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Nuclear Physics B 835 (2010) 1-28

www.elsevier.com/locate/nuclphysb

A stringy origin of M2 brane Chern-Simons theories

Mina Aganagic

Center for Theoretical Physics, University of California, Berkeley, CA 94720, United States Received 16 July 2009; received in revised form 8 October 2009; accepted 8 January 2010 Available online 15 January 2010

Abstract

We show that string duality relates M-theory on a local Calabi-Yau fourfold singularity X4 to type IIA string theory on a Calabi-Yau threefold X3 fibered over a real line, with RR 2-form fluxes turned on. The RR flux encodes how the M-theory circle is fibered over the IIA geometry. The theories on N D2 branes probing X3 are the well-known quiver theories with N = 2 supersymmetry in three dimensions. We show that turning on fluxes, and fibering the X3 over a direction transverse to the branes, corresponds to turning on N = 2 Chern-Simons couplings. String duality implies that, in the strong coupling limit, the N D2 branes on X3 in this background become N M2 branes on X4. This provides a string theory derivation for the recently conjectured description of the M2 brane theories on Calabi-Yau fourfolds in terms of N = 2 quiver Chern-Simons theories. We also provide a new N = 2 Chern-Simons theory dual to AdS4 x Q1'1'1. Type IIA/M-theory duality also relates IIA string theory on X3 with only the RR fluxes turned on, to M-theory on a G2 holonomy manifold. We show that this implies that the N M2 branes probing the G2 manifold are described by the quiver Chern-Simons theory originating from the D2 branes probing X3, except that now Chern-Simons terms preserve only N = 1 supersymmetry in three dimensions. © 2010 Elsevier B.V. All rights reserved.

1. Introduction

Duality provides powerful tools to understand string theory. AdS/CFT correspondence defines quantum gravity in AdS backgrounds in terms of the dual conformal field theory. At the same time, gravity on AdS space describes the conformal theory at large N . Similarly IIA/M-theory duality provides a way to understand the strong coupling limit of type IIA string theory in terms of 11-dimensional supergravity. Conversely, in compactifications of M-theory on a circle, IIA string theory defines the still mysterious quantum theory that underlies M-theory.

E-mail address: mina@math.berkeley.edu.

0550-3213/$ - see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2010.01.007

One of the mysteries of M-theory is how to describe the theory on N coincident M2 branes. At low energies, this should be the conformal field theory dual to M-theory on AdS4 x M7 where M7 is seven-dimensional Einstein manifold [1]. The theories on N M2 branes in M-theory are dual to the theory on N D2 branes in IIA. The latter theory is a gauge theory, with the dimensionfull coupling constant proportional to gs. In the strong coupling limit, the D2 brane theory should describe M2 branes. However, the resulting theory is apparently always strongly coupled, so the description does not seem useful.

A way around this was proposed in [2]. The idea is that, at low energies, the theory on the M2 branes may be a Chern-Simons theory, with matter. The Chern-Simons level provides a dimensionless coupling constant needed to define the gauge theory. Moreover, Chern-Simons interaction is scale invariant, being independent of the metric, so the theory could also be con-formal. Recently, [3] constructed a Chern-Simons theory corresponding to N M2 branes on Calabi-Yau fourfold X4 = C4/Zk, where k gets identified with the level. This theory is conformal, and dual, at large N to M-theory on AdS4 x S1 /Zk. Moreover, for k = 1 it should correspond to M2 branes on flat space, with maximal supersymmetry in three dimensions. Various authors [4-8] proposed generalizations of this construction to Chern-Simons theories describing M2 branes on local Calabi-Yau fourfolds. A check of these proposals is that for a single M2 brane, the moduli space is indeed the Calabi-Yau fourfold X4. However, in general, there is more than one such Chern-Simons theory that one can write down which would still give the same moduli space [9-11]. It is possible that there are dualities relating some of these theories, but we cannot a-priori exclude that they flow to different theories at long distance.

The aim of this paper is to provide a stringy derivation of the Chern-Simons theory on N M2 branes probing a local Calabi-Yau fourfold X4.1 To begin with, we show that, M-theory on a local toric Calabi-Yau fourfold X4 is dual to IIA string theory on a Calabi-Yau threefold X3 fibered over R. The Kahler moduli of X3 vary over R and moreover, there are RR 2-form fluxes on X3 x R. This is a consequence of a purely geometrical observation: The toric Calabi-Yau fourfold X4 is a circle fibration over X3 x R. Moreover, the viewing the coordinate on the S1 as a section of a U(1) bundle on X3 x R the curvature of the corresponding connection is easy to compute. When we compactify M-theory on X4, the IIA/M theory duality relates it to IIA on X3 x R, where the curvature of the U(1) bundle gets identified with the RR 2-form flux in IIA.

The theory on N D2 branes in IIA probing the local Calabi-Yau threefold X3 is well known, in the limit where X3 develops a singularity. These are N = 2 quiver gauge theories in three dimensions, with superpotential. More precisely, the theories on Dp branes probing X3 were studied extensively for D3 branes and DO branes.2 However, given X3, the theories on N D-branes probing it are the same, at least classically, in any dimension — differing only by compactifications on tori, and dimensional reduction. This is because the theory on the branes is determined by internal CFT on a disk — and the answers can be formulated in terms of the geometry of X3 and fractional branes wrapping it.

1 Such a derivation was proposed in [3,6] for the theories studied there. While the string realizations are dual to each other, our proposal is simpler, and the relevant string backgrounds are better understood. Moreover, we provide a unified derivation, applicable for a large class of toric Calabi-Yau singularities.

2 The former were studied as examples of conformal field theories in four dimensions with four supercharges (see for

example [12-15]) while the latter were studied in the context of 4d N = 2 black holes (see for example [16,11]). Detailed derivation of the quiver theory from mirror symmetry [18] and the classical geometry of intersecting three-cycles was presented in [12]. An alternative derivations starting from the theory of branes on orbifolds c3/zk x z m, which have a free worldsheet CFT description and following its deformations was developed in [19] and many followup works.

Next, we consider deforming the theory by turning on RR fluxes and fibering X3 over R, so that the type IIA string theory becomes dual to M-theory on a local Calabi-Yau fourfold singularity X4. On the one hand, the deformation has to preserve N = 2 supersymmetry on the D2 brane world volume, since the theory lifts to M2 branes on the Calabi-Yau fourfold. On the other hand, turning on RR fluxes on X3 corresponds to turning on Chern-Simons couplings in the gauge theory. Namely, at the Calabi-Yau singularity, the D2 branes split into fractional branes, wrapped on the vanishing cycles. The Wess-Zumino terms on the brane world-volume, in the presence of RR flux, induce Chern-Simons couplings in three dimensions. By supersymmetry then, these become the N = 2 preserving Chern-Simons terms. This is exactly what is needed to describe D2 branes on X3 fibered over R. Namely, N = 2 Chern-Simons terms imply that Fayet-Iliopoulos terms vary as a function of the real scalar field in the N = 2 vector multiplet. The later parameterizes the direction in R3,1 transverse to the D2 branes. Since Fayet-Iliopoulos terms in the gauge theory correspond to Kahler moduli in the Calabi-Yau, this is exactly what is needed to describe N D2 branes on X3 fibered over R.

Consequently, string duality implies that N = 2 quiver theory on N D2 branes probing X3, with N = 2 Chern-Simons terms turned on lifts to the theory on N M2 branes probing X4. At strong coupling, the gauge kinetic terms vanish, and the theory becomes a Chern-Simons quiver theory. These are precisely the theories proposed in [3,6,4,5,7,8] to describe N M2 branes on various local Calabi-Yau fourfold singularities. The relation to the theories on the D3 branes probing X3 was noticed there, but its physical meaning was missed. As mentioned above, one test of the proposals in the literature is that the moduli space of a single M2 brane should be X4. What we are proposing here is stronger, since the theories on D2 branes probing X3 x R, with fluxes, can be derived from string theory, as we explained. This should be contrasted with the theories "without 4d parents" in [11,9,10] .3

It is natural to expect that the quiver Chern-Simons theory of X4 we wrote down is a con-formal field theory in three dimensions. This has already been shown in cases studied in [3,6]. It should hold more generally, since on general grounds, local Calabi-Yau fourfold singularities X4's should be cones over Einstein-Sasaki 7-manifolds y7 [21]. We have shown that the quiver Chern-Simons theory describes N M2 branes at the tip of this cone. In the near horizon limit, the branes deform the geometry to AdS4 x y7, so at least in the large N limit, the conformal symmetry of the theory is manifest.4

Finally, similar ideas can be used to find the theories on N M2 branes probing local G2 holonomy manifolds.5 Namely IIA/M-theory duality relates IIA on a Calabi-Yau threefold X3 with RR fluxes turned on, to M-theory on a G2 holonomy manifold. Turning on fluxes in this case corresponds to deforming the N = 2 quiver gauge theories on N D2 branes probing X3 by N = 1 Chern-Simons terms. At low energies, these theories become the theories on N M2 branes on the corresponding G2 holonomy manifolds. Since the gauge kinetic terms vanish in this limit, the theories describing the M2 branes are N = 1 quiver Chern-Simons theories. We do not expect these theories to be conformal, as G2 x R is not a cone, and no corresponding Einstein 7-manifolds are known [21].

The paper is organized as follows. In Section 2 we show that a toric Calabi-Yau fourfold can be thought of as a circle fibration over X3 x R. We show how to find the first Chern class of the fibration. This implies that M-theory compactified on X4 is dual to IIA on X3 x R with fluxes

Some of these theories do pass some stringent tests. See for example [20,10].

The cases where the explicit cone metrics are known are limited. See for example [22] and references therein. The Chern-Simons theories on 1 M2 brane probing a local G2 holonomy manifold were first discovered in [23].

turned on. In Section 3 we derive from string theory the theories on N D2 branes probing X3, in these backgrounds. For one brane, we show that the moduli space becomes X4, similarly to [5,1,24]. Moreover, X4 is fibered over X3 x R in a manner consistent with the Chern-Simons terms turned on. Namely, in IIA/M-theory duality the center of mass U(1) gauge field on the D2 brane is dual, via scalar vector duality in three dimensions, to the compact scalar field parameterizing the M2 brane position on the M-theory circle. We will show that in the presence of Chern-Simons terms, this scalar picks up a charge. This has the interpretation of having the M-theory circle be fibered non-trivially over X3. Moreover, we will show that the Chern class of this fibration, is precisely such to correspond to RR 2-form fluxes that generated the Chern-Simons terms on the D2 brane in the first place. In Sections 4-8 we give some examples. The examples of Sections 4-1 have already appeared in literature. We are providing a string theory derivation of the corresponding Chern-Simons quiver theories and an interpretation, in the context of IIA/M-theory duality. The example in Section 8 is new, providing a new quiver Chern-Simons theory that should be dual, based on IIA/M-theory duality to N M2 branes on a cone over Q1,1,1.6 Finally, in Section 9 we use IIA/M-theory duality to derive the N = 1 quiver Chern-Simons theories on M2 branes probing local G2 holonomy manifolds.

2. Calabi-Yau fourfolds and M-theory/IIA duality

In this section from purely geometric considerations, we will derive a duality between M-theory compactification on a Calabi-Yau fourfold X4 and type IIA on a Calabi-Yau threefold X3 fibered over a real line R, with fluxes turned on. Namely, we will show here that a Calabi-Yau fourfold X4 can be described as a circle fibration over a base. The base is a Calabi-Yau 3-fold X3 fibred over a real line R. The first Chern Class of the circle fibration is typically non-vanishing. This implies that when we compactify M-theory on X4, there will be RR 2-form fluxes turned on, on X3 x R. Later on, we will argue that the string duality then implies that M2 brane probes of X4 are dual to D2 brane probes of the IIA geometry, in the strong coupling limit of the later.

2.1. Geometric considerations

A convenient way to describe the Calabi-Yau geometry [26] is as a moduli space of a linear sigma model. To describe a d-dimensional Calabi-Yau manifold Xd we start with a copy of CN+d, for some N, parameterized by complex coordinates

i = 1,...,N + d,

Next, for each №, we pick a set of charges Q'a, for a = 1,...,N satisfying

Y.Q1 = o. (2.1)

The Calabi-Yau manifold is than obtained by setting

£ Qa № |2 = (2.2)

6 This theory was discussed recently in [25], but this connection was not made.

and dividing by the gauge group

fa ^ fa exp(ikaQ{a)■ (2.3)

The parameters ra correspond to Kahler moduli of the Calabi-Yau manifold. Mathematically, the above construction is called a symplectic quotient. As explained in [26], symplectic quotient construction can be realized physically as a Higgs branch of a (2, 2) supersymmetric linear sigma model in two dimensions, with fa corresponding to the bottom components of chiral fields , and with gauge group U(1)N. Closer to our purposes, it can also be interpreted in terms of a theory in one dimension higher, i.e. in terms of a Higgs branch of an Abelian gauge theory in three dimensions, with four supercharges.

We will now show that we can view the d-dimensional Calabi-Yau threefold Xd as an S1 fibration over a base, which itself is a fibration of a (d - 1)-dimensional Calabi-Yau Xd-i over a line R. To see this, consider adding a complex variable

r0 + i&0

and pick a set of charges Qi0, which also satisfy the Calabi-Yau condition,

Eq0=0.

To avoid changing the manifold, we introduce an additional constraint

XQ0 Ifa I=r0 (2.4)

(note that r0 is variable now) and an additional gauge symmetry

00 ^ 00 + k, fa ^ exp(iQ0k)fa ■ (2.5)

Namely, using (2.4) we can solve for r0 and gauge away 00 using (2.5). So, as expected, we have just written the original Calabi-Yau Xd in a different way.7

However, this gives us a way to think about Xd as a circle fibration. Namely, the S1 fiber of the Calabi-Yau is 00. To find the base, we consider a projection that "forgets" 00. At fixed r0, the base is simply a (d - 1)-dimensional Calabi-Yau manifold Xd-1, given by solving (2.2), (2.4)

J2Qa ifai i2=ra, EQ0ifa-1=r0 (2.6)

and dividing by N + 1 U(1) gauge transformations (2.3) and (2.5). Allowing r0 to vary, the lower-dimensional Calabi-Yau manifold Xd-1 is a fibration over the real line parameterized by r0, where as we vary r0, the Kahler moduli of Xd-1 vary according to (2.6).

7 We could also consider replacing r0 by kr0 while giving 00 charge k in (2.5), corresponding to sending 00 to 00 + kk instead. In this case we do change the manifold, from Xd to a Zk quotient Xd/Zk, since gauge transformations by k = 2ni/k leave 00 invariant.

We can easily find the first Chern class of the fibration. To begin with, note that

is not gauge invariant because dO0 transforms under (2.5). The gauge invariant combination is dOo + ^ iqj d<j/<j

T,qjQ° = 1 HqjQ) = o, «=

Then Arr = j qj d4>j/4>j is the connection on the circle bundle over X3 x R whose curvature Frr = dARR = ^2 qjaj (2.7)

where wj = 82(<j)d<j A d<j is a two-form with 5-function support on the divisor Dj corresponding to setting <j = 0. We can write

[FRR] = J2 qj [Dj], (2.8)

for the cohomology classes [FRR] and [Di ] of FRR and wi, respectively. Consider now the flux of FRR through a compact 2-cycle C in the geometry,

There is a correspondence between generators of H2(Xd-\) and the U(1) gauge groups in the linear sigma model. For a curve C, let

£eC |2 = rc

denote the corresponding D-term. In particular, rC is the Kahler class of the curve. It is a standard result in toric geometry (see for example [27] for a recent review) that

#(Dj n C) = QC,

and using this, we can evaluate

f Frr = £ qj # (Dj n C) = £ q}QC}. C

By the usual IIA/M-theory duality, it follows that M theory, compactified on circle fibered Xd, is dual to IIA on the base of the circle fibration. Moreover the curvature of the circle fibration, Frr above, becomes the RR 2-form flux on Xd_i x R. In the present context, we are interested in this duality where d = 4, and we are relating M-theory on Calabi-Yau fourfold X4 to type IIA on Calabi-Yau threefold X3, with flux, fibered over R,

M-theory on X4 ^ IIA on X3 x R.

Note that sometimes, the IIA dual would involve D6 branes.8 There is a-priori nothing wrong with this, but for our purposes, we are interested in those cases without the explicit D6 branes. To avoid this, we will use IIA string theory as a guide. In the next sections, we will construct the linear sigma models for X3 as low energy effective field theories on the D2 brane probes in IIA string theory. We will explain how turning on fluxes and fibering X3 over R modifies the linear sigma model. Moreover, we will show that the resulting manifolds can be viewed as a Calabi-Yau fourfold X4 fibered over X3 x R.

3. String duality and D-branes on local Calabi-Yau

Consider the theory on N D2 branes probing a local toric Calabi-Yau X3, near the place where X3 develops a singularity. Before turning on fluxes or fibering, this turns out to be a well-known quiver theory gauge in three dimensions, preserving N = 2 supersymmetry. As explained in the introduction, the theory is the same as the theory on D3 branes probing X3, dimensionally reduced to three dimensions. The latter was studied in numerous places, for example [12-15]. After reviewing this, we consider turning on RR fluxes through the vanishing cycles of X3. We will show that these generate Chern-Simons terms for the 3d gauge theory. To preserve N = 2 supersymmetry on the branes, than turning on fluxes has to be accompanied with fibering the Calabi-Yau threefold X3 over the one spatial direction transverse to the branes. This corresponds to turning on N = 2 Chern-Simons terms on the D2-branes. These are precisely the three-fold fibrations that we showed are dual to M-theory on Calabi-Yau fourfolds X4! In particular, IIA/M-theory duality implies that, at strong coupling the D2 brane gauge theory goes over to theory on N M2 branes probing X4. Moreover, at strong coupling the kinetic terms on the gauge fields vanish, so the natural description of this is a quiver Chern-Simons theory with N = 2 supersymmetry.

As a further test, the IIA/M-theory duality implies that the center of mass U(1) gauge field on one D2 brane probing X3 is dual to a compact scalar describing the position of the M2 brane on the M-theory circle. For N = 1, we can explicitly dualize the center of mass U(1) field on the D2 brane probe of our geometry, and show that the result is a linear sigma model on X4, as expected. Moreover, we can show that the circle is fibered over X3 in a manner consistent with the RR fluxes turned on, thus closing the loop of correspondences.

3.1. D2 brane gauge theory on X3

Before turning on fluxes, or fibering, the local Calabi-Yau manifold X3 preserves 8 supercharges. The N D2 branes probing this break half of the supersymmetries, preserving four supercharges, or N = 2 supersymmetry in three dimensions. The theory on the brane probe is a N = 2 quiver gauge theory with superpotential. The data of the theory can be decoded from the internal geometry. In particular this means that, at the classical level, the quiver for N D2 brane probes is the same as the quiver for N D0 branes or N D3 branes probing the same geometry - it is only the dimension of the D-branes that changes. The fact that the kinematics of the theories is

8 A simple example of this is the simplest case of our construction. Namely, take d = 2 and consider c2, with coordinates xi, X2 as a fibration over c x r where we introduce a D-term constraint |x2|2 — |xj |2 = tq, and divide by (do,x\, X2) ~ (So + X, x\e'x,x2e—,X). It is easy to see that Frr = &(xi)dxi A dxi has flux through the two sphere surrounding the origin of c x r. In other words, there is a D6 brane sitting at the origin. This is as expected, since the quotient by Sq corresponds precisely to viewing c2 as Taub-Nut.

the same, is a consequence of T-duality in the non-compact dimensions. This fact was also used recently in [28], in a different context.

When X3 is smooth, the theory on N D2 brane probes, at low enough energies, becomes the same as the theory on N D2 branes probing flat space - so it is the maximally supersymmetric U(N) gauge theory in three dimensions. At a singularity, however, the D2 branes split into a collection of fractional branes, carrying charges of D6, D4 and D2 branes wrapping 4, 2 and 0-dimensional cycles shrinking at the singularity (see for example, [29,18,12]). For each such fractional brane on a vanishing cycle Aa, we get a node of the quiver. If the D2 brane splits into a collection of n cycles

we get a

U(d1N) x ••• x U(dnN)

quiver theory on the branes at the singularity. Moreover, there are nap arrows from the node a to node p corresponding to bifundamental fields ^p,

: a ^ P,

in (daN, dpN) representation, with a = 1,...,nap (see next section for examples). There is a gauge invariant superpotential

W(<P),

which can only depend on the complex structure moduli of X3. Finally, the Kahler moduli of X3 enter the D-terms of the theory. In particular, the Fayet-Iliopoulos parameters ra for the U(1) factors of the gauge group depend on them, as

Y^ra Tr Da.

Each vanishing cycle carries definite D-brane charge. We can describe the charge of branes wrapping Aa in terms of its homology cycles in Heven(X^) as

[Aa] = £ Q6 [Da] + Qi [Cj] + Q2 [pt]

where Q6 denotes the D6 brane charge and D the 4-cycle class the D6 branes wrap, Q4 the D4 brane charge, and C a 2-cycle class, and Q2 the D2 brane charge. The charges are such that daAa in (3.1) has the charge of one D2 brane, and nothing else. Note that some of the brane charges appearing must be negative, since da are all non-negative. At the singularity the appropriately chosen combinations of D-branes and anti D-branes can be mutually supersymmetric. This is precisely what allows the D2 brane to split into fractional branes in the first place.

3.2. D2 branes in IIA on X3 with flux

Consider turning on RR 2-form on X3. We can write

Frr = £ qjuf (3.2)

where qj are integers and rn) ' are two-forms with 8-function support on divisors Dj in X3 (FRR is given as a form here, not a class on X3, so the divisors Dj need not all be compact). We would like to know what effect this has in the D2 brane gauge theory at the singularity.

Consider N D4 branes wrapping a vanishing 2-cycle C in X3. The RR field enters the theory on the branes via a Wess-Zumino coupling

Arr Tr F A F.

Cxi2'1

Note that, even though C is a shrinking cycle, we are allowed to use the geometric description above, since the coupling is topological, and the metric does not enter. This coupling gives rise to the Chern-Simons term on the D-brane world volume. Integrating by parts, we can rewrite this as

J FRR Tr «cs = k J Tr «cs

Cxi2,1 R2,1

where mcs = A a dA + 3 A A A A A is the Chern-Simons form, and

¡frr.

To compute k, we use the description of FRR in terms of forms with 8 function support in (3.2) to express k in terms of intersection numbers of C with divisors Di,

k = qj # (Dj n C). (3.3)

This fractional brane wrapped on C could also carry D2 brane charges. Since these would have entered as a first Chern class of the gauge field on the D4 brane, they cannot affect the Chern-Simons coupling above.

Similarly, if we had a D6 brane wrapping a 4-cycle D the Chern-Simons coupling would be given by

k= Frr a Tr F.

This reduces to (3.3), where the role of C is now played by the two-cycle class dual to Tr F inside the divisor D. This is where the D4 brane charge is induced. Note that in the wrapped D6 brane case, the fluxes of RR four-form GRR can contribute to the level as well, by

8k = j Grr . D

We can similarly represent GRR by a 4-form

Grr = ^2 PA^f' A

in terms of 4-forms with 8-function support on curves CA, and then

8k = J2 PA # (Ca n D).

In summary, turning on RR fluxes corresponds to turning on Chern-Simons terms. It is easy to see from the above formulas, that the generic combination of 2- and 4-form fluxes corresponds to choosing generic Chern-Simons couplings ka for the nodes of the quiver, subject to

ka = 0. (3.4)

This is a consequence of the fact that the net D6 brane and D4 brane charges of the quiver sum up to zero, by construction.

Now, if we were to turn on just fluxes in the Calabi-Yau, the fluxes would break the supersym-metry on the brane to N = 1. We will return to this case in the last section of the paper. Suppose however that we want to preserve N = 2 supersymmetry. This is the same amount of super-symmetry preserved by an M2 brane on a Calabi-Yau fourfold. In this case, the Chern-Simons coupling induced by the flux has to be accompanied with the additional terms, corresponding to

kj d4e V£(V)

where V is the vector multiplet, and £ is the related linear multiplet.9 The bosonic components of the above are

k(Tr vcs(A) + 2Tr Da)

where a is the real scalar field in the vector multiplet, and D is the D-term.10

We see from the above that, if we want to preserve N = 2 supersymmetry, in the presence of RR fluxes (some combination of) Kahler moduli of the Calabi-Yau X3 are no longer parameters in the theory — but they become variable. More precisely, the vanishing of the potential in 2 + 1 dimensions requires [4,5]

aa®lp = Kpap.

To see what this means, consider for simplicity, the case of a single D2 brane, N = 1 on X3. Then, on the Higgs branch, the above is solved by

aa = rolda, a = 1,...,n (3.5)

for some variable r0. In particular, this means that the Fayet-Iliopoulos term for the gauge group on node Aa is given by

ra = karo. (3.6)

Since a's parameterize the positions of the branes along the one R direction in R3'1 transverse to the D2 branes, this means two things: first we should identify r0 with coordinate on R,

r0 ^ coordinate on R.

9 S is defined in terms of V by S = /g dt D(etV De-tV). In the Abelian case, we have simply S = DDV. 10 The vector multiplet has the expansion V = -2i99a + Wy^ê + 62Ô2D.

Second, it means that X3 is fibered over R in a manner that correlates with the RR fluxes turned on. For N D2 branes we just get the symmetric product version of this, with oa = op describing N branes at different positions on R.

Now, we showed that the IIA compactification on a local Calabi-Yau threefold X3 where we turn on RR 2-form fluxes and fiber the manifold over R by allowing the Kahler moduli to vary is dual to M-theory on Calabi-Yau fourfold X4. This implies that the gauge theory describing D2 branes on X3 in this background, flows, in the IR to theory on M2 branes probing X4. It is natural to conjecture — based on string/M-theory duality — that the theory on N M2 branes probing X4 is the quiver Chern-Simons theory obtained from the above by setting the gauge kinetic terms to zero, since these vanish in the IR on dimensional grounds.11

3.3. One D2/M2 brane

As additional evidence for the proposed duality, consider again the case of a single D2 brane in IIA. It is well known that the U(1) gauge field Ao on the D2 brane probe is dual, via the scalar-vector duality in three dimensions to the compact scalar 0o related to A0 via

dAo = *d6o. (3.7)

This scalar describes the position of the M2 brane on the M-theory circle. Based on the IIA on X3/M theory on X4 duality, we expect that when we turn on N = 2 Chern-Simons terms on the branes, 0o should be fibered over the base X3 x R in such a way so as to give a Calabi-Yau fourfold X4. Moreover, the circle fibration should have the right first Chern class on X3 to correspond to RR 2-form flux we had started with. We will see in detail that this is indeed the case in the examples we study in the next section. For now, we will just sketch how this comes about.

For most singularities, and all the singularities we will study in this paper, da = 1, for all a.12 In this case the theory is Abelian, with gauge group

for a quiver with n nodes.

The moduli space is the space of solutions to minima of F- and D-term potentials, modulo the gauge transformations. Before turning on fluxes, the moduli space of this theory, the Higgs branch, is the Calabi-Yau manifold X3 x R. As above, the R direction is parameterized by one massless combination of o's. On the Higgs branch, all components of the gauge fields with charged matter become massive. However, there is one decoupled U(1) gauge field, under which nothing is charged. This corresponds to the diagonal combination

Ao = -) Aa.

This component becomes the U(1) gauge field on the D2-brane at a generic point in X3. Dualizing Ao to the compact scalar 0o via (3.7), the moduli space of the M2 brane is a direct product

11 Note that we can also relax the condition (3.4). This corresponds to turning on RR zero form flux, which couples to the D2 brane charge. This agrees with the proposal recently in [3o,31], for Chern-Simons theories dual to massive IIA backgrounds.

12 For examples where not all da = 1 see [12].

S1 x X3 x R. Moreover, the Fayet-Iliopoulos terms in the gauge theory are identified with the Kahler moduli of X3, and they are independent of where in R we are.

Now consider turning on 2- and 4-form fluxes. This generates Chern-Simons terms

d ^ kaAa A

where Ya ka = 0, as we explained. The effect of this on the moduli space is twofold. On the one hand, turning on fluxes modifies the Fayet-Iliopoulos terms in the vacuum corresponding to fibering the X3 over R as in (3.6). The effect on the gauge transformations is as follows. Put, Aa = A0 + Aa, where Ya Aa = 0. Then A0 enters the Chern-Simons terms via the coupling13

/ d 3x^kaA a A dA0. (3.8)

Dualizing A0 to the compact scalar 60, replaces dA0 by d0o. Presence of (3.8) means that 60 picks up a logarithmic charge ka under the gauge group on node a. In particular, while d0o is no longer well defined, the combination

d&o + i^kaA a

is well defined.

We can view this in two different ways. Firstly, this implies that the moduli space is Calabi-Yau fourfold X4. It is obtained by not dividing by the U(1) gauge transformations generated by

^ ^ ka A a ?

these we can use to set 60 to zero.14 This is related by supersymmetry to the one linear combination of O-terms we need not impose: we can use them to solve for r0, the superpartner of A0 and 60.

Second, this allows us to view the Calabi-Yau fourfold X4 as a circle fibration over X3 x R, with 60 parameterizing the fiber. We will now argue that we can think of Y,a kaFa as the RR flux itself, or more precisely that

[Frr] = J2ka [Da], (3.9)

where Da is the divisor class in X3 corresponding to the U(1) on the node a.15 To see this [23] recall first of all that on the D2 brane there is a Wess-Zumino coupling

13 Note that the other Chern Simons terms vanish on the Higgs branch, since all the components of the gauge fields under which matter is charged are massive.

14 When the greatest common denominator of the ka's is not 1, this leaves a discrete subgroup of the gauge symmetry unbroken, this leaves over a discrete subgroup of the gauge symmetry unbroken. The moduli space is then an orbifold by this symmetry.

15 Da corresponds to setting to zero a product of linear sigma model variables with coupling with charge 1 to Aa, and

charge zero to the rest.

j d3x Frr A Ao.

In the presence of the above flux, the action of a D2 brane wrapped on a curve C would get shifted by£a #(kaDa n C) times / Ao.

This is exactly what (3.8) implies. Namely, configurations of D2 brane wrapped on a curve C in the linear sigma model carries non-zero vortex charge [26]. In particular, a D2 brane wrapped on C carries vortex charge

f Fa = #(Da n C)

for the U(1) on node a. Correspondingly, in the presence of the coupling (3.8) the action of a D2 brane wrapped on C gets modified precisely by the same amount, as it does in the presence of the RR flux (3.9), as we claimed. In the next section, we will work this out explicitly in some examples.

The N = 1 case gives a physical realization to the linear sigma models in the previous sec-tion.16 It was noted in [32], that D-branes on Calabi-Yau manifolds tend to see only the geometric phases of the theory, and the non-geometric phases are absent. Apparently, the same is true for M2 branes. In the next sections we present some examples of the theories we discussed above. All of the examples below have appeared in the literature, we are just giving them a new interpretation, in terms of the IIA/M-theory duality.

4. Example 1: Conifold and the ABJM theory

The Calabi-Yau X3 is the total spaceof O(-1) © O(-1) bundle over P1.TheP1 corresponds to the one shrinking 2-cycle of this geometry. There are two fractional branes,

A1, A2, whose D-brane charges are given by

[A 1] = [pt] — [P1], [A2l = [P1]. (4.1)

In other words, A2 corresponds to a D4 brane wrapping the P1, and A1 to the anti-D4 brane bound to one unit of D2 brane. The latter can be thought of as turning on /p1 F = — 1 on the anti-D4 brane. Thus, one D2 brane probing X3 corresponds to

[A1] + [A2].

The quiver is given by U(N) x U(N), N = 2 gauge theory in 3 dimensions, with two pairs of bifundamentals A1,2 in (N, N), and anti-bifundamentals B12 in (N, N), and superpotential

W = MTr A1B1A2B2 — Tr A1B2A2B1).

16 A-priori, the linear sigma model in question differs from the ones in Section 2 by the presence of superpotentials. This is a technicality. In fact, as we'll see in the next section, the solutions to F-term equations can be often rewritten as minima of an effective linear sigma model without superpotential [32]. This fact was used extensively in [19]. In such cases the theories on the D-branes are called toric. There are also non-toric examples, related to the toric ones by Seiberg dualities.

The Higgs branch of the theory describes N D2 branes probing X3 x R. Namely, for N = 1, the superpotential vanishes. The potential for the o1j2 vanishes by setting

01 = 02 = ro, (4.2)

where we view ro as a coordinate on R inside R3,1. The D-term potential vanishes for

| A112 +|A2|2 — |B112 — |B2|2 = ru (4.3)

where r1 is the Fayet-Iliopoulos term for the first U(1). For the second U(1) the charges are opposite, and r2 = —r1. Dividing by the gauge group, the moduli space is the conifold, X3 times R. While the off-diagonal gauge fields are Higgsed, the diagonal U(1) survives, since nothing is charged under it. It is identified with the center of mass U(1) gauge field on the D2 brane at a generic point in the moduli space. For general N, the Higgs branch is the symmetric product SymN(X3 x R) describing N identical D2 branes moving about on X3 x R.

Consider now deforming the theory by N = 2 Chern-Simons terms for the two gauge groups,

k J d40 (Tr V1£( V1) — Tr V2S( V2)) (4.4)

where V1j2 are the vector multiplets corresponding to the two nodes. This is precisely the theory studied in [3].17 Turning on N = 2 Chern-Simons terms in the D-brane theory means that we are deforming the bulk IIA theory by turning on RR 2-form flux in through the P1, and fibering X3 over R. To see this, consider again the moduli space of the theory for N = 1. In this case, the Chern-Simons interaction makes Fayet-Iliopoulos parameters dynamical, with

r1 = kro = —r2

where ro is the vev of the o fields in (4.2). We still have to divide by the gauge group. Dualizing the center of mass gauge field to a compact scalar 0o, the latter picks up a logarithmic charge k under the off diagonal U(1). Thus the gauge symmetry acts as

0o ^ do + kk, A1,2 ^ A1,2eik, B^2 ^ BU2e—ik. (4.5)

Solving the D-term constraint for ro, and using (4.5) to set do to zero, the moduli space is a copy of C4, parameterized by A1j2, and B12. We still have to divide by the discrete gauge transformations that leave do invariant, corresponding to k = 2n/k. Correspondingly, the moduli space is the fourfold

X4 = C4/Zk,

as explained in [3]. On the other hand, this allows us to view X4 as an S1 fibration over X3 x R. Namely, projecting to the base, by forgetting do, and fixing a point ro in R, we get a copy of X3. Now consider the first Chern class of the fibration. While ddo itself is not well defined, for example

ddo + ikdB1/B1

is well defined over X3 x R. Hence, we can identify

Arr = kdB1/B1.

17 More precisely, it is the same theory provided we tune k = 1/k, when the theory has an enhanced supersymmetry.

The corresponding curvature FRR = dARR is Frr = k«B1

with o>b1 = iS2(B1)dB1 A dB 1. The cohomology class of FRR is the same as k copies of divisor DB1 corresponding to setting B1 = 0,

[FRR]=k[DB1 ]. (4.6)

As we explained in the previous section, the RR fluxes turn on Chern-Simons terms on the brane. Given the D-brane charges (4.1), we have

k2 = J Frr = -k1. p1

Consistency requires that the Chern-Simons terms induced by the flux agree with (4.4). As we explained, given (4.6) we have

f Frr = k # (DB1 n P1). p1

We can read off the intersection number from (4.3). Namely, (4.3) is the D-term constraint corresponding to the class of the P1, so the intersection of the divisor DB1 with the P1 is just the charge of B1 under the corresponding U(1). This is -1, so

#(Db1 n P1) = -1, giving k1 = k = —k2, as we had expected.

5. Example 2: C2/Z2 x C singularity

The Calabi-Yau X3 = C2/Z2 x C

is an orbifold, whose resolution is an O(—2) © O bundle over a P1. Correspondingly, there is one vanishing two-cycle, the P1 in the base. The theory on the D-branes on X3 was discovered in [24]. There are two fractional branes,

A1, A2,

[A1] = [pt] — [P1], [A2] = [P1].

The quiver is given by U(N) x U(N) gauge theory with a pair of bifundamental chiral multiplets A1, B1 in (N, N), anti-bifundamentals A2, B2, in (N, N) and an additional pair of adjoints of the two groups. The superpotential18 is

W = Tr 01A1A2 — B1B2) — Tr 02(A2 A1 — B2B1).

18 In fact the conifold theory in the previous subsection above was obtained as a deformation of the orbifold theory. In the gauge theory, the deformation corresponds to adding a mass term m (Tr #\ — Tr #2), with m = 1/X.

This theory was studied in the present context in [5]. For a single D2 brane on X3, the moduli space is X3 x R. The R direction is parameterized by a1 = r0 = a2, as before. X3 emerges by setting the F- and the D-terms to zero and dividing by the gauge group. Setting the F-terms to zero we have = which parameterize a copy of C. In addition, we have

A1A2 = B1B2. (5.1)

The D-term for the first U(1) gives

|Ai|2 +|Bi|2-|A2|2-|B2|2 = ri,

and similarly for the second U(1) with r1 = -r2, and signs of all the charges reversed. It is useful to rewrite the F-term constraints as D-term constraints, so that we get a linear sigma model without superpotential. We can solve (5.1) introducing four new variables, and putting

A1 = X1X0, A2 = X2X3, B1 = X1X0, B2 = X2 X3.

There is a redundancy inherent in this, which we can remove by simultaneously introducing a new U(1) gauge field U(1)aux under which A's and B's are neutral, and x1,2 have charge +1, and x0 3 charge -1. This introduces a new D-term constraint

|X1|2 + |X2|2 -|xo|2 -|X3|2 = 0. (5.2)

The FI term for this auxiliary U(1) has to be set to zero for this to be equivalent to the solutions of F-term equations. The fields Xi also carry charges under the original gauge fields, since the A's and the B's did. The original D-terms translate to

|X3|2 -|X0|2 = n. (5.3)

For r1 > 0, we can use (5.3) and the corresponding gauge symmetry to solve for x3, so the moduli space is simply

|X1|2 + |X2|2 - 21X012 = r1, (5.4)

modulo U(1). This, is a copy of (the resolution of) C2/Z2, and together with a copy of C parameterized by the adjoints, it gives X3.

As an aside, note that the orbifold phase is absent. This would have corresponded to taking r1 < 0. However, we are not allowed to do that, since we can no longer solve (5.3) in the manner we did before. Instead, what we need to do is exchange the roles of x0 and x3, and then we recover a geometric phase again.

Consider now turning on N = 2 Chern-Simons couplings for the two gauge groups of the quiver

kj d4e(Tr V1S(V1) - Tr v2Z(V2)). The flux has two effects. The FI parameter of (5.4) becomes dynamical, r1 = kr0 = -r2

and, dualizing the center of mass gauge field A0 = 1 (A1 + A2) to a dual scalar e0, the corresponding gauge transformation becomes

e0 ^ e0 + kX, X3 ^ X3elk, X0 ^ X0e~lkX0

with x1,2 invariant. This allows us to solve for r0,00 an drop these equations. Their only remnant is a discrete gauge symmetry that takes

x3,x0 ^ e2ni/kx3,e—2ni/kx0.

The D-term and gauge symmetry corresponding to the auxiliary U(1)aux in (5.2) are unaffected. The moduli space is a Calabi-Yau fourfold X4 which is Zk orbifold of the conifold times a copy of C [5,7].

Now, let us view X4 as an S1 fibration over X3 x R. We will show that the first Chern class of the fibration is just what is needed for the RR flux to be the origin of the Chern-Simons terms. Namely, it is easy to see that

d00 + ik(dx2/x2 + dx0/x0)

is invariant under the U(1)2 x U(\)aux gauge group. Thus, compactifying M-theory on X4 and interpreting 00 as the M-theory circle, we get IIA on X3 x R with RR flux turned on corresponding to

[FRR]=k[D2]+k[D0]

where Di is a divisor corresponding to setting xi = 0. Now, the Chern-Simons level should be given by

k2 = —k1 = J Frr. p1

We have

Frr = k # ((D2 + D0) n P1) p1

and we would like to compute what this is on X3. Since the D-term constraint corresponding to the class of the P1 is (5.4) the intersection numbers are simply the charges of x2, x0 under it. This gives

#(D2 n P1) = 1, #(D0 n P1) = —2 we see that we recover k1 = k = — k2, as expected.

6. Example 3: The suspended pinch point

In this case, the X3 geometry has two vanishing P^s, P1 with normal bundle O © O(—2) bundle, and P^ with normal bundle O(—1) © O(—1). There are now three fractional branes.

[A1] = [p1], [a2] = [p1], [A3] = — [p1] — [P2] + [pt]. (6.1)

For N D2 branes probing X3 we get a

U(N) x U(N) x U(N)

A3 "B,

Fig. 1. Quiver corresponding to the N D2 branes on the Suspended Pinch Point singularity.

quiver theory with bifundamental matter as in Fig. 1. Since the brane in class A1 is free to move in the O direction, (the normal bundle has one holomorphic section) there is one adjoint chiral multiplet 0 corresponding to that. The superpotential is

W = Tr 0(A1A2 — C1C2) + Tr B1B2C2C1 — Tr B2B1A2 A1.

Consider the moduli space for one D2 brane on X3. As usually, there is one direction R parameterized by oa = r0, for all a. The F-term equations set

A1A2 = C1C2, 0 = B1B2. We can write these as D-term equations by putting

A1 = a1b1, A2 = a2b2, C1 = a1b2, C2 = a2&1,

and introducing an additional U(1)aux gauge symmetry, under which a1,2 have charge +1 and b1,2 have charge —1. All in all, we have U(1)4 gauge invariance, and four D-term equations

where r1 + r2 + r3 = 0. It is easy to show, solving the D-term equations and dividing by the gauge group, that the moduli space is precisely X3 with r1 corresponding to the size of P11 and r2 to P2. Namely, using the first equation, and the corresponding gauge symmetry to solve for a1, the moduli space can be written simply as

where the first equation now corresponds to the cohomology class of the Pi and the second to Pi.

Consider now turning on N = 2 Chern-Simons couplings. Given the D-brane charges (6.1) the Chern-Simons couplings should be related to fluxes as

M2 - |a2|2 = ri, \U2\2 - \bi\2 - |Bi|2 + IB2I2 = r2,

M2 - |b2|2 + |Bi|2 - |B2|2 = rs, -|ai|2 - + |bi|2 + |b2|2 = 0

|bi|2 + |b2|2 - M2 = ri, ^i2 - |bi|2 - |Bi|2 + |B2|2 = r2

6.1. Case (k1 ,k2,k3) = (k, -k, 0)

The Fayet-Iliopoulos terms r1, r2 vary over R parameterized by r0 as

r1 ~ r0, r2 r—r0,

while r3 is a constant. We can view X4 obtained by just dropping the first two equations in (6.2) and the corresponding U(1)'s, and dividing by a Zk action that sends a1, a2 to e2ni/ka1, e-2ni/ka2. The moduli space is the Zk quotient of the fourfold D3 given in [5,7].

On the other hand, we can view X4 as a circle fibration over X3 x R by dualizing the center of mass U(1) to a compact scalar field e0. Then, e0 is invariant under the third and fourth U(1) in (6.2), and transforms under the first two with logarithmic charge k, and -k, respectively. We can pick, for example

de0 - k(da1/a1 + db1/b\) as the invariant one form on X3 x R. This implies [FRR]=k[Da1 + Db1 ]

where Da1, Db1 are the divisors corresponding to setting a1, and b1 to zero. From (6.2), and the charges of a1, b1 we can read off that

#( Da1 n p1) = 0, #( Da1 n P2) = 0, #( Db1 n p1) = 1, #( Db1 n ?!) = -1

while r2 is a constant. We can view X4 obtained by dropping the first three equations and replacing them by one linear combination of them which is independent of r0, say k times the first equation minus m times the second. In addition, we divide by the corresponding U(1) gauge symmetry. The result is a Calabi-Yau fourfold X4.

Now, let's view X4 as a circle fibration over X3 x R. We are want to show that the corresponding circle fibration has the correct Chern class. The compact scalar e0 transforms with logarithmic charges k, m and -k - m under the first three U(1)'s in (6.2) and not at all under the last one. The invariant one form is

de0 - (kda1/a1 - mdb1/b1 + (k + m) db2/b2). From this it follows that

so that

as expected.

6.2. The general case (k1,k2,k3) = (k, m, -k - m)

The Fayet-Iliopoulos terms r1, r3 vary over R parameterized by r0 as

r1 ~ mr0, r2 ~ kr0, r3 (k + m)r0,

[Frr] = k[Dfl1 ] - m[Db1 ] + (k + m)[Db2 ].

Fig. 2. Quiver corresponding to the N D2 branes on c3/z3. There are three bifundamentals connecting each pair nodes.

We know the intersection numbers of Dai ,bj already. We have in addition

#D n Pj) = 1, #(Db2 n P2) = 0. Adding this all up, we find so that

Jfrr = k, J

p1 p1 p1 p2

Frr = m,

as expected, given the D-brane charges (6.1). 7. Example 4: C3/Z3

In this case, the resolution of the singularity is O(-3) ^ P2, so there is both a shrinking four cycle, the P2 and a shrinking 2-cycle, the P1 inside the P2. There are three vanishing cycles A1j2,3. The D-brane charges of vanishing cycles are [12]

Ai = -2[P2] + [P1] - I[pt], A2 = [P2], A3 = [P2] - [P1] - 1 [pt]. The gauge group on N D2 branes is U(N) x U(N) x U(N),

with nine chiral multiplets connecting them, as in Fig. 2. In addition, there is a superpotential

w =J2 jTr AtBjCk.

Consider now the moduli space for a single D2 brane probe. This problem was solved explicitly in [32]. As in the previous examples, we can redefine variables to be able to rewrite the F-term equations as D-term equations. This example was worked out in detail in [32]. At the end of the day, we have an effective linear sigma model with six fields Pi, i = 0,1,..., 5, and U(1)4 gauge symmetry

|P112 + IP212 + |P3|2 - | P0 | 2 - | P4 | 2 - |P5 |2 = 0, | P4 | 2 + | P5 | 2 = H,

I P0 | 2 -|P5 | 2 = r2, - |P0 | 2 + |P4 | 2 = r3 (7.1)

where ri is the FI parameter of the i th node, and r1 + r2+r3 = 0. For any choice of FI parameters, the moduli space is always X3 = O(—3) ^ P2, i.e. the orbifold phase is absent. For example, for r2 < 0 < r3, we can use the last two equations above to "solve" for p4,5 in terms of p0, and the moduli space is simply the solutions to

|pi|2 + |P212 +|P3|2 - 3|po|2 = r3 - r2

which is X3 with Kahler class r3 - r2.

Consider now turning on Chern-Simons couplings for the branes on the three nodes with

(ki,k2,k3) = (k, —k — m, m).

It is easy to work out what the corresponding Calabi-Yau fourfold X4 is. This corresponds to letting the FI parameters vary as

ri a kro, r2 a—(m + k)ro, r3 a mro,

moreover, the compact scalar 0o has charges k, —(k + m) and —m under the three U(1)'s corresponding to the three nodes of the quiver, and is neutral under the auxiliary U(1) (corresponding to first equation in (7.1)). The invariant one form can be written as

d6o + i(mdpo/po — k dp5/p5 + (m — k) dp1/p1j. This implies

[Frr] = —m[Do] — (m — k)[D1]+k[D5]. From above, we can read off,

#(Do n P1) = —3, #(D1 n P1) = 1, #(D5 n P1) = o

J Frr = 2m + k. p1

Now, consider the contribution of this flux to the Chern-Simons levels. For the three fractional branes A1, A2, A3, the RR 2-form flux gives:

(k1,k2,k3) ^ J Frr, o, — J Frr J = (2m + k, o, —2m — k).

Now, recall that we are also free to turn on RR four-form flux on the P2 in IIA. From the charges of the wrapped D-branes, we see that this could shift the levels by

(Sk1,Sk2,Sk3) = (^—2 J Grr, j Grr, j Grr^j,

p2 p2 p2

which is exactly what we need, provided

J Grr = m. p2

Note that, for r0 < 0, we get a different assignment of fluxes. As we pass through the singularity at r0 = 0, and continue on to negative r0, the fractional branes are permuted by the quantum Z3 symmetry, that takes Ai to Ai-1. This also acts on the homology in the corresponding way, namely, ([P2], [P1]) go to ([P2] - [P1], 3[P2] - 2[PJ]). For this reason, a given set of 2-form and four-form fluxes turned on X3 have different periods for r0 positive and negative. The same Z3 action acts on the Chern-Simons couplings as well.

7.1. Case (k1,k2,k3) = (0, -m,m)

For example, consider the case where k = 0. In this case, the Calabi-Yau fourfold X4 is an Zm quotient of an O(-2) © O(-1) bundle over P2. Namely, we can use the last equation and the corresponding gauge symmetry (7.1) to "solve" for r0, and 00. The two left-over, independent D-term constraints are the first two. We can solve the latter one for p5, and the manifold becomes

|P112 + |P2|2 +|P3|2 -|P0|2 - 2|P4|2 = r1

modulo the corresponding ^(1). This is the O(-1) © O(-2) bundle over P2. This agrees with [5,7]. More precisely, X4 is a quotient of this

P0,P4 ^ e2ni/mp0,e-2ni/mp4

which is a discrete gauge symmetry left over from solving for 00.

We claim that the quiver gauge theory in this case describes M2 branes on X4. While the two-form flux is geometrized in M-theory, the four-form flux is not. Instead, the four-form flux in IIA should be dual to a four-form flux on X4.

It was proposed in [4,22] that, at large N, the theory at hand should be dual to M-theory on AdS4 x Ym,2m (P2). However, in the supergravity solution of [4,22], the four-form flux vanishes.19 It would be very interesting to study the dynamics of the theory at hand in more detail. Secondly, with fluxes turned on, some Kahler moduli in M-theory will be lifted, correspondingly, only for some values of FI terms the full brane plus bulk theory will preserve supersymmetry. To begin with, one should identify the four-form flux in M-theory that this IIA background lifts to. Given an explicit 4-form flux in IIA, the flux in M-theory is simply the pullback. However, we have only specified the cohomology class of the flux. Because of the subtlety with the Z3 action on the periods of X3 in going from r0 positive to negative, even the cohomology class of the four-form flux in M-theory cannot be simply read off from knowing the cohomology class of the flux in IIA.

8. Example 5: Local F0 and the cone over Q1'1'1

This case is interesting, since as we will see, it will provide a new Chern-Simons theory that should be dual to N M2 branes probing a cone over Q111. The later is a very old example of an Einstein-Sasaki manifold. The conformal field theory dual to AdS4 x Q1,1,1 has been sought for a long time. Here we will provide a string theory derivation of the dual CFT. Other proposals have been made in [9] and studied in [20] recently.

The geometry of the Calabi-Yau threefold X3 is a line bundle over

F0 = P1 x P1.

19 We thank J. Sparks for discussions on this issue.

Fig. 3. Quiver corresponding to the N D2 branes on local Fq. There are two bifundamentals connecting each pair nodes.

In this case there are four vanishing cycles. There are two different assignments of charges we can make where the superpotential is still toric (in the sense of the footnote 14). We will pick one of these, for simplicity. The D-brane charges are [12]

Ai = [Fo] - [?1] - [?1] + [pt], A2 = [Fo] - [?1],

A3 = -[Fo]+2[Pl], A4 = -[Fo] + [Pl ]• The theory on N D2 branes is a

U(N) x U(N) x U(N) x U(N) quiver theory corresponding to Fig. 3, with superpotential

W = eik€jl Tr AiBjCkDi. (8.1)

Consider the moduli space for N = 1. Before turning on superpotential, the moduli space is R x S1, parameterized by oa = r0 and 0o, times X3. The latter is a space of solutions to F-term equations

AC = A2C1, B1D2 = B2D1

and D-term constraints modulo U(1)4. We can rewrite the F-term constraints as D-term constraints of an auxiliary linear sigma model, by introducing 8 new variables

x, y, z, w, x, y, z, w such that

A1 = xy, A2 = xw, C1 = zy, C2 = zw,

B1 = xy, B2 = xw, D1 = z y, D2 = z w.

The D-term constraints now become

|x |2 -|z |2 = r1,

|x|2 -|x|2 = r2,

|x |2 + |z|2-|y|2 - |w|2 = 0,

|z|2 -|^|2 = r3,

|^|2 -|z|2 = r4,

|^|2 + |^|2-|y|2-|»^|2 = 0

and the corresponding U(1) actions can be read off from there. The last two D-terms serve to remove the redundancies inherent in (8.2). The first four D-terms correspond to the four U(1)'s of the quiver. Note that only three of these are independent. There is one constraint on the Fayet-Iliopoulos terms,

r1 + r2 + r3 + r4 = 0.

For, say r1,2,3 > 0, we can solve for x, z, x in terms of Z, and the moduli space is manifestly the line bundle over local F0, obtained by setting

|2 + M2 = 21Z|2 + r1 - r4, |y|2 + |w|2 = 21Z|2 + r1 + r2

modulo the corresponding U(1)'s.

Consider now turning on a specific combination of Chern-Simons terms,

(k1,k2,k3,k4) = (0,k,0, -k).

This corresponds to setting

r2 ~ kr0, r4 ~ -kr0,

while the others are constant. We can use the second D-term in (8.3) to solve for r0, and the corresponding gauge symmetry,

90 ^ 90 + kl, x ^ xe-il, X^Xeil (8.4)

to set 00 to zero. The first and the third D-term express x and z in terms of x and Z. The moduli space is thus

modulo the corresponding U(1)2 action. For k = 1, and r1 = 0 = r2, this gives a moduli space which is a cone over Q1,1,1!

For general k, we still have to divide by the discrete gauge symmetry left over, corresponding to l = 2ni/k in (8.4). A symmetry under which x and x transform as (x, x) ^ (e2ni/kx, e-2ixi/kx), while the rest of coordinates are neutral is gauge equivalent to one where x and Z transform as

It is easy to see from above, that this corresponds to IIA string theory on X3 x R with

where to reproduce the Chern-Simons terms, an additional four-form flux must be turned on, J Grr = k.

As noted in the previous section, the RR 4-form flux in IIA lifts to G-flux in M-theory. Because of the monodromies acting on the cycles in IIA and as we go from positive to negative r0, we don't know how to directly read of the four-form flux in M-theory, even in cohomology. We leave this problem for future work.

In summary, IIA/M-theory duality predicts that the theory dual to the cone on AdS4 x Q1,1,1/Zk should be the U(N)4 Chern-Simons theory corresponding to the quiver in Fig. 3 with superpotential (8.1), and Chern-Simons levels (0,k, 0, -k).20 Since two of the Chern-Simons

|2 +M2 = |x|2 + |Z|2 + r1 + r3, m2 +M2 = |x|2 + |Z|2

(x,Z) ^ (e2ni/kx,e-2ni/kZ).

20 The quotient Q1'1'1 /Zk is also called Yk,k(p1 x p1) in [4,22].

levels vanish, it is not obvious the theory has a weak coupling expansion at large k. The gauge fields associated to nodes 1 and 3 act as constraints, setting the corresponding currents to zero. Since this provides a very simple example of AdS/CFT correspondence, it would be nice to study this theory in detail. The theory may have a dual description in terms of Chern-Simons theories in [9,20], whose direct string theory realization we do not know.

9. N M2 branes on G2 holonomy manifolds

In this section we conjecture a Lagrangian description of N M2 branes probing a local G2 holonomy manifold. For a single M2 brane, the corresponding low energy effective theory was proposed in [23]. We will explain here how this generalizes to arbitrary N. To begin with, consider IIA string theory compactified on a Calabi-Yau threefold X3. The theory has N = 2 supersymmetry in the bulk. Turning on RR 2-form fluxes through the two-cycles of X3 breaks supersymmetry to N = 1 in four dimensions. The IIA string theory on X3 with RR fluxes is dual M-theory on a G2 holonomy manifold. We can get an explicit description of the G2 holonomy manifold [23] along the lines of what we did in Section 2.

Consider again, as we did in Section 2, a Calabi-Yau threefold described by a linear sigma model with N + 4 chiral fields fa and N + 1 U(1) gauge fields with charges Q", a = 0,...,N. The Calabi Yau is the space of solutions to D-term equations

Y.QQI fa |2 = ra, a = 0,...,N i

modulo the gauge transformations. Consider now turning on RR 2-form flux [Frr] = J2 ka [Da ]

where Da is a divisor corresponding to setting to zero a product of variables with charge 1 under the a-th U(1), and zero under the rest. Type IIA string theory in this background should lift to a G2 holonomy manifold in M-theory. It is easy to write down the corresponding G2 manifold, at least topologically. The fluxes imply that the G2 holonomy manifold is a non-trivial S1 fibration over X3. Borrowing the results of Section 2, the G2 manifold is a Hopf-fibration: under the U(1)N+1 gauge transformations that take

fa ^ fa exp(iQaXa) the compact scalar 0o parameterizing the circle transforms as

0o ^ 0o +

The difference with respect to Section 2 is that the Calabi-Yau threefold is no longer fibered over R in a non-trivial way. This implies that the Calabi-Yau fourfold geometries of Section 2 can also be viewed as G2 holonomy fibrations over the real line. We'll return to this point below.

Consider now N D2 brane probing X3. As we reviewed in Section 3, before turning on fluxes, in the limit where X3 develops a singularity, the theory on the branes is an N = 2 quiver gauge theory in three dimensions with superpotential. As we discussed in Section 3, turning on RR fluxes turns on Chern-Simons terms on the D-branes. In fact, in the present context, they correspond to turning on N = 1 Chern-Simons terms. This is because the fluxes break half the

supersymmetry, so the theory on the branes with fluxes should have N = 1 supersymmetry in three dimensions. The corresponds to deforming the action by

where a denotes the nodes on the quiver, and

Scs(Aa) = Tr Mcs(Aa) - Tr XaXa

denotes the N = 1 Chern-Simons term in the notation of [2]. The Chern-Simons levels get related to RR 2-form fluxes and the RR 4-form fluxes through the vanishing cycles, just as in the N = 2 case, and moreover satisfy

For a single brane probe, the center of mass gauge field on the D2 brane can be dualized to a compact scalar B0 parameterizing the position of the M2 brane on the M-theory circle. It is easy to see that, in the presence of N = 1 Chern-Simons terms, the S1 becomes fibered non-trivially over X3 in such a way that the moduli space becomes G2 x R.

For example, take X3 to be the conifold, discussed in Section 4. The theory on N D2 branes on X3 is the Klebanov-Witten type quiver theory described there. Now consider turning on RR 2-form flux through the P1 of the conifold,

but not fibering X3 over R, as we did there. This turns on N = 1 Chern-Simons terms with k1 = k = -k2. Consider now the moduli space, for one D2 brane. The potential for a's vanishes for a1 = r0 = a2 as before, but this no longer affects the FI terms. The D-term potential vanishes for

where r1 = r = -r2 is a constant Fayet-Iliopoulos term. We still need to divide by the gauge group. In the presence of the Chern-Simons coupling, the 00 scalar dual to the center of mass gauge field picks up a charge, so the gauge transformations act as

00 ^ 00 + kl, A1,2 ^ A1,2eil, B1,2e-il.

We can use these to gauge away 00, so the moduli space is a 7-manifold described by the locus (9.1) in C4 parameterized by A1,2, and B12. For k = 1, this is precisely the G2 manifold discussed in [33]. For k = 1, we still have to divide this by the discrete gauge symmetry left over, generated by l = 2n/k so the manifold is a Zk quotient. It is related by the "second" M-theory flop discovered in [34] to the Zk actions considered in [33].

Similarly, for all other examples we wrote down in Sections 5-8, we get pairs of G2 manifolds and the corresponding N = 1 quiver Chern-Simons theories. Namely, as we mentioned above, the Calabi-Yau fourfolds we studied in the N = 2 context can be viewed as G2 holonomy manifolds fibered over R. Turning on RR fluxes in IIA, but not fibering the Calabi-Yau threefold over R corresponds to replacing the N = 2 Chern-Simons couplings in Sections 5-8 by N = 1

J2k* = °

|Ai|2 +|A2|2-|Bi|2 | B212 = r

Chern-Simons couplings. The type IIA/M theory duality then relates this to N M2 branes on a G2 holonomy manifold obtained by fixing a point in the base R of the corresponding Calabi-Yau four-fold X4. Moreover, the RR 2-form fluxes are geometrized in M-theory, but RR 4-form fluxes become G-fluxes on the G2 manifold.

String duality now implies that, in the IR, the theory on N D2 branes in these IIA backgrounds becomes the theory on N M2 branes probing the corresponding G2 holonomy manifolds. Since the gauge kinetic terms vanish in this limit, the theory should naturally become the N = 1 quiver Chern-Simons theory. This provides a Lagrangian description of the low energy theory on the N M2 branes probing a G2 holonomy manifold. Note that there are no known Einstein manifolds which are cones over G2 holonomy manifolds times R [21], so we don't expect these theories to flow to conformal field theories in three dimensions. It will be interesting to see if some insights into the dynamics of these theories could still be obtained.

Acknowledgements

We would like to thank C. Beem and Y. Nakayama for fruitful discussions, and collaboration on an earlier related project. We are also grateful to O. Ganor for valuable discussions. We thank C. Vafa for collaboration on [23] which was crucial for this paper. The author is indebted to A. Hanany for a beautiful talk at BCTP which initiated this work, and discussions at the early stage of the project.

The research of M.A. is supported in part by the UC Berkeley Center for Theoretical Physics and the NSF grant PHY-0457317.

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