we can think of the massless matter fields simply as fundamentals of
U Sp (2 l ), since the O (1) gauge group is just a discrete gauged Z . It is enough to analyzethe case of l = 1, since the analysis for general l will just be l copies of this.The F-term equations imply that A T A J A T = A T A J A T , but since A is just avector we have that A J A T = 0 for any A . Thus A J A T = 0 (or A = 0), and similarly A J A T = 0 (or A = 0). We conclude that A and A must be proportional to each other, A = αA for some α ∈ C .It is obvious that σ O (1) = 0, since there is no O (1) Lie algebra, while σ USp is equal to σ USp = A † A + J A T A ∗ J + A † A + J A T A ∗ J . The constraint from the σ -term of the firstfield is then 0 = A σ USp = A A † A + A A † A = (1 + | α | )( A A † ) A , (A.8)since A is proportional to A , and A J A T = 0. This is impossible unless A = A = 0,since (1 + | α | )( A A † ) is positive. Thus, the only classical solution to all the constraintsis A = A = 0. 26e conclude that there is no moduli space associated to this residual theory, in spiteof the presence of massless fundamentals at its unique classical vacuum.27 eferences [1] J. Bagger and N. Lambert, “Comments On Multiple M2-branes,” JHEP ,105 (2008) [arXiv:0712.3738 [hep-th]]; J. Bagger and N. Lambert, “Gauge Symme-try and Supersymmetry of Multiple M2-Branes,” Phys. Rev. D , 065008 (2008)[arXiv:0711.0955 [hep-th]]; J. Bagger and N. Lambert, “Modeling multiple M2’s,”Phys. Rev. D , 045020 (2007) [arXiv:hep-th/0611108].[2] J. H. Schwarz, “Superconformal Chern-Simons theories,” JHEP , 078 (2004)[arXiv:hep-th/0411077].[3] A. Basu and J. A. Harvey, “The M2-M5 brane system and a generalized Nahm’sequation,” Nucl. Phys. B , 136 (2005) [arXiv:hep-th/0412310].[4] A. Gustavsson, “Algebraic structures on parallel M2-branes,” arXiv:0709.1260 [hep-th]; A. Gustavsson, “Selfdual strings and loop space Nahm equations,” JHEP ,083 (2008) [arXiv:0802.3456 [hep-th]].[5] O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, “N=6 superconformalChern-Simons-matter theories, M2-branes and their gravity duals,” arXiv:0806.1218[hep-th].[6] D. Gaiotto and E. Witten, “Janus Configurations, Chern-Simons Couplings, And TheTheta-Angle in N=4 Super Yang-Mills Theory,” arXiv:0804.2907 [hep-th].[7] K. Hosomichi, K. M. Lee, S. Lee, S. Lee and J. Park, “N=4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets,” arXiv:0805.3662 [hep-th].[8] K. Hosomichi, K. M. Lee, S. Lee, S. Lee and J. Park, “N=5,6 Superconformal Chern-Simons Theories and M2-branes on Orbifolds,” arXiv:0806.4977 [hep-th].[9] J. A. Minahan and K. Zarembo, “The Bethe ansatz for superconformal Chern-Simons,” arXiv:0806.3951 [hep-th]; D. Gaiotto, S. Giombi and X. Yin, “Spin Chainsin N=6 Superconformal Chern-Simons-Matter Theory,” arXiv:0806.4589 [hep-th];D. Bak and S. J. Rey, “Integrable Spin Chain in Superconformal Chern-Simons The-ory,” arXiv:0807.2063 [hep-th].[10] G. Arutyunov and S. Frolov, “Superstrings on AdS x CP as a Coset Sigma-model,”arXiv:0806.4940 [hep-th]; B. Stefanski, “Green-Schwarz action for Type IIA strings on AdS × CP ,” arXiv:0806.4948 [hep-th]; G. Grignani, T. Harmark and M. Orselli, “TheSU(2) x SU(2) sector in the string dual of N=6 superconformal Chern-Simons theory,”arXiv:0806.4959 [hep-th]; G. Grignani, T. Harmark, M. Orselli and G. W. Semenoff,“Finite size Giant Magnons in the string dual of N=6 superconformal Chern-Simonstheory,” arXiv:0807.0205 [hep-th]; N. Gromov and P. Vieira, “The AdS4/CFT3 alge-braic curve,” arXiv:0807.0437 [hep-th]; C. Ahn and P. Bozhilov, “Finite-size effectsof Membranes on AdS × S ,” arXiv:0807.0566 [hep-th]; N. Gromov and P. Vieira,“The all loop AdS4/CFT3 Bethe ansatz,” arXiv:0807.0777 [hep-th]; B. Chen and28. B. Wu, “Semi-classical strings in AdS * CP ,” arXiv:0807.0802 [hep-th]; D. Astolfi,V. G. M. Puletti, G. Grignani, T. Harmark and M. Orselli, “Finite-size corrections inthe SU(2) x SU(2) sector of type IIA string theory on AdS x CP ,” arXiv:0807.1527[hep-th]; C. Ahn and R. I. Nepomechie, “N=6 super Chern-Simons theory S-matrixand all-loop Bethe ansatz equations,” arXiv:0807.1924 [hep-th]; B. H. Lee, K. L. Pani-grahi and C. Park, “Spiky Strings on AdS × CP ,” arXiv:0807.2559 [hep-th]; C. Ahn,P. Bozhilov and R. C. Rashkov, “Neumann-Rosochatius integrable system for stringson AdS × CP ,” arXiv:0807.3134 [hep-th]; T. McLoughlin and R. Roiban, “Spinningstrings at one-loop in AdS × P ,” arXiv:0807.3965 [hep-th]; L. F. Alday, G. Aru-tyunov and D. Bykov, “Semiclassical Quantization of Spinning Strings in AdS × CP ,”arXiv:0807.4400 [hep-th]; C. Krishnan, “ AdS /CF T at one loop,” arXiv:0807.4561[hep-th].[11] T. Kitao, K. Ohta and N. Ohta, “Three-dimensional gauge dynamics from braneconfigurations with (p,q)-fivebrane,” Nucl. Phys. B , 79 (1999) [arXiv:hep-th/9808111]; O. Bergman, A. Hanany, A. Karch and B. Kol, “Branes and supersymme-try breaking in 3D gauge theories,” JHEP , 036 (1999) [arXiv:hep-th/9908075];K. Ohta, “Supersymmetric index and s-rule for type IIB branes,” JHEP , 006(1999) [arXiv:hep-th/9908120].[12] J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk and H. Verlinde, “A MassiveStudy of M2-brane Proposals,” arXiv:0807.1074 [hep-th].[13] A. Hanany and E. Witten, “type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics,” Nucl. Phys. B , 152 (1997) [arXiv:hep-th/9611230].[14] E. Witten, “Supersymmetric index of three-dimensional gauge theory,” arXiv:hep-th/9903005.[15] H. C. Kao, K. M. Lee and T. Lee, “The Chern-Simons coefficient in supersymmet-ric Yang-Mills Chern-Simons theories,” Phys. Lett. B , 94 (1996) [arXiv:hep-th/9506170].[16] E. Witten, “SL(2,Z) action on three-dimensional conformal field theories with Abeliansymmetry,” arXiv:hep-th/0307041.[17] E. R. Sharpe, “Analogues of discrete torsion for the M theory three-form,” Phys. Rev.D , 126004 (2003) [arXiv:hep-th/0008170].[18] S. Sethi, “A relation between N = 8 gauge theories in three dimensions,” JHEP ,003 (1998) [arXiv:hep-th/9809162].[19] A. Hanany and B. Kol, “On orientifolds, discrete torsion, branes and M theory,” JHEP , 013 (2000) [arXiv:hep-th/0003025].[20] J. Distler, S. Mukhi, C. Papageorgakis and M. Van Raamsdonk, “M2-branes on M-folds,” JHEP , 038 (2008) [arXiv:0804.1256 [hep-th]].[21] N. Lambert and D. Tong, “Membranes on an Orbifold,” arXiv:0804.1114 [hep-th].[22] E. Witten, “Dyons Of Charge E Theta/2 Pi,” Phys. Lett. B , 283 (1979).2923] A. Armoni and A. Naqvi, “A Non-Supersymmetric Large-N 3D CFT And Its GravityDual,” arXiv:0806.4068 [hep-th].[24] E. Witten, “Baryons and branes in anti de Sitter space,” JHEP , 006 (1998)[arXiv:hep-th/9805112].[25] N. J. Evans, C. V. Johnson and A. D. Shapere, “Orientifolds, branes, and duality of4D gauge theories,” Nucl. Phys. B , 251 (1997) [arXiv:hep-th/9703210].[26] K. A. Intriligator and N. Seiberg, “Duality, monopoles, dyons, confinement and obliqueconfinement in supersymmetric SO(N(c)) gauge theories,” Nucl. Phys. B , 125(1995) [arXiv:hep-th/9503179].[27] M. Schnabl and Y. Tachikawa, “Classification of N=6 superconformal theories ofABJM type,” arXiv:0807.1102 [hep-th].[28] D. R. Morrison and M. R. Plesser, “Non-spherical horizons. I,” Adv. Theor. Math.Phys. , 1 (1999) [arXiv:hep-th/9810201].[29] J. P. Gauntlett, G. W. Gibbons, G. Papadopoulos and P. K. Townsend, “Hyper-Kaehler manifolds and multiply intersecting branes,” Nucl. Phys. B , 133 (1997)[arXiv:hep-th/9702202].[30] A. Hanany, N. Mekareeya and A. Zaffaroni, “Partition Functions for Membrane The-ories,” arXiv:0806.4212 [hep-th].[31] J. de Boer, R. Dijkgraaf, K. Hori, A. Keurentjes, J. Morgan, D. R. Morrison andS. Sethi, “Triples, fluxes, and strings,” Adv. Theor. Math. Phys.4