Dparticles on orientifolds and rational invariants
HJE
Dparticles on orientifolds and rational invariants
SeungJoo Lee 0 1 3
Piljin Yi 0 1 2
0 Seoul 02455 , Korea
1 Blacksburg , VA 24061 , U.S.A
2 School of Physics, Korea Institute for Advanced Study
3 Department of Physics , Robeson Hall, Virginia Tech , USA
We revisit the D0 bound state problems, of the M/IIA duality, with the Orientifolds. The cases of O4 and O8 have been studied recently, from the perspective of fivedimensional theories, while the case of O0 has been much neglected. The computation we perform for D0O0 states boils down to the Witten indices for N = 16 O(m) and Sp(n) quantum mechanics, where we adapt and extend previous analysis by the authors. The twisted partition function Ω, obtained via localization, proves to be rational, and we establish a precise relation between Ω and the integral Witten index I, by identifying continuum contributions sector by sector. The resulting Witten index shows surprisingly large numbers of threshold bound states but in a manner consistent with Mtheory. We close with an exploration on how the ubiquitous rational invariants of the wallcrossing physics would generalize to theories with Orientifolds.
Dbranes; Differential and Algebraic Geometry; MTheory; Supersymmetry

and Duality
1 Introduction
3.1
3.2
3.3
Witten index and Mtheory on R9/Z2
Toward rational invariants for orientifolds
A Elliptic Weyl elements and rational invariants
A.1 ΩGN =16 with simple and connected G
A.2 Common building blocks for orthogonal and sympletic groups
A.3 ΩGN =16 for Dparticles on an orientifold point
B Integrand for the O−(2N )
SU(N ) theories [6] and some attempts made for other gauge groups [7, 8], but there are often
issues with a contour choice in the last stage of such computations. The new derivations
– 1 –
obviate this last uncertainty as they actually derive rigorously what the contour should be.
For SU(N ), one finds in the end the twisted partition function [5]
tuning. For SU(N ) theory in question, the classical vacua form a cone R9(N−1)/SN , and
the planewavelike states can also contribute to the relevant path integral. The correct
interpretation here is to identify the first term “1” as the index while the rest are attributed
to various continuum sectors. In fact, the other “1”’s in the sum are also nothing but the
Witten index of the SU(N/p) subsectors. This interpretation was pioneered in ref. [2],
where the nonequivariant version of Ω was computed for SU(2), and has been generalized
to all SU(N ) rather convincingly [6, 10].
Thus one question that has to be resolved if one is to repeat the problem for more
complicated spacetime is how to separate the continuum contribution from true Witten
index systematically. This does not seem to admit a universal answer, as there are numerous
cases where continuum sectors can conspire to contribute a net integral piece to Ω [5]. At
present, extraction of I from Ω, when the theory involves gapless asymptotic directions, is
more of an art than a science.
Ref. [5], nevertheless, noted how the main feature of N = 16 SU(N ) generalizes
straightforwardly to other N = 16 theories and also to N = 4 nonprimitive quiver
theories with bifundamental matters only. The various continuum contributions to ΩGN =16
have been physically understood, identified, and catalogued. Naturally, this opens up the
possibility of computing true Witten indices for Dparticle binding to Orientifold points.
In fact, the results of ref. [5] almost suffice, except for the case of O0− orientifold. In this
note, we wish to place the last missing piece in the problem and compute Witten indices
for all D0O0 bound states.
Section 2 will give a general discussion on the twisted partition function versus the
Witten index, with emphasis on what the localization procedure actually computes.
Section 3 will review the recent results for N = 16 YangMills quantum mechanics, which
we will generalize in section 4 to O(m) gauge groups. This will lead us to the Witten
indices that count bound states between D0’s and any one of four types of the orientifold
point and to a known Mtheory interpretation, adding yet another strong and rather direct
confirmation of M/IIA duality. In the final section, we comment on new type of rational
expressions we found along the way and propose them as building blocks for the rational
invariants suitable for Orientifolded theories.
– 2 –
HJEP07(21)46
Index I vs. twisted partition function Ω
For supersymmetric quantum theory, one of the useful and accessible quantities that probe
the ground state sector is the Witten index [9],
β→∞
I = lim tr h(−1)F e−βH i
.
The chirality operator (−1)F can be replaced by any operator that anticommutes with the
supercharges. One often wishes to compute the equivariant version by inserting chemical
potentials, x, associated with global symmetries, F ,
I(y, x) ≡ lim tr h(−1)F yRxF e−βQ2 i
.
However, as is well known, this quantity may not be amenable to straightforward
computations.
If the dynamics is compact, i.e., with a fully discrete spectrum, βdependence can be
argued away based on the naive argument that I is topological. Under such favorable
circumstances, one is motivated to consider instead
(2.1)
(2.2)
(2.3)
(2.4)
and compute the other limit, which tends to reduce the path integral to a local expression,
with the anticipation that Ω is independent of β so that I = Ibulk.
For theories with continuum sectors, however, this naive expectation cannot hold in
general; I is by definition integral, while Ω need not be integral and thus can differ from
I. If the continuum has a gap, E ≥ Egap > 0, its contribution is suppressed as
e−βEgap ,
so we may have an option of scaling Egap up first and then taking β → 0 afterward, leaving
behind the integral index I only [4, 11].
When the continuum cannot be gapped, or when a gap can be introduced only at the
expense of qualitative modification of the asymptotic dynamics, however, we are often in
trouble. The resulting bulk part Ibulk differs from the genuine index. For such theories,
isolating I hidden inside Ibulk requires a method of computing yet another piece, known
as the defect term,
Ω(y, x, β) ≡ tr h(−1)F yRxF e−βQ2 i ,
Ibulk(y, x) ≡ lim Ω(y, x, β) ,
β→0
−δI ≡ Ibulk − I .
– 3 –
This program depends on particulars of the given problem and, in particular, on the
boundary conditions.1 As far as we know there is no general theory for δI.
For a large class of gauged dynamics, the localization procedure has been applied
successfully to reduce the path integral representation of Ω to a formulae involving
rankmany contour integrations. For N ≥ 2 gauged quantum mechanics [4] and for d = 2 elliptic
genera [15, 16], in particular, reasonably complete and reliable derivations exist. At the
end of such computations, one finds that βdependence is absent. When the dynamics is
not compact and Ω is expected to be βdependent, the question is exactly which β limit of
Ω one has computed.
One key trick here is to scale up the gauge kinetic term by sending e2 → 0, as the
posed above is β → 0; the dimensionless combination of the two is
βe2/(4−D) ,
so e2 → 0 is equivalent to β → 0 for D ≤ 3. Another typical dimensionful parameters
that could be present are FayetIliopoulos constants ζ, but, for a sensible results, one often
must take a limit of ζ first [4]. This raises a gap Egap along certain Coulomb directions to
infinity, if not all, so we expect that, again, the β → 0 limit of Ω is computed effectively at
the end of the localization procedure. After all, one finds a local expression, at the end of
such processes, involving zero mode integrals only, which is impossible at the other limit
of β → ∞.
As such, we will define for this note, Even after a successful localization computation of Ω(y, x), one is often left with an even more difficult task of identifying the continuum contribution, −δI, inside Ω if one wishes to compute I.
1A canonical example is the supersymmetric nonlinear sigma models onto a manifold with boundary.
as delineated in ref. [5], there exists classes of d = 1 supersymmetric gauged linear sigma
models for which this problem may be dealt with honestly. One such is adjointonly
YangMills quantum mechanics, and another is N = 4 nonprimitive quiver theories with compact
classical Higgs vacuum moduli space. In the next section, we recall this phenomenon for
N = 4, 8, 16 pure YangMills quantum mechanics with connected simple group G.
For gauged linear sigma model with at least two supersymmetries, the localization
procedure gives a JeffreyKirwan residue formulae [4],
Ω(y, x) =
1
W 
JKResη Q t
s s
g(t)
drt ,
where (t1, . . . , tr) parameterize the r bosonic zero modes living in (C∗)r, that usually scan
the Cartan directions but can be further restricted in topologically nontrivial holonomy
sectors. The determinant g(t) is due to massive modes in the background of t’s. In this
note, we use N = 4 notations for supermultiplets, and as such, g(t) takes the general form,
(3.1)
(3.2)
g(t) =
1
charges needs a little bit of thought. For N = 8, one more chemical potential x can be
turned on, associated with the natural U(1) rotation of the chiral field, and R = 0 is
assigned to the adjoint chiral. No superpotential is possible under such assignments. For
N = 16, with three adjoint chirals, a trilinear superpotential term is needed, so at most
two flavor chemical potentials are allowed, say, x and x˜ associated with F and F˜. We
can for example assign R = (2, 0, 0), F = (2, −1, −1), and F˜ = (0, 1, −1) that allow only
trilinear superpotential as required by N = 16. In actual N = 16 formula below xF should
be understood as the product, xF x˜F˜ , over the two flavor chemical potentials.
One thing special about the pure gauge theories is that we are instructed to ignore the
poles located at the boundary of the zero mode space (C∗)r [5]. This is a property which
holds generally for theories with the total matter content in a real representation under
the gauge group.
– 5 –
Although (3.1) and (3.2) are seemingly phrased in terms of the Lie algebra data only,
the global properties of the gauge group G should matter in general. The main
computation of this note, in section 4, addresses precisely such a difference, SO(m) vs. O(m).
However, ΩG = ΩG/Γ must hold if the field content is invariant under Γ, as no local physics
would know about such Γ’s. How is this reflected in (3.1)? Note that the integrations
in (3.1) implicitly assume 2πi period for each log(ts), independent of one another, and are
accompanied by a naive inverse volume 1/(2π)r. This potential mistreatment of the Cartan
torus is automatically corrected by the degeneracies of poles in g(t), which are, in turn,
determined by the field content and hence by the largest compatible Γ. This way, for
example, ΩG = ΩG/ZG with the center ZG is naturally built into (3.1) for pure YangMills [18].
See ref. [18] for a more complete discussion and related subtleties.
3.1
This gives us an unambiguous procedure of computing the twisted partition functions ΩGN
for all possible G and N . There are some further computational issues, such as how to deal
with the degenerate poles, which complicates the task but still allows us to go forward.
We will not give too much details here and instead refer the readers to ref. [5] for pure
YangMills cases, and to ref. [4] for general gauged quantum mechanics.
It turns out that, after a long and arduous computerassisted computation of JK
residues, the twisted partition functions for pure N = 4, 8 Ggauged quantum mechanics,
can be organized into purely algebraic quantities. For N = 4, one finds
ΩGN =4(y) =
1
′
X
y−N+1 + y−N+3 + · · · + yN−3 + yN−1 ,
ΩSNU=(N4)(y) =
ΩSNO=(44)(y) =
ΩSNO=(54)(y) = ΩSNp=(24)(y) =
ΩSNO=(74)(y) = ΩSNp=(34)(y) =
1
N
1
1
8
·
·
1
48
·
where each term can be associated with a sum over conjugacy classes of the same cyclic
HJEP07(21)46
For pure N = 8 Ggauged quantum mechanics, obtained by adding to the N = 4
theory an adjoint chiral, we can include a flavor chemical potential x of the adjoint after
assigning a unit flavor charge without loss of generality. With R = 0 for the adjoint chiral,
we also have the universal formula,
ΩGN =8(y, x) =
1
′
X
WG w
1
and the pattern generalizes to higher rank cases in an obvious manner.
The reason why the result can be repackaged into such a simple algebraic formulae has
been explained both for nonequivariant form [2, 10, 19, 20] and for equivariant form [5].
Consider −δI. This part of Ω has to arise from the continuum and, because of this,
depends only on the asymptotic dynamics. The latter becomes a nonlinear sigma model
on an orbifold
HJEP07(21)46
(3.5)
(3.6)
(3.7)
(3.8)
so that the δI of the two theories must agree with each other. On the other hand, we
expect no quantum mechanical bound state localized at the orbifold point, so
which implies [2]
The right hand side of (3.6) has been evaluated using the Heat Kernel regularization, when
y = 1 and x = 1, for SU(2) case in ref. [2], and more generally in refs. [10, 19], with the
result
What we described above in (3.3) and in (3.4), individually confirmed by direct localization
computation, are the equivariant uplifts of this expression for N = 4, 8 respectively.
With this, the origin of ΩGN =4,8 is abundantly clear. They come entirely from the
asymptotic continuum states spanned by the free Cartan dynamics, modulo the orbifolding
by the Weyl group; the pathintegralcomputed ΩGN =4,8 has no room for a contribution from
threshold bound states. Therefore, the true enumerative part I inside Ω has to be null,
O(G)N =4,8 = R3r/W or R5r/W ,
INO(=G4),8 bulk
+ δINO(=G4),8 = 0
−δING=4,8 =
INO(=G4),8 bulk
planes, and the Witten index of these theories must vanish. This physical expectation
dovetails with the above structure nicely.
The same principle generalizes to N = 16 cases. However, their asymptotic dynamics
will no longer be captured by analog of O(G) alone; the presence of threshold bound states
implies that the continuum sectors ΩGN =16 will no longer be that simple. There could be
additional sectors involving partial bound states tensored with continuum of remaining
asymptotic directions. We turn to this next.
3.2
On N = 16 continuum sectors
The same kind of continuum sectors as the above N = 4, 8 examples should exist for
N = 16, with the asymptotic dynamics of the form,
O(G)N =16 = R9r/WG ,
and we can easily guess the contribution to ΩGN =16 from this sector to take the form,
ΔGN =16 ≡
1
′
X
as a straightforward generalization of N = 4, 8 expressions. Here, a labels the three adjoint
chirals. Indeed, as we will see below, each ΩGN =16, computed via localization, is seen to
have an additive piece of this type.
The difference for N = 16 is, however, that threshold bound states are expected in
general. For all SU(N ), e.g., a single threshold bound state must exist for Mtheory/IIA
theory duality to hold. Since such states can also occur for subgroups of G as well and
since they can explore the remaining asymptotic directions, a far more complex network
of continuum sectors exist. Generally a product of subgroups
⊗AGA < G
correspond to a collection of oneparticlelike states, each labeled by A. When this
subgroup equals the Cartan subgroup of G, the corresponding continuum sector contributes
the universal ΔGN =16 to ΩGN =16. When at least one of GA is a simple group, the
corresponding partial bound state(s) can contribute a new fractional piece to ΩGN =16. The relevant
continuum sector is the asymptotic Coulombic directions where the “particles” forming the
bound state associated with GA moves together. In other words, the asymptotic Coulombic
directions are parameterized by a subalgebra
of the Cartan of G, where ⊗AGA is the centralizer of h[⊗AGA].
of WG that leaves h[⊗AGA] invariant yet act faithfully. Contribution to Ω would arise from
generalized elliptic Weyl elements of W ′,
det 1 − w
′
6= 0 ,
where the determinant is now taken in the smaller representation over h[⊗AGA]. In a slight
abuse of notation, it turns out that the continuum contribution from W ′ to ΩGN =16 can be
expressed as a product of the form,
state whatsoever.
simple form,
on the right hand side, with the coefficient 1, representing the sector with no partial bound
Ref. [5] showed that this is indeed the case, even though such a pattern is hardly
visible at the stage of JKresidue computations. For SU(N ), the result takes a particularly
HI is a simple subgroup of G whose Weyl group is a subgroup factor of W ′.
where ΔNHI=16 are defined for some subgroups HI of G in the same manner as (3.10). Each
3.3
ΩGN =16 can also be directly computed using the HKY procedure [4]. One then searches for
a unique decomposition as sum over such continuum pieces as
ΩGN =16 = ING=16 +
X
⊗GA<G
G
n{GA}
Y ΔNHI=16 ,
I
with nonnegative integral factor, n{GGA}. Furthermore, there should be a term
Then, the argument leading to (3.6) can be adapted to this slightly more involved case;
a continuum contribution from this sector would be associated with a subgroup
The rational contributions come from the continuum directions, h[⊗AGA], parameterized as
diag(v1, . . . , v1; v2, . . . , v2; · · · ; vp, . . . , vp)
with each eigenvalue repeated (N/p)times, and P
of partial SU(N/p) bound states form, continuum states of which contribute ΔSNU=(p1)6; the
relevant Weyl subgroup is the permutation group that shuffles v’s, so can be naturally
A vA = 0. In this sector, p number
labeled as H = SU(p). In the end, this implies
W ′
Y ΔNHI=16
I
1 · ΔGN =16
INSU=(1N6) = 1
– 9 –
(3.11)
(3.12)
(3.13)
for all N . The nonequivariant limit of the same decomposition
ΩSNU=(N16)
y→1;x→1
y→1
= ΩSNU=(N16)
= 1 +
X
pN;p6=1
1
p2
,
has been computed and understood early on [2, 6, 10] along this line of reasoning.
The authors have also computed twisted partition functions for more general simple
groups, up to rank 4, and decomposed the resulting ΩGN ’s in this manner [5]. See
appendix A.1 for the results. The main lesson is again that we can read off the true Witten
index I from such a decomposition of each Ω; all the rational pieces have to be part of
−δI, sector by sector. The only integral part, the first terms on the right hand sides, may
be interpreted as the Witten index, giving us
HJEP07(21)46
INSO=(146) =
INSU=(126) 2
= 1 ,
INSO=(156) = INSp=(21)6 = 1 ,
INSO=(166) = INSU=(146) = 1 ,
INSO=(176) = 1 ,
INSp=(31)6 = 2 ,
INSO=(186) = 2 ,
INSO=(196) = 2 ,
INSp=(41)6 = 2 ,
(3.14)
as well as IG2
4
D0O0
−
for Dparticles on an Orientifold point.
N =16 = 2. In the next section, we will adopt and extend some of these results
Let us come to the main problem of this note. Just as the Witten index for N = 16 SU(N )
theory confirms existence of Mtheory circle, hidden in IIA theory, one may ask what this
Mtheory circle will predict in the presence of IIA orientifold planes. For O8 and O4,
Dparticle states bound to the orientifold planes require additional Dbranes: eight D8’s for
O8, since otherwise Mtheory lift does not exist [21], and more than one D4’s for O4. See
refs. [22, 23] for recent computations of twisted partition functions in the presence of O4/O8
orientifolds. This leaves O0, namely Orientifold points. While it is, a priori, unclear why
there should be Dparticles trapped at O0, our computation of nontrivial Witten indices
for N = 16 SO and Sp theories suggests that there should be such states after all. An
orientifold projection R9/Z2 can give either Sp(n) or O(m) gauge groups. For O0+’s, the Sp
computation above suffices. For O0−’s, however, one must supplement SO(m) computation
by taking into account Z2 = O(m)/SO(m). In this section, we generalize SO(m) to O(m)
theories, for Dparticles bound to O0−’s.
Physically, the difference between the two is whether we demand the physical states
be invariant under the gaugeparity operation, which we call P, in addition to the local
Gauss constraint. So if a twisted partition function for SO(m) theory has the form,
tr h(−1)F · · · e−βH i ,
its O(m) counterpart must have the operator insertion,
tr (−1)F · · · e−βH ·
1 + P
2
,
where P is the parity operator in O(m)/SO(m). In the end, the twisted partition function
of an O(m) theory is the average of two terms,
HJEP07(21)46
ΩON(m)(y, x) =
ΩN
O+(m)(y, x) + ΩN
O−(m)(y, x)
2
O+(m)(y, x) = ΩSNO(m)(y, x) has already been computed, while the second
term needs to be computed with the insertion of P as
ΩN
O−(m)(y, x) ≡ tr h(−1)F yRxF e−βQ2 Pi
localization
First, we turn to O(2N ) for 2N ≥ 4. For ΩON−=(42,N8,1)6, we made an explicit JKresidue
evaluation as in the previous section. The insertion of P can be represented by a Z2
holonomy along the Euclidean time circle,
whereby the zero mode space shrinks by one dimension, so r = N − 1 for O−(2N ). The
reduced zero modes, t1,2,...,N−1 = e2πiu1,2,...,N−1 , parameterize O−(2N ) holonomy as
(4.1)
(4.2)
(4.3)
e2πiσ2u1
0
· · ·
0
0
e2πiσ2u2 · · ·
· · ·
0
0
0
· · ·
· · ·
0
0 e2πiσ2uN−1 0
0
0
0
0
0
0
0 ,
σ3
which sets tN = 1 in g(t). The N th Cartan elements in all multiplets become massive,
instead, and now contribute factors with the signs flipped, e.g., one of the N overall y −y−1
factors in the denominator for the Cartan is flipped to y + y−1. See appendix B. However,
we must caution against viewing this as a spontaneous symmetry breaking of the dynamics.
Consider very long (Euclidean) time β. The “symmetry breaking effect” becomes diluted
arbitrarily, as the size of the timelike gauge field scales with 1/β. Moreover, at each time
slice, this A0 can be gauged away, locally, and thus will not alter the dynamics. It is only
when we are instructed to perform the trace, this P makes a difference.
Finally, one needs to be careful about the usual division by the Weyl group when
computing O−(2N ) contributions. Recall that the Weyl group of O+(2N ) = SO(2N ) is
WSO(2N) = SN ⋉ (Z2)N−1 ,
with the latter factor representing the even number of sign flips. For the O−(2N ) sector of
the path integral, the N th zero mode is turned off and hence, the nontrivial permutation
reduces to SN−1 while the effective number of signflips remains the same. We thus need
to divide by
SN−1 ⋉ (Z2)N−1 = 2N−1 · (N − 1)! ,
instead of dividing by WSO(2N) = 2N−1 · N !. We warn the readers not to confuse these
groups with the Weyl group of O(2N )
WO(2N) = SN ⋉ (Z2)N ,
which will enter the continuum interpretation of the rational pieces below. Just as in
O+(2N ) = SO(2N ), the results for the twisted partition function for O(2N ) can be
organized physically, in terms of planewavelike states that explore the classical vacua. These
plane waves will see all N Cartan directions as flat, even though in the localization
computations one must regard the N th as massive. This means that the continuum contributions
to ΩO(2N) will take a similar form as those to ΩSO(2N) with WO(2N) replacing WSO(2N).
However, WO(2N) itself does not enter the residue computation of ΩN
O−(2N) directly.
4.1.1
As in section 3, we present N = 4, 8 results first, and motivate how N = 16 continuum
sectors should look like. This will enable us to decompose uniquely N
= 16 results into
the integral part and the rational parts, in much the same way as ΩN =16
decomposed. Having computed ΩON−=(42,N8) by a direct path integral evaluation, we again find
O+(2N)=SO(2N)’s were
the results can be all organized into the following simple expressions,
(4.4)
(4.5)
(4.6)
(4.7)
HJEP07(21)46
ΩN =4
O−(2N)(y) =
1
′′
X
The sum is now over the Weyl elements of SO(2N ) such that det (1 − w˜P ) 6= 0 , where P inside the determinant
P = diagN×N (1, 1, . . . , 1, −1)
is the representation of P on the weight lattice of SO(2N ). In this note, we will call these
w˜’s the twisted Elliptic Weyl elements.2
Why this happens is fairly clear in view of the heuristic arguments in section 3. The
origin of ΩGN =4,8 was understood as a result of the orbifolding of the asymptotic Cartan
dynamics by the Weyl action, or equivalently via the insertion of the Weyl projection
operator in the Hilbert space trace for O(G),
1
W  σ∈W
X σ .
1
′
X
Only the elliptic Weyl elements w with det(1 − w) 6= 0 contribute to Ω, and produce
For O−’s, the operator P multiplies on the right, so the only difference is that the Weyl
projection for O−(2N ) is now shifted to
WSO(2N) σ∈W
1
X σP .
This leads to the modified sum (4.7), where w is replaced by w˜ · P . See appendix A for
more details on Elliptic Weyl elements and twisted Elliptic Weyl elements.
Although we computed O±(2N ) sector contributions separately, the total partition
function
can be more succinctly written as ΩON(=2N4)(y) = Ξ(NN=)4 with
1
2
ΩON(=2N4)(y) =
ΩSNO=(24N)(y) + ΩN =4
O−(2N)(y)
Ξ(NN=)4 ≡
1
′
X
,
where the sum is now over elliptic Weyl elements of O(2N ) and, likewise, W (N) =
WO(2N). This follows from the fact that P is a Weyl element of O(2N ) which generates
WO(2N)/WSO(2N). The universal role played by elliptic Weyl elements is evident here again.
2As an illustration, we list the first few for ΩON−=(42N)(y),
ΩON−=(44)(y) = 21 · y−21+ y2
,
ΩON−=(46)(y) = 214 y−3 + y3 +
8
ΩON−=(48)(y) = 116 y−4 + y4 +
4
1
(y−1 + y)3 ,
1
(y−2 + y2)(y−1 + y)2 .
(4.9)
(4.10)
(4.8)
As in the previous section, N = 8 is a straightforward extension of this, with additional
factors from the single adjoint chiral multiplet,
ΩN =8
O−(2N)(y, x) =
1
′′
X
the simplest of which is
1
Again, we can write the total partition function as
ΩON(=2N8)(y, x) = Ξ(NN=)8 ≡
1
′
X
where the sum is over elliptic Weyl elements of O(2N ).
After computing ΩON−=(126N), we again wish to decompose it into the integral part and other
are new types of continuum contributions that can enter ΩON−=(126N), of the form
rational parts from various continuum sectors. Our findings for N = 4, 8 imply that there
· Y3 det xFa/2yRa/2−1 − x−Fa/2y1−Ra/2 · w˜P
det xFa/2yRa/2 − x−Fa/2y−Ra/2 · w˜P
, (4.11)
(4.12)
, (4.13)
(4.14)
,
(4.15)
(4.16)
HJEP07(21)46
where the sum is over the twisted elliptic Weyl elements of SO(2N ). For ΩON−=(126N), we can
also have continuum contributions constructed from,
The reason for the equality is explained in next subsection.
and (A.1) are found for O−(2N ) as follows
Upon direct computations of the twisted partition functions, the analogs of (3.12)
ΩON−=(146) = 1 + ΔON−=(146) ,
ΩON−=(166) = 1 + 3ΔON−=(136) + 2ΔSNO=(31)6 · ΔON−=(136) + ΔON−=(166) ,
ΩON−=(186) = 2 + 2ΔON−=(136) + ΔON−=(156) + ΔSNO=(31)6 · ΔON−=(146) + ΔON−=(186) .
Note that the decomposition is unique.3 The fact that each term on the right hand side
has only one of the latter type factor is also reasonable, as at most one subgroup H would
see the projection operator P.
3Up to the accidental identity, ΔON−=(126) = 2ΔON±=(136) . See the subsection 4.3.
As with N = 4, 8, the full partition function of O(2N ) gauge theory can also be
expressed in terms of the elliptic Weyl sums,
Ξ(NN=)16 ≡
as follows
1
′
X
The partition functions of SO(2N ) theories do not equal those of O(2N ) theories,
yet we observe that the integral pieces that enumerate threshold bound states do agree
between O(2N ) and SO(2N ),
Explicit computations have shown this latter identity for up to rank 4, and we believe this
holds for all N .4
4.2
One can similarly compute ΩN
diag(2N+1)×(2N+1)(−1, −1, . . . , −1) .
On representations with an even number of vectorlike indices, such as the adjoint
representation or symmetric 2tensors, the action of P is trivial. Neither the determinants nor
the zero modes are affected by P, so we find
for all N and all N = 4, 8, 16. Consistent with this is the fact that the twisted elliptic Weyl
elements w˜ are in fact ordinary elliptic Weyl elements for the case of O(2N + 1). This,
from the trivial action of P on the Cartan of SO(2N + 1), implies that the decomposition
into continuum sectors are also intact under the projection, leading us from (4.21) to
O−(2N+1) for N ≥ 1 via HKY procedure, but in the end finds
O+(2N+1). Perhaps the simplest way to understand this is to use a different
ΩON(=2N16) 6= ΩSNO=(21N6) ,
INO(=21N6) = INSO=(126N) .
ΩON(2N+1) = ΩSNO(2N+1) ,
N
IO(2N+1) = ISO(2N+1) .
N
,
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
ΩN =4
O−(2) =
ΩN =8
O−(2) =
ΩON−=(126) =
y−1 + y
y−1 + y
1
1
1
y−1 + y
= 2Ξ(N1)=4
·
x1/2y−1 + x−1/2y
and, in view of (4.23),
(4.23)
(4.24)
(4.25)
(4.26)
(4.27)
(5.1)
(5.2)
for each N = 4, 8, 16. Since Ξ’s are inherently of continuum contributions, this implies
that not only for N = 4, 8 but also for N = 16, ΩON(2), the integral index vanishes,
Finally, O(1) means a single D0 trapped in O0. As such, even though the theory is empty
literally, it still makes sense to assign,
as the counting of a IIA quantum state. This, together with higher rank computations
above, completes O(m) cases. This result may look a little odd in that, of all orientifold
theories, the O(2) theory proves to be the only case with null Witten index. In the next
section, we will explain this from a simple and elegant Mtheory reasoning.
5
Witten index and Mtheory on R9/Z2
Combining results of the previous two sections, and with help of some foresight [24], we
end up with the following, rather compelling expressions as the generating functions,
Let us close with two exceptional cases of O(2) and O(1). In the O+(2) = SO(2) sector,
the twisted partition function vanishes
O+(2)
ΩN =4,8,16 = 0 ,
as all fields are chargeneutral and the determinant g(t) is independent of the gauge variable
t; the relevant JKresidue sum has to vanish identically, since we are supposed to pick up
residue only from physical poles for these pure YangMills quantum mechanics [5].
For the O−(2) sector, however, t no longer appears as a zero mode, so there is no final
residue integral to perform. The localization merely reduces to a product of determinants,
ΩON(2) = Ξ(N1)
INO(=21)6 = 0 = INSO=(126) .
INO(=11)6 = 1 ,
n≥1
m≥1
1 + X z2n INSp=(n1)6 =
1 +
X zm INO(=m1)6 =
k=2,4,6,...
Y
Y
k=1,3,5,...
(1 + zk) ,
(1 + zk) .
The two generating functions count the number of partitions of 2n and m into, respectively,
distinct even natural numbers and distinct odd natural numbers. Our pathintegral
computation confirmed this formulae up to 2n = 8 and m = 9, that is, up to nine Dparticles
in the covering space. Recall that O(2) is the only Orientifold theory with no bound states,
IO(2)
N =16 = 0. We find the manner in which (5.2) realizes this m = 2 result, quite compelling
and elegant: m = 2 is the only positive integer that cannot be expressed as a sum of
distinct odd natural numbers.
A further evidence in favor of these generating functions can be found in ref. [19], which
counted classical isolated vacua of massdeformed theories instead. The mass deformation
is easiest to see when N = 16 theory is viewed as N
= 4 with three adjoint chirals
and a particular trilinear superpotential W. Adding a quadratic mass term to W, one
finds certain “distinguished” classical vacua which are cataloged by su(2) embedding, with
trivial centralizers so that the solution is isolated. Kac and Smilga proposed the counting
of such special subsets of classical vacua equals the true Witten index of the undeformed
theory. Interestingly, this drastic approach had previously produced the desired results of
INSU=(1N6) = 1 [25].
Extending this to SO and Sp groups, Kac and Smilga found numbers which can be
seen to be consistent with the generating functions as above. Since SO(m) theories and
O(m) theories are different, one further needs to check ISO(m) = IO(m) for all m, but
this equality follows easily: the classical vacua for the massdeformed SO(m) theory can
be thought of as a triplet of m × m matrices forming a su(2) representation [19]. The
defining representation of SO(m) is real, so only integral spins can enter, while the absence
of centralizer demands these spins be distinct. Each partition of m into distinct odd natural
HJEP07(21)46
numbers,
theories.
m =
X ks ;
ks + 1 ∈ 2Z+,
ks 6= ks′ if s 6= s′
then gives a solution where the three adjoints are blockdiagonal with ks × ks blocks. The
action of P on such solutions is trivial, up to possible shift along SO(m) orbits, regardless
of even or odd m, for the same reason as P acts trivially on SO(2N + 1) pure YangMills
It has been observed by Hanany et al. [24] that spectrum of type (5.1) and (5.2)
have a simple explanation in Mtheory. For this, we must first go back to the story of
Mtheory on T4p+1/Z2 originally due to Dasgupta and Mukhi [26]. p = 0 is the
wellknown HoravaWitten [27], while p = 1 is relevant for Dtype (2,0) theories and anomaly
inflow thereof [28–30]. The lesserknown case of p = 2 was also discussed, however, where
the authors noted that the net anomaly after the projection can be canceled by a single
chiral fermion supported at each fixed point. As first proposed in ref. [24], this implies
certain spectrum of Dparticle states at the Orientifold point R9/Z2. Upon a further S1
compactification, the fixed point will become a IIA orientifold point, and at this point the
chiral fermion will generate infinite towers of harmonic oscillators, with either integral or
halfintegral KK momenta, depending on a choice of the spin structure.
With the antiperiodic spin structure, we have fermionic harmonic oscillators bk/2, b†k/2
with odd k’s. The Hilbert space built out of these, with positive KK momenta k/2 has
the partition function of the second type above, i.e., (5.2). An even number of oscillators
corresponds to O(2N ) cases of O0− while an odd number of oscillators corresponds to
O(2N + 1) cases of Og0−. With the periodic boundary conditions, we have bk/2, b†k/2 with
even k’s, instead, so this would lead to partition function of the first type, i.e., (5.1).
With periodic spin structure, the zero mode b0, b†0 also appear, meaning that there are
actually two towers, built on either the vacuum 0i or on b†00i. It looks reasonable that
we associated these two towers with O0+ and Og0+, respectively.5 The correspondence is
complete once we recall that 2n and m are the Dparticle charges in the covering space and
must be divided by 2. These four towers also explain neatly the four possible types of O0’s.
There are a few noteworthy facts. First, apart from the antiD0 towers due to oscillators
with negative k’s, there are additional states with positive and negative k oscillators mixed.
These correspond to mixture of D0 and antiD0 from the standard M/IIA duality, and a
pair annihilation must occur to reduce them to collection of either D0 and antiD0 only.
The relevant coupling involves the closed string multiplet in the bulk, as the energy must
be radiated away to transverse space. With nothing that prevents the necessary couplings,
the above four towers we reproduced from D0O0 perspective are the only stable states
from these free fermions.
Second, each of these stable states is, for any such collection of k’s of the same sign, a
single quantum state rather than a supermultiplet. Although this may sound strange given
the extensive supersymmetry, there is really no contradiction as these states are strictly
onedimensional. Supersymmetry does not always imply an onshell supermultiplet for
quantum mechanical degrees of freedom. Recall that the usual D0 problem in the flat
IIA case is governed by U(N ) = U(1) × SU(N ), and U(1) is responsible for R9 center of
mass degrees of freedom and the BPS multiplet structure of 256. In the orientifold analog,
this U(1) is projected out, which is consistent with the fact that O0 breaks the spatial
translational invariance completely.
Finally, the number of states at a given large Dparticle quantum number k seems to
grow pretty fast with k. For example, the number of threshold bound states in Sp(n) case
equals to the number of distinct partitions of n, with the known asymptotic formula [32],
1
4 · 31/4 · n3/4 exp πpn/3 + · · · .
(5.3)
This exponential growth is a straightforward consequence of the single chiral fermion along
the Mtheory circle at the origin of the IIA theory.
Whether this has other physical consequences remains to be explored.
5For D0O0 quantum mechanics and indices thereof, the distinction is clearly lost. The distinction
between DpOp+ and DpOgp+ survives the field theory limit for p = 4, only thanks to π4(Sp(n)) = Z2,
while already for p = 3 the two field theories are continuously connected via a 2π shift of the θangle [31].
HJEP07(21)46
For N = 4 quiver theories based on U(N )type gauge groups,6 it has been observed that
there is a universal relationship between Ω’s and I’s of the form,
ΩΓ(y) =
X 1
NΓ
N
y−1 − y
·
y−N − yN · IΓ/N (yN )
where the sum is over possible divisor N of the quiver Γ [5], in the sense that Γ/N is the
same quiver except the rank vector is divided by N . Not only is this structure evident
in the final answers but also in the computational middle steps as well, and is thus quite
ubiquitous in counting problems in the wallcrossing [20, 33, 34]. The object of type (6.1),
prior to being identified as the twisted partition functions [5], was also known as the rational
invariants for the obvious reason. Note that the universal factor
HJEP07(21)46
1
N
·
y−1 − y
y−N − yN
(6.1)
(6.2)
in this expression coincides with ΩSNU=(N4), and carries the continuum contribution from a
planewave sector of N identical 1particlelike states. This is because the continuum sector
in question resides in the Coulomb branch, and, as such, any other N = 4 U(N ) type
quiver theory with Coulombic flat directions can receive the same type of contributions.
Universality of this begs for the question whether there is an analog of this rational structure
for Dbrane theories with Orientifolds.
Indeed, one of the most tantalizing outcome is the “orientifolded” version of (6.2)
Ξ(NN)
precisely defined in (4.10), (4.13), and (4.17), as building blocks for ΩGN for orthogonal
and symplectic groups. These functions Ξ(NN) appear universally for these theories, simply
because O(2N ), O(2N + 1), and Sp(N ) share a common Weyl group;
WO(2N) = WO(2N+1) = WSp(N) = W (N) ≡ SN ⋉ (Z2)N .
One difference of Ξ(NN=)4 from the above U(N ) version (6.2) is that Ξ(NN) has increasing
large number of linearly independent terms, due to large number of contributing
conjugacy classes. Another complication is that, as we saw in various N = 16 Orientifolded
theories, the continuum sectors are no longer constrained to sectors with identical partial
bound states.
We note here that at least the first issue has a simple and elegant solution; Ξ(NN),
even though they look individually quite complicated, can be all constructed from a single
function Ξ(N1). Introducing
χ(Nn)(y, · · · ) ≡ Ξ(N1)(yn, · · · )
(6.3)
6This has been extensively tested in the class of quivers where 1cycles and of 2cycles are absent,
meaning absence of adjoint chirals and of complex conjugate pairs.
where the ellipsis on the left hand side denotes other possible equivariant parameters, while
the one on the right hand side denotes the same parameters raised to the nth power, Ξ(NN)
can be seen to be sums of products of χ(n) with contributing n’s sum to N . One then finds
the generating functions,
∞
N=1
1 +
X qN · Ξ(NN) = Exp
X∞ q
k=1
k
k χ(k)
N
!
= P.E. hq · χ(N1)i
for all N , where P.E. is the Plethystic Exponential [35]. We expect that these quantities,
term by term in qexpansion, should play a role similar to (6.2), now for Orientifolded
We are not aware of a general answer to the second complication, yet. Trivial
examples, in this sense, are N = 4, 8 Orientifold theories, partition functions of which can be
quiver theories.
paraphrased as
and
(6.4)
(6.5)
(6.6)
∞
N=1
1 +
X qN ΩGNN=4(y) = P.E.
q
2(y−1 + y)
,
∞
N=1
1 +
X qN ΩGNN=8(y, x) = P.E.
"
q
2(y−1 + y)
·
x1/2y−1 + x−1/2y #
,
common for GN = O(2N ), O(2N + 1), or Sp(N ). But the analog of (6.1) for general
Orientifolded quiver theories, which may have nontrivial ground states, is yet another
matter. Even for N = 16 theories computed in this note, we are yet to find a closed form
of generating functions, inclusive of all ranks. We wish to come back to the problem of
finding generic Orientifold version of the rational invariants in near future.
Acknowledgments
We would like to thank Chiung Hwang and Joonho Kim, for discussions on their work
involving other types of Orientifold planes, Amihay Hanany for bringing our attention to
his old work on Orientifold points, and Matthew Young for illuminating discussions on the
quiver stability. SJL is grateful to Korea Institute for Advanced Study for hospitality. The
work of SJL is supported in part by NSF grant PHY1417316.
A
Elliptic Weyl elements and rational invariants
An elliptic element w of Weyl group W is defined by absence of eigenvalue 1 in the canonical
representation of W on the weight lattice.
For SU(N ), the Weyl group SN is a little special because the rank is actually N − 1.
The only elliptic Weyl’s are the fully cyclic ones, say, (123 · · · N ) and all of these belong to
a single conjugacy class. For SO(2N ), SO(2N + 1), and Sp(N ) groups, the Weyl groups
are SN semidirectproduct with (Z2)N−1, (Z2)N , and (Z2)N , respectively. The elements
can be therefore represented as follows
where dots above a number indicate a sign flip. For example (12 3˙) represents the element,
σ = (abc˙d˙ . . . )(klm n˙ . . . ) · · ·
In this form, the above (Z2)N−1 for SO(2N ) means that the total number of sign flip has to
be even. Since the determinant factorizes upon the above decomposition of w, this should
be true for each cyclic component. It is fairly easy to see that this requires each cyclic
component of w to have an odd number of sign flips.
Let us list the conjugacy classes of elliptic Weyl elements for classical groups, for some
low rank cases, from which the pattern should be quite obvious,
HJEP07(21)46
• SO(5) and Sp(2)
• SO(7) and Sp(3)
• SO(
9
) and Sp(4)
( 1˙ 2˙ 3˙), (12 3˙), (1 2˙)( 3˙), ( 1˙)( 2˙)( 3˙)
( 1˙ 2˙ 3˙)( 4˙), (12 3˙)( 4˙), (1 2˙)(3 4˙), ( 1˙)( 2˙)( 3˙)( 4˙)
We may classify the twisted elliptic Weyl elements, w˜, for O(m)’s, similarly. We take
this to be defined by absence of eigenvalue 1 in w˜ · P where w˜ is an element of WSO(m).
One immediate fact is that the underlying action of P is trivial on the root lattice of
SO(2N + 1), so for SO(2N + 1), the elliptic Weyl elements coincide with the twisted elliptic
Weyl elements. This is, in retrospect, another reason behind why ΩO−(2N+1) = ΩO+(2N+1)
and hence ΩO(2N+1) = ΩSO(2N+1). For O(2N ), however, P flips an odd number of Cartan’s,
Using the same notation as above, we can then classify the conjugacy classes of w˜ · P
(123 · · · N )
( 1˙)( 2˙)
(1 2˙), ( 1˙)( 2˙)
(1 2˙)( 3˙)
• O−(6)
• O−(8)
• O−(10)
Note that P is in fact nothing but the generator of WO(2N)/WSO(2N) = Z2. Therefore, one
can also think of w˜ · P as elliptic Weyl elements of O(2N ) which are not in WSO(2N). In
particular, this means that WO(2N) = WO(2N+1) = WSp(N) and the the respective elliptic
Weyl elements also coincide.
A.1
ΩGN =16 with simple and connected G
We list results for twisted partition functions with N = 16, from ref. [5];
ΩN =16 = 1 + 2ΔSNO=(31)6 + ΔSNO=(41)6 ,
SO(4)
SO(5)
ΩN =16 = 1 + 2ΔN =16
SO(3)=Sp(1) + ΔN =16
SO(5)=Sp(2)
= ΩN =16 ,
Sp(2)
ΩGN2=16 = 2 + 2ΔSNU=(21)6 + ΔGN2=16 ,
ΩN =16 = 1 + ΔSNO=(31)6 + ΔSNO=(61)6 ,
SO(6)
ΩN =16 = 1 + 3ΔSNO=(31)6 +
SO(7)
Sp(3)
ΩN =16 = 2 + 3ΔN =16 +
Sp(1)
ΩN =16 = 2 + 4ΔSNO=(31)6 + 2 ΔN =16
SO(8)
+
+ 2ΔSNO=(51)6 + ΔN =16 · ΔSNO=(51)6 + ΔSNO=(71)6 + ΔSNO=(91)6 ,
SO(3)
Sp(4)
ΩN =16 = 2 + 5ΔN =16 + 2 ΔN =16
Sp(1)
+ 2ΔN =16 + ΔSNp=(11)6 · ΔN =16 + ΔN =16 + ΔN =16 ,
Sp(2) Sp(2) Sp(3) Sp(4)
where Δ’s are defined in (3.10). As with SU(N ) case in (3.12), these decompositions are
unique.
• rank 1
• rank 2
• rank 3
• rank 4
• rank 5
1
48 y−3 + y3 +
8
48
1
384 y−4 + y4 +
1
1
2 y−1 + y
2
1
8 y−2 + y2 +
1
Common building blocks for orthogonal and sympletic groups
Since the Weyl groups of O(2N ), O(2N + 1), and Sp(N ) coincide, the quantities defined
in (4.10), (4.13), and (4.17) are common to all three classes of the gauge groups. These can
be classified by the rank alone, without reference to the type of orientifolding projection,
suggesting universal building blocks for continuum contributions. Here we list a few low
rank examples of Ξ(NN=)4(y) of (4.10);
(A.2)
+
(y−4 + y4)(y−1 + y)
(y−3 + y3)(y−2 + y2)
+
+
80
20
(y−3 + y3)(y−1 + y)2 +
(y−2 + y2)(y−1 + y)3 +
60
(y−2 + y2)2(y−1 + y)
1
(y−1 + y)5
1
(y−1 + y)4
160
Elevating these to building blocks of N = 8, 16 orientifolded theories is a matter of attaching
chiral field contributions to each linearlyindependent rational pieces, as in (4.13) and
in (4.17). ΩN =4,8 and ΔN =16’s are related simply to these as
and
Ξ(NN=)4,8 = ΩON(=2N4,8) = ΩON(=2N4,8+1) = ΩSNO=(24N,8+1) = ΩSNp=(N4,)8 ,
Ξ(NN=)16 = ΔON(=2N16) = ΔON(=2N16+1) = ΔSNO=(21N6+1) = ΔSNp=(N16) .
Although there is a universal form (4.17) of continuum contributions to N = 16 theories
with an Orientifold point, the actual partition functions and the indices differ among
O(2N ), O(2N + 1), and Sp(N ) groups.
Here we list all three series, for comparison, although O(2N + 1) and Sp(N ) cases were already shown in section A.1 in a different notation;
ΩN =16 = 0 + Ξ(N1)=16 ,
O(2)
ΩN =16 = 1 + Ξ(N1)=16 + Ξ(N2)=16 ,
O(4)
ΩN =16 = 1 + 2Ξ(N1)=16 +
O(6)
ΩN =16 = 2 + 3Ξ(N1)=16 +
O(8)
Ξ(N1)=16
Ξ(N1)=16
ΩN =16 = 1 + Ξ(N1)=16 ,
O(3)
ΩN =16 = 1 + 2Ξ(N1)=16 + Ξ(N2)=16 ,
O(5)
+ Ξ(N3)=16 ,
+ 2Ξ(N2)=16 + Ξ(N1)=16 · Ξ(N2)=16 + Ξ(N4)=16 ,
gO+(2N)(t) =
×
×
Y
α
Y
α
=
a
· Y
a
a α
1 − tα
tαy−1 − y
· Y Y y−(Ra/2−1) − tαxFa yRa/2−1
tαxFa yRa/2 − y−Ra/2
ΩN =16 = 1 + 3Ξ(N1)=16 +
O(7)
Ξ(N1)=16
+ Ξ(N2)=16 + Ξ(N3)=16 ,
ΩN =16 = 2 + 4Ξ(N1)=16 + 2 Ξ(N1)=16
O(
9
)
+ 2Ξ(N2)=16 + Ξ(N1)=16 · Ξ(N2)=16 + Ξ(N3)=16 + Ξ(N4)=16 ,
ΩN =16 = 1 + Ξ(N1)=16 ,
Sp(1)
ΩN =16 = 1 + 2Ξ(N1)=16 + Ξ(N2)=16 ,
Sp(2)
ΩN =16 = 2 + 3Ξ(N1)=16 +
Sp(3)
Ξ(N1)=16
+ Ξ(N2)=16 + Ξ(N3)=16 ,
ΩN =16 = 2 + 5Ξ(N1)=16 + 2 Ξ(N1)=16
Sp(4)
+ 2Ξ(N2)=16 + Ξ(N1)=16 · Ξ(N2)=16 + Ξ(N3)=16 + Ξ(N4)=16 .
B
Integrand for the O−(2N )
The determinant gO−(2N)(t) that appears in the localization formula (3.1) for the twisted
partition function of the O−(2N ) pure YangMills theory can be obtained by modifying
the following O+(2N ) counterpart,
2
2
2
2
2
2
(A.9)
(A.10)
(A.11)
(B.1)
x−Fa/2y−(Ra/2−1) − xFa/2yRa/2−1 !N
a α
y−(Ra/2−1) − xFa yRa/2−1 !N
· Y Y t−α/2x−Fa/2y−(Ra/2−1) − tα/2xFa/2yRa/2−1
tα/2xFa/2yRa/2 − t−α/2x−Fa/2y−Ra/2
so that the parity action is appropriately taken into account. Here, α’s are the roots of
SO(2N ) and a’s label the 0, 1, and 3 adjoint chiral multiplets for N = 4, 8, and 16 theories,
respectively. With the parity represented as in eq. (4.3),
gO−(2N)(t) = gO+(2N−2)(t) ·
y + y−1 · Y y−(Ra/2−1) + xFayRa/2−1
1
xFayRa/2 + y−Ra/2
x−1x˜ − 1
1 − t1
1 − t1x2
t1y−1 − y · t1y−1 + y · t1−1y−1 − y · t1−1y−1 + y
t1x2y − y−1 · t1x2y + y−1 · t1−1x2y − y−1 · t1−1x2y + y−1
1 + t1x2
1 − t1−1x2
1 + t1−1x2
1 + t1
x−1x˜ + 1
1 − t1−1
x−1x˜−1 − 1
x−1x˜−1 + 1
1 + t1−1
1
1
1
1
1
·
1
1 − ti−1
1 + ti−1
(B.2)
(B.3)
N−1
× Y
1 − ti
1 + ti
a
N−1
· Y
i=1 tiy−1 − y tiy−1 + y
i=1 ti−1y−1 − y ti−1y−1 + y
a i=1
a i=1
× Y NY−1 y−(Ra/2−1) − tixFayRa/2−1 y−(Ra/2−1) + tixFayRa/2−1
tixFayRa/2 − y−Ra/2
tixFayRa/2 + y−Ra/2
× Y NY−1 y−(Ra/2−1) − ti−1xFayRa/2−1 y−(Ra/2−1) + ti−1xFayRa/2−1
,
ti−1xFayRa/2 − y−Ra/2
ti−1xFayRa/2 + y−Ra/2
the N th zero mode is frozen to tN = 1 and some of the oneloop determinants relevant
to the N th Cartan U(1) undergo appropriate sign flips as described in the paragraph
including eq. (4.3). The determinant gO−(2N)(t) is then a function of the N − 1 zero
modes, t = {t1, . . . , tN−1}, and can be written as
where the expression for gO+(2N−2)(t) can be read from eq. (B.1).
For an illustration, we list below the determinants for the O−(4) theories with N = 4,
8, and 16:
gON−=(44)(t1) =
y − y−1 · y + y−1
1 − t1
1 + t1
t1y−1 − y · t1y−1 + y · t1−1y−1 − y · t1−1y−1 + y
,
1 − t1−1
1 + t1−1
gON−=(84)(t1) =
y − y−1 · y + y−1 ·
y − xy−1 y + xy−1
·
x − 1
x + 1
gON−=(146)(t1) =
y − y−1 · y + y−1 · x2y − y−1 · x2y + y−1
t1x − 1
t1x + 1
1 − t1−1
1 − t1
1 + t1
t1y−1 − y · t1y−1 + y · t1−1y−1 − y · t1−1y−1 + y
y − t1xy−1 y + t1xy−1 y − t1−1xy−1 y + t1−1xy−1
1 + t1−1
·
t1−1x + 1
1 − x2
t1−1x − 1
1 + x2
y − x−1x˜y−1 y + x−1x˜y−1 y − x−1x˜−1y−1 y + x−1x˜−1y−1
t1x−1x˜ − 1
t1x−1x˜ + 1
t1x−1x˜−1 − 1
·
y + t1x−1x˜−1y−1
t1x−1x˜−1 + 1
·
t−1x−1x˜ + 1
y − t1−1x−1x˜−1y−1
t1−1x−1x˜−1 − 1
·
y + t1−1x−1x˜−1y−1
t1−1x−1x˜−1 + 1
,
where Rcharges and flavor charges have been assigned as R = 0 and F = 1 to the adjoint
chiral multiplet of the N = 8 theory and as R = (2, 0, 0), F = (2, −1, −1) and F˜ = (0, 1, −1)
to the three adjoint chirals of the N = 16 theory.
As a final remark, the determinant formula (B.3) has the following subtlety in sign. It
is natural to expect that the massive Cartan factors in the first line of eq. (B.3) each come
with an additional minus sign,7 just like they do in the O+ theory,
y−(Ra/2−1) − xFa yRa/2−1
.
(B.4)
If true, the formula would have an incorrect overall sign for N = 8 and 16 cases as there
exist one and three such massive Cartan factors, respectively. However, we propose that
they do not come with an expected minus sign and eq. (B.3) is correct as it is. For a
consistency check, let us consider N = 4 O−(2N ) theory with an adjoint chiral multiplet,
to which R = 1 and F = 0 are assigned. Since this theory admits a mass term for the
chiral field, it should flow to pure N = 4 O−(2N ) theory and hence, the twisted partition
functions of the two theories must agree, with the same overall sign.
We have indeed confirmed this for N = 2 and 3 based on the oneloop determinants (B.3).
Open Access.
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