VT-IPNAS 10-20

Sums over Topological Sectors
and Quantization of Fayet-Iliopoulos Parameters
Simeon Hellerman1 and Eric Sharpe2
1

Institute for Physics andMathematics of the Universe (IPMU)
5-1-5 Kashiwanoha
Kashiwa, 277-8583, Japan2

Physics Department
Virginia TechBlacksburg, VA 24061

simeon.hellerman.1@gmail.com, ersharpe@vt.edu

In this paper we discuss quantization of the Fayet-Iliopoulos parameter in supergravity
theories with altered nonperturbative sectors, which were recently used to argue a fractional
quantization condition. Nonlinear sigma models with altered nonperturbative sectors are
the same as nonlinear sigma models on special stacks known as gerbes. After reviewing theexisting results on such theories in two dimensions, we discuss examples of gerby moduli‘spaces’ appearing in four-dimensional field theory and string compactifications, and theeffect of various dualities. We discuss global topological defects arising when a field or
string theory moduli space has a gerbe structure. We also outline how to generalize resultsof Bagger-Witten and more recent authors on quantization issues in supergravities fromsmooth moduli spaces to smooth moduli stacks, focusing particular attention on stacks thathave gerbe structures.
December 20101

Contents
1 Introduction

3

2 Review of two-dimensional theories with altered topological sectors

4

3 Four-dimensional physics

9

4 Examples, duality in gerby moduli ‘spaces’

13

4.1

Field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

4.2

String theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

5 Topological defects and gerby moduli spaces

21

6 Consistency conditions on classical supergravity

24

6.1

Review of standard supergravity case . . . . . . . . . . . . . . . . . . . . . .

24

6.2

Generalization to smooth Deligne-Mumford stacks . . . . . . . . . . . . . . .

25

7 Conclusions

32

8 Acknowledgements

34

A Four-dimensional decomposition conjecture34

B Two-dimensional BF theory and cluster decomposition

35

References

47

2

1

Introduction

Recently there has been much progress in understanding Fayet-Iliopoulos parameters insupergravity, generalizing work of e.g. Bagger-Witten [1], see for example [2, 3, 4, 5, 6,
7, 8, 9]. In particular, the recent paper [7] argued that in the special case of linearlyrealized group actions, Fayet-Iliopoulos parameters could be interpreted as charges for aU(1) gauge symmetry, and so are quantized. This result was generalized in [8] to the morenearly generic case of nonlinearly-realized group actions, by demonstrating that the FayetIliopoulos parameters determine the lift of the group action to the Bagger-Witten [1] line
bundle. As such lifts of group actions are quantized, the Fayet-Iliopoulos parameters are
therefore also quantized.
This paper will focus on another aspect of [7], specifically, a proposal for Fayet-Iliopoulos
quantization when the moduli space is defined by two-dimensional models with a restriction
on allowed instantons. Two-dimensional sigma models with restrictions on allowed instantons
have been discussed previously in e.g. [10, 11, 12, 13, 14, 15, 16], and are the same as sigma
models on gerbes, special kinds of stacks.
Schematically, smooth stacks are “manifolds paired with automorphisms.” Stacks admit
metrics, spinors, and all the other structures appearing in classical field theories. The original interest in stacks in the physics community revolved around using them to form new
string compactifications, new conformal field theories, and applying them to give a more
fundamental understanding of certain existing compactifications.
Previous work on consistency conditions in supergravity theories has assumed that themoduli space is a smooth manifold. However, in mathematics, moduli ‘spaces’ are usually
stacks, and not manifolds, so to have a broad understanding of classical consistency conditions on supergravity theories, one must understand cases in which the moduli ‘space’ ofthe supergravity is a stack. This paper is a step in a program of understanding consistency
conditions for such more general cases.
To be more specific, in this paper we will discuss generalizations of consistency conditions
on supergravities from moduli spaces that are manifolds to moduli ‘spaces’ that are smooth
Deligne-Mumford stacks, focusing particular attention on stacks that are gerbes over manifolds. That said, in typical examples arising in string compactifications, the moduli stackhas singularities, so our generalization to stacks will still not describe all cases pertinent tostring compactifications, but is a step towards a directly pertinent treatment.
We begin in section 2 by reviewing two-dimensional sigma models on stacks, focusing inparticular on gerbes over manifolds. Two-dimensional sigma models on gerbes over manifolds
look like sigma models on the underlying manifolds but with a restriction on topologicalsectors. These have been discussed in considerable detail in both the mathematics andphysics literature, as we review.
3In section 3 we discuss analogous four-dimensional theories. There are some significant
differences between two-dimensional and four-dimensional cases, including issues aroundpresentation dependence, and (on R4 ) a lack of nonperturbative sectors in gerbe theories.
In section 4 we discuss particular examples of both field and string theories whose moduli‘spaces’ are gerbes over manifolds. In particular, previous work on gerbe structures in supergravity moduli spaces [7] did not give any examples of string compactifications in whichsuch structures would arise, which we remedy here. We discuss the physical impact of suchgerbe structures, and also discuss the action of duality groups.
In section 5 we discuss global topological defects in theories with gerby moduli spaces.
Topological defects are classified by homotopy of the moduli space, and gerbe structurescontribute nontrivially to the homotopy. We discuss whether the contributions to homotopy
from gerbe structures have physical meaning.
In section 6 we outline how to generalize consistency conditions on classical supergravitiesin [1, 8] to moduli ‘spaces’ that are smooth Deligne-Mumford stacks, focusing in particularon the case of stacks that are gerbes. In particular, we discuss the case of Bagger-Witten [1]line bundles that are ‘fractional’ over the gerby moduli space.
In appendix A we discuss a four-dimensional analogue of the ‘decomposition conjecture’[14] that plays a vital role in understanding two-dimensional sigma models on gerbes. In thisfour-dimensional analogue, we restrict sums over four-dimensional instantons – as a result,the four-dimensional version is not directly relevant to four-dimensional sigma models on
gerbes, but nevertheless we thought it appropriate to discuss here.
Finally, in appendix B we discuss two-dimensional BF theory and analogues of gerbe
structures and decomposition statements there.
While this work was being completed, the paper [9] appeared, which has nontrivial overlap.

2

Review of two-dimensional theories with altered topological sectors

The recent paper [7] discussed theories defined by restricting sums over instantons to a subsetof all instantons. In this section we briefly review some of the previous work done on suchtheories.
In the case of two-dimensional nonlinear sigma models, a nonlinear sigma model in which
the sum over worldsheet instantons is restricted to a subset of all instantons is the same asa string on a gerbe, a special kind of stack, as is discussed in the physics literature in for4example [10, 11, 12, 13, 14, 15, 16, 17, 18] and reviewed in conference proceedings including[19, 20, 21]. (There is also a significant mathematics literature on Gromov-Witten invariantsof stacks and gerbes; see for example [22, 23, 24, 25] for a few representative examples.)Briefly, a stack is a manifold “paired with automorphisms.” (See e.g. [26, 27, 28] for amore technical definition.) At the same level of brevity, a gerbe is a stack in which one has
the same automorphisms everywhere. Mathematically, a gerbe can be thought of locally as
covered by patches of the form [U/G] where U is an open set and G acts trivially on U.
Stacks keep track of even trivial group actions, and so [U/G] is distinguished (as a stack)from just U.
One of their properties that plays a role in this paper is that if G is a gerbe over a manifold
M, then maps into G are equivalent to maps into M with a restriction on their degree, as
discussed in for example [13]. Briefly, a map from any space X into a gerbe G over M isequivalent1 to a map f : X → M into the underlying space, together with a trivialization off ∗ G. For a Zk gerbe, say, f induces a map
f ∗ : H 2 (M, Zk ) −→ H 2 (X, Zk )which maps the characteristic class of the gerbe G (an element2 in H 2 (M, Zk )) to the characteristic class of f ∗ G, which should vanish (otherwise f ∗ G would not admit a trivialization).
For example, if X = P1 and M = PN −1 , then f is characterized by an integer, its degree. In
this case, f ∗ acts by multiplication by the degree of f , so if we let n denote the characteristicclass of the gerbe (mod k), and d the degree of f , then we have the constraintd (n mod k) = 0 mod k.
In other words, dn must be a multiple of k, a constraint on the allowed degrees of maps f .
(Note the constraint depends upon the characteristic class of the gerbe – for example, for atrivial gerbe, n ≡ 0 mod k, and so there is no constraint.)
In mathematics, moduli ‘spaces’ are usually stacks, hence one should not be surprised
to find stack structures arising in moduli spaces of interest to physicists. Indeed, in this1

There is a closely related statement for bundles. Given a map g : X → E for some bundle π : E → M ,we can compose with π to produce a map f : X → M . Furthermore, we can define a trivialization of f ∗ E
canonically: recallf ∗ E ≡ {(x, e) ∈ X × E | f (x) = π(e)}so we can define a trivialization X → f ∗ E by x 7→ (x, g(x)) for x ∈ X. Conversely, If E is a bundle over aspace M , and we are given a map f : X → M and a trivialization of f ∗ E, then since there is a canonical map
f ∗ E → E, the trivialization X → f ∗ E can be composed with that canonical map to form a map g : X → E,
whose composition with the projection is f again.
2
In general, the characteristic class of a G-gerbe on a manifold X is an element of H 2 (X, C ∞ (G)). Here,
since G is assumed finite, C ∞ (G) = G. For U (1) gerbes,
H 2 (X, C ∞ (U (1))) = H 3 (X, Z)
and in this fashion one recovers the usual physics description in terms of the curvature H of the B field.

5paper we shall discuss examples of moduli ‘spaces’ with nontrivial stack structures arisingin physics, and their analysis.
In previous work e.g. [11, 12, 13], sigma models on smooth Deligne-Mumford stacks
were defined physically by using the fact that essentially3 all such stacks can be described
as quotients [X/G] for X some space and G some group acting on X. The special caseof a gerbe corresponds to a subgroup of G acting trivially on X. (We shall review how
physics keeps track of even trivial group actions.) To such a quotient we associate a Ggauged sigma model on X. A given stack can admit multiple presentations of this form; we
associate universality classes of renormalization group flow to particular stacks. Much effort
was expended in previous work to check presentation-independence of universality classes.
A standard example4 of a gerbe is a Zk gerbe on a projective space, defined by a supersymmetric Pn−1 model, a linear gauge theory, with n chiral superfields Φi each of charge
k instead of charge 1. As discussed in [11], in two dimensions such theories are nonperturbatively distinct from the ordinary Pn−1 model. On a noncompact worldsheet, this canbe seen by thinking about periodicity of the two-dimensional theta angle – such theories
can be distinguished by the existence of massive minimally-charged objects, which alter theperiodicity.
Let us work through this argument in more detail. As described in [11][section 3], sincein two dimensions the θ angle couples to Tr F , we can determine the instanton numbers
through the periodicity of θ. Suppose we have the physical theory described above, namelya GLSM with Higgs fields of charge k, plus two more massive fields, of charges +1 and −1. Ina two-dimensional theory, the θ angle acts as an electric field, which can be screened by pair
production, and that screening determines the periodicity of θ. If the only objects we could
pair produce were the Higgs fields of charge k, then the theta angle would have periodicity
2πk, and so the instanton numbers would be multiples of 1/k. However, since the space isnoncompact, and the electric field fills the entire space, we can also pair produce arbitrary
numbers of the massive fields, which have charges ±1, and so the θ angle has periodicity
2π, so the instantons have integral charges. In particular, even if the masses of the massive
fields are beyond the cutoff scale, the theta angle periodicity can still know about them, andso they can still help determine the low-energy effective field theory.
We can phrase this more simply as follows. In a theory with only Higgs fields of chargek, the instanton numbers are multiples of 1/k, and so the resulting physics is equivalent tothat of a GLSM with minimal charges. However, if we add other fields of charge ±1, thenthe instanton numbers are integral, and if those fields become massive, and we work at anenergy scale below that of the masses of the fields, then we have a theory with Higgs fields of3

See [12] for a discussion of rare counterexamples, and their apparent lack of physical relevance.
This Zk gerbe has characteristic class −1 mod k, so from the previous analysis, maps into the gerbe
are maps into the underlying projective space of degree divisible by k, as should also be clear from thedescription of the gauge theory.
46

charge k, and integral instanton numbers, giving us the physics that corresponds to a gerbe
target. (This argument was, to our knowledge, first developed by J. Distler and R. Plesser
at an Aspen summer meeting in 2004, used with their permission in [11][section 3] and thenalso described much more recently in [7].)On a compact worldsheet, this distinction between minimal and nonminimal charges isa consequence of how matter couplings are defined globally (i.e. as sections of bundles). In
detail, to uniquely define the theory nonperturatively on a compact space, we must specify,
by hand, the bundles that the Higgs fields couple to. If the gauge field is described by aline bundle L, then coupling all of the Higgs fields to L⊗k is a different prescription fromcoupling all of the Higgs fields to L. As a result, the spectrum of zero modes differs between
the two theories, hence correlation functions and anomalies differ between the two theories.
Some of the structure above – such as the theta angle argument – is specific to twodimensions, but some will generalize. Later, we will argue that gerbes are relevant to fourdimensional physics when either the four-dimensional spacetime is topologically nontrivial,or there are massive states which are not invariant, mirroring aspects of the two-dimensional
story above.
In any event, here are a few physical consequences of this distinction between minimallyand nonminimally-charged theories in this two-dimensional example [13]:• The axial U(1)A of the supersymmetric Pn−1 model is broken to Z2kn by instantons
instead of Z2n , when the chiral superfields have charge k.
• The quantum cohomology ring is C[x]/(xkn − q) instead of C[x]/(xn − q), reflecting thefact that A model correlation functions differ. (The paper [13] provided GLSM-based
physical computations of these quantities, as well as a mirror symmetry computation
verification, and in addition there are also mathematical definitions and matching
results; see for some representative examples [23, 22, 24, 25].)• The (Toda) Landau-Ginzburg mirror can be described by a superpotential with a fieldΥ valued in kth roots of unity,
W = eY1 + eYn−1 + Υe−Y1 −···−Yn−1 ,reflecting both the fact that the theory on a gerbe is equivalent to a theory on a disjoint
union of spaces, and the fact that mirror symmetry dualizes nonperturbative effects
into perturbative ones. (This result was physically derived in [13] from duality for
GLSM’s ala [29, 30], and also independently derived in e.g. [25] from mathematicalconsiderations.)
So far we have outlined how noneffective continuous group actions can lead to new physics;
the same is true of finite group actions. For example, consider the orbifold [T 6 /D4 ] where7

D4 is an eight-element group that projects onto Z2 × Z2 :1 −→ Z2 −→ D4 −→ Z2 × Z2 −→ 1.
To specify the orbifold, we must specify the action of D4 on T 6 . Let us take the Z2 center to
act trivially, so that the D4 acts by first projecting to Z2 × Z2 , and then act with a standard
Calabi-Yau action of Z2 × Z2 on T 6 , as described in e.g. [31]. Since the Z2 center acts
trivially, one might naively assume that the [T 6 /D4 ] orbifold would be physically equivalent
to a [T 6 /Z2 × Z2 ] orbifold. Instead, one computes that at one-loop, for example, [14][section
5.2]


a

Z [T 6 /D4 ] = Z [T 6 /Z2 × Z2 ] [T 6 /Z2 × Z2 ]d.t. ,where the subscript indicates the presence of discrete torsion in one of the two factors. We
therefore see explicitly that, in this example, the string orbifold knows about the triviallyacting Z2 subgroups. Many additional examples have been described in detail in the references. Thus, string orbifolds know about trivially-acting subgroups, just as we saw in
two-dimensional gauge theories.
Technically, in (2,2) supersymmetric cases, these two-dimensional theories (in which a
trivially-acting group have been gauged) do not obey cluster decomposition. (This is animmediate consequence of Weinberg’s ancient argument for theories with any restriction on
instanton degrees, and can also be seen for CFT cases by, for example, computing massless
spectra and observing multiple dimension zero operators.) This would be a problem were
it not for the fact that they are equivalent to nonlinear sigma models on disjoint unions ofordinary spaces [14], a result described there as the ‘decomposition conjecture.’ The latteralso do not obey cluster decomposition, but are obviously nevertheless under good control, inthe sense that we can renormalize by local counterterms, and so forth. Thus, they are sumsof theories which obey cluster decomposition, and so at least morally are “local” theories.
One of the original reasons for interest in these gerbe theories was the idea that they might
define new SCFT’s, new string compactifications. Because of the decomposition conjecture,that is not really the case in (2,2) theories, as one gets sums of existing theories. In (0,2)
theories, on the other hand, the story seems to be somewhat more complex; an example isoutlined in [32][section 3.2], and a more complete description will appear in [16].
We can understand the decomposition conjecture schematically as follows. Consider anonlinear sigma model on a space X, for simplicity with H 2 (X, Z) = Z, with a restriction onworldsheet instantons to degrees divisible by k. We can realize that restriction in the path
integral by inserting a projection operator

 Z

k−1
2πn
1X
∗
exp i φ
ω
,
k n=0
k

where ω is the de Rham image of a generator of H 2 (X, Z). Inserting this operator into a8partition function is equivalent to working with a sum of partition functions with rotating
B fields, and this is the essence of the decomposition conjecture.
One of the applications of the result above is to Gromov-Witten theory, where it has
been checked and applied to simplify computations of Gromov-Witten invariants of gerbes,
see [33, 34, 35, 36, 37, 38]. Another application is to gauged linear sigma models [15], where
it answers old questions about the meaning of the Landau-Ginzburg point in a GLSM for acomplete intersection of quadrics, as well as corrects old lore on GLSM’s.

3

Four-dimensional physics

Now, let us turn to four-dimensional theories, and discuss how the physics differs from twodimensional cases.
To sharply compare with the two-dimensional cases outlined in the previous section, let
us work through a toy example. Consider a U(1) gauge theory in supergravity5 , defined overspacetime R4 , with supergravity moduli space C2n+2 , describing 2n + 2 complex scalars, on
which the U(1) acts as follows: n fields of charge k, n fields of charge −k, one field χ ofcharge +1, one field χ̃ of charge −1. Let us furthermore assume that there is a superpotential6
W = mχχ̃, giving a mass m to the two fields of charge ±1. The upshot of this construction isthat at low energies, one has a U(1) gauge theory with nonminimal charges, closely analogous
to those discussed in the last section realizing sigma models on gerbes.
Now, one might worry that at at low energies, below the cutoff scale, perhaps all thestates of the theory have charges that are a multiple of k. In such a case, the fact that theelectron charges above are nonminimal would be physically irrelevant; at low energies, thetheory would be physically equivalent to a theory in which all fields had charge 1, not k. To
settle this issue, we need to understand the correct electric charge quantization in this theory.
In two dimensions, we could distinguish a theory with nonminimal charges from a theory
with minimal charges via nonperturbative effects, invoking the theta angle periodicity to ‘see’
states with masses beyond the cutoff. Here, by contrast, note that since we are describinga U(1) gauge theory on R4 , there are no U(1) instantons. However, because this theory is
coupled to gravity, we can appeal to the existence of Reissner-Nordström black holes. We
can use them to determine the correct electric charge quantization in the theory at places in
its moduli space where the U(1) is unbroken, and then appeal to continuity to understandthe rest.
5

The analysis presented here is due to J. Distler, and we thank him for allowing us to present it here.
Since we are working in supergravity, the superpotential is a section of the Bagger-Witten [1] line bundle,
but since the supergravity moduli space is C2n+2 , the Bagger-Witten line bundle is necessarily the trivial
line bundle O.
69

First, suppose that the mass m < MPl . When the U(1) is unbroken, there are electricallycharged Reissner-Nordström black holes. Since m < MPl , microscopic black holes can Hawking radiate χ, χ̃, and so even if one started with a black hole of charge a multiple of k, it could
Hawking radiate down to charge 1. Thus, at least at points where the U(1) is unbroken, thenonminimal charges of the Higgs fields are physically relevant. At more generic points on
the moduli space, where the U(1) is Higgsed to Zk , we need there to be excitations on which
the Zk acts nontrivially, and at least for small Higgs vev, the Reissner-Nordström black holes
should7 become such excitations.
If m > MPl , then the Hawking radiation process above can not happen, but demandingthat physics be continuous in m leads us to believe that the electric charges of black holes
are still multiples of 1 rather than k. Thus, again, the fact that the massless fields in thegauge theory have nonminimal charges, is physically relevant.
So far we have discussed a four-dimensional analogue of the two-dimensional theta-angleperiodicity argument for the relevance of nonminimal charges, using black holes rather thantheta angles. In the rest of this section we shall discuss some subtleties specific to fourdimensional theories, and their relevance to four-dimensional sigma models on stacks.
First, let us examine more closely the (lack of) nonperturbative sectors in field theories
on R4 not coupled to gravity. One of the significant properties of two-dimensional sigma
models on gerbes was that the nonperturbative sector is altered: one sums over only someinstantons, not all of them. However, in four dimensions, on R4 , there is no constraint on
nonperturbative sectors:• We have already seen the example of a U(1) gauge theory with nonminimal charges
on R4 . Instantons in such a theory would be described by principal U(1) bundles
on S 4 (the one-point compactification of R4 , taken to encode compact-support issues), and there are no nontrivial principal U(1) bundles on S 4 , because RH 2 (S 4 , Z)
vanishes. (Moreover, on R4 or S 4 , for a U(1) gauge theory the quantity TrF ∧ Fvanishes identically for all gauge field configurations appearing in the path integral(i.e. compactly-supported ones), not just saddle points.)• Next, consider a nonlinear sigma model of maps R4 → G for G some G-gerbe (for finiteG) over a smooth manifold M. Again for compact-support reasons we can replace R4
by S 4 for the purposes of mathematical computations. As discussed in the last section,a map X → G for any space X is the same as a map f : X → M together with atrivialization of f ∗ G, and demanding that f ∗ G be trivializable restricts possible maps
f . In the present case, however, since H 2 (S 4 , Z) vanishes, the pullback f ∗ G is always
7

In effect, we are appealing to a continuity argument. As a potential loophole, we should mention that
it is known from work on wall-crossing that the spectrum of BPS states in a supersymmetric theory does
not always behave so simply. We do not need to assume the states here are BPS, but, it is possible that
analogous processes may apply.

10

trivializable for any G, and so demanding that f ∗ G be trivial is no longer a constraint
on possible maps.
In particular, unlike two dimensions, on R4 there is no reason to believe that cluster decomposition will necessarily be violated, and there is no analogue of the decomposition conjecture[14] for gerbe theories.
Next, let us consider a technical point in the renormalization-group flow of the low-energy
effective gauged sigma models arising in this and analogous examples. (We would like tothank J. Distler for patient explanations of this point.) Schematically, if v is the scale of the
Higgs vev, and g the coupling, then the low-energy effective action is an expansion in powersof E/v. However, Higgsed gauge fields have masses which scale as gv, and so for weakcoupling g, generate E/(gv) effects which can be stronger than low-order effective action
terms. Put another way, W bosons are light relative to the natural mass scale defining themetric. As a result, the effective field theory arising in the infrared from a gauged sigma
model often can not be the same as a nonlinear sigma model. There can still be a moduli
space, a metric on the moduli space, and many other features consistent with nonlinear sigma
models, (as happens with e.g. Narain moduli spaces in toroidally-compactified heterotic
strings,) but strictly speaking, the infrared limit of a low-energy effective gauged sigma
model in four dimensions need not be physically equivalent to a nonlinear sigma model.
This result implies an issue of presentation-dependence in four-dimensional theories, that
does not exist in two dimensions. In two dimensions, we identify universality classes ofrenormalization group flow with stacks: a given stack can have multiple presentations withdifferent UV physical descriptions (a nonlinear sigma model, a gauged sigma model, an
orbifold) which mathematically correspond to the same stack. Physically, it is conjectured
that those different presentations lie in the same universality class, that the renormalizationgroup ‘washes out’ all details of the presentation, leaving physics that only depends upon thestack and not how it is described or presented. In particular, typically we are only interested
in conformal field theories arising at endpoints of renormalization group flow, so the detailsof a physical presentation of a massive UV theory are irrelevant.
Part of the point of the observation on four-dimensional low-energy effective field theories
is that the presentation-independence we enjoyed in two dimensions no longer applies in four
dimensions. We can also see this from another perspective, involving the gauge kinetic
terms. A sum over maps from a space into a stack presented as [X/G] involves a sum
over G bundles with connection – a sum over G-gauge fields. In two dimensions, gauge
kinetic terms are generated dynamically, so we could effectively ignore them and identify anonlinear sigma model on [X/G] with a G-gauged sigma model on X – the sum over maps
includes the sum over gauge fields, and the gauge kinetic term comes for free. By contrast, infour dimensions, gauge kinetic terms are not generated dynamically. We can describe some
aspects of a G-gauged sigma model on X with the stack [X/G], but we do not get a gauge
kinetic term automatically, the stack does not even determine a classical value of the gauge
11coupling. Hence, in four dimensions, merely specifying a stack does not uniquely determine
the physics.
In two dimensional cases, we associated stacks with universality classes of renormalizationgroup flow. Instead, in four dimensions we will use stacks as ‘universal’ objects from which
various different physical presentations can be associated. The details of those presentations,the presentation-dependence, will no longer be physically irrelevant, unlike two dimensions;nevertheless, some (not all) of the physics will be determined by the stack. It is in thissense that we will associate stacks with low-energy effective four-dimensional (gauged) sigma
models and related theories.
Before going on, let us summarize the circumstances under which gerbes will be physicallymeaningful in a four-dimensional theory. One way for gerbes to be physically meaningful
in a four-dimensional theory is if the four-dimensional spacetime is topologically nontrivial,with nonzero H 2 (Z). In this case, one would have nontrivial nonperturbative sectors in thecases above. Just as in two dimensions, to uniquely define Higgs fields one would need to
specify the precise bundle the field couples to, and that choice would be reflected in zeromode spectra, hence in anomalies and so forth. A second way gerbes can be physicallymeaningful is if there are massive non-invariant states. We have only argued this above intheories coupled to gravity; however, we shall also see examples later where some aspectsof gerbes manifest even in four-dimensional theories on R4 not coupled to gravity. Bothof these cases had analogues in two dimensions – for example, the theta-angle-periodicity
argument in two dimensions relied on the existence of massive minimally-charged states. We
shall see examples of both cases in the next section.
Partly with an eye towards nonabelian applications we shall discuss later in this paper,
let us summarize the conclusions of this and the previous section in the following slogan:
Perturbative physics is determined by the Lie algebra of the gauge group, but
nonperturbative physics is determined by the Lie group, not just the algebra.
Just as an asymptotic series expansion does not uniquely determine the function it is expanding, so too does the perturbative physics not uniquely determine the nonperturbative
physics.
In the rest of this paper, we will outline gerbes in four dimensional field theories andstring compactifications.

12

4

Examples, duality in gerby moduli ‘spaces’

Gerby moduli ’spaces’8 seem to appear in both four dimensional field theory and in string
compactifications, as we shall outline in this section. Briefly, a gerbe looks locally like aquotient by a trivially-acting group – although the group acts trivially, both mathematics
and, at least sometimes, physics nevertheless knows about the group action. As sigma models
on gerbes can be viewed as sigma models on spaces or effective quotients with a restriction
on nonperturbative sectors, these are precisely the examples discussed recently in [7].
In this section we shall discuss examples of gerby moduli spaces appearing in both fieldand string theories, and also discuss how the gerbiness behaves under field and string theoretic dualities.

4.1

Field theory

At a purely mathematical level, it is easy to generate examples of four dimensional field
theories with gerbe structures over their moduli spaces. As the moduli space of a field
theory is typically of the form [V /G], where V is a vector space spanned by matter vevs
and G is the gauge group, whenever any subgroup of G acts trivially on all of the masslessmatter, mathematically one could associate a gerbe structure to that moduli space. For
example, in Yang-Mills theory with adjoint matter, the maximal torus of the gauge group
acts trivially on matter vevs. Thus, if r is the rank of the gauge group, then in such theories
there is formally a U(1)r gerbe structure generically9 . The physical content of that gerbe
structure is another matter. Morally, if not literally10 , a sigma model on U(1)r gerbe ought
to be a U(1)r gauge theory, which certainly arise in Yang-Mills theories with only adjoint
matter.
However, we need a bit more structure (such as massive noninvariant matter, or a
topologically-nontrivial spacetime four-manifold) before we believe such gerbe structures
are physically meaningful. In addition, in this paper we will focus on finite gerbe structures(corresponding to Deligne-Mumford stacks, rather than Artin stacks). In the examples weshall discuss in this section, the gerbe structure will arise by focusing on the center of the
8

Strictly speaking, if there is a gerbe structure, then the moduli ‘space’ is actually a stack, not a space,but because the language of stacks is as yet unfamiliar to many physicists, we will call them “gerby spaces”
in much of this paper.
9This stabilizer changes over the moduli space; for example, at the origin where all vevs vanish, thestabilizer is all of G. Since the stabilizer changes, this is not, strictly speaking, a gerbe, but rather is a moregeneral stack, that only looks like various gerbes on specific strata
10
A stack with non-finite stabilizers is known as an Artin stack. The geometric interpretation of Artinstacks is somewhat more complicated than that of Deligne-Mumford stacks, which the analysis of [10, 11,
12, 13, 14, 15, 19, 20, 21, 16] focused on. In this paper we also almost exclusively focus on Deligne-Mumford
stacks.

13

gauge group. If we return again to Yang-Mills theories with adjoint matter, this means we
consider the gerbe structure on the moduli space arising from the fact that the center acts
trivially on the matter.
Our first physical example will involve a topologically-nontrivial spacetime four-manifold.
Consider N = 4 supersymmetric theories arising in recent work on the geometric Langlands
program [39]. There, one compactifies a four-dimensional N = 4 theory along a Riemannsurface to get a two-dimensional theory, a nonlinear sigma model whose target space is the
Hitchin moduli space on the compactification curve. The authors of [39] observed that said
moduli space has a number of components. An alternative way of understanding that fact
is to utilize the finite gerbe story outlined above. If we start with a G gauge theory in four
dimensions, then following the ansatz above, the moduli space of the four-dimensional theory(and hence the target of the compactified two-dimensional sigma model) has a Z(G) gerbe
structure, where Z(G) denotes the center of G. Application of the decomposition conjectureof [14] to the two-dimensional sigma model on the gerbe then quickly reproduces the multiple
component structure worked out more painfully by [39], as discussed in [14, 40].
One lesson of the example from geometric Langlands above is that these formal gerbe
structures on moduli spaces do have physical content – the disconnectedness of the target of
the two-dimensional sigma model is a consequence of a gerbe structure on the moduli space.
That said, duality often does not preserve centers of gauge groups: for example, S-duality
in N = 4 maps SU(n) gauge theories to SU(n)/Zn gauge theories. In effect, the center ofthe gauge group is being exchanged for extra characteristic classes, disconnectedness in thetwo-dimensional target moduli space. Hence, gerbe structures are not duality-invariant.
We are often used to moduli spaces being invariant under duality operations – this is,after all, one of the standard checks of a duality. What is going on here is that the underlying
space is unchanged, only the automorphisms that are paired with the space are changing.
Therefore, the number of flat directions, the geometry of the flat directions is unchanged,
only the automorphisms differ. Since it is only the number and geometry of the flat directions
that must necessarily be preserved by duality, the fact that gerbe structures change does notcontradict duality.
Let us examine this N = 4 duality in greater generality. Geometric Langlands exchanges
the center Z(G) with the dual of π1 (L G), where L G denotes the Langlands dual to G. The
center Z(G) encodes a gerbe structure, and π1 (L G) describes how the moduli space breaks
into components (indexed by a characteristic class in H 2 (X, π1 (L G))). We can see how Z(G)and π1 (L G)∗ are exchanged as follows. Let M denote the weight lattice of the Lie group G.
It is a sublattice of the weight lattice of the corresponding Lie algebra, which we shall denoteΛ. (M is determined by the representations of the Lie group, instead of the Lie algebra.) If
we let R denote the root lattice, then in general
R ⊆ M ⊆ Λ.
14

(1)

The action of Langlands duality is to dualize each of these three lattices:
R 7→
M 7→
Λ 7→

L

R ≡ Hom(Λ, Z),
M ≡ Hom(M, Z),LΛ ≡ Hom(R, Z),
L

and it is straightforward to see from (1) thatL

R ⊆ L M ⊆ L Λ.

In this language, the center and π1 of G are determined by the lattices above, as follows:
Z(G) = (M/R)∗ = Hom (M/R, Z) ,π1 (G) = (Λ/M)∗ = Hom (Λ/M, Z)(the first equality comes from the fact that Z(G) is the kernel of the adjoint action, whose
weights generate the root lattice), which should make it clear that
Z(G) = π1 (L G)∗ , π1 (G) = Z(L G)∗ .
In other words, Langlands duality exchanges the center of a group G with (the dual of) π1
of the Langlands dual group L G.
The Hitchin moduli stack, the target of the two-dimensional sigma model, is a Z(G)gerbe over a disconnected space with multiple components. One has different components
corresponding to the fact that there is a characteristic class in H 2 (X, π1 (G)), and the components are indexed by the value of that characteristic class. The effect of Langlands duality
is to exchange Z(G) gerbiness with π1 (L G) disconnectness (see e.g. [41] for a more detailed
discussion). One might ask if there is an alternative description as some Z(G)×Z(L G) gerbe
over another space, giving a duality-invariant stack, but we are told [42] such a construction
does not exist.
Let us next consider some examples of gerbe structures appearing in the field theories discussed in [43, 44, 45, 46]. These papers discuss examples in which an N = 1 supersymmetricgauge theory with a gerbe structure on its moduli space is (Seiberg-)dual to another N = 1
supersymmetric gauge theory which has monopoles. The massive, non-invariant matter on
the gerbe side is dual to the monopoles. Just as in the geometric Langlands story above, thegerbe structure is not preserved by duality.
The prototype for these examples is discussed in [44]. That paper argued that a Spin(8)gauge theory with Nf fields in the 8V and one field in the 8S is dual to a chiral SU(Nf − 4)theory with a symmetric tensor and Nf fields in the antifundamental representation. Whenthe 8S is given a mass, the dual SU(Nf −4) theory is Higgsed to SO(Nf −4) with Nf vectors.
Moreover, that SO(Nf − 4) theory admits a monopole, since π2 (SU(Nf − 4)/SO(Nf − 4)) =15

π1 (SO(Nf − 4)) = Z2 . The perturbatively massive spinor in the Spin(8) theory is dual tothe monopole in the SO(Nf − 4) theory.
In the original (unHiggsed) dual pair, on neither side does the moduli space admit agerbe structure: no part of the center of Spin(8) acts trivially on both 8V and 8S , and thecenter of SU(Nf − 4) does not act trivially on the antifundamentals. After Higgsing, a Z2
subgroup of the center of Spin(8) acts trivially on the remaining 8V fields, hence that branchof the moduli space (formally) admits a Z2 gerbe structure. (Its dual still does not have agerbe structure on its moduli space.)
The upshot is that we have two dual theories, one with a gerbe structure on its moduli space and a massive spinor, dual to a theory without a gerbe structure on its moduli
space, but with a massive monopole instead. For example, a Wilson loop in the spinor
representation of Spin(8) is mapped to the ’t Hooft loop in the magnetic Z2 [45].
Just as in geometric Langlands, we see that gerbe structures are not preserved by duality. This interpretation is reiterated (albeit without explicitly naming a gerbe structure) in[45][section 2], [46] in terms of screening effects, and further examples of the same generalform are given in [45, 46].
For completeness, note that the presence of massive nonminimally charged matter plays
an important role in this story, just as it did in two-dimensional examples of theories withgerbe structures.

4.2

String theory

Just as in field theory, one can also (formally) associate gerbe structures to various moduli
spaces, whenever there is a subset of the low-energy gauge group that acts trivially on
massless fields (and nontrivially on at least one massive field). In this section, we will
outline examples of gerby moduli spaces appearing in string compactifications.
The first example is the Narain moduli space of toroidally-compactified heterotic string
theories. Just as with Yang-Mills theories with adjoint matter, there are at least two naturalways to formally add a stack structure to such moduli spaces, both of which revolve aroundthe fact that part of the Narain moduli space describes flat connections on a torus. If wetake the moduli stack of flat G-connections to be[Hom(π1 , G)/G]
then we have a stack which along strata has variable gerbe structures along strata (though
as the stabilizer varies across strata, it is not considered globally to be a gerbe, unless G has
a nontrivial center). For example, at the point on the moduli space where low-energy adjoint
scalars vanish, one has a G gerbe; at more nearly generic points, where only a maximal torus
16

T commutes with adjoint scalars, one has a T gerbe. The mathematical interpretation ofsuch structures is just as in the field theory discussion previously.
In the case of field theories, we observed that a different stack structure may have greater
physical relevance, involving only finite centers of stabilizers rather than the entire stabilizer.
This structure also varies across strata, giving rise along any one stratum to a variety ofpossible gerbe structures. Globally, the entire stack would have a Z(G) gerbe structure,
where Z(G) is the center of G, which can be enhanced over various strata. In the case ofgeometric Langlands, this was the gerbe structure that gave rise to the disconnectedness ofthe target Hitchin moduli space of the two-dimensional theory.
Phrased more simply, ordinarily we think of toroidally-compactified heterotic strings ashaving a Narain moduli space (or rather, more generally [47], a moduli space with several
components, one of which is the Narain moduli space). Here we are observing that the
Narain moduli space (and other components) carry additional structure, at least formally,
namely that of a gerbe. The moduli stack of toroidally-compactified heterotic strings is agerbe over a stack with, in general, several components, one of which is the Narain moduli
space, plus enhanced gerbe structures on various strata. (A more formal discussion of suchphenomena in the context of Hitchin moduli spaces can be found in e.g. [41].)
In the case of a Spin(32)/Z2 heterotic string compactification, the moduli space of thetoroidally-compactified string theory generically (and formally) has a Z2 gerbe structure,since the center of Spin(32)/Z2 is Z2 . As described elsewhere, for such a gerbe structure to
be meaningful for a theory on R4 , we also need massive states which are not invariant under
the group. In the present case, the Spin(32)/Z2 heterotic string has at its first massive levelstates transforming in the spinor representation of Spin(32)/Z2 , which is not invariant under
the Z2 ([48][section 6.3.1], [49][section 2.3]), exactly as needed for a gerbe description of themoduli space to be physically relevant.
On the ten-dimensional heterotic string worldsheet, this proposed Z2 gerbe structure on
the CFT moduli space manifests itself as the quantum symmetry11 [51] associated with theleft-moving GSO analogue that defines the Spin(32)/Z2 string in its RNS presentation. (Thecenter of Spin(32) is Z2 × Z2 , and the GSO analogue itself is responsible for the Z2 quotientin Spin(32)/Z2 . As the action of the center is being expressed on associated vectors, noton the group itself, it manifests in terms of orbifolds and quantum symmetries. There is a
closely analogous story for the Spin(16)/Z2 in E8 and RNS constructions of E8 ×E8 heterotic
strings.) In particular, all of the massless ten-dimensional states arise from a left-moving
NS sector, and the only charged states are adjoints; all of the left-moving R sector states
11

The symmetry we are describing leaves the NS sector states invariant, but multiplies the R sector statesby a sign. A Z2 quantum symmetry leaves the untwisted sector invariant, and multiplies the twisted sector
by a sign, which is consistent with the symmetry in this case if one remembers that we are using R, NS todescribe states on the cylinder, but the quantum symmetry is defined by (un)twisted sectors on the complex
plane, and the conformal transformation between the two exchanges R and NS sectors [50][section 7.1].

17

are massive. The quantum symmetry leaves the left-moving NS sector states invariant andmultiplies the left-moving R sector states by a phase, which matches the effect of the Z2
gerbe structure.
For completeness, let us also consider the string-dual type I theory in ten dimensions. The
gauge group of the type I string is SO(32), different from that of its dual ten-dimensional
heterotic string. We have seen that under dualities, gauge groups will change – this is atypical property of Langlands duality, for example. The massless spectrum is invariant under
the Z2 center of SO(32), suggesting a gerbe structure; however, all of the massive states arealso invariant, as the perturbative spectrum of the type I string in ten dimensions contains
only symmetric and antisymmetric 2-tensors [49][section 2.3], compatible in principle with agauge group
SO(32)/Z2 = Spin(32)/ (Z2 × Z2 ) .

For this reason, we do not identify a gerbe structure on the moduli space of compactified
type I strings. Furthermore, in close analogy with our discussion of e.g. [43, 44, 45, 46] inthe previous subsection, there exists a particle-type topological defect in type I string theory,
arising from an element of π8 (SO(32)), which transforms as a spinor of the Lie algebra [52],the same property as a massive perturbative state in the dual heterotic Spin(32)/Z2 theory.
Along special loci this gerbe structure can be enhanced, as expected on general grounds
from our discussion of moduli stacks of flat connections. Consider Higgsing a toroidallycompactified E8 × E8 heterotic string, for example. There is no gerbe structure over theentire moduli space (as E8 has no center). Now Higgs one of the E8 ’s to a Spin(16)/Z2
subgroup. As all of the adjoint-valued scalars in the theory are derived by dimensionallyreducing a ten-dimensional gauge field, Higgsing the E8 should lift all components of thosescalars that are not adjoints under Spin(16)/Z2 . The center of Spin(16)/Z2 is Z2 , and it acts
trivially on the adjoints, the surviving massless matter. However, it does not act trivially on
all of the string modes. The adjoint representation of E8 decomposes as
248 = 120 + 128
where 120 is the adjoint representation of Spin(16)/Z2 , and 128 is a spinor. By Higgsing
the E8 to Spin(16)/Z2 , we give a mass to the 128, which is not invariant under the centerof Spin(16)/Z2 . Thus, we have, at low energies, a gauge group with nontrivial (Z2 ) center
that acts trivially on massless matter, but nontrivially on massive matter.
Note that we can construct examples with N − 1 supersymmetry in four dimensions
and gerbe structures on their moduli spaces by compactifying a Spin(32)/Z2 heterotic string
on a nontrivial Calabi-Yau threefold. For simplicity, let us consider such a heterotic string
compactification with the standard embedding.
One way to see the existence of the gerbe structure on the moduli space is from worldsheet
considerations. Just as in the ten-dimensional theory, all of the massless states arise fromleft-moving NS sectors; the left-moving R sectors contribute only massive states. As a result,18the quantum symmetry associated to the left-moving GSO analogue (which leaves left NS
sectors invariant, but acts by a phase on left R sectors) leaves the massless states invariant,but acts by a phase on massive states.
We can also see the gerbe structure on the moduli space in the low-energy effective
field theory. Consider for simplicity a Spin(32)/Z2 heterotic string compactification on a
nontrivial Calabi-Yau threefold with the standard embedding. The low-energy gauge groupis12
Spin(26) × U(1).
Z4The U(1) factor is typically anomalous and Higgsed via a four-dimensional version of theGreen-Schwarz mechanism [54, 55], closely related to a (field-dependent, hence not directly
relevant to this paper) Fayet-Iliopoulos parameter. The remaining Z2 center of Spin(32)/Z2
descends to part of the center of the group above, and the massless states are all invariant under this Z2 , as the massless states all descend from invariant representations of Spin(32)/Z2 .
In the context of heterotic compactifications on elliptically-fibered Calabi-Yau’s, the moduli stack of G-bundles for a group G has, in essence, directions corresponding to moduli of
the spectral cover and directions corresponding to the moduli of a line bundle on the spectral cover. The latter has, at least formally, a Z(G)-gerbe structure. In the dual F theory
compactification, such moduli dualize to moduli of G flux, suggesting [42] that the moduli
of G fluxes has a gerbe structure. In fact, naively not only do moduli spaces of F theory
compactifications admit gerbe structure, but at least sometimes there is evidence that duals
to some heterotic string compactifications are F theory compactifications on gerbes [42].
12

We can compute this as follows. We are embedding SU (3) into a Spin(6) = SU (4) subgroup, so we
begin by observing that Spin(32) has the subgroup [53][appendix A]
Spin(26) × Spin(6).
Z2Since the center of both Spin(26) and Spin(6) is Z4 , there is only one diagonally-acting Z2 subgroup. We
can describe the center of the group above as generated by a, b, subject to the relations a4 = b4 = 1, a2 = b2 .
Now, we want the subgroup of Spin(32)/Z2 , and after taking the second Z2 quotient we could have either aZ2 × Z2 or Z4 quotient of Spin(26)× Spin(6), corresponding to quotienting either a2 or ab, respectively. We
can distinguish them as follows. For simplicity, replace Spin(26) by Spin(6), to form subgroups of Spin(12),and use the fact that Spin(6) = SU (4), the 4, Alt3 4 = 4 are the spinor representations, and Alt2 4 = 6 thevector. The Z2 quotient (originally of Spin(32), now Spin(12)) should flip the sign of the 12 = (6, 1) ⊕ (1, 6),
and preserve only one of the two spinor representations. Since a, b both act by multiplying the 4 by a fourthroot of unity, it is straightforward to check that a2 leaves the vector representation invariant, whereas ab
flips the sign of the vector representation. (Both preserve only one spinor of Spin(12).) Thus, we should
quotient by ab, and hence the correct subgroup of Spin(32)/Z2 isSpin(26) × Spin(6).
Z4Embedding SU (3) into SU (4) = Spin(6) leaves us with the maximal commutant shown. We would like tothank A. Knutson for a useful discussion of this issue.

19Specifically, it has been observed [42] that the multisection structures appearing in [56] inthe F theory duals of heterotic CHL strings have an alternative interpretation in terms ofelliptic fibrations over Z2 gerbes. Part of the point is that a heterotic compactification on an
elliptic fibration with multisection is described by a spectral cover in a gerbe over the relative
Jacobian, together with a (possibly fractional, in a sense we describe later) line bundle over
the restriction of the gerbe to the spectral cover. The G fluxes then behave as a torsor under
the appropriate Deligne cohomology group. We shall not pursue such F theory structures
further here.
We do not expect such gerbe structures to always appear in CFT moduli spaces. For
example, consider a heterotic E8 × E8 string compactified on a simply-connected Calabi-Yau
threefold, with the standard embedding. Although one has embedded an SU(3) bundle, andSU(3) has a nontrivial center (Z3 ), it has been embedded into an E8 , which has no center.
We can determine the existence of a gerbe structure by looking at the charged matter inthe low-energy effective field theory, To get a gerbe structure, the matter would all have to
be invariant under the center of the low-energy effective gauge group (in this case, E6 , withcenter Z3 ). Neither 27’s nor 27’s are invariant under the center of E6 [57], hence we do not
expect to get a gerbe structure on the CFT moduli space, since there is not a gerbe structureon the field theory moduli space.
More generally, it is worth emphasizing that many moduli spaces which do not have gerbe
structures globally will still have gerbe structures on subvarieties. For a simple example,
the quotient stack [C2 /Z2 ] looks like the quotient space C2 /Z2 everywhere except at theorigin, where there is a copy of the classifying stack BZ2 inserted, which mathematicallydesingularizes the quotient space. In that example, one has a Z2 gerbe over the origin(as BZ2 is a gerbe over a point), but nowhere else. We have already seen such structuresin moduli stacks of flat connections, and they can also arise in moduli stacks of spaces: forexample, the moduli stack of elliptic curves admits special points which are locally quotients,and so have finite gerbe structures. (The elliptic curves at those points have automorphisms
not possessed by generic elliptic curves.) Sometimes (though not always) a gerbe structure
at a subvariety on a moduli space will reflect an orbifold structure. For example, the moduli
space of K3 surfaces contains a Z2 orbifold point, at which the K3 is represented by [T 4 /Z2 ].
In this example, the orbifold structure on the moduli space reflects the quantum symmetryof the orbifold theory (though as already noted, this is not always the case).
In passing, we should also mention that there may be further examples of string theories
with gerby moduli spaces implicit in [58], which ‘geometrically engineers’ four-dimensional
theories with nonabelian gauge groups from type II compactifications on singular spaces.

205Topological defects and gerby moduli spaces

Recall stable topological defects are classified by the homotopy groups of the moduli space
M: cosmic strings13 arise from π1 (M), monopoles from π2 (M), textures from π3 (M). Homotopy groups can be defined for moduli stacks (see e.g. [60] and references therein), andin particular for moduli spaces with gerbe structures, and are not quite the same as thehomotopy groups of the underlying spaces. In this section we will outline such homotopy
and discuss their potential application to topological defects.
Let us begin by outlining some pertinent facts about homotopy of gerbes. For M a
G-gerbe14 over M, there is a homotopy long exact sequence· · · −→ πn (BG) −→ πn (M) −→ πn (M) −→ πn−1 (BG) −→ · · · .
In the sequence above, BG denotes the “classifying stack” of G (so named because of formal
similarities with the classifying space of G). Technically, the classifying stack is defined as


point
.
BG ≡
G
If we think of a gerbe as being analogous to a fiber bundle, then the fibers are copies of
BG. In terms of homotopy groups, it can be shown that πn (BG) = πn−1 (G). In particular,
for finite G, π1 (BG) = G. (As maps from a space X → BG are defined by principal Gbundles over X, and principal G-bundles over S 1 are classified by conjugacy classes in G,
not elements of G, this also means we should be careful about interpreting π1 as a group ofmaps from a circle.)Now, let us apply the description above to cosmic strings, and discuss whether topologicaldefects should be classified by homotopy groups of the gerbe or the underlying space.
Consider, for example, a moduli space with a Zn gerbe structure, call it M. If we denotethe underlying (technically, “coarse”) moduli space by M, then topological defects would
ordinarily be computed by homotopy of M. The effect of the gerbe is to add a BZn fiber
over each point of M. Over any point of M, therefore, is a copy of BZn , which has π1 = Zn .
13

For example, the stringy cosmic strings of [59] arise from the fact that π1 of the moduli stack of elliptic
curves is SL(2, Z). (This stack should be distinguished from its Deligne-Mumford compactification. Thatcompactification maps onto S 2 , hence its homotopy groups have all of the complexity of the homotopy groupsof S 2 and more [42].) However, the higher homotopy groups all vanish, so from compactifications on elliptic
curves, the only topological defects one can get are stringy cosmic strings.
14
In the special case that the gerbe is trivial, i.e. M = M × BG = [M/G] for trivially-acting G, M is a
G-bundle over M, and so there is an additional long exact sequence· · · −→ πn (G) −→ πn (M ) −→ πn (M) −→ πn−1 (G) −→ · · · .

21

If topological defects are classified by homotopy groups of the moduli stack, not theunderlying moduli space M, then we would get a cosmic string defined by a loop around
BZn fibers, which may or may not be globally stable depending upon global properties15of the gerbe. This would be some new type of cosmic string, as ordinary cosmic strings
arise from π1 (M). In this new type of cosmic string, the moduli space scalars would beunchanged as one walks around the string, except that the theory would undergo some Zn
gauge transformation around such a loop. Only massive noninvariant fields would see thatgauge transformation.
Let us now turn to physical examples. The hypothetical cosmic string above sounds very
similar to the Zn cosmic string discussed in e.g. [61][section 4.2.2]. There, one has an SU(2)gauge theory with a pair of triplet-valued Higgs fields which are required (by virtue of apotential term) to be orthogonal. Giving the first Higgs triplet a vev breaks SU(2) to U(1);giving the second an (orthogonal) vev breaks U(1) to Z2 . After both symmetry breakings
have occurred, one has Z2 cosmic strings, as π1 (SU(2)/Z2 ) = Z2 . In such a theory, themoduli space of possible Higgs vevs has a natural Z2 gerbe structure, and the cosmic strings
described by [61] seem to naturally coincide with the cosmic strings we have outlined above
arising from homotopy of the gerbe. In fact, our homotopy considerations would appear togive a new perspective on the Zn cosmic strings of [61], as they are discussed there only as
homotopy of group cosets, and here we seem to have found the same structure in homotopy
of a moduli stack.
Unfortunately, further analysis does not seem to bear out this perspective. One seeming
counterexample arises in [62][section 4.2]. That reference also describes Zn cosmic strings,
though in that case, the adjoints act primarily as spectators, and the cosmic string solution
naturally involves winding of vevs of massless fundamentals, with a potential fixing theirvevs to be nonzero. In the present case, for physical relevance of a gerbe structure, we neednoninvariant fields, albeit massive noninvariant fields. If the noninvariant fields are massive,then its vev vanishes, and any sort of winding solution of the form outlined in [62][section4.2], unlikely.
Here is a more convincing counterexample. Consider an SU(2) gauge theory containing
only a single Higgs triplet, the SU(2) would only be broken to U(1), and although theresulting theory has monopoles (as π2 (SU(2)/U(1)) = Z), it does not have cosmic strings(as π1 (SU(2)/U(1)) = 0). Thus, in this case, the homotopy of the gerbe gives a misleading
result.
15

Suppose, for example, the gerby moduli space M is the nontrivial Zn gerbe over P1 defined by taking
two homogeneous coordinates x, y to have weight n under C× . The space S 3 is a circle bundle over thisgerbe, so we have a homotopy long exact sequence of the form
· · · −→ πn (S 1 ) −→ πn (S 3 ) −→ πn (M) −→ πn−1 (S 1 ) −→ · · · .
In particular, since π0 (S 1 ) ∼= π0 (S 3 ) and π1 (S 3 ) = 0, we have that π1 (M) = 0, and so the gerbe M is simply
connected. Thus, our hypothetical cosmic string would not be globally stable.

22One potential fix to the counterexample above is to replace Deligne-Mumford stacks
with more general Artin stacks (which are not required to have finite stabilizers). In theexample above, an SU(2) gauge theory with a single Higgs triplet, an Artin moduli stackwould naturally have a U(1) gerbe structure. Now, π2 (BU(1)) = π1 (U(1)) = Z, so the same
homotopy analysis of the gerbe would imply the existence of monopoles in this example,
matching the physical result. For that matter, as π1 (BU(1)) = π0 (U(1)) = 0, there is no
prediction of cosmic strings, also matching the physics. On the other hand, as the gerbe
structure would only see the unbroken part of the gauge group, not the original gauge group,
it seems unlikely that the example above would generalize to give accurate results in othercases.
Our tentative conclusion is that, at least for Deligne-Mumford moduli stacks, the homotopy of the gerbe is misleading, the extra elements of π1 that one encounters do not reflect
physically meaningful new cosmic string solutions, and that topological defects should becounted by homotopy of the underlying space. This then begs the question of how tounderstand cosmic strings and other topological defects when the moduli space is a more
complicated stack.
For completeness, let us also formally discuss higher defects in Deligne-Mumford stacks,
though as already established, their physical relevance may not be significant. To be specific,
consider the Zn gerbe over P1 defined by taking two homogeneous coordinates x, y to haveweight n under C× , rather than weight 1. Call this gerbe M. We shall consider hypotheticalmonopoles arising from the moduli stack M. From the long exact sequence for homotopy,
we have
π2 (BZn ) −→ π2 (M) −→ π2 (P1 ) −→ π1 (BZn ) −→ π1 (G).
Now, π2 (BZn ) = 0, and it can be shown π1 (M) = 0, so we have
0 −→ π2 (M) −→ π2 (P1 ) −→ Zn −→ 0.
Thus, the total number of stringy monopoles arising from this gerby moduli space would becountable, just as for an ordinary projective space, but note that not every monopole arisingfrom P1 arises when the moduli stack is a gerbe over P1 , closely mirroring the fact that in
two-dimensional sigma models on gerbes there is a restriction on degrees of allowed maps.
We leave for future work a detailed discussion of global topological defects for moregeneral moduli stacks. Our results here suggest that global gerbe structures may not be
relevant, at least for Deligne-Mumford moduli stacks. It is possible that this is ultimately
a reflection of subtleties in low-energy effective actions discussed in section 3. We shallnot attempt to address the relevance of homotopy of gerbe structures that exist only oversubvarieties, or homotopy of Artin moduli stacks.
In passing, we should mention that [9][section 4.2] speculated on the existence of cosmic
strings of the form above in cases with gerby moduli spaces.
23

6Consistency conditions on classical supergravity

In this section we will discuss consistency conditions on classical supergravities. We begin
by reviewing results [1, 7, 8] for the case that the moduli space is a smooth manifold, andthen we generalize to smooth Deligne-Mumford stacks, focusing on gerbes over manifolds.

6.1

Review of standard supergravity caseFirst, let us recall the argument of Bagger and Witten [1] that the Kähler class of the moduli
space of scalars of a supergravity theory is quantized, in the case that that moduli space
is a smooth manifold. First, recall that across coordinate patches on the moduli space, theKähler potential K transforms as
K 7→ K + f + f,
where f is a holomorphic function of moduli, which must be accompanied by a rotation of
the gravitino ψµ and the superpartners χi of the scalar fields on the moduli space:




i
i
i
i
χ 7→ exp + Im f χ , ψµ 7→ exp − Im f ψµ .
(2)
22Consistency of the rotations (2) across triple overlaps (even within classical physics) implies
that the f ’s define a line bundle with even c1 . If we denote that line bundle by L⊗2 , thenthe gravitino is a spinor-valued section of T X ⊗ φ∗ L−1 , where X is the four-dimensional
low-energy effective spacetime and φ : X → M the boson of the four-dimensional nonlinear
sigma model on the compactification moduli space M, and that the fermions χi are spinorvalued sections of φ∗ (T M ⊗L). Similarly, one shows that the superpotential is a holomorphic
section of L⊗2 , and, in order to have a positive-definite metric, L−2 (whose c1 matches thecohomology class of the Kähler form) must be ample16 .
The recent paper [8] extended the analysis of [1] to gauged group actions. If we gauge theaction of some group G on the target space of the nonlinear sigma model in the supergravity
theory, then we have to lift that group action to the Bagger-Witten line bundle L⊗2 inorder to define the gauging globally. We can see this explicitly in the supergravity gaugetransformations. Under an infinitesimal group action
δφi = (a) X (a)i
16Specifically, positive-definiteness of the metric implies that every closed analytic subvariety of the moduli
space M has positive volume with respect to c1 (L−2 ), i.e. for Y ⊂ M closed of dimension p,
Z
c1 (L−2 )p > 0Ywhich is equivalent to L

−2being ample.

24

where X (a) is a holomorphic Killing vector describing the infinitesimal group action, thesuperpartners χi , gaugino λ(a) , and gravitino ψµ transform asi∂X (a)i j
(a) i
i
(a)
χ + Im F χ ,δχ = 
∂φj2i
δλ(a) = f abc (b) λ(c) − (a) Im F (a) λ(a) ,2i (a)δψµ = −  Im F (a) ψµ ,
2
(a)
(a)
(a)
where F
= X K + iD (K the Kähler potential), and F (a) is easily checked to beholomorphic. For real (a) , the Kähler potential undergoes a standard Kähler transformation
δK = (a) F (a) + (a) F(a),hence in the gauge transformations above, terms proportional to Im F (a) are precisely encoding the Kähler transformations on fermions given in equation (2). Thus, the gaugetransformation terms proportional to Im F (a) are encoding an infinitesimal lift of the group
action to L.
To define the gauge theory, we must extend the infinitesimal action encoded in supergravity to an action of the group, not just the Lie algebra. In general, lifts of group actions
to line bundles need neither exist nor be unique. The existence issue provides a constraint on
possible consistent supergravities. (Lack of) uniqueness is encoded in the Fayet-Iliopoulos
parameter, as it is argued in [8] that implicit in the supergravity is the statement that the
Fayet-Iliopoulos parameter determines different lifts of the action of G to L⊗2 . As such lifts
are quantized, the Fayet-Iliopoulos parameter is necessarily quantized, and corresponds toan element of Hom(G, U(1)) for G the gauge group. Just as D-terms are understood in rigid
supersymmetry in terms of symplectic quotients, the paper [8] argues that the structure
above in supergravity can be understood in terms of ‘geometric invariant theory’ quotients(see e.g. [63, 64, 65]), the algebro-geometric analogue of symplectic quotients. In particular,
in a geometric invariant theory quotient, the analogue of the Fayet-Iliopoulos parameter isquantized, because it is realized as a lift of a group action to a line bundle.
In the rest of this section we shall extend the analysis of [1] and [8] to smooth DeligneMumford stacks, focusing on gerbes over manifolds.

6.2

Generalization to smooth Deligne-Mumford stacks

The original work of Bagger-Witten [1] and followups [8], reviewed above, only considered
supergravity theories in which the moduli space is a smooth manifold. However, moduli
spaces which are smooth manifolds are vanishingly rare – more typically, they have singularities and/or various stack structures, and a generalization of [1, 8] to such cases would be
useful.
25

Formally, generalizing [1] and [8] to moduli ‘spaces’ that are smooth17 Deligne-Mumford
stacks is very straightforward – the analysis of [1], [8] applies with only minimal modification. The main caveat is that specifying a “nonlinear sigma model on a stack” does notuniquely specify the physics; we must choose a presentation of the stack, and we could get
different physics according to the choice. Put another way, there are multiple distinct physical theories, components of possibly multiple supergravities, that can be interpreted as anonlinear sigma model on a single fixed stack. In four dimensions, we use stacks to providea ‘universal’ four-dimensional object for which any given physical realization corresponds toa presentation.
Let us outline the analysis in two different presentations:• Deligne-Mumford stacks have coverings by open sets, so first consider a presentation inwhich the atlas is such a collection. Then, we can work patch-by-patch. Physically, thismeans we have a nonlinear sigma model on each open set, with perhaps a discrete gauge
group. Just as for spaces, transformations of the Kähler potential across coordinate
patches imply that there is a line bundle L over the moduli space, to which the gravitino
ψµ and chiral superpartners χi couple. In other words, just as for the case that themoduli stack M is a space, the gravitino is a spinor-valued section of T X ⊗ φ∗ L−1 (Xthe four-dimensional spacetime, φ : X → M), and the superpartners χi are spinorvalued sections of φ∗ (T M ⊗ L). The superpotential is a section of L⊗2 , and L−1 must
be a positive bundle. The Fayet-Iliopoulos parameter is a choice of lift of group action
to L, and such choices, possible values of the Fayet-Iliopoulos parameter are elementsof Hom(G, U(1)) for G the gauge group.
• Now, let us consider another presentation. Nearly all (see [12] for a discussion of
exceptions) smooth Deligne-Mumford stacks can be presented as global quotients ofordinary smooth manifolds by (not necessarily finite) groups, whose actions need not
be effective. To such presentations we can associated gauged supergravities, to which
we can immediately apply [8]. To be specific, suppose the moduli stack is presented
as [Y /G] for some smooth manifold Y and some group G, corresponding physically toa supergravity theory with moduli space Y and gauged18 G action. In this case, theBagger-Witten line bundle on the cover Y with a G-equivariant structure (specified
when one defines the gauge theory [8]) is equivalent to a (Bagger-Witten) line bundle
on [Y /G]. Other results follow analogously. For example, in this presentation, quantization of the Kähler form on the stack [Y /G] follows from both quantization of theKähler form on Y [1] and from quantization of Fayet-Iliopoulos parameters [8]. In allcases, generalizing Bagger-Witten [1] to this presentation intertwines the analyses and17

Experts should note that since we are implicitly working over the complex numbers, ‘smooth’ implies,
for example, that there are no nonreduced scheme structures [42].
18
The stack does not specify a classical gauge coupling; again, in four dimensions, we associate stacks tophysics but not physics to stacks.

26

results of [1] and [8]. (The toy example of [7] was realized by a presentation of thisform.)
There exist more types of presentations of stacks (e.g. groupoid quotients), and so possibly
more physical theories; in this paper, we shall discuss only the presentations above. As
emphasized in section 3, even in the IR these presentations can be physically distinct.
Regarding metric positivity, notions of ampleness and corresponding constraints on stacks
are discussed in [60]; we assume, but have not carefully checked, that they are pertinent here.
In other words, formally, the results of [1, 8] carry over more or less immediately tosmooth Deligne-Mumford stacks, at least in presentations of the form above. The only
significant differences are as follows:• A technical point is that cohomology of stacks more naturally lives in a different stack,the “associated inertia stack,” not the stack itself. Thus, the analysis of [1] still implies
that the cohomology class of the Kähler form on the moduli stack should match thecohomology class of c1 of the Bagger-Witten line bundle on the stack, but although
the Kähler form and Bagger-Witten line bundles themselves live on the stack, thecohomology lives in the associated inertia stack, and must be compared there. This
adds no essential physical constraint.
• Because the stack is, roughly, a space with (finite) automorphisms, coordinate patches
need match only up to (finite) automorphisms. Hence, for bundles on stacks, transition
functions on triple overlaps need only close up to finite automorphisms. This means
an honest bundle on a stack can be a ‘twisted’ or ‘fractional’ bundle on a space –objects which are not bundles in the ordinary sense. (We shall define these below.)
Put another way, there are more bundles on stacks than on underlying spaces, andmany things on spaces that are not quite bundles, become honest bundles on stacks.
Therefore, we need to carefully examine possible Bagger-Witten line bundles on stacks
for possible physical subtleties.
Let us examine the second issue above, in the special case of smooth Deligne-Mumford
stacks that have a (finite) gerbe structure over a smooth manifold. A twisted bundle on a
space (see e.g. [66, 67, 68, 69]) is a bundle in which the transition functions close only up toa higher cocycle; schematically:gαβ gβγ gγα = hαβγ
for some Cech cocycle hαβγ , where the gαβ are transition functions. Consistency requires thatthe rank of a twisted bundle be related to the order of the cohomology element defined by(hαβγ ); since we are interested in line bundles, no nontrivial twisted bundles can contribute.
Therefore, we need only consider the possibility that the Bagger-Witten line bundle might
be a fractional line bundle.
27

To explain fractional line bundles, which will play a crucial role in this section, let us givean explicit example. Consider19 a Zk gerbe on Pn defined by n + 1 homogeneous coordinatesof weight k, i.e.

 n+1
C
−0
,
C×
where the C× acts as
(x0 , · · · , xn ) 7→ λk x0 , · · · , λk xn .

On this gerbe one can define line bundles with arbitrary weight under the C× . For example,
a line bundle of weight m has total space20
 n+1

(C
− 0) × C,
C×where the C× acts as
(x0 , · · · , xn ; y) 7→ λk x0 , · · · , λk xn ; λm y .

When m is divisible by k, this is the pullback of an honest line bundle on Pn , namelyO(m/k). More generally, the line bundle above on the gerbe is sometimes (ambiguously)
denoted O(m/k) even when m is not divisible by k. In such cases, one has a ‘fractional’ line
bundle. (See [16] for a more complete description of fractional line bundles.)In passing, the properties of fractional branes at orbifold points are not unrelated to
fractional bundles. Ultimately the reason for the relationship is that in an orbifold, there isa gerbe structure appearing over orbifold points, which has the effect of desingularizing theorbifold.
It should now be clear that these fractional line bundles on the gerbe are precisely what
is being described in the example of [7], and more generally whenever one has “fractional
Fayet-Iliopoulos parameters.” If one were to pick a different physical presentation of the
19This example could not arise physically because of anomalies. We give it here as a purely mathematicaldemonstration and explanation of fractional line bundles, no more.
20
Curiously, total spaces of fractional line bundles over gerbes often have the property that they are honest
spaces with orbifolds, instead of gerbes, as is implicit in the expression given. Despite the existence of theorbifold structure along the zero section, one still has a notion of local trivializations; over the gerbe, thetotal spaces of the fractional bundles above have a local description of the form U × BZk × C for U an open
patch on Pn and BZk the classifying stack of Zk . The point is that for any vector space V , the quotient[V /G] is the same stack as the total space of a vector bundle of fiber V over BG. The difference between
the two descriptions might be described as distinguishing fibers over ‘gerby points’ from fibers over ‘variety
points’: in the former case, one speaks of a vector bundle over BG, whereas in the latter, one speaks of[V /G]. Put another way, the (representable) projection map to the gerbe on Pn has two types of fibers: thefiber over a point of Pn is ×

Ck × Cm
C×(with subscripts indicating weights), whereas the fiber over a BZk is just C.

28

gerby moduli ‘space,’ say as a nonlinear sigma model with a restriction on nonperturbative
sectors rather than as a gauged linear sigma model, then the line bundle L of which thesuperpotential is a section would just be taken to be a fractional line bundle from the outset.
Now that we have explained fractional line bundles, let us return to our discussion of themas possible Bagger-Witten line bundles arising when the moduli stack possesses a (finite)gerbe structure, and discuss possible consistency conditions. We see two possibilities:
1. One possibility is that allowed Kähler forms can have cohomology classes matching the(image of the) first Chern class of any line bundle on the gerbe, including any fractionalline bundle.
2. Another possibility is that allowed Kähler forms can have cohomology classes matching
the (image of the) first Chern class only of line bundles which are pullbacks of line
bundles on the underlying space – no fractional line bundles allowed.
The recent paper [7] argued the former case, that if the moduli ‘space’ of the supergravity
theory were actually a gerbe over an ordinary space then the quantization condition of
Bagger-Witten should be modified, and fractional values of the Fayet-Iliopoulos parametershould be allowed. We shall now study this claim in detail.
Let us re-examine the example21 of a U(1) gauge theory coupled to supergravity discussed
in section 3, in the spirit of [7]. In this theory, the supergravity moduli space is C2n+2 , andunder the gauged U(1), the fields have the following charges: n fields φi of charge k, n fields
φ̃i of charge −k, one field χ of charge +1, one field χ̃ of charge −1. Furthermore, the fields
of charge ±1 have mass m.
Restricting to the massless fields, the D-term condition has the formXX
k|φi |2 −k|φ̃i |2 = riiwhere r is the Fayet-Iliopoulos parameter. As discussed elsewhere [8, 7], in supergravity r isconstrained to be an integer, so [7] observed that when we divide by the common factor ofk, the D-term condition becomesXX|φi|2 −
|φ̃i |2 = r/kiiformally giving a fractional Fayet-Iliopoulos parameter (albeit normalized in such a way as
to make that fact less explicit).
Naively, the model above appears to describe a loophole in the analysis of Bagger-Witten[1], by allowing for fractionally quantized metrics. However, as discussed in section 3, the21

The supersymmetric CPn model would be marginally simpler to describe, but is also anomalous.

29

infrared limit of a four-dimensional gauged sigma model (in effective field theory) need not
be the same as a four-dimensional nonlinear sigma model. Thus, the example above is
not describing a loophole in Bagger-Witten, as it does not RG flow to a theory of the form
analyzed by Bagger-Witten. Rather, it is giving a quantization condition on a different theory
than is considered by Bagger-Witten. (We would like to thank J. Distler for emphasizing
this point to us.)
The physics above maps to a stack, a Zk gerbe over a space C2n //C× , and a fractionalline bundle that extracts the ‘universal’ aspects of the physics above. The fractional line
bundle over the gerbe corresponds to the equivariant structure implicit in the choice of
Fayet-Iliopoulos parameter. We cannot consistently construct a low-energy effective field
theory by integrating out the Higgsed gauge field, and so there is no regime in which we canconsistently talk about a fractionally quantized metric; however, we can nevertheless apply
stacks to give a ‘universal’ object encoding some essential aspects of the physics, and thegauge theory in question would be described mathematically by a fractional line bundle ona gerbe.
So far we have discussed the interpretation of certain choices of equivariant structures
on Bagger-Witten line bundles. It remains to understand whether those choices that lead tofractional line bundles on gerbes are physically consistent.
In particular, let us examine the kinetic terms for the gravitino ψµ and superpartners χI
more systematically. Recall the gravitino ψµ is a spinor-valued section of T X ⊗ φ∗ L−1 , andthe fermions χI are spinor-valued sections of φ∗ (T M ⊗ L), where X is the four-dimensional
low-energy effective spacetime, and φ : X → M is the bosonic map in the four-dimensional
nonlinear sigma model in the supergravity. If the moduli space admits a gerbe structure,and the Bagger-Witten line bundle L is fractional, then there are some potential issues:• First, fractional line bundles have no smooth (or even continuous) single-valued sections.
• Second, as noted earlier, seen as bundles over the underlying space, fractional bundles
have orbifold singularities in their fibers, making a metric on those fibers potentiallysingular. As that metric appears in the fermion kinetic terms, this is potentially ahazard.
In principle, both of these problems are solved by the fact that if the moduli space M has agerbe structure, then the path integral only sums over maps φ with degrees satisfying certain
divisibility properties – this is one of the defining properties of a nonlinear sigma model ona gerbe. (That said, if the four-dimensional spacetime X is R4 , then as discussed before
this matter is somewhat trivial, but let us describe the most general case here.) That samedivisibility criterion ensures that φ∗ L is an honest line bundle, not anything fractional. As a30

result, even when L is fractional, the gravitino ψµ and fermions χI do exist as single-valued
objects, coupling to honest bundles, with smooth fiber metrics.
In passing, let us mention another potential issue. If the SCFT moduli space admits agerbe structure, and the Kähler form arises from a fractional line bundle, then there is aninteresting structure on the worldsheet operators over SCFT moduli space (see [70] for a
discussion for ordinary moduli spaces). Specifically, as we walk around the SCFT moduli
space, some of the worldsheet operators (including the spectral flow operator) acquire phasesfrom the (fractional) line bundle, and hence are necessarily multi-valued over the SCFT
moduli space. This is at least odd, though not necessarily a physical contradiction. For
example, the SL(2, Z) transforms in monodromies on the u-plane in Seiberg-Witten theory
tell us that the low-energy effective action there is really only globally-defined on an SL(2, Z)Riemann surface covering the u-plane, not the u-plane itself. The situation here is closely
analogous.
As indirect consistency checks that these theories with fractional Bagger-Witten line
bundles are consistent, let us point out some closely related (and consistent) examples:• One example occurs in two-dimensional (0,2) SCFT’s, describing heterotic strings on
gerbes. If one compactifies a heterotic string on a gerbe with a fractional or twisted
gauge bundle (i.e. a bundle on the gerbe that is not a pullback from the underlying
space), the result looks like a sigma model on a space with a non-honest bundle, anda restriction on degrees of maps such that the pullbacks of non-honest bundles arehonest. These will be discussed in detail in [16]. (Note the left-moving worldsheet
fermions in this example are closely analogous to the four-dimensional gravitino and soforth we have been discussing – both couple to pullbacks of fractional bundles.) One
way to construct examples is through asymmetric orbifolds, that act ineffectively onright-movers but effectively on left-movers. Examples can also be constructed in (0,2)
GLSM’s, such as the (anomaly-free, fractional) bundle
0 −→ E −→ O(1)⊕9 −→ O(9) −→ 0
over P3[2,2,2,4] [10], a Z2 gerbe over P3[1,1,1,2] [5]. Other two-dimensional examples havebeen constructed by dimensional reduction of twisted four-dimensional N = 2 theories,
as in [32]. These examples all seem to be consistent.
• It is perhaps worth observing that nonlinear sigma models on total spaces of fractional
bundles are well-behaved. Consider a (2,2) supersymmetric gauged linear sigma model
describing
 2(C − 0) × C
C×where the C× acts on (C2 − 0) with weight 2, and on C with weight 1, say. This is
the total space of the (fractional) line bundle O(1) over a Z2 gerbe on P1 ; it is also31

a modification of the exceptional set away from the weighted projective stack P2[1,2,2] .
This is a consistent (2,2) supersymmetric theory.
• A four-dimensional gauge theory can also be constructed with closely analogous properties. Consider an SU(n) gauge theory with matter in the fundamental of SU(n).
We can interpret this as the Feynman diagrams of SU(n)/Zn gauge theory with asubset of the SU(n)/Zn instantons (omitting fractional instantons), restricted so as
to make the fundamental matter always well-defined. (We cannot precisely call thisan SU(n)/Zn gauge theory with a restriction on instantons, because the SU(n)/Zngauge transformations are not well-defined on the matter fields. For this reason, boththe SU(n) and the SU(n)/Zn gauge theories obey cluster decomposition.) After all,perturbatively an SU(n) and SU(n)/Zn gauge theory are identical (same Lie algebra,same Lagrangian, same Feynman diagrams), the difference between the data given isthat an SU(n)/Zn has additional (‘fractional’) instantons not present in the SU(n)/Zn
theory. One could imagine splitting an SU(n) instanton into SU(n)/Zn instantons,
but if one does so, one would have to introduce topological defects in order to allow
the matter in the fundamental representation to be well-defined globally.
This last example perhaps best exemplifies the slogan
Perturbative physics is determined by the Lie algebra of the gauge group, but
nonperturbative physics is determined by the Lie group, not just the algebra.
mentioned in section 3.
Our tentative conclusion is that “fractional Fayet-Iliopoulos parameters” are consistentin supergravity theories in which the moduli stack is a gerbe, and are a reflection of stacky
subtleties arising in more general supergravity theories. One should be careful about asserting that this implies a loophole in Bagger-Witten’s old quantization result [1], as the infrared
limit of a four-dimensional gauged sigma model need not be the same as a nonlinear sigma
model. We leave a more detailed analysis of consistency conditions in supergravity theories
with moduli stacks to future work.

7

Conclusions

In this paper we have reviewed recent discussions of quantization of the Fayet-Iliopoulos
parameter in supergravity theories. We began this paper by reviewing previous work on
two-dimensional theories with restrictions on nonperturbative sectors – equivalently, sigma
models on gerbes – and more general aspects of two-dimensional sigma models on stacks,

32

followed by a discussion subtleties appearing in four-dimensional analogues. We gave examples in both field and string theory of models with gerbe structures on their moduli spaces,and discussed the action of duality. We discussed global topological defects when the moduli
space is a stack, focusing on stacks that are gerbes over smooth manifolds. We then discussed
consistency conditions on classical supergravity theories for moduli spaces that are smooth
Deligne-Mumford stacks, after reviewing the state-of-the-art for smooth manifolds.
In the text we listed a number of interesting possible followups. Another direction thatwould be interesting to pursue is sigma model anomalies, in the sense of Moore-Nelson[71, 72, 73], in cases where the target space is a gerbe or other stack.
Yet another direction concerns deformation issues. Briefly, stacks and underlying spaces
do not always admit the same deformations. To illustrate the principle, consider a local
quotient stack structure resolving an orbifold singularity on a Calabi-Yau. (Moduli spaces
are typically not Calabi-Yau, but this will provide a simple example of the deformation theory
issue.) Although quotient spaces often admit Calabi-Yau blowups, corresponding quotient
stacks do not. (In string compactifications on stacks, this leads to an apparent mismatchin moduli which was discussed in [12].) Notions of blowup still exist, but are usually not
Calabi-Yau. For moduli stacks appearing in field theory and string theory, then, a natural
question to ask is whether the existence of a quotient stack structure ‘resolving’ an orbifold
singularity on the moduli space reflects any obstruction to resolution or deformation of thesingularity. It would be interesting to understand if this deformation-theoretic mismatch
had any applications in either field or string theory.
There are several other potential applications of such gerbe and stack structures in field
theory moduli spaces that we can imagine. For example, it would be interesting to understand whether ‘stacky’ resolutions of quotient singularities on moduli spaces, i.e. [Cn /G]versus Cn /G, convey any additional information about the theory, such as properties of lightparticles. It would also be interesting if gerbe structures could be used to help disentangle
confusing potential Seiberg duals. Examples of such are discussed in, for example, [74], andthere is a gerbe structure on some of the moduli spaces of the field theories discussed there.
Similarly, it would be interesting to understand the three-dimensional ‘mirrors’ [75] to theories with nonminimally-charged electrons. In two dimensions, such mirrors turned out to
involve either discrete-valued fields [13] or, equivalently, disconnected targets [14].
It would also be interesting to understand if the ideas in this paper could be applied tounderstand the distinctions between SU(2) and SO(3) Donaldson and related mathematicalinvariants, see e.g. [76] and references therein.

33

8

Acknowledgements

Some of the ideas in this paper have germinated over the last decade and been discussed withnumerous people, more than we can list here. In particular, we would like to thank J. Distler
for numerous discussions of cluster decomposition and supergravity moduli space issues,
A. Knutson for many discussions of group theory pertinent to heterotic strings, T. Pantev
for many years of discussions and collaboration on strings propagating on gerbes and stacks,and M. Strassler for insights and references on four-dimensional gauge theoretic analogues
provided during a visit to the University of Washington in spring 2005. We would also liketo thank P. Argyres, A. Shapere, and M. Unsal for more recent useful conversations.
The work of S.H. was supported by the WorldPremier International Research Center
Initiative, MEXT, Japan, and by a Grant-in-Aid for Scientific Research (22740153) from theJapan Society for Promotion of Science (JSPS). E.S. was partially supported by NSF grants
DMS-0705381 and PHY-0755614.

A

Four-dimensional decomposition conjecture

In this appendix we will discuss a four-dimensional analogue of the decomposition conjecture
for two-dimensional CFT’s discussed in [14]. This will arise via restricting four-dimensional
instantons (mathematically, c2 ’s, not c1 ’s), and so will not be directly relevant for the gerbes
studied elsewhere in this paper.
Consider a four-dimensional SCFT obtained from a gauge theory, e.g. N = 4 SU(n)SYM, or N = 2 SU(n) SYM with 2n hypermultiplets in the fundamental, or one of theN = 1 SCFT’s.
In that gauge theory, restrict the nonperturbative sector to instantons of degree divisible by k. Note that the resulting theory will not be associated to gerbes – we are here
imposing a restriction on Pontryagin classes of bundles, visible to four-dimensional theta
angles, whereas gerbe structures would only affect analogues of the first Chern class. This
theory automatically violates cluster decomposition, by Weinberg’s ancient argument; weshall describe how it can be written formally as a sum of other theories with rotating theta
angle.
In this theory, since the instantons have degrees divisible by k, the Chern-Simons vacua
split into k separate sets. The allowed instantons define tunnelling only between ChernSimons vacua within the same set. In this fashion, one recovers k separate zero-energy
ground states. Under the assumption that when the gauge field is extended flatly in extra
dimensions, the Chern-Simons number is cobordism invariant, the Chern-Simons number isconserved modulo k.
34

Using the state-operator correspondence for conformally-invariant theories, one can build
k different zero-energy states, which for the reasons above obey the same multiplicative rules
as twist fields in the two-dimensional theories discussed in [14], and hence can be used todefine projection operators.
Thus, we conjecture that the four-dimensional SCFT above with theta angle θ, can be
decomposed into a sum of k SCFT’s, and we further conjecture that those k SCFT’s arecopies of the SCFT with theta angles θ + 2πn/k for n = 0, · · · , k − 1, where θ has period 2π
in the theory where instanton number 1 configurations are allowed. (This sum has the effectof cancelling out gauge field configurations in the path integral whose instanton degrees are
not multiples of k.)
For the two-dimensional decomposition conjecture pertinent to sigma models on gerbes,
there is now abundant evidence, including all-genera partition function computations in orbifold examples [14], checks in mirror symmetry and quantum cohomology [14], applications
to gauged linear sigma models [15], and now checks of predictions for Gromov-Witten invariants [33, 34, 35, 36, 37, 38]. By contrast, in the four-dimensional case above, we have no
independent evidence, no examples, only the arguments above.

B

Two-dimensional BF theory and cluster decomposition

In this section we will examine BF theory in two dimensions, as an example of a manifestly
local theory that does not obey cluster decomposition. Let B be a circle-valued scalar,i.e. identified under B 7→ B + 2π. Let A be an abelian gauge field with the usual gaugetransformation, so that locallyA →7
A + dχ,F ≡ dA,
where χ is a circle-valued gauge parameter: χ ≡ χ + 2π. Then the field strength F thensatisfies the Dirac quantization condition,ZF ∈ 2πZ.
The action for BF theory is

and the Euclidean action is

k
S =
2π

Z

BF

ik
SE =
2π

Z

BF.

35

This theory is simple enough that it can be solved exactly and explicitly. To do this we solve
for the dimension of the Hilbert space of states on a spatial slice S 1 and for the action of
the operator algebra on that Hilbert space. The particular point to which we draw attention
is the absence of cluster decomposition: inside the local operator algebra of the theory is apair of local operators O±1 that disobey the condition for cluster decomposition, in the sensethat
lim hO1 (x)O−1 (0)i =6 lim hO1 (x)ihO−1 (0)i.
x→∞

x→∞

Hilbert space of the BF theory
First, we compute the overall dimension of the Hilbert space of states on S 1 . To do this,we compute the partition function on a spatial circle at finite temperature β −1 :X
Z(β) =
exp {−βEk } .
(3)kSince the two-dimensional metric does not appear in the BF action, we expect that thetheory is topological and that the energies Ek should all vanish identically, and that thepartition function Z(β) is therefore independent of β. We will see that this is indeed thecase.
To compute the partition function at finite temperature β −1 , we perform the path integral
βin Euclidean signature with Euclidean time compactified with a radius of r2 ≡ 2π
. We also
compactify the spatial direction with radius r1 , so that the Hilbert space becomes manifestly
separable. We have then reduced the finite-temperature partition function to a path integral
over a discretely infinite set of variables, the Fourier modes of the B field and the U(1)gauge connection. The path integral over the nonzero modes is purely Gaussian, and can be
performed straightforwardly so long as we divide appropriately by the measure for the local
U(1) gauge group. The path integral over the zero modes we perform separately.
Path integral measure on a finite torus
Path integrals in finite volume require a bit of care in order to get the overall normalization
correct – we mostly follow the method of [77], deviating from the presentation there only indetails particular to the application here.
Define the measure – for the gauge group, the gauge field, and the B field – as in [77], ina local way. To do this, decompose the fields and the gauge parameter into normal modes:X
Ai (x) ≡a(M ) φi,(M ) (x)M36

B(x) ≡
χ(x) ≡

Xb(N ) φ(N ) (x)NXc(N ) φ(N ) (x)N

where φ(N ) is a set of unit-orthonormalized real eigenfunctions of the scalar Laplacian, andφi,(N ) is a set of unit-orthonormalized real vector eigenfields of the vector Laplacian:
Z
Z
√
2 √
d x g φ(N ) (x)φ(N 0 ) (x) = δN 0 N ,d2 x g g ij φi,(M ) (x) φj,(M 0 ) (x) = δM 0 M ,(4)and the a(M ) , b(N ) , c(N ) are mode amplitudes, the Fourier transforms of the dynamical fields.
Then define the path integral measuresYDA ≡da(M )
(M )

DB ≡
Dχ ≡

Y

db(N )(N )Y

dc(N )(N )with unit normalization.
Concretely, for the torus, we can let N run over the values 0 and ([p], re), where ([p], im).
Here, the symbol [p] represents a pair {pi , −pi } of equal and opposite nonzero momenta
obeying appropriate quantization conditions, and re and im represent the real and imaginaryparts of the mometum eigenfunction.
As for M, we let it run over nonzero modes ([p], re, ⊥), ([p], im, ⊥), ([p], re, k) and ([p], im, k), and also over zero modes labelled (0, I). Here [p] means the same as it does for the scalar
eigenmodes, i.e., a pair of equal and opposite nonzero momenta on the two-torus, and reand im represent the real and imaginary parts of a plane wave. The symbols ⊥ and k denotethe transverse and longitudinal polarizations for the nonzero modes, and the I labelling thezero modes (Wilson lines) runs over the two directions of the torus.
Path integral over nonzero modes
Let us now perform the path integral over nonzero modes. For a given pair [p], wehave two multiplicative contributions to Z(r1 , r2 ): First, we have the Gaussian path integral
over the nonzero modes b([p],re) , b([p],im) and a([p],re,⊥) , a([p],im,⊥) . And then, we also have theJacobian determinant of the gauge transformation of a([p],re,k) , a([p],im,k) by c([p],re) , c([p],im) .

37

First, we combine the real and imaginary parts of the plane waves into the natural complex
combinations:
Φi,([p],⊥) ≡Φi,([p],k) ≡Φ[p] ≡
a([p],⊥) ≡a([p],k) ≡b[p] ≡
c[p] ≡1√ (φi,([p],re,⊥) + i φi,([p],im,⊥) ),2
1√ (φi,([p],re,k) + i φi,([p],im,k) ),2
1√ (φ([p],re) + i φ([p],im) ),2
1√ (a([p],re,⊥) + i a([p],im,⊥) ),21√ (a([p],re,k) + i a([p],im,k) ),21√ (b([p],re) + i b([p],im) ),21√ (c([p],re) + i c([p],im) ).
2

(5)
(6)
(7)
(8)
(9)
(10)
(11)

In terms of the complex combinations above, the measure isdare daim = 2 d2 a = 2 d(Re a) ∧ d(Im a),
dbre dbim = 2 d2 b = 2 d(Re b) ∧ d(Im b),
dcre dbim = 2 d2 c = 2 d(Re c) ∧ d(Im c),(12)where we have suppressed the indices [p], ⊥ and k, and we have used the standard conventionid2 z ≡ dz ∧ dz̄ = d(Rez) ∧ d(Imz)2(13)

for the measure on a complex variable z.
The expansion of Ai (x), B(x) and χ(x) in eigenmodes takes the form
Ai (x) = (zero modes)
(14)
X
∗
∗
∗
∗
+
a([p],⊥) Φi,([p],⊥) + a([p],k) Φi,([p],k) + a([p],⊥) Φi,([p],⊥) + a([p],k) Φi,([p],k) ,[p]

B(x) = (zero mode) +Xb[p] Φ[p] + b∗[p] Φ∗[p] ,[p]

χ(x) = (zero mode) +X

c[p] Φ[p] + c∗[p] Φ∗[p] .

[p]

The orthonormality conditions for the complex normal modes are
Z
Z
Z
√
∗
ij ∗
2 √
2 √
d x g g Φi,([p],⊥)Φj,([p0 ],⊥) =d2 x g g ij Φ∗i,([p],k) Φj,([p0],k)d x g Φ[p] Φ[p0 ] =
38

(15)

Z

d2 x

√

g g ij= δ[p],[p0] ,
Z

√
∗
∗
d2 x g g ij Φi,([p],⊥)Φj,([p0 ],k) = 0,Φi,([p],⊥)Φj,([p0 ],k) + Φi,([p0 ],k) Φj,([p],⊥) =
Z
Z
√
2 √
d x g Φ[p] Φ[p0 ] =d2 x g g ij Φi,([p],⊥)Φj,([p0 ],k) = 0.

and the transversality conditions areg ij ∂i Φj,([p],⊥) = ij ∂i Φj,([p],k) = 0 .
A natural choice for the normal modes isr1
Φ[p] =
exp {ip · x} ,
vol2
s
r
1
|g|
ik g kl pl exp {ip · x} ,Φi,([p],⊥) =mn
vol2 g pm pnrr
11
Φi,([p],k) =pi exp {ip · x} .
vol2 g mn pm pn

(16)

(17)

Now we want to write the Euclidean action in terms of the complex normal modes. The
action is of course a topological invariant, but since the normalization conditions of thenormal modes are written in terms of the metric, it is useful to write the action as
Z
p
ik
ij
SE = +
d2 x |g| p B Fij ,
(18)
2π
2 |g|
In terms of the complex normal modes the action is
X
SE =
S[p]
[p]


k
|p| b[p] a∗([p],⊥) − b∗[p] a([p],⊥)
2π


b[p]
∗
∗
= (b[p] a([p],⊥) ) O,a([p],⊥)

S[p] ≡ +where O is the anti-Hermitean matrix
O≡ −ik|p| 2
·σ ,
2π

|p| ≡

p
g ij pi pj .

(19)

Since the operator O has imaginary eigenvalues, we can define the path integral by theprescription
O → O +  |p|2 · 12×2 ,
39

(20)

and let  → 0+ be a positive real parameter approaching zero from above. With this definition
the action is still local and reduces in the  → 0+ limit to the undeformed action, but thepath integral over each set of modes is convergent.
For an operator O with positive definite real part, the Gaussian path integral over a
vector of complex variables A is given by  −1Z Y
O2†(2 d A(N ) ) exp −A O A = det
,
(21)
2π
(N )

so, letting
A≡



b[p]



,
a([p],⊥)
ik|p| 2
σ ,
O≡−
2π

(22)
4

π
we find that the path integral over b[p] and a([p],⊥) is k162 |p|
2 . The other contributing factor,

from equal and opposite nonzero momenta [p], is the Fadeev-Popov determinant
The gauge transformation of a([p],k) isδ a∗([p],k) = −i |p| · δ c∗[p] ,δ a([p],k) = i |p| · δ c[p] ,d2 a([p],k).
d2 c[p](23)

so the Jacobian of the gauge transformation is given byd2 a([p],k)= |p|2 .
2
d c[p](24)

Thus for each equal and opposite pair [p] of nonzero momenta there is a cancellation of|p|-dependence between the dynamical Gaussian path integral and the Fadeev-Popov deter4
minant, leaving a factor of + 16π
for each [p].
k2Formally, then, the path integral over nonzero modes, modulo the volume of the groupof gauge transformations at nonzero momentum, isX
Znonzero = exp {Fnonzero } ,Fnonzero =F[p] ,[p]F[p] = ln



16π
k2

4



= −ln



k2
16π 4



.

(25)

The set of equal and opposite pairs [p] in the sum above is indexed by a set of half the
momenta. Since the summand is invariant under p → −p, it is easier to halve the summandand let the sum run over all momenta:
 2


X
4π
k
1
Fnonzero =Fp ,Fp ≡ F[p] = +ln
= −ln
.
(26)
2
k
4π 2
p
40

Formally, then, the path integral over the nonzero modes B and A, dividing out by thevolume of gauge group at nonzero momenta, is given bykZnonzero = exp {−F · ln
},
4π 2
X
F ≡
1.
(27)
p6=0

The quantity F awaits regularization and renormalization. For the moment we assert thatF = −1 + q · vol2 for any local renormalization procedure, where vol2 is the volume 4π 2 r1 r2 =2πβr1 of the two-torus, and q is a counterterm adjusting the effective vacuum energy density:


k
.
(28)
q = ρΛ /ln
4π 2
The choice of q (or equivalently ρΛ ) is a local counterterm and the magnitude of its finite pieceis inherently ambiguous in the absence of some symmetry principle to determine it. Since
the classical action is scale invariant, we are motivated to choose the value of q that restores
scale-invariance, namely q = 0. The magnitude of the non-extensive piece of F cannot be
absorbed into a local counterterm, is unambiguously determined, and should be the samefor any local renormalization procedure to define F . Thus we will choose a renormalization
proceure of the general form
F → Fregulated + (∆q) · vol2 ,(29)

where (∆q) is always chosen so that the extensive piece of F for whatever regulator wechoose:Fregulated∆q = − lim.
(30)
vol2 →∞
vol2
Then we exploit the fact that the summand is just a constant 1, and thus infrared-finite
for small |p|, to write F as −1 plus a counterterm plus a factorized sum, where each sum
depends only on the momentum in a single direction:
Fregulated = −1 + (∆q) vol2 + F1,regulated F2,regulated .

(31)

For any local regulator characterized by a scale Λ, the sums F1,regulated and F2,regulated vanish
up to a UV divergent piece proportional to Λ r1 (resp. Λr2 ), as well as terms vanishing more
quickly than Λ−1 . Thus for appropriately chosen ∆q, the sum Fregulated goes to −1 plus
terms that vanish at least as quickly as a negative power of Λ when Λ is sent to ∞.
by

The sum Fregulated is also −1 in zeta-function regularization: the term F1,regulated is defined
F1,regulated ≡Xµ2s (g 11 (p1 )2 )−s = (µr1 )2s

p1

X
n1

41

n1−2s



= (µr1 )2s 1 + 2

X
n≥ 1



n1−2s  = (µr1 )2s (1 + 2ζ(2s)) ,(32)

where ζ is the Riemann zeta function. This sum is convergent for s > 21 but is defined
uniquely by analytic continuation for all values s 6= 1. Removing the regulator corresponds
to evaluating s = 0. Since ζ(0) = − 12 we have F1,regulated = 0, and likewise F2,regulated = 0.
Thus there is no need for counterterms in zeta-function regularization: ∆q = 0 and F = −1.
So we have found that the appropriately renormalized value of F is simply −1, and thus
the renormalized path integral over nonzero modes, dividing appropriately by the nonzero
mode gauge group measure, isZnonzero =

k
.
4π 2

(33)

The partition function over nonzero modes alone is nonlocal in all directions, does not havea Hilbert space interpretation and need not be integer, which is as it should be. To derive anappropriate Hilbert space interpretation of the vacuum amplitude, we need to include thecontributions of the zero modes.
Integral over the zero modes
We now compute the volume of the zero modes of Ai and B, and divide byQthe volume of
the zero mode of χ. The measure for the dynamical zero modes is just given by I=1,2 da0,I db0
and the measure for the χ zero mode is dc0 . For both integrals the integrand is 1 and allthat remains is to compute the region of integration.
The zero mode pieces of the dynamical fields Ai , B and the gauge parameter χ areX
Ai |zero mode =φi,(0,I) a(0,I)
I

B|zero mode = φ0 b0
χ|zero mode = φ0 c0 ,(34)

where φ0 and φ(0,I) are zero modes of the Laplacian satisfying the orthonormality conditions.
We take
r
gii
1
.
(35)
,
φi,(0,I) = δiI
φ0 = √
vol2
vol2
To determine the fundamental region for the Wilson lines a(0,I) , recall that the Wilson lines
are identified under large gauge transformations,Ai |zero mode ∼ Ai +
42

2π
,
∆xi

(36)

where ∆xi is the extent of the coordinate xi . In terms of the orthonormalized zero modes,
this translates into√√
2π δiI vol22π δiI vol2a(0,I) ∼ a(0,I) + √,(37)=
gii ∆xi
Li√
where Li ≡ gii ∆xi is the physical length of the cycle in the xi direction.
As for the zero modes of B and χ, both are identified under 2π, and their zero modes
have the same normalization, so their zero modes have the same identifications:ppc0 ∼ c0 + 2π vol2 .
(38)
b0 ∼ b0 + 2π vol2 ,So the total partition function for the zero modes is

Za0 Zb0.
Zc 0

Zzero =

(39)

The b0 and c0 zero mode integrals are the same,
Z √2π
vol2
Zb0 =
d b0 = Zc0

(40)

0

so their ratio is unity, and we are left with contributions from the a0 zero modes:
Zzero = Za0 =

Z 2π
0

√

vol2
L1

Z 2π

√

vol2
L2

d2 a(0,I) =0

4π 2 vol2
.
L1 L2

(41)

The product of the lengths of cycles of a rectangular torus is equal to the volume, L1 L2 = vol2 ,soZzero = 4π 2 .

(42)Thus the total partition function over the torus is the product of the zero mode andrenormalized nonzero mode path integrals:
Z = Zzero · Znonzero = (4π 2 ) ·

k
= k,
4π 2

(43)

independent of r1 and r2 . So we conclude that the BF theory has exactly k quantum states,
all of the same energy, which can be set exactly to zero by a choice of counterterm for thetwo-dimensional vacuum energy density. We see that the BF theory at level k ≥ 2 is theminimal Lagrangian realization of a quantum field theory with k degenerate vacua – it isminimal in the sense that it contains only the degenerate vacuua, and nothing else.
Hilbert space interpretation of the vacuum amplitude
43

The vacuum amplitude is purely topological: with the appropriate choice of vacuum
energy density counterterm, the partition function is independent of r1 and r2 . Interpretingthe Euclidean vacuum amplitude on the torus as a thermal partition function at temperature1, we see that the dimension of the Hilbert space is k, and all states have exactly
β −1 = 2πr2zero energy. Note, however, that even if we had chosen a different value for ∆q, we would
have had k degenerate states, with common energy E = 2πr1 ρΛ = 2πr1 (∆q)/ln 4πk 2 .
Spectrum and commutation relations of local operators and line operators
The operators in question are
On (x) ≡ : exp (inB(x)) :which clearly obey On · Om = On+m . We also have the Wilson line operators I Wn ≡ : exp in A : ,with the line integral taken over a spatial cycle. As we shall show shortly, the Wilson line
operators obey simple equal-time commutation relations with the local operators Om (x):Wn Om = ξ nm Om Wn
for ξ = exp(−2πi/k). As a result, W and O commute like clock and shift operators.
Given the dimension of the Hilbert space and the fact that the operators W and O are
invertible, it follows that the operators W and O not only commute but actually act asclock and shift operators in the standard k-dimensional representation. The commutation
relations define an algebra whose smallest nontrivial representation is k-dimensional. Since
the operators W and O are invertible, their representation on the Hilbert space cannot act aszero identically, so their representation must be nontrivial, and the standard k-dimensional
representation is the only one that is sufficiently small.
Commutation relations
Now we shall compute the commutation relations between the Wilson line operators andthe On . We will work in timelike gauge: A0 = 0. In this case, we can take the action to beZk
B∂0 A1S =2πhence the conjugate momenta are
πA1 =

kB
kA1
, πB = −2π2π
44so we have the equal-time commutators[B(x1 ), A1 (x2 )] = −i2πδ(x1 − x2 ).
k

From this one immediately derives[A1 (x), f (B(y))] =DefineL =so thatI

2πi 0
f (B(y))δ(x − y).
kdxA1 (x)[L, f (B)] =

2πi 0
f (B)
k

so in particular
2πα iαBeL, eiαB = −kfor any constant α. It is then straightforward to compute that


2πiαβ
iβL iαB −iβL
eiαB
e e e
= exp −kfor any constants α, β. Thus, in particular,


2πimn
Wn Om = exp −Om Wn .
k

We have already seen that the dimension of the Hilbert space of states in this theory is
k. Now we see that this k-dimensional Hilbert space carries a minimum-dimensional representation of the finite-dimensional analog of the Heisenberg algebra, generated by clock andshift operators at level k, which are generated by natural local operators and line operatorsof the theory.
Concretely, the operator On acts on the Hilbert space as
1 0 0 ···0
 0 ξn 0 · · ·0 0 0 ξ 2n · · ·0On ∼ 
.
 .. ..

..
..
 . .

.
.
(k−1)n0 0 0 ··· ξ
Note also that the On exhaust the set of linearly independent local operators, as opposed
to line operators, in the theory, and therefore that the state-operator correspondence holds in45

this theory, despite its unfamiliar features: the dimension of the Hilbert space is k, matching
the number of independent On .
Having established the action of the On on the k-dimensional Hilbert space, it is thenpossible from linear combinations of the On to construct k independent projection operatorsthat are also local operators, which confirms the decomposition hypothesis that we haveconjectured to hold in general for two-dimensional conformal theories not satisfying cluster
decomposition [14]. For example, the operator P0 projecting onto states invariant under thecontinuous B → B +  symmetry is given by
k−1

P0 =

1X
Oi .
k i=0

Of course projection operators always exist in any finite-dimensional (or even separable)
Hilbert space. The existence of the projectors has special significance, indicating the failureof cluster decomposition, only because they are local operators.
Direct demonstration of non-cluster-decomposition
We can also deomonstrate the failure of cluster decomposition directly in this theory,
without considering Wilson lines; we can simply compute the correlation function of two
local O-operators made from B, and note that the correlation function is not equal to the
product of one-point functions, even when the spacelike separation between the operators
becomes arbitrarily large. Correlation functions containing only B’s are particularly simple,
receiving only contributions from the zero modes of B – nonzero modes do not contribute.
This is because B is a field that enters only linearly in the Lagrangian. In terms of Feynman
diagrams, the propagator is purely anti-diagonal between B and A:
A

B

and there are no interaction vertices. Thus, the expectation value of a set of B nonzeromodes is equal to its classical value, i.e. the value where all nonzero modes of B are set tozero, because in the absence of A modes the external B lines have nowhere to terminate. This
argument does not apply to zero modes of B because these modes do not have a well-definedpropagator and there is no diagrammatic calculation of their correlation functions.
One implication is that On · Om = On+m . In particular, this means that
hOn i = hO1n i
and from integrating over the circle of B zero modes, we find thathO1n i = h1iδn,0 mod k .
46

The result above shows directly that cluster decomposition does not hold in the BF
theory at level k ≥ 1. If cluster decomposition were to hold, it would mean thathO1 (x)ihO−1 (0)i
.
x→∞
h1i

lim hO1 (x)O−1 (0)i = lim

x→∞

(44)

On the other hand, from the results above, we know thathO1 (x)O−1 (0)i = h1i
hO±1 (x)i = 0(unless k = 1)

so the property (44) does not hold for k ≥ 2: the operators O±1 are correlated with one
another at arbitrary spacelike separation, as expected from a summation over multiple degenerate vacua labeled by expectation values of the O±1 .

References[1] E. Witten, J. Bagger, “Quantization of Newton’s constant in certain supergravity
theories,” Phys. Lett. B115 (1982) 202-206.
[2] Z. Komargodski, N. Seiberg, “Comments on the Fayet-Iliopoulos term in field theory
and supergravity,” JHEP 0906 (2009) 007, arXiv: 0904.1159.
[3] Z. Komargodski, N. Seiberg, “From linear SUSY to constrained superfields,” JHEP
0909 (2009) 066, arXiv: 0907.2441.
[4] Z. Komargodski, N. Seiberg, “Comments on supercurrent multiplets, supersymmetricfield theories and supergravity,” JHEP 1007 (2010) 017, arXiv: 1002.2228.
[5] D. Butter, “Conserved currents and Fayet-Iliopoulos terms in supergravity,” arXiv:
1003.0249.
[6] T. Dumitrescu, Z. Komargodski, M. Sudano, “Global symmetries and D-terms in supersymmetric field theories,” arXiv: 1007.5352.
[7] N. Seiberg, “Modifying the sum over topological sectors and constraints on supergravity,” arXiv: 1005.0002.
[8] J. Distler, E. Sharpe, “Quantization of Fayet-Iliopoulos parameters in supergravity,”
arXiv: 1008.0419.
[9] T. Banks, N. Seiberg, “Symmetries and strings in field theory and gravity,” arXiv:
1011.5120.
47

[10] S. Katz, T. Pantev, E. Sharpe, “D-branes, orbifolds, and Ext groups,” Nucl. Phys.
B673 (2003) 263-300, arXiv: hep-th/0212218.
[11] T. Pantev, E. Sharpe, “Notes on gauging noneffective group actions,” arXiv:
hep-th/0502027.
[12] T. Pantev, E. Sharpe, “String compactifications on Calabi-Yau stacks,” Nucl. Phys.
B733 (2006) 233-296, arXiv: hep-th/0502044.
[13] T. Pantev, E. Sharpe, “GLSM’s for gerbes (and other toric stacks),” Adv. Theor.
Math. Phys. 10 (2006) 77-121, arXiv: hep-th/0502053.
[14] S. Hellerman, A. Henriques, T. Pantev, E. Sharpe, M. Ando, “Cluster decomposition,T-duality, and gerby CFTs,” Adv. Theor. Math. Phys. 11 (2007) 751-818, arXiv:
hep-th/0606034.
[15] A. Caldararu, J. Distler, S. Hellerman, T. Pantev, E. Sharpe, “Non-birational twisted
derived equivalences in abelian GLSMs,” Comm. Math. Phys. 294 (2010) 605-645,
arXiv: 0709.3855.
[16] T. Pantev, E. Sharpe, “Heterotic strings on gerbes,” to appear.
[17] R. Karp, “On the Cn /Zm fractional branes,” J. Math. Phys. 50 (2009) 022304, arXiv:
hep-th/0602165.
[18] C. Herzog, R. Karp, “On the geometry of quiver gauge theories: Stacking exceptional
collections,’ arXiv: hep-th/0605177.
[19] E. Sharpe, “Derived categories and stacks in physics,” contribution to the proceedingsof the ESI research conference on homological mirror symmetry (Vienna, Austria, June
2006), arXiv: hep-th/0608056.
[20] E. Sharpe, “Landau-Ginzburg models, gerbes, and Kuznetsov’s homological projective
duality,” to appear in the proceedings of Topology, C∗ algebras, string duality (Texas
Christian University, May 18-22, 2009).
[21] E. Sharpe, “GLSM’s, gerbes, and Kuznetsov’s homological projective duality,” contribution to the proceedings of Quantum theory and symmetries 6, arXiv: 1004.5388.
[22] W.-M. Chen, Y.-B. Ruan, “A new cohomology theory for orbifold,” Comm. Math.
Phys. 248 (2004) 1-31, arXiv: math/0004129.
[23] D. Abramovich, T. Graber, A. Vistoli, “Gromov-Witten theory of Deligne-Mumford
stacks,” Amer. J. Math. 130 (2008) 1337-1398, arXiv: math.AG/0603151.
[24] T. Coates, A. Corti, Y. Lee, H. Tseng, “The quantum orbifold cohomology of weighted
projective spaces,” Acta. Math. 202 (2009) 139-193, arXiv: math.AG/0608481.
48

[25] E. Mann, “Orbifold quantum cohomology of weighted projective spaces,” J. Alg. Geom.
17 (2008) 137-166, arXiv: math.AG/0610965.
[26] A. Vistoli, “Intersection theory on algebraic stacks and on their moduli spaces,” Inv.
Math. 97 (1989) 613-670.
[27] T. Gomez, “Algebraic stacks,” Proc. Indian Acad. Sci. Math. Sci. 111 (2001) 1-31,
arXiv: math.AG/9911199.
[28] G. Laumon, L. Moret-Bailly, Champs algébriques, Springer, 1999.
[29] K. Hori, C. Vafa, “Mirror symmetry,” arXiv:

hep-th/0002222.

[30] D. Morrison, R. Plesser, “Towards mirror symmetry as duality for two-dimensionalabelian gauge theories,” Nucl. Phys. Proc. Suppl. 46 (1996) 177-186, arXiv:
hep-th/9508107.
[31] C. Vafa, E. Witten, “On orbifolds with discrete torsion,” J. Geom. Phys. 15 (1995)
189-214, arXiv: hep-th/9409188.
[32] A. Kapustin, “Holomorphic reduction of N = 2 gauge theories, Wilson-’t Hooft operators, and S-duality,” arXiv: hep-th/0612119.
[33] E. Andreini, Y. Jiang, H.-H. Tseng, “On Gromov-Witten theory of root gerbes,”
arXiv: 0812.4477.
[34] E. Andreini, Y. Jiang, H.-H. Tseng, “Gromov-Witten theory of product stacks,”
arXiv: 0905.2258.
[35] E. Andreini, Y. Jiang, H.-H. Tseng, “Gromov-Witten theory of etale gerbes, i: root
gerbes,” arXiv: 0907.2087.
[36] H.-H. Tseng, “On degree zero elliptic orbifold Gromov-Witten invariants,” arXiv:
0912.3580.
[37] A. Gholampour, H.-H. Tseng, “On Donaldson-Thomas invariants of threefold stacks
and gerbes,” arXiv: 1001.0435.
[38] X. Tang, H.-H. Tseng, “Duality theorems of étale gerbes on orbifolds,” arXiv:
1004.1376.
[39] A. Kapustin, E. Witten, “Electric-magnetic duality and the geometric Langlands program,” arXiv: hep-th/0604151.
[40] E. Witten, “Mirror symmetry, Hitchin’s equations, and Langlands duality,” arXiv:
0802.0999.

49[41] R. Donagi, T. Pantev,
math.AG/0604617.

“Langlands duality for Hitchin systems,”

arXiv:[42] T. Pantev, private communication.
[43] P. Pouliot, “Chiral duals of non-chiral susy gauge theories,” Phys. Lett. B359 (1995)
108-113, arXiv: hep-th/9507018.
[44] P. Pouliot, M. Strassler, “A chiral SU(N) gauge theory and its non-chiral Spin(8)dual,” Phys. Lett. B370 (1996) 76-82, arXiv: hep-th/9510228.
[45] M. Strassler, “Duality, phases, spinors and monopoles in SO(n) and Spin(n) gauge
theories,” JHEP 9809 (1998) 017, arXiv: hep-th/9709081.
[46] M. Strassler, “On phases of gauge theories and the role of non-BPS solitons in field
theory,” arXiv: hep-th/9808073.
[47] J. de Boer, R. Dijkgraaf, K. Hori, A. Keurentjes, J. Morgan, D. Morrison, S. Sethi,“Triples, fluxes, and strings,” Adv. Theor. Math. Phys. 4 (2002) 995-1186, arXiv:
hep-th/0103170.
[48] M. Green, J. Schwarz, E. Witten, Superstring theory, volume I, Cambridge University
Press, Cambridge, 1987.
[49] D. Gross, J. Harvey, E. Martinec, R. Rohm, “Heterotic string theory I: the free heterotic string,” Nucl. Phys. B256 (1985) 253-284.
[50] P. Ginsparg, “Applied conformal field theory,” pp. 1-168 in Fields, strings andcritical phenomena (Les Houches 1988), North-Holland, Amsterdam, 1990, arXiv:
hep-th/9108028.
[51] C. Vafa, “Quantum symmetries of string vacua,” Mod. Phys. Lett. A4 (1989) 16151626.
[52] E. Witten, “D-branes and K theory,”
hep-th/9810188.

JHEP 9812 (1998) 019,

arXiv:[53] J. Distler, E. Sharpe, “Heterotic compactifications with principal bundles for general groups and general levels,” Adv. Theor. Math. Phys. 14 (2010) 335-398, arXiv:
hep-th/0701244.
[54] M. Dine, N. Seiberg, E. Witten “Fayet-Iliopoulos terms in string theory,” Nucl. Phys.
B289 (1987) 589-598.
[55] J. Atick, L. Dixon, A. Sen, “String calculation of Fayet-Iliopoulos D-terms in arbitrary
supersymmetric compactifications,” Nucl. Phys. B292 (1987) 109-149.

50

[56] M. Bershadsky, T. Pantev, V. Sadov, “F-theory with quantized fluxes,” arXiv:
hep-th/9805056.
[57] A. Knutson, private communication.
[58] C. Vafa, “Geometric origin of Montonen-Olive duality,” arXiv:

hep-th/9707131.

[59] B. Greene, A. Shapere, C. Vafa, S.-T. Yau, “Stringy cosmic strings and noncompact
Calabi-Yau manifolds,” Nucl. Phys. B337 (1990) 1-36.
[60] A. Kresch, “On the geometry of Deligne-Mumford stacks,” pp. 259-271
in Algebraic geometry (Seattle, 2005), Proc. Symp. Pure Math. 80, part
1, Amer. Math. Soc., Providence, Rhode Island, 2009, also available at
http://www.math.uzh.ch/fileadmin/user/kresch/publikation/geodm.pdf.
[61] A. Vilenkin, E. P. S. Shellard, Cosmic strings and other topological defects, Cambridge
University Press, 1994.
[62] M. Shifman, A. Yung, Supersymmetric solitons, Cambridge University Press, 2009.
[63] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, third edition,
Springer-Verlag, 1994.
[64] P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute ofFundamental Research, Springer-Verlag, Berlin, 1978.
[65] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Princeton
University Press, 1984.
[66] A. Căldăraru, “Derived categories of twisted sheaves on Calabi-Yau manifolds,” Cornell
University Ph.D. thesis, 2000. Available at http://www.math.upenn.edu/~andreic[67] A. Căldăraru, “Derived categories of twisted sheaves on elliptic threefolds,” arXiv:
math/0012083.
[68] A. Căldăraru, “Non-fine moduli spaces of sheaves on K3 surfaces,” arXiv:
math/0108180.
[69] A. Căldăraru, S. Katz, E. Sharpe, “D-branes, B fields, and Ext groups,” Adv. Theor.
Math. Phys. 7 (2004) 381-404, arXiv: hep-th/0302099.
[70] J. Distler, “Notes on N = 2 sigma models,” pp. 234-256 in String theory and quantum
gravity 1992 (Trieste, 1992), arXiv: hep-th/9212062.
[71] G. Moore, P. Nelson, “Anomalies in nonlinear sigma models,” Phys. Rev. Lett. 53(1984) 1519-1522.

51

[72] G. Moore, P. Nelson, “The Etiology of sigma model anomalies,” Comm. Math. Phys.
100 (1985) 83-132.
[73] A. Manohar, G. Moore, P. Nelson, “A comment on sigma model anomalies,” Phys.
Lett. B152 (1985) 68-74.
[74] J. Brodie, P. Cho, K. Intriligator, “Misleading anomaly matchings?,” Phys. Lett. B429
(1998) 319-326, arXiv: hep-th/9802092.
[75] K. Intriligator, N. Seiberg, “Mirror symmetry in three-dimensional gauge theories,”
Phys. Lett. B387 (1996) 513-519, arXiv: hep-th/9607207.
[76] A. Malmendier, “Donaldson invariants of CP1 × CP1 and mock theta functions,”
arXiv: 1008.0175.
[77] J. Polchinski, “Evaluation of the one loop string path integral,” Commun. Math. Phys.
104 (1986) 37-47.

52