hep-th/9903005, IASSNS-HEP-99-20

Supersymmetric Index

Of Three-Dimensional Gauge Theory

Edward Witten

School of Natural Sciences, Institute for Advanced Study

Olden Lane, Princeton, NJ 08540, USA

In super Yang-Mills theory in three spacetime dimensions, with a simple gauge group and a Chern-Simons interaction of level , the supersymmetric index can be computed by making a relation to a pure Chern-Simons theory or microscopically by an explicit Born-Oppenheimer calculation on a two-torus. The result shows that supersymmetry is unbroken if (with the dual Coxeter number of ) and suggests that dynamical supersymmetry breaking occurs for . The theories with large are massive gauge theories whose universality class is not fully described by the standard criteria.

To appear in the Yuri Golfand memorial volume.

February, 1999

1. Introduction

If a -dimensional supersymmetric quantum field theory is quantized on (with understood as space and parametrizing the time), the spectrum is often discrete. If so, one can define a supersymmetric index , the number of zero energy states that are bosonic minus the number that are fermionic. The index is invariant under smooth variations of parameters (such as masses, couplings, and the flat metric on ) that can be varied while preserving supersymmetry. For this reason, it often can be computed even in strongly coupled theories [1].

When is nonzero, there are supersymmetric states for any volume of , and hence the ground state energy is zero regardless of the volume. When one has a reasonable control on the behavior of the theory for large field strengths (to avoid for example the possibility that a supersymmetric state goes off to infinity as the volume goes to infinity), it follows that the ground state energy is zero and supersymmetry is unbroken in the infinite volume limit. Conversely, if , this gives a hint that supersymmetry might be spontaneously broken in the quantum theory, even if it appears to be unbroken classically.

There are interesting examples of theories (e.g., nonlinear sigma models in two dimensions, and pure supersymmetric Yang-Mills theory in four dimensions) in which a nonzero value of has been used to show that supersymmetry remains unbroken even for strong coupling. But there are in practice very few instances in which vanishing of this index has served as a clue to spontaneous supersymmetry breaking. One reason for this is that many interesting supersymmetric theories have a continuous spectrum if compactified on a torus, making difficult to define, or have a nonzero value of , so that supersymmetry cannot be broken. In other examples, is defined and equals zero, but does not give a useful hint of supersymmetry breaking because this phenomenon is either obvious classically or is obstructed by the existence, classically, of a mass gap, or for other reasons.

The present paper is devoted to a case in which the index does seem to give a clue about when supersymmetry is dynamically broken. This example is the pure supersymmetric gauge theory in three spacetime dimensions, with simple compact gauge group . The theory can be described in terms of a gauge field and a gluino field (a Majorana fermion in the adjoint representation). We include a Chern-Simons interaction, so the Lagrangian with Euclidean signature reads

The parameter is quantized topologically [2]. If denotes the dual Coxeter number of , then the quantization condition is actually that should be an integer, as was pointed out in [3,4], using a mechanism of [5,6]. The situation will be reviewed in section 2.

Let denote the supersymmetric index as a function of . We will show that for , but for . For example, for , we have , and

So vanishes precisely if .

From this it follows that supersymmetry is unbroken quantum mechanically for . But we conjecture that in the “gap,” , supersymmetry is spontaneously broken. For this we offer two bits of evidence beyond the vanishing of the index. One is that an attempt to disprove the hypothesis of spontaneous supersymmetry breaking for by considering an theory (instead of ) fails in a subtle and interesting way. The second is that, as we will see, if the theory is formulated on a two-torus of finite volume, spontaneous breaking of supersymmetry occurs. Of course, these considerations do not add up to a proof, but they are rather suggestive.

The paper is organized as follows. In section 2, we compute the index for sufficiently large by using low energy effective field theory and the relation [7] of Chern-Simons gauge theory to two-dimensional conformal field theory. In the process, we also review the anomaly that sometimes shifts to half-integer values, and we explain the failure of a plausible attempt to disprove the hypothesis of symmetry breaking in the gap via gauge theory. In section 3, we make a more precise microscopic computation of the index, and show that for finite volume symmetry breaking does occur in the gap. Finally, in section 4, we consider three-dimensional Chern-Simons theories in the light of the familiar classification [8] of massive phases of gauge theories, and show that such massive phases are not fully classified by the usual criteria. This is true even in four dimensions, but the full classification of massive phases is particularly rich in three dimensions.

For other recent results on dynamics of supersymmetric Chern-Simons theories in three dimensions, see [9].

2. Computation Via Low Energy Effective Field Theory

The index can be computed very quickly if is sufficiently large. At the classical level, the theory has a mass gap for [2]. The mass is of order , which if is much greater than the scale set by the gauge couplings. So for , the classical computation is reliable, the theory has a mass gap, and in particular (as there is no Goldstone fermion) supersymmetry is unbroken.

Moreover, we can compute the index for sufficiently large using
low energy effective field theory.
For large enough , the mass gap implies
that the fermions can be integrated out to give
a low energy effective action that is still local.
Integrating out the fermions
gives a shift in the effective value of . The shift can be computed
exactly at the one-loop level.^{†}^{†} There are many ways
to prove this. For example, the -loop effective action
for is the integral of a gauge-invariant local density, which the
Chern-Simons functional is not, so a renormalization of the effective
value of can only come at one loop. Alternatively, an -loop
diagram is proportional to , and so can only renormalize
an integer if .
In fact, integrating out the fermions
shifts the effective value of in the low energy effective field
theory to

where is the sign of [3]. (The shift in is proportional to the sign of , because this sign determines the sign of the fermion mass term.) So for example if is positive, as we assume until further notice, then . For the low energy theory to make sense, must be an integer, and hence must be congruent to modulo . So if is odd, then is half-integral, rather than integral [3]. For example, for , and is half-integral if is odd.

Since the factor of will be important in this paper, we pause to comment on how it emerges from Feynman diagrams. The basic parity anomaly [5,6] is the assertion that for an gauge theory with Majorana fermions consisting of two copies of the two-dimensional representation, the one-loop shift in is . (We must take two copies of the of , not one, because the is a pseudoreal representation, but Majorana fermions are real.) For any other representation, the one-loop shift is scaled up in proportion to the trace of the quadratic Casimir for that representation. For three Majorana fermions in the adjoint representation of , the trace of the quadratic Casimir is twice as big as for two ’s, so the shift in is , which we write as , with for . The result is universal, since is the group theory factor in the one-loop diagram for any group. This argument does not explain the minus sign in the formula , which depends on some care with orientations. This sign can be seen in Feynman diagrams [3], and also has a topological meaning that we will see in section 3.

Now, for very large , though the theory has a mass gap, it is not
completely trivial at low energies. Rather, there is a nontrivial
dynamics of zero energy states governed by the Chern-Simons theory
at level .
At low energies, we can ignore the Maxwell-Yang-Mills term in the action,
and approximate the theory by a “pure Chern-Simons” theory, with
the Chern-Simons action only. This is a topological field theory
and in fact is a particularly interesting one.
In general, if the pure Chern-Simons theory at level
is formulated on a Riemann surface of genus , then
[7] the number of zero energy states equals the number of
conformal blocks of the WZW model of at level . Moreover,
these states are all bosonic.^{†}^{†} Or they are all fermionic.
In finite volume, there is a potentially arbitrary sign choice
in the definition of the operator , as we will see in more
detail in section 3.
For our present
application, and the genus is 1. In this case, the
number of conformal blocks is equal to the number of representations
of the affine Lie algebra at level . This number is
positive for all , and for large is of order ,
with the rank of . For more detail on the canonical
quantization of the Chern-Simons theory, see [10-12].

The following paragraph is aimed to avoid a possible confusion. In Chern-Simons theory at level , many physical results, like expectation values of products of Wilson loops, are conveniently written as functions of . From the point of view of Feynman diagram calculations, this arises because a one-loop diagram with internal gauge bosons shifts to [13,14]. In a Hamiltonian approach to Chern-Simons gauge theory without fermions, one sees in another way that if the parameter in the Lagrangian is , many physical answers are functions of [10,11]. (This Hamiltonian approach is much closer to what we will do in section 3 for the theory with fermions.) In asserting that the effective coefficient of the Chern-Simons interaction is , we are referring to an effective Lagrangian in which the fermions have been integrated out, but one has not yet tried to solve for the quantum dynamics of the gauge bosons.

Since the pure Chern-Simons theory is a good low energy description for sufficiently large , the index of the supersymmetric theory at level can be identified for sufficiently large with the number of supersymmetric states of the pure Chern-Simons theory at level :

For example, suppose . The representations of the affine algebra at level have highest weights of spin ; there are such representations in all. As for , we get

at least for sufficiently big where the effective description by Chern-Simons theory at level is valid.

The formula, however, has a natural analytic continuation for all , and we may wonder if (2.3) holds for all . We will show this in the next section by a microscopic computation, but in the meantime, a hint that this is so is as follows. The sign reversal is equivalent in the Chern-Simons theory to a reversal of spacetime orientation, so one might expect . Actually, in general, the sign of the operator in finite volume can depend on an arbitrary choice, as in some examples in [1]. If a parity-invariant choice of this sign cannot be made in general, then we should expect only . This is consistent with (2.3), which gives . We will see in section 3 that the general formula, for a gauge group of rank , is

In (2.3), we can also see the claim made in the introduction: for , and for . For , as , this is equivalent to the statement that vanishes precisely if . We thus learn that for , supersymmetry is unbroken for all , and we conjecture that it is spontaneously broken for . (If this is so, then in particular there is a Goldstone fermion for , and the pure Chern-Simons theory on which we have based our initial derivation of (2.3) is not a good low energy description for .)

A similar structure holds for other groups. For example, for one has

(One way to compute this formula – and its generalization to other groups – will be reviewed in section 3.) When expressed in terms of , this gives the formula for already presented in the introduction:

We see the characteristic properties and for .

2.1. Microscopic Derivation Of Parity Anomaly

The shift in the effective value of – namely – has played an important role in this discussion. As we have already noted, the existence of this shift implies – since the effective Chern-Simons coupling must be an integer – that is congruent modulo to . When is odd – for example, for with odd – it follows that is not an integer and in particular cannot be zero. Such a phenomenon in three-dimensional gauge theories is known as a parity anomaly [5,6], the idea being that the theory conserves parity if and only if vanishes, so the non-integrality of means that parity cannot be conserved.

The derivation of the parity anomaly from the shift in the effective value of is valid for sufficiently large – where there is an effective low energy description as a Chern-Simons theory – but is not valid for small . One would like to complement this low energy explanation by an explanation at short distances, in terms of the elementary degrees of freedom, that does not depend on knowledge about the dynamics at long distances.

We will now review how this is done [5,6]. In this discussion, we assume to begin with that the gauge group is simply-connected (and connected), so that the gauge bundle over the three-dimensional spacetime manifold is automatically trivial. For most of the discussion below, the topology of does not matter, but for eventual computation of , one is most interested in or .

The path integral in a three-dimensional gauge theory with fermions has two factors the definition of whose phases requires care. One is the exponential of the Chern-Simons functional. The other is the fermion path integral. As the fermions are real, the fermion path integral equals the square root of the determinant of the Dirac operator . (When we want to make explicit the dependence of the Dirac operator on a gauge field , we write it as . Note that we consider the massless Dirac operator. The topological considerations of interest for the moment are independent of the mass.) Thus, the factors that we must look at are

First let us recall the issues in defining .
The operator is hermitian, so its eigenvalues are real.
Moreover, in three dimensions, for fermions taking values in a real
bundle such as the adjoint bundle, the eigenvalues are all
of even multiplicity. This follows from the existence of an
antiunitary symmetry analogous to CPT in four dimensions.
^{†}^{†} Use standard gamma matrix conventions such that, in a local
Lorentz frame, the gamma matrices are the Pauli spin
matrices, which are
real and symmetric or imaginary
and antisymmetric.
The Dirac operator then commutes with the antiunitary transformation
.
Since is antiunitary and , and
are always linearly independent, so the eigenstates of the Dirac
operator come in pairs.
The determinant of the Dirac operator is defined roughly as

where the infinite product is regularized with (for example) zeta function or Pauli-Villars regularization. Note in particular that the determinant is formally positive – there are infinitely many negative ’s, but they come in pairs – and this positivity is preserved in the regularization. Now consider the square root of the determinant, which is defined roughly as

where the product runs over all pairs of eigenvalues and the symbol means that (to get the square root of the determinant) we take one eigenvalue from each degenerate pair. This infinite product of course needs regularization. Since has already been defined, to make sense of we must only define the sign. For this we must determine, formally, whether the number of negative eigenvalue pairs is even or odd; it is here that an anomaly will come in.

It suffices to determine the sign of up to an overall
-independent
sign (which cancels out when we compute correlation functions).
For this, we fix an arbitrary connection (chosen generically
so that the Dirac operator has no zero eigenvalues),
and declare that
is, say, positive. Then to determine the sign
of for any other connection on the same bundle,
we interpolate from to
via a one-parameter family of connections , with ,
and .^{†}^{†} For example,
we can take the family .
We follow the spectrum of as evolves
from to 1, and denote
the net number of eigenvalue pairs that change sign from positive
to negative as the spectral flow . (If is a generic
one-parameter family, then there are no level crossings for ,
and the spectral flow for is as follows: every eigenvalue
pair flows upwards or downwards by units.)
Then we define the sign of
to be , the intuitive idea being that the sign
of the product in (2.9) should change whenever an eigenvalue
pair crosses zero. The only potential problem with this
definition is that it might depend on the path from to .

A problem arises precisely if there is a path dependence in the value of
modulo 2. There is such path dependence if and only if
there is a closed
path, in the space of connections modulo gauge transformations,
for which the spectral flow is odd. To determine whether this
occurs, we proceed as follows. Let , for , be a family of gauge fields such that is gauge-equivalent
to by a gauge transformation .
Such an is classified by its “winding number”
which takes values in .^{†}^{†} The winding number completely
specifies the topology of because we are taking
to be connected and simply-connected. Otherwise, depending
on the topology of spacetime,
may have additional topological invariants.
In this situation,
there is a nice formula for the spectral flow.
Each is a connection on a trivial bundle over .
The family , , can be fit together
to make a connection on a trivial bundle over , where
is the closed unit interval. Gluing together the endpoints
of to make a circle – and identifying the gauge bundles
over the boundaries of using the gauge transformation
– one can reinterpret the family as a connection on
a possibly nontrivial bundle over . This bundle
has instanton number , determined by the topological twist
of . The spectral flow is then

This relation between spectral flow and the topology of the bundle [15], which is important in instanton physics [16], is proved roughly as follows using the index theorem for the four-dimensional Dirac operator on . We call that operator and let and be the restrictions of to spinors of positive or negative chirality, namely

where is a positive real number that one can introduce by scaling the metric on . For small , the Dirac equation can be studied in terms of the -dependent spectrum of . If are the eigenstates of with eigenvalues , then one can approximately solve the four-dimensional Dirac equation with the formula

where the sign in the exponent is to give zero modes of . The sum over has been included to ensure . Different that are related by spectral flow (that is by ) give the same , so for generic spectral flow there are linearly independent four-dimensional solutions of this kind. For to be square integrable, the exponential factor in (2.12) must vanish for , so must be negative for and also for . This determines that the chirality of the solutions is the same as the sign of the spectral flow . The upshot is that the index of equals . (The factor of 2 arises because we defined by counting pairs of eigenvalues; each pair contributes two four-dimensional zero modes.) On the other hand, the index theorem for the Dirac operator gives . Combining the formulas for gives (2.10).

Now we put our results together. Under the gauge transformation , or in other words in interpolating from to , the sign of changes by . In view of (2.10), this factor is . On the other hand, the change in the Chern-Simons functional under a gauge transformation of winding number is . So under the gauge transformation , the dangerous factors (2.7) in the path integral pick up a factor

Gauge invariance of the theory amounts to the statement that this factor must be an integer for arbitrary integer , and this gives us the restriction on :

2.2. The Theory

Now we have assembled the ingredients to put the hypothesis of dynamical supersymmetry breaking for to an apparently rather severe test. The discussion is most interesting for the case , so we focus on that case.

The idea is to consider for an theory on . The key difference between and is that any bundle on is trivial, but an bundle on is characterized by a “discrete magnetic flux” that takes values in . (For , , and the discrete flux is the second Stieffel-Whitney class of the bundle.) An example of a bundle with any required value of is as follows. Consider a flat bundle whose holonomies and around the two directions in , if lifted to , obey

Such a flat bundle has .

The computation of for this theory can be made
very easily in case and are relatively prime, for instance .
(The computation can be done for any by using the relation
to the WZW model of , along the lines of section 2.1
above, or more explicitly using the techniques of section 3.)
The idea is simply [1] that zero energy quantum states
are obtained, for weak coupling, by quantizing the space of zero
energy classical states (including possible bosonic or fermionic
zero modes). A zero energy classical configuration of the gauge
fields is a flat connection. For and relatively prime,
a flat connection – that is a pair of matrices and obeying
(2.15) – is unique up to gauge transformation. Moreover, in expanding
around such a flat connection, there are no bosonic or fermionic zero modes.
Hence, the quantization is straightforward: quantizing a unique, isolated
classical state of zero energy, with no zero modes, gives a unique
quantum state.^{†}^{†} There is actually a potential subtlety in this statement,
though it is inessential in the examples under discussion. One must
verify that the one state in question obeys Gauss’s law, in other words
that it is invariant under the gauge symmetries left unbroken
by the classical solution that is being quantized.
The index is therefore (with the sign possibly
depending on a choice of sign in the definition of the operator
).

Note that plays no role in this argument. Hence, for any for which the theory exists, this theory, if formulated on a bundle with prime to , has a supersymmetric vacuum state for any volume of . Taking the limit of infinite volume, it follows that the theory, for any such , has zero vacuum energy and hence unbroken supersymmetry.

But in infinite volume, the and theories
are equivalent.^{†}^{†} Except for questions of which operators one
chooses to probe them by; such questions are irrelevant for the present
purposes. Hence for any for which the theory is
defined, the theory has unbroken
supersymmetry.

Does this not disprove the hypothesis that the theory has spontaneously broken supersymmetry in the “gap,” that is for ? In fact, there is an elegant escape which we will now describe.

The allowed values of were determined for by requiring that

should equal 1 for all integer values of the instanton number . (We have rewritten (2.13) using the fact that for .) For , there is a crucial difference: the instanton number is not necessarily an integer, but takes values in [17]. (For example, setting , an bundle on that has unit magnetic flux in the 1-2 and 3-4 directions and other components vanishing has instanton number modulo . In fact, on a four-manifold that is not spin, the instanton number takes values in , but for our present purpose – as the supersymmetric theory has fermions – only spin manifolds are relevant.) Hence gauge invariance of the theory requires that (2.16) should equal 1 not just for all , but for all . This gives the relation

Thus, for , cannot be in the “gap” , and the behavior of the theory in finite volume cannot be used to exclude the hypothesis that in the gap supersymmetry is dynamically broken. Though this does not prove that supersymmetry is broken in the gap for , the elegant escape does suggest that that is the right interpretation.

3. Microscopic Computation Of The Index

In this section, we will make a microscopic computation of in the supersymmetric pure gauge theory in three spacetime dimensions. We consider first the case that the gauge group is simply connected.

We thus formulate the theory on a spatial torus (times time) and look for zero energy states. As in [1], the computation will be done by a “Born-Oppenheimer approximation,” quantizing the space of classical zero energy states, and is valid for weak coupling or small volume of . To be more exact, we work on a torus of radius , and let and denote, as before, the gauge and Chern-Simons couplings. Particles with momentum on have energies of order , while the fermion and gauge boson bare mass is . We work in the region

We will write an effective Hamiltonian that describes states with energies of order (or less) but omits states with energies of order .

A zero energy classical gauge field configuration
is a flat connection and is determined up to gauge transformation
by its holonomies around the two directions in . These
holonomies, since they commute, can simultaneously be conjugated
to the maximal torus of ,^{†}^{†} The analogous statement
can fail – see the
appendix of [18]
– for the case of three commuting elements
of .
in a way that is unique up to
a Weyl transformation. The moduli space of flat
-connections on is thus a copy of , where
is the Weyl group.
Concretely, a flat connection on can be represented by
a constant gauge field

where the are a basis of the Lie algebra of and the can be regarded as constant abelian gauge fields on . The flat metric on determines a complex structure on ; it also determines a complex structure on in which the complex coordinates are the components of the one-forms . The are defined modulo shifts.

To construct the right quantum mechanics on , we must also look at the fermion zero modes. Actually, by “zero modes” we mean modes whose energy is at most of order , rather than . In finding these modes, we can ignore the fermion bare mass and look for zero modes of the massless two-dimensional Dirac operator . We then will write an effective Lagrangian that incorporates the effects of the fermion bare mass. Let and be the gluino fields of positive and negative chirality on . (They are hermitian conjugates of each other.) For a diagonal flat connection such as we have assumed, the equation has a very simple structure. For generic and (or equivalently for generic ), the “off-diagonal” fermions have no zero modes, while the “diagonal” fermions have “constant” zero modes. In other words, the zero modes of are given by the ansatz

with being anticommuting constants.

Now let us discuss quantization of the fermions. Quantization of the nonzero modes gives a Fock space. Quantization of the zero modes is, as usual, more subtle. The canonical anticommutation relations of the ’s are, with an appropriate normalization,

Thus, we can regard the and as creation and annihilation operators. For example, we can introduce a state annihilated by the , and build other states by acting with ’s; or we can introduce a state annihilated by the ’s, and build the rest of the Hilbert space by acting with ’s. The relation between the two descriptions is of course

Now let us try to define the operator . It is clear how we want to act on the Fock space built by quantizing the nonzero modes of the fermions: it leaves the ground state invariant and anticommutes with all nonzero modes of . The only subtlety is in the action on the zero mode Hilbert space. There is in general no natural choice for the sign of . If we pick, say , then (3.5) implies that . Thus, if is even, we can fairly naturally pick both of the states to be bosonic. But if is odd, then inevitably one is bosonic and one is fermionic; which is which depends on a completely arbitrary choice. Now we can explain an assertion in section 2, namely that

Changing the sign of the Chern-Simons level is equivalent to a transformation that reverses the orientation of . Such a transformation exchanges with , and so exchanges with . This exchange reverses the sign of the operator if is odd, and that leads to (3.6).

The Hilbert space made by quantizing the fermion zero modes has a very natural interpretation. The quadratic form (3.4) has the same structure as the metric of , so the ’s can be interpreted as gamma matrices on . Hence the Hilbert space obtained by quantizing the zero modes is the space of spinor fields on , with values in a line bundle that we have not yet identified. (Such a line bundle may appear because, for example, for a given point on , the state is unique up to a complex multiple, but as one moves on , it varies as the fiber of a not-yet-determined complex line bundle.)

Because is a complex manifold, spinor fields on have a particularly simple description. Let be the canonical line bundle of , and assume for the time being that there exists on a line bundle . Then the space of spinors on is the same as the space of -forms on (for ) with values in . In this identification, we regard as a -form on (with values in a line bundle), and we identify a general state in the fermionic space

as a form on . From this point of view, we identify with the form , and with the “contraction” operator that removes the one-form from a differential form, if it is present. (Of course, by exchanging the role of and , we could instead regard the spinors on as -forms, with values in a line bundle.)

The relation of spinors on to )-forms has the following consequence. The Dirac operator acting on sections of a holomorphic line bundle over a complex manifold has a decomposition

where is the operator acting on -forms with values in , and is its adjoint. These operators obey

where . If we identify and with the two supercharges and with the Hamiltonian, then (3.9) coincides with the supersymmetry algebra of a -dimensional system with supersymmetry, in a sector in which the momentum vanishes. In the present discussion the momentum vanishes because the classical zero energy states that we are quantizing all have zero momentum. This strongly suggests that, in the approximation of quantizing the space of classical zero energy states, the supercharges reduce to (a multiple of) and .

It is not difficult to show this and at the same time identify the line bundle . In canonical quantization of the Yang-Mills theory with Chern-Simons coupling, the momentum conjugate to is

Writing formally , we have

where is a “covariant derivative in field space,”

The object is a connection on a line bundle over the space of connections. The connection form of is . Requiring that the curvature form of should have periods that are integer multiples of gives a condition that is equivalent to the quantization [2] of the Chern-Simons coupling (see [10-12] for more on such matters), so if we set the line bundle that we get is the most basic line bundle over the phase space, in the sense that it has positive curvature and all other line bundles over the phase space are of the form for some integer . The factor of in (3.12) means that the line bundle over the phase space is . So the states are spinors with values in , or equivalently -forms with values in .

As for the supercharges, they are

To write an effective formula in the space of zero energy states, we set the spatial part of to zero. The supercharges of definite two-dimensional chirality then become

Evaluating this expression in the space of zero modes,
the ’s become gamma matrices (or raising and lowering operators)
on spinors over the moduli space ; and and
are holomorphic and antiholomorphic covariant derivatives on .
Altogether, the supercharges and
reduce to times the and operators
on spinors valued in .^{†}^{†} The factor of arises because
the kinetic energy in the original Lagrangian was
, so is a canonically
normalized fermion. As the supercharges are properly normalized
as and , the Hamiltonian
is .

In this discussion, we have not incorporated explicitly the fermion bare mass . But that bare mass is related by supersymmetry to the Chern-Simons coupling, which we have incorporated, so the supersymmetric effective Hamiltonian that we have written inevitably includes the effects of the fermion bare mass. This arises as follows: because there is a “magnetic field” on proportional to (with connection form on the right hand side of (3.12)), the operator , if written out more explicitly, contains a term . This coupling is the bare mass term, written in the space of ’s.

3.1. Calculations

Now we will perform calculations of . First we consider the case that .

For , the moduli space is a copy of .
^{†}^{†} For example, for , the maximal torus is a circle
and the Weyl group is , so ,
which is an orbifold version of . For general ,
the standard proof that can be found,
for example, in section 2.1 of [19].
The basic line bundle over is , the bundle
whose sections are functions of degree one in the homogeneous coordinates
of . The canonical bundle of is
.

The quantum Hilbert space found in the above Born-Oppenheimber approximation is the space of spinors with values in , or equivalently -forms with values in . Since only integral powers of are well-defined as line bundles over , we get the restriction

This is the restriction found in [3] and reviewed in section 2; we have now given a Hamiltonian explanation of it.

Since the supersymmetry generators are the and operators, the space of supersymmetric states, in this approximation, is

The supersymmetric index is

This can be computed by a Riemann-Roch formula, which implies in particular that is a polynomial in of order .

However, for a more precise description – and in particular to see supersymmetry breaking in the “gap,” – we wish to compute the individual cohomology groups, and not just the index. For this computation, see for example [20]. For , one has for all . Hence, for

there are no zero energy states at all in the present approximation. Thus, for this range of , supersymmetry is spontaneously broken if the theory is formulated on a two-torus with sufficiently weak coupling that our analysis is a good approximation. (Because the ground state energy in finite volume is a real analytic function of the volume, it also follows that supersymmetry is unbroken for any generic volume on .) This hints but certainly does not prove that also for infinite volume, supersymmetry is spontaneously broken if .

For , for , and is the space of homogeneous polynomials of degree in the homogeneous coordinates of . The dimension of this space is Setting and interpreting this dimension as the supersymmetric index , we get the formula for that was stated in the introduction:

Finally, Serre duality determines what happens for in terms of the results for . In particular, for , the cohomology vanishes except for , and is dual to . From this, we get a formula for with which coincides with (3.19). Note that for , all supersymmetric states are bosonic, and for , all supersymmetric states have statistics . Serre duality gives directly .

Generalization To Other Groups

We will now more briefly summarize the generalization for an arbitrary simple, connected and simply-connected gauge group of rank .

First of all, the moduli space is a weighted projective space , where the weights are 1 and the coefficients of the highest coroot of . This is a theorem of Looijenga; for an alternative proof see [19]. In particular, the weights obey . The basic line bundle over is , characterized by the fact that sections of for any are functions of weighted degree in the homogeneous coordinates of . The canonical bundle of is . So, in the Born-Oppenheimer approximation, the low-lying states are spinors valued in . Integrality of the exponent gives again the result that must be integral.

The space of supersymmetric states is, again,

A weighted projective space has certain properties in common with an ordinary projective space. One of these is that for all if . This implies (in finite volume) supersymmetry breaking in the “gap,” . For , the cohomology groups vanish except in dimension 0, and is the space of polynomials homogeneous and of weighted degree in the homogeneous coordinates of . In particular, for , and supersymmetry is unbroken. Serre duality asserts that is dual to , and relates the region to . In particular, for , the only nonzero cohomology group is in dimension , the states have statistics , and the index is determined by and so is in particular nonzero.

3.2. Orbifolds And Anyons

Here, we make a few miscellaneous comments on the problem.

The moduli space is an orbifold , a quotient of a flat manifold by a finite group. However, we have not used this fact in computing the index. The reason is that although the moduli space is an orbifold, the quantum mechanics on is not orbifold quantum mechanics, that is, it is not obtained from supersymmetric free particle motion on by imposing -invariance. Rather, the quantum mechanics on depends on the line bundle .

One can ask whether there are values of at which the quantum mechanics on reduces to orbifold quantum mechanics. We will approach this as follows. We begin with a system consisting of -forms on with the Hamiltonian being simply the Laplacian (relative to the flat metric on ). In orbifold quantum mechanics, we want -invariant states of zero energy.

A zero energy state must have a wave function that is invariant under translations on . This means that the bosonic part of the wave-function, being a constant function, is -invariant. Hence, -invariance must be imposed on the fermionic part of the wave function, which as we recall takes values in a fermionic Fock space with basis obtained by acting with creation operators on a vacuum or .

The states and transform as one-dimensional representations of , since the condition that a state be annihilated by all ’s (or by all ’s) is Weyl-invariant. The group has two one-dimensional representations: the trivial representation; and a representation in which each elementary reflection acts by . Since

and the product is odd under every elementary reflection, the two states and transform oppositely: one transforms in the trivial representation of , and the other transforms as .

Suppose that we take the action such that transforms trivially. Then itself (times a constant function on ) is a -invariant state of zero energy, and is in fact the only one. To prove the uniqueness, one can use the fact that the action on the fermion Fock space is the same as that on the -forms on . The -invariant and translation-invariant states on can therefore be identified with the cohomology group , where is a trivial holomorphic line bundle. This cohomology is one-dimensional for and vanishes for , since .

Now let us compare this orbifold quantum mechanics to the Born-Oppenheimer quantization of the gauge theory. In the latter, at general level , we identified the supersymmetric states with elements of , where . This agrees with the orbifold answer if and only if , so that must be the correct value of corresponding to orbifold quantum mechanics with assumed to be Weyl-invariant. The other possibility, that is Weyl-invariant, is obtained by reversal of orientation, which is equivalent to ; so this other orbifold quantum mechanics should correspond to .

These arguments strongly suggest that the low energy quantum mechanics is just orbifold quantum mechanics for these special values of . As we discuss below and in section 4, these are apparently the values for which the theory is confining.

Anyons

It is perhaps surprising that the “simple” orbifold cases correspond not to the obvious case but to . Let us see instead consider what happens for . For simplicity, we take , so that is a circle and .

Even though is an orbifold, the quantum mechanics is, as we have seen, not orbifold quantum mechanics for . To measure the failure, let us see what happens near the orbifold singularities of . For example, we can consider the singularity associated with the trivial flat connection, where the introduced in (3.2) all vanish. (For , the index takes only one value, so we write the just as .) The Weyl group acts as , and there is a singularity at . How can we best understand this singularity? Near , it is more illuminating to consider not compactification from dimensions to – as we have done so far – but dimensional reduction to dimensions, in which one starts with -dimensional super Yang-Mills thoery and by fiat one requires the fields to be invariant under spatial translations. Dimensional reduction and compactification differ in that the compactified theory also has modes of non-zero momentum along , and has periodic identifications of the under . These are irrelevant for studying the singularity near .

In the dimensionally reduced theory, there is a symmetry under rotations of the plane. (The compactified theory only has a discrete subgroup of this symmetry; that is one of the main reasons to consider the dimensionally reduced theory in the present discussion.) We will call the generator of this the angular momentum. The fermion Fock space has, for , only the two states and (as there is only one creation operator and one annihilation operator). Since a fermion creation operator, of spin , maps one to the other, their angular momenta are and for some . But the dimensionally reduced theory has also a parity symmetry exchanging these two states and reversing the sign of the angular momentum. Hence , and the two states have angular momenta . This contrasts with orbifold quantum mechanics on , where the spins are half-integral. Thus a precise measure of the difference of the system from an orbifold is that the system generates “anyons,” states whose angular momentum does not take values in .

Such anyons may arise in the compactification of Type IIB superstring theory to three dimensions on a seven-manifold of holonomy. Consider a system of parallel sevenbranes wrapped on . This system is governed by -dimensional super Yang-Mills theory with two supercharges, dimensionally reduced to dimensions. This suggests that the wrapped sevenbranes may be anyons of spins , but to be certain, one would need to look closely at the definition of angular momentum for these string theory excitations.

3.3. Discrete Electric And Magnetic Flux

We will briefly discuss the generalization of the computation to incorporate discrete electric and magnetic flux. (We will consider only the simplest versions of electric or magnetic flux. It is of course possible to mix the two constructions.)

Including magnetic flux simply means taking the gauge group not to be simply connected and working on a non-trivial gauge bundle over . The moduli space of zero energy gauge configurations is now the moduli space of flat connections on . In certain cases, mentioned in section 2.2 above, is a single point, and the quantization is then completely straightforward. In general, is always a weighted projective space (which can be constructed using the technique in [19]), and the quantum ground states are always, as above, . In particular, the index is always nonvanishing for .

Including electric flux means that one goes back to the case that is simply connected. One considers a gauge transformation that, in going around, say, the first circle in , transforms as

where is an element of the center of . This transformation is a symmetry of the theory. If, for example, is of order , then generates a gauge transformation that is homotopic to the identity, and on all physical states. The eigenvalues of are thus of the form where is an integer called the discrete electric flux.

A simple way to determine the action of on the space of supersymmetric ground states of our supersymmetric gauge theory is to use the relation of pure Chern-Simons theory at level to the WZW model, also at that level. The Hilbert space of the pure Chern-Simons theory in quantization on has a basis that can be described as follows. Regard as the boundary of , where is a two-dimensional disc. Consider the Chern-Simons path integral on , with an insertion of a Wilson line operator

Here is a representation of , and is a circle of the form , with being a point in the disc . For any given , the path integral on with insertion of gives a state in the Chern-Simons Hilbert space on at level , and this space, as we have argued, is the same as . As ranges over the highest weights of integrable representations of the affine algebra at level , the furni