Modifying the Sum Over Topological Sectors and Constraints on Supergravity
aa r X i v : . [ h e p - t h ] J u l Modifying the Sum Over Topological SectorsandConstraints on Supergravity
Nathan Seiberg
School of Natural SciencesInstitute for Advanced StudyEinstein Drive, Princeton, NJ 08540The standard lore about the sum over topological sectors in quantum field theory is thatlocality and cluster decomposition uniquely determine the sum over such sectors, thusleading to the usual θ -vacua. We show that without changing the local degrees of freedom,a theory can be modified such that the sum over instantons should be restricted; e.g.one should include only instanton numbers which are divisible by some integer p . Thisconclusion about the configuration space of quantum field theory allows us to carefullyreconsider the quantization of parameters in supergravity. In particular, we show thatFI-terms and nontrivial K¨ahler forms are quantized. This analysis also leads to a newderivation of recent results about linearized supergravity.April 2010 . Introduction This note addresses two issues. The first topic is purely field theoretic and the secondtopic involves supergravity.Our field theory analysis was motivated by supergravity considerations. We wereinterested in topological constraints on supergravity theories which lead to quantization ofthe parameters in the Lagrangian. The proper understanding of the configuration spaceturns out to affect the correct quantization condition of these parameters.For concreteness we will limit ourselves to N = 1 supergravity in four dimensions.We will find it convenient to distinguish two classes of theories:1. Supersymmetric field theories coupled to supergravity. Here we assume that the fieldtheory has no parameter of order the Planck scale M P lanck – the K¨ahler potential andthe superpotential are independent of M P lanck . All the dependence on the Planck scalearises either from the coupling to supergravity or from nonrenormalizable operatorswhich have no effect on the low energy dynamics. Here we exclude theories withmoduli whose target space is of order the Planck scale.2. Intrinsic supergravity theories. Here some couplings of the non-gravitational fieldsare fixed to be of order M P lanck ; i.e. they cannot be continuously varied. In particu-lar, these couplings cannot be parametrically smaller than M P lanck . Therefore, suchtheories do not have a (rigid) field theory limit.In the first class of theories the field theory dynamics decouples from gravity andwe can study supergravity theories by first analyzing the rigid limit. The coupling tosupergravity is determined in the linearized approximation by the energy momentum tensorand supercurrent of the rigid theory.The investigation in [1,2] was limited to theories in the first class. A careful analysis ofthe supersymmetry current has shown that Abelian gauge theories (which include chargedfields) with an FI-term ξ and theories whose target space has non-exact K¨ahler form ω can be coupled to standard supergravity only if the theory has an exact continuous globalsymmetry. However, as emphasized in [1], the absence of global continuous symmetriesin a gravitational theory makes such theories less interesting. Such theories with FI-terms were originally studied in [3-6] and more recently, following [1,2],in [7-9]. To the best of our knowledge the analogous situation with nontrivial topology was notstudied before [2]. (The authors of [10] studied theories in the second class.) ξ or non-exact ω . Followingstandard gauging procedure of the supersymmetry current of these theories has led toa larger supergravity multiplet similar to the one in [11,12]. As emphasized in [13] andelaborated in [2], such theories can be reinterpreted as ordinary supergravity theoriescoupled to a modified matter system which has an additional chiral superfield. The latterfixes the problems of the original field theory by making ξ “field dependent” [14] and byruining the topology underlying the non-exact ω .In this note we examine theories of the second class above in which ξ ∼ M P lanck and the target space has a nontrivial topology with R ω ∼ M P lanck . Such theories areinherently gravitational.In section 2 we discuss a purely field theoretic problem. In many field theories theconfiguration space splits into disconnected sectors labeled by the topological charge n ∈ Z .It is commonly stated that in order to satisfy locality and cluster decomposition one mustsum over all these sectors with a weight factor e inθ . In a Hamiltonian formalism thiscorresponds to considering θ -vacua rather than n -vacua. Section 2 argues that such theoriescan be deformed, without adding local degrees of freedom, such that the instanton summust be modified. For example, in some cases we should sum only over values of n whichare divisible by some integer p .Section 3 is devoted to the constraints on supergravity theories in the second classwhich are intrinsically gravitational. Here we argue that the FI-term is quantized ξ = 2 N M P , N ∈ Z . (1 . H are constrained. However, becauseof the subtleties discussed in section 2, these constraints are weaker than in [10]. Insteadof studying the most general K¨ahler manifold, we focus on CP with the metric ds = f π d Φ d Φ(1 + | Φ | ) (1 . f π is constrained to satisfy f π = 2 Np M P , p, N ∈ Z . (1 . We use notation G N = M P lanck = πκ = 8 πM P ; i.e. M P is the reduced Planck mass. p is the one we find in section 2.We should emphasize that our discussion is incomplete for several reasons:1. We ignore the possible back-reaction of the spacetime metric. As pointed out in[15,16], when topological objects of codimension two are present and a deficit angle inspacetime is generated, there can be additional constraints on the allowed parameters.2. We focus on the classical theory. Perturbative quantum considerations can lead tofurther restrictions like the requirement of anomaly cancelation.3. Nonperturbative quantum effects are also important. For example, the incompatibilityof global continuous symmetries with gravity leads to additional conditions.4. Finally, it is quite likely that there are other more subtle consistency conditions whichwe are not yet aware of.In section 4 we study theories of the first class – rigid supersymmetric field theoriescoupled to supergravity. Here we use the results of section 3 to derive the known resultsabout these theories. In particular, we show that Abelian gauge theories with an arbitraryFI-term ξ can be coupled to supergravity only if the gauge group is noncompact; i.e. it is R rather than U (1). Furthermore, if the rigid theory includes charged fields, it should have acontinuous global R-symmetry. Section 4 also considers theories with a target space whoseK¨ahler form ω is not exact and its periods are arbitrary. Such theories can be coupledto minimal supergravity only if the theory has a continuous global R-symmetry and thetotal wrapping number of spacetime over the target space is constrained. In these twosituations of nonzero ξ and arbitrary ω the resulting supergravity theory has an exactcontinuous global symmetry (and therefore such a theory is not expected to arise from afully consistent theory of quantum gravity).
2. The sum over topological sectors
This section addresses the sum over topological sectors in quantum field theory. In-stead of presenting a general abstract theory, we will discuss simple examples.
As a warmup we review the situation in two dimensional Abelian gauge theories,emphasizing points which will be important below.3e start with the pure gauge U (1) theory on a Euclidean compact spacetime. Theconfiguration space splits to “instantons” labeled by the first Chern class12 π Z F ∈ Z . (2 . e i θ π R F (2 . θ ; i.e. θ ∼ θ + 2 π . Wecan easily add charged particles to this system. Their charges must be quantized.The Hamiltonian interpretation of this system is obtained when spacetime is S × R and we view S as space and R as time. As is well known, the parameter θ is interpreted asa background electric field [17]. The Hamiltonian interpretation of these θ -vacua involvestwo different elements which should not be confused:1. The different values of θ in (2.2) label distinct superselection sectors. Wilson lineoperators exp (cid:0) i H A (cid:1) where the integral is around our S space change the backgroundelectric field by one unit and shift θ by 2 π . Hence the superselection sectors are labeledby − π < θ ≤ π .2. A given superselection sector labeled by θ can include several stable states with dif-ferent value of the background electric field. Unlike the previous point which dependsonly on the configuration space, this is a more detailed issue, which depends also onthe dynamical charges in the system and on the Hamiltonian. If charge p particles arepresent, the background electric field can be screened [17] to be between − p and p .This happens by creating a particle-antiparticle pair, moving one of them around the S space and then annihilating them. Therefore, when p = 1 each superselection sec-tor labeled by − π < θ ≤ π includes p stable states with different background electricfield. If there are several different charged particles with charges p i , the stable valuesof the background electric field in each superselection sector are determined by theirsmallest common factor.If the gauge group is noncompact, the previous situation is modified. On a compactspacetime the condition (2.1) becomes R F = 0 and hence there is no θ parameter. TheHamiltonian formalism interpretation of this fact is that the system does not have su-perselection sectors – Wilson line operators exp (cid:0) ir H A (cid:1) with arbitrary real r can set thebackground electric field to any value. As above, the stability of states with background4lectric field is a more detailed question which depends on the dynamical charges. If thereare no dynamical charges, every state is stable. If there are at least two charges whoseratio is irrational, every background electric field can be screened.Finally, as emphasized, e.g in [18], if spacetime is taken to be R , then the only notionof θ -vacua is a stable state with background electric field. This depends both on the gaugegroup and on the set of dynamical charges. CP model Our second example is the two-dimensional CP sigma model. The Lagrangian of thesystem is L = f π ∂ µ Φ ∂ µ Φ(1 + | Φ | ) . (2 . CP by adding the point at infinity (which is at finitedistance in the metric (2.3)). The patch around Φ = ∞ is related to the other patch bythe transformation Φ → / Φ . (2 . S . Then the configuration space is dividedinto classes labeled by the wrapping number – instanton number I = Z ν ∈ Z , (2 . ν is proportional to the pull back of the K¨ahler form on CP to spacetime and isnormalized such that (2.5) is satisfied. Correspondingly, we can add to the Lagrangian(2.3) a θ term iθν . (2 . θ has period 2 π .It is often stated that we cannot restrict I to any fixed value. This would amount tostudying the “ n -vacua” rather than the “ θ -vacua” and would be in conflict with clusterdecomposition and locality. Here we would like to reexamine this statement.We add to the Lagrangian based on (2.3) and (2.6) δ L = iλ (cid:16) ν − p π F (cid:17) + i b θ π F (2 . p . Here λ ( x ) is a Lagrange multiplier and the twoform F is the field strength of a U (1) gauge theory normalized as in (2.1). Shifting λ bya constant we can set b θ = 0 and obtain a new θ term θ → θ + b θ/p .Has the addition of the terms (2.7) changed the theory? The equation of motion ofthe gauge field of F sets λ to a constant. The Lagrange multiplier λ sets ν = p π F (2 . CP model in (2.7).However, even though the original theory and the theory with (2.7) have the sameset of local degrees of freedom, the two theories are actually different. First, when ourspacetime is compact, the constraint (2.8) leads to I = Z ν ∈ p Z . (2 . p = 1 this condition is uninteresting. For integer p = 1 (2.9) states that the totalinstanton number must be a multiplet of p . Therefore, θ in (2.6) has period 2 π/p . Al-ternatively, we could use the freedom in shifting λ to set θ = 0 and label the vacua by b θ in (2.7) with period 2 π . Second, when our spacetime is S we can solve (2.8) by setting thegauge field equal to the K¨ahler connection up to a gauge transformation. But when ourspacetime has nontrivial one cycles, e.g. when it is S × R or a compact Riemann surface,the constraint (2.8) determines F but leaves freedom in a nontrivial flat gauge field whichshould be integrated over. Correspondingly, this theory has additional operators whichthe underlying sigma model does not have W m = e im H A , m ∈ Z (2 . A is the gauge field and the integral is over any nontrivial cycle. Note that if theintegral is over a topologically trivial cycle, or when m is a multiple of p , the operator W m can be expressed in terms of the sigma model variable and was present before the theorywas modified.Considering the simple case where spacetime is S , the partition function of the theoryis related to the usual partition function of the p = 1 theory as follows Z p ( θ ) = p − X n =0 Z p =1 (cid:18) θ + 2 πnp (cid:19) . (2 . If p = mn is rational (with m and n coprime), I = R ν ∈ m Z and if p is irrational, I = R ν = 0.
6n conclusion, the modified theory is locally the same as the original sigma model, butglobally it is different. In particular, the instanton sum is performed differently in the twotheories.We see that the standard lore about instantons in the CP theory corresponds to p = 1.However, for generic integer p we find a theory with exactly the same local structure, butwith the constraint (2.9) on the total instanton number. Since we added to the originalLagrangian (2.3)(2.6) the local term (2.7), it is clear that the resulting theory is local!We would like to make several comments:1. In the p → ∞ limit the total instanton number I = R ν must vanish. One way to seeit is by first rescaling the gauge field thus turning the U (1) gauge theory into an R gauge theory in which R F = 0. The same result is obtained for irrational p in (2.7).2. We are used to studying standard field theories based on local degrees of freedom inwhich the global structure does not matter much. We are also familiar with topolog-ical field theories which have no local degrees of freedom and whose entire dynamicsdepends on the global structure. The theories we study here can be viewed as standardlocal theories coupled to topological theories. In our example above the topologicaltheory is a BF-theory [19], where the Lagrange multiplier λ plays the role of the B field and also couples to the sigma model variables.3. We can get further insight into the role of the gauge field in (2.7) by adding to thesystem a massive field ϕ which couples to the gauge field with charge q ∈ Z . Workingon S we can easily integrate out λ and the gauge field to find that the massive field ϕ couples to the massless mode Φ. For p = 1 it couples to the K¨ahler connection ofthe CP and it ends up being a section of a line bundle on the CP . However, when p = 1 and q is not a multiple of p such an interpretation is not possible. Yet, whenthe constraint (2.9) is satisfied the massive field ϕ is single valued and well defined onour S spacetime.4. We can add to the system several massive fields ϕ with various charges and U (1) gaugeinvariant interactions. When the Lagrange multiplier λ is integrated out, it eliminatesthe dynamical gauge field and the effective theory has a global U (1) symmetry. Thisglobal symmetry will play an important role below.5. As in our warmup discussion in section 2.1, the Hamiltonian interpretation of thissetup is as follows. We study the system on S × R and view S as space and R as time. The new operators (2.10) which wind around our space change b θ by 2 πm ,7r equivalently, they change θ of the underlying CP theory by πmp . Hence, as inthe original CP theory, the Hilbert space includes distinct states with − π < θ ≤ π ,but the different superselection sectors are labeled by − πp < θ ≤ πp . The sum in theright hand side of (2.11) can be interpreted as a sum over the p different values ofthe background field in the same superselection sector. When the system includesadditional charged fields ϕ with q = 1, only one of the p different states labeled by − π < θ ≤ π in the same superselection sector is stable. The others can decay throughpair production and annihilation of the ϕ particles as in [17]. The construction above has a number of obvious generalizations.We can repeat this construction with any two-dimensional nonlinear sigma modelon any target space. The total instanton number associated with any two-cycle can beconstrained by adding an additional gauge field and a Lagrange multiplier as in (2.7).Another obvious generalization is to a higher dimensional spacetime. Clearly, we canconstrain the total winding number around any cycle. Here λ in (2.7) is a two form (as inBF-theories) and b θ = 0.A somewhat more interesting generalization is to non-Abelian Yang-Mills theory infour dimensions. The total instanton number can be constrained to be a multiple of p byadding to the Lagrangian iλ ( ν − p π F (4) ) + i b θ π F (4) (2 . λ is a Lagrange multiplier, the four form ν is the Pontryagin density normalized suchthat R ν ∈ Z , and F (4) is a four form field strength of a three form gauge field normalizedsuch that R F (4) ∈ π Z . The periodic variable b θ ∼ b θ + 2 π takes the role of the ordinary θ angle which now has period 2 π/p . With irrational p or with the gauge group of F (4) beingnoncompact we have R F (4) = 0 which leads to R ν = 0. Hence this theory does not havedistinct superselection sectors labeled by θ . The construction (2.7) might look contrived. Therefore, we now present a more fa-miliar theory which leads to the same effect, but the added topological degrees of freedom This does not lead to a solution to the strong CP problem. Instead, as in the discussion insection 2.1, the gauge theory of F (4) can have a background “electric field” which plays the roleof θ . But since our system does not include dynamical charged 2-branes which couple to the threeform gauge field, this background electric field cannot be screened and it is stable. z i =1 , which are charged under a U (1) gauge field A . Normally, they are taken to have charge one, but we take them to have charge p . The scalars are subject to a potential with a minimum along the space | z | + | z | = f π .The low energy theory can easily be found. If z = 0 we can parameterize it by the gaugeinvariant (inhomogeneous) coordinate Φ = z z , (2 . z = f π e iα p | Φ | . (2 . pA = (cid:18) i Φ ∂ Φ − Φ ∂ Φ2(1 + | Φ | ) − ∂α (cid:19) dx (2 . CP model and the winding number is constrained by (2.9).However, globally these two systems are different. In section 2.2 we added a flat U (1) gaugetheory, while here it is a Z p gauge theory. This Z p is the unbroken part of the underlying U (1) gauge theory when z i get nonzero vevs. As in section 2.2, we can add to this theorymassive charged fields ϕ which can induce transition between different, otherwise stable,“ θ -vacua” in the same superselection sector. In section 2.2 the local degrees of freedomwere the CP fields and the massive fields ϕ with a global U (1) symmetry (which wasassociated with the constrained U (1) gauge theory). Here we have the same variables buttheir Lagrangian has only a Z p symmetry.For use in later sections we consider now a supersymmetric version of this theory.There are two charged chiral superfields z , with a K¨ahler potential which includes anFI-term K = | z | e pV + | z | e pV − ξV . (2 . p, ξ > z z . (2 . Such a system was considered from a more mathematical point of view in [20-22]. V . Its equation of motion p ( | z | + | z | ) e pV = ξ (2 . V = − p log(1 + | Φ | ) + chiral + chiral (2 . K eff = ξp log(1 + | Φ | ) . (2 . f π = ξp . (2 . U (1), then the total wrapping number must bea multiple of p . And if the gauge group is R , the total wrapping number must vanish.
3. Constraints on Supergravity Theories
As a preparation for our discussion, we recall the well known fact that under theK¨ahler transformation (here and elsewhere in this section we set M P = 1) K → K + Ω + Ω (3 . W , the matter fermions χ j , the gauginos λ , and the gravitino ψ µ trans-form as W → e − Ω W ,χ j → e (Ω |− Ω | ) χ i ,λ → e − (Ω |− Ω | ) λ ,ψ µ → e − (Ω |− Ω | ) ψ µ , (3 . | denotes the θ = θ = 0 component of Ω.10 .1. Theories with FI-terms We start by studying theories with FI-terms where K = ... − ξV . (3 . V → V + Λ + Λ (3 . − ξ Λ . (3 . W → e ξ Λ W , which meansthat the gauge symmetry is an R -symmetry under which the superpotential has charge − ξ , and hence the supersymmetry coordinate θ has charge − ξ and the gravitino ψ µ hascharge − ξ . If a superfield Φ j has charge q j (i.e. it transforms as Φ j → e − q j Λ Φ j ), thefermion χ j has charge q j + ξ . (As a check, use Wess-Zumino gauge where the remaininggauge freedom is Λ = iα with real α . Then use Ω = − iξα in (3.2) to find the charges ofthe various fields.)Is this compatible with the gauge symmetry of the problem? Let us first assume thatthe gauge symmetry is U (1) such that Λ in (3.4) is identified with Λ + 2 πi . Then, chargequantization clearly implies that the scalars have integer charges; i.e. q j ∈ Z . Examiningthe charges of the fermions we learn that the FI-term must be quantized ξ = 2 N with N ∈ Z . (3 . R rather than U (1), no condition like (3.6) isrequired.We would like to make three comments about these theories:1. One might question the applicability of the condition (3.6) in supergravity when itis viewed as the low energy approximation of some more complete quantum gravitytheory. Then one might not want to consider coupling constants which are of orderthe Planck scale.2. The theory includes charged fermions and one must make sure that all the anomaliesare properly canceled. For a recent discussion of anomalies in such theories see [25-28].3. To the best of our knowledge no example of theories satisfying (3.6) were constructedin string theory. This suggests that perhaps a deeper consistency condition might ruleout some or even all of them. Quantization of ξ was considered by various people including [23,24]. .2. Theories with a nontrivial K¨ahler potential Next, we consider theories in which the K¨ahler form of the target space is not exact.As a typical example, we study the CP model. Following the discussion around (2.16) weconstruct it in terms of a linear model of two charged chiral superfields z , with charge p . Again, if the gauge group is U (1), we have the condition (3.6). Combining this with(2.21) we have f π = ξp = 2 Np with p, N ∈ Z . (3 . p = 1 equation (3.7) is the condition of Witten and Bagger [10]. However, we findthat there is freedom in an arbitrary integer p . As explained in section 2, it correspondsto a Z p gauge theory. In this case of a supergravity theory this Z p symmetry is an R-symmetry. Our fermions and gravitino transform under this Z p symmetry and they playthe role of the massive field ϕ in section 2. This allows the p = 1 theory to be consistent,despite the fact that the condition of [10] is not satisfied.It is interesting to consider the p → ∞ limit in which f π in (3.7) can be arbitrary. Inthis limit the discrete symmetry Z p becomes Z . One way to analyze this limit is to rescale V . This effectively makes the gauge group noncompact. As we remarked after (3.6), if thegauge group is R , the FI-term ξ is arbitrary and therefore f π is also arbitrary.One might object to using an effective Lagrangian with f π ∼ M P . However, as iscommon in string constructions, there are examples where the entire low energy theory isunder control when various moduli change over Planck scale distances.Finally, we would like to stress again that our condition (3.7) ignores various additionalconsiderations. For example, we might want to examine the consistency of the functionalintegral only for configurations which are close to solutions of the equations of motion.In other words, when studying topological objects (like the instantons considered here)we might want to take into account the back reaction on the metric. This can changethe underlying spacetime and lead to different consistency conditions from the ones wediscussed here.
4. Recovering the Rigid Limit
In this section we study theories in which the rigid limit M P → ∞ leads to super-symmetric field theories with a nonzero FI-term or with a nontrivial K¨ahler form. (Forthat we restore the dimensions by appropriate factors of M P .) This allows us to connect12ith the results of [2] and earlier references (see also the recent paper [29] and referencestherein).We start by considering an Abelian gauge theory with an FI-term ξ . Equation (3.6)does not allow us to find a smooth rigid limit ( M P → ∞ ) with finite ξ . Therefore, theAbelian gauge group must be noncompact.Furthermore, the fact that for finite M P the gauge group acts as an R-symmetry ( θ hascharge − ξ M P ) puts interesting restrictions on the theory. Consider first the rigid theoryand assume that it has some matter fields Φ j with gauge charges q j . When gravitationalcorrections are turned on the charges of the bosons can shift q j → q j − r j ξ M P + O (cid:18) ξ M P (cid:19) (4 . r j . At the same time θ becomes charged and hence the su-perpotential must carry gauge charge − ξM P . Consider a typical term in the superpotential W ⊃ Φ j Φ j · · · (4 . q j + q j + ... = 0 (4 . r j + r j + ... = 2 . (4 . j has R-charge r j . Equivalently, the supergravity theory has a global continuous non-R-symmetry under which Φ j has charge q j . We conclude that, if a theory with nonzero FI-term is to have a rigid limit, its gaugegroup must be R and the rigid theory should have a global R-symmetry [4-6]. Furthermore,the supergravity theory has a continuous global symmetry [1]. More precisely, for this conclusion to be valid we need to make two assumptions. First, forthe global symmetry to be nontrivial, we need to assume that at least one chiral superfield has q j = 0. Second, we ignore the singular possibility of including terms in the Lagrangian in whichthe number of fields Φ j diverges in the rigid limit like a power of 2 M P /ξ .
13e should emphasize that this conclusion about the presence of a global continuoussymmetry follows from our assumption in this section that ξM P is parametrically small.In the context of the discussion in section 3 we can easily find supergravity theories withno global symmetries. For example, let ξ = 2 M P (i.e. N = 1 in (3.6)) and consider atheory with two chiral superfields Φ ± with gauge charges ±
1. Then, the superpotential W = Φ − ( a + a (Φ + Φ − ) + a (Φ + Φ − ) + ... ) with come constants a i carries the desiredgauge charge without additional global symmetry.Next we discuss theories with a nontrivial K¨ahler form. Here we want to considerthe rigid limit M P → ∞ with fixed f π . Constructing such theories using gauged linearmodels we can use the result above that the gauge group must be R and the rigid theoryshould have a global continuous R-symmetry. Alternatively, we can use (3.7) and take the M P → ∞ limit together with p → ∞ .We conclude that if we are willing to consider supergravity theories with continuousglobal symmetries, not only can we have theories with FI-terms, we can also have sigmamodels with nontrivial K¨ahler forms [2]. An alternate way to construct these supergravitytheories is to consider the “new minimal” auxiliary fields of supergravity [30,31]. Thisamounts to gauging the R-multiplet rather than the Ferrara-Zumino multiplet [2] (seealso [7,8]).We should stress, however, that a consistent theory of quantum gravity cannot haveany global continuous symmetries. Therefore such supergravity theories cannot be real-ized [1,2]. This is an example of a point we have made a number of times above thatour classical considerations lead only to necessary conditions and it is quite possible thatadditional, more subtle considerations put further restrictions on the theories studied here.
Acknowledgements
We would like to thank Z. Komargodski and M. Rocek for participation in early stagesof this project and for many useful comments. We have also benefitted from useful discus-sions with O. Aharony, N. Arkani-Hamed, J. Distler, M. Douglas, D. Freed, J. Maldacena,G. Moore, S. Shenker, Y. Tachikawa, and E. Witten. The work of NS was supported inpart by DOE grant DE-FG02-90ER40542. Any opinions, findings, and conclusions or rec-ommendations expressed in this material are those of the author(s) and do not necessarilyreflect the views of the funding agencies. The authors of [32] argued that globally this theory differs from the more standard “oldminimal” theory. eferences [1] Z. Komargodski and N. Seiberg, “Comments on the Fayet-Iliopoulos Term in FieldTheory and Supergravity,” JHEP , 007 (2009) [arXiv:0904.1159 [hep-th]].[2] Z. Komargodski and N. Seiberg, “Comments on Supercurrent Multiplets, Supersym-metric Field Theories and Supergravity,” arXiv:1002.2228 [hep-th].[3] D. Z. Freedman, “Supergravity With Axial Gauge Invariance,” Phys. Rev. D , 1173(1977).[4] R. Barbieri, S. Ferrara, D. V. Nanopoulos and K. S. Stelle, “Supergravity, R InvarianceAnd Spontaneous Supersymmetry Breaking,” Phys. Lett. B , 219 (1982).[5] R. Kallosh, L. Kofman, A. D. Linde and A. Van Proeyen, “Superconformal symmetry,supergravity and cosmology,” Class. Quant. Grav. , 4269 (2000) [Erratum-ibid. ,5017 (2004)] [arXiv:hep-th/0006179].[6] G. Dvali, R. Kallosh and A. Van Proeyen, “D-term strings,” JHEP , 035 (2004)[arXiv:hep-th/0312005].[7] K. R. Dienes and B. Thomas, “On the Inconsistency of Fayet-Iliopoulos Terms inSupergravity Theories,” arXiv:0911.0677 [hep-th].[8] S. M. Kuzenko, “The Fayet-Iliopoulos term and nonlinear self-duality,” arXiv:0911.5190[hep-th].[9] S. M. Kuzenko, “Variant supercurrent multiplets,” arXiv:1002.4932 [hep-th].[10] E. Witten and J. Bagger, “Quantization Of Newton’s Constant In Certain Supergrav-ity Theories,” Phys. Lett. B , 202 (1982).[11] G. Girardi, R. Grimm, M. Muller and J. Wess, “Antisymmetric Tensor Gauge Po-tential In Curved Superspace And A (16+16) Supergravity Multiplet,” Phys. Lett. B , 81 (1984).[12] W. Lang, J. Louis and B. A. Ovrut, “(16+16) Supergravity Coupled To Matter: TheLow-Energy Limit Of The Superstring,” Phys. Lett. B , 40 (1985).[13] W. Siegel, “16/16 Supergravity,” Class. Quant. Grav. , L47 (1986).[14] M. Dine, N. Seiberg and E. Witten, “Fayet-Iliopoulos Terms in String Theory,” Nucl.Phys. B , 589 (1987).[15] B. R. Greene, A. D. Shapere, C. Vafa and S. T. Yau, “Stringy Cosmic Strings AndNoncompact Calabi-Yau Manifolds,” Nucl. Phys. B , 1 (1990).[16] S. Ashok and M. R. Douglas, “Counting flux vacua,” JHEP , 060 (2004)[arXiv:hep-th/0307049].[17] S. R. Coleman, “More About The Massive Schwinger Model,” Annals Phys. , 239(1976).[18] T. Banks, M. Dine and N. Seiberg, “Irrational axions as a solution of the strongCP problem in an eternal universe,” Phys. Lett. B , 105 (1991) [arXiv:hep-th/9109040]. 1519] G. T. Horowitz, “Exactly Soluble Diffeomorphism Invariant Theories,” Commun.Math. Phys. , 417 (1989).[20] T. Pantev and E. Sharpe, “Notes on gauging noneffective group actions,” arXiv:hep-th/0502027.[21] T. Pantev and E. Sharpe, “GLSM’s for gerbes (and other toric stacks),” Adv. Theor.Math. Phys. , 77 (2006) [arXiv:hep-th/0502053].[22] A. Caldararu, J. Distler, S. Hellerman, T. Pantev and E. Sharpe, “Non-birationaltwisted derived equivalences in abelian GLSMs,” arXiv:0709.3855 [hep-th].[23] E. Witten, unpublished[24] J. Distler and B. Wecht, Unpublished, mentioned in http://golem.ph.utexas.edu/ ∼ distler/blog/archives/002180.html [25] A. H. Chamseddine and H. K. Dreiner, “Anomaly Free Gauged R Symmetry In LocalSupersymmetry,” Nucl. Phys. B , 65 (1996) [arXiv:hep-ph/9504337].[26] D. J. Castano, D. Z. Freedman and C. Manuel, “Consequences of supergravity withgauged U(1)-R symmetry,” Nucl. Phys. B , 50 (1996) [arXiv:hep-ph/9507397].[27] P. Binetruy, G. Dvali, R. Kallosh and A. Van Proeyen, “Fayet-Iliopoulos termsin supergravity and cosmology,” Class. Quant. Grav. , 3137 (2004) [arXiv:hep-th/0402046].[28] H. Elvang, D. Z. Freedman and B. Kors, “Anomaly cancellation in supergravity withFayet-Iliopoulos couplings,” JHEP , 068 (2006) [arXiv:hep-th/0606012].[29] T. Kugo and T. T. Yanagida, “Coupling Supersymmetric Nonlinear Sigma Models toSupergravity,” arXiv:1003.5985 [hep-th].[30] V. P. Akulov, D. V. Volkov and V. A. Soroka, “On The General Covariant TheoryOf Calibrating Poles In Superspace,” Theor. Math. Phys. , 285 (1977) [Teor. Mat.Fiz. , 12 (1977)].[31] M. F. Sohnius and P. C. West, “An Alternative Minimal Off-Shell Version Of N=1Supergravity,” Phys. Lett. B , 353 (1981).[32] N. D. Lambert and G. W. Moore, “Distinguishing off-shell supergravities with on-shellphysics,” Phys. Rev. D72