D-Particles on Orientifolds and Rational Invariants
KKIAS-P17009
D-Particles on Orientifolds and Rational Invariants
Seung-Joo Lee ∗ and Piljin Yi †∗ Department of Physics, Robeson Hall, Virginia Tech,Blacksburg, VA 24061, U.S.A. † School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
Abstract
We revisit the D0 bound state problems, of the M/IIA duality, with theOrientifolds. The cases of O4 and O8 have been studied recently, from theperspective of five-dimensional theories, while the case of O0 has been muchneglected. The computation we perform for D0-O0 states boils down to theWitten indices for N = 16 O ( m ) and Sp ( n ) quantum mechanics, where weadapt and extend previous analysis by the authors. The twisted partitionfunction Ω, obtained via localization, proves to be rational , and we establisha precise relation between Ω and the integral Witten index I , by identifyingcontinuum contributions sector by sector. The resulting Witten index showssurprisingly large numbers of threshold bound states but in a manner consistentwith M-theory. We close with an exploration on how the ubiquitous rationalinvariants of the wall-crossing physics would generalize to theories with Orien-tifolds. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] F e b ontents I vs. Twisted Partition Function Ω
33 Rational Ω G N and Integral I G N N = 4 , N = 16 Continuum Sectors . . . . . . . . . . . . . . . . . . . . . 103.3 N = 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 − O (2 N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.1 N = 4 , N = 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 O (2 N + 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 O (2) and O (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 R / Z
216 Toward Rational Invariants for Orientifolds 24A Elliptic Weyl Elements and Rational Invariants 27
A.1 Ω G N =16 with Simple and Connected G . . . . . . . . . . . . . . . . . . 29A.2 Common Building Blocks for Orthogonal and Sympletic Groups . . . 29A.3 Ω G N =16 for D-Particles on an Orientifold Point . . . . . . . . . . . . . 31 B Integrand for the O − (2 N ) Introduction
One of the earliest BPS bound state counting problems in the context of superstringtheory is that of multi-D0 threshold bound states. M theory/IIA theory dualityanticipates supersymmetric bound states of N D-particles, for all natural numbers N [1]. This problem was given a lot of attention since its first inception by Witten,and obviously, N = 2 case, i.e., N = 16 SU (2) quantum mechanics, has been dealtwith the most rigor [2, 3], while higher N cases have lead to many new insights overthe years.This problem was given a fresh treatment recently via the localization technique[4, 5]. Previously, computation of twisted partition functions had been performed for N = 16 SU ( N ) theories [6] and some attempts made for other gauge groups [7,8], butthere are often issues with a contour choice in the last stage of such computations.The new derivations obviate this last uncertainty as they actually derive rigorouslywhat the contour should be. For SU ( N ), one finds in the end the twisted partitionfunction [5] Ω SU ( N ) N =16 = 1 + (cid:88) p | N ; p> · ∆ SU ( p ) N =16 , (1.1)with rational functions ∆ whose precise form for a general Lie Algebra can be foundin Eq. (3.10).This Ω, being non-integral, is certainly not the same as the Witten index [9].Such is usually a symptom of having asymptotic flat directions that cannot be liftedby a parameter tuning. For SU ( N ) theory in question, the classical vacua form acone R N − /S N , and the plane-wave-like states can also contribute to the relevantpath integral. The correct interpretation here is to identify the first term “1” as theindex 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 Ωwas 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 formore complicated spacetime is how to separate the continuum contribution from trueWitten index systematically. This does not seem to admit a universal answer, as2here are numerous cases where continuum sectors can conspire to contribute a netintegral piece to Ω [5]. At present, extraction of I from Ω, when the theory involvesgapless asymptotic directions, is more of an art than a science.Ref. [5], nevertheless, noted how the main feature of N = 16 SU ( N ) generalizesstraightforwardly to other N = 16 theories and also to N = 4 non-primitive quivertheories with bifundamental matters only. The various continuum contributions toΩ G N =16 have been physically understood, identified, and catalogued. Naturally, thisopens up the possibility of computing true Witten indices for D-particle binding toOrientifold points. In fact, the results of Ref. [5] almost suffice, except for the case ofO0 − orientifold. In this note, we wish to place the last missing piece in the problemand compute Witten indices for all D0-O0 bound states.Section 2 will give a general discussion on the twisted partition function versus theWitten index, with emphasis on what the localization procedure actually computes.Section 3 will review the recent results for N = 16 Yang-Mills quantum mechanics,which we will generalize in Section 4 to O ( m ) gauge groups. This will lead us to theWitten indices that count bound states between D0’s and any one of four types of theorientifold point and to a known M-theory interpretation, adding yet another strongand rather direct confirmation of M/IIA duality. In the final section, we commenton new type of rational expressions we found along the way and propose them asbuilding blocks for the rational invariants suitable for Orientifolded theories. I vs. Twisted Partition Function Ω For supersymmetric quantum theory, one of the useful and accessible quantities thatprobe the ground state sector is the Witten index [9], I = lim β →∞ tr (cid:2) ( − F e − βH (cid:3) . The chirality operator ( − F can be replaced by any operator that anti-commuteswith the supercharges. One often wishes to compute the equivariant version byinserting chemical potentials, x , associated with global symmetries, F , I ( x ) ≡ lim β →∞ tr (cid:2) ( − F x F e − βH (cid:3) . Q which commutes with a linear combination of R -symmetry generators, call it R , andone of the F ’s, resulting in a fully equivariant Witten index, I ( y , x ) ≡ lim β →∞ tr (cid:104) ( − F y R x F e − β Q (cid:105) . (2.1)However, as is well known, this quantity may not be amenable to straightforwardcomputations.If the dynamics is compact, i.e., with a fully discrete spectrum, β -dependencecan be argued away based on the naive argument that I is topological. Under suchfavorable circumstances, one is motivated to consider insteadΩ( y , x, β ) ≡ tr (cid:104) ( − F y R x F e − β Q (cid:105) , (2.2)and compute the other limit, which tends to reduce the path integral to a localexpression, I bulk ( y , x ) ≡ lim β → Ω( y , x, β ) , (2.3)with the anticipation that Ω is independent of β so that I = I bulk .For theories with continuum sectors, however, this naive expectation cannot holdin general; I is by definition integral, while Ω need not be integral and thus can differfrom I . If the continuum has a gap, E ≥ E gap >
0, its contribution is suppressed as e − βE gap , so we may have an option of scaling E gap up first and then taking β → I only [4, 11].When the continuum cannot be gapped, or when a gap can be introduced onlyat the expense of qualitative modification of the asymptotic dynamics, however, weare often in trouble. The resulting bulk part I bulk differs from the genuine index.For such theories, isolating I hidden inside I bulk requires a method of computing yetanother piece, known as the defect term, − δ I ≡ I bulk − I . (2.4)4his program depends on particulars of the given problem and, in particular, on theboundary conditions. As far as we know there is no general theory for δ I .For a large class of gauged dynamics, the localization procedure has been appliedsuccessfully to reduce the path integral representation of Ω to a formulae involvingrank-many contour integrations. For N ≥ d = 2 elliptic genera [13, 14], in particular, reasonably complete and reliablederivations exist. At the end of such computations, one finds that β -dependence isabsent. When the dynamics is not compact and Ω is expected to be β -dependent,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 e →
0, asthe term is often BRST-exact for the spacetime dimension D less than three. In theabsence of other dimensionful parameters of the theory, the only obvious answer tothe question we posed above is β →
0; The dimensionless combination of the two is βe / (4 − D ) , so e → β → D ≤
3. Another typical dimensionful parametersthat could be present are Fayet-Iliopoulos constants ζ , but, for a sensible results, oneoften must take a limit of ζ first [4]. This raises a gap E gap along certain Coulombdirections to infinity, if not all, so we expect that, again, the β → β → ∞ .As such, we will define for this note,Ω( y , x ) ≡ Ω( y , x, β ) (cid:12)(cid:12)(cid:12)(cid:12) localization , (2.5)whereby, according to the above scaling argument, we may identify I bulk = Ω( y , x ) . (2.6)We will call this quantity the twisted partition function, although, strictly speaking, A canonical example is the supersymmetric nonlinear sigma models onto a manifold with bound-ary. If one adopt the so-called APS boundary condition, δ I is then computed by the eta-invariant,leading to the Atiyha-Patodi-Singer index theorem [12]. This boundary condition, however, doesnot in general translate to L condition on the physical space. y , x, β ) may yet differ from Ω( y , x ). This bringsus to a general statement I ( y , x ) = Ω( y , x ) + δ I ( y , x ) . (2.7)Even after a successful localization computation of Ω( y , x ), one is often left with aneven more difficult task of identifying the continuum contribution, − δ I , inside Ω ifone wishes to compute I .There appears to be no single universal relationship between I and Ω, but surpris-ingly, as delineated in Ref. [5], there exists classes of d = 1 supersymmetric gaugedlinear sigma models for which this problem may be dealt with honestly. One suchis adjoint-only Yang-Mills quantum mechanics, and another is N = 4 nonprimitivequiver theories with compact classical Higgs vacuum moduli space. In the next sec-tion, we recall this phenomenon for N = 4 , ,
16 pure Yang-Mills quantum mechanicswith connected simple group G . Ω G N and Integral I G N For gauged linear sigma model with at least two supersymmetries, the localizationprocedure gives a Jeffrey-Kirwan residue formulae [4],Ω( y , x ) = 1 | W | JK-Res η g ( t ) (cid:81) s t s d r t , (3.1)where ( t , . . . , t r ) parameterize the r bosonic zero modes living in ( C ∗ ) r , that usuallyscan the Cartan directions but can be further restricted in topologically nontrivialholonomy sectors. The determinant g ( t ) is due to massive modes in the backgroundof t ’s. In this note, we use N = 4 notations for supermultiplets, and as such, g ( t )takes the general form, g ( t ) = (cid:18) y − y − (cid:19) r (cid:89) α t − α/ − t α/ t α/ y − − t − α/ y × (cid:89) i t − Q i / x − F i / y − ( R i / − − t Q i / x F i / y R i / − t Q i / x F i / y R i / − t − Q i / x − F i / y − R i / . (3.2)6ere, α runs over the roots of the gauge group and i labels the individual chiralmultiplets, with the gauge charge Q i and the flavor charge F i under the Cartans ofthe gauge group and of the flavor group, respectively. Finally, W is the Weyl group ofthe gauge group and η is a choice of r auxiliary parameters. For detailed definition ofthe JK residue [15], the condition on the auxiliary parameters η , and the derivationof the above formula, we refer the reader to the section 4 of Ref. [4]. We will refer tothis general procedure as HKY.For pure N = 4 theories, the computation admits the R-charge chemical potential y only. For N = 8 ,
16, we have additional adjoint chirals, and the assignment of globalcharges needs a little bit of thought. For N = 8, one more chemical potential x can beturned on, associated with the natural U (1) rotation of the chiral field, and R = 0 isassigned to the adjoint chiral. No superpotential is possible under such assignments.For N = 16, with three adjoint chirals, a trilinear superpotential term is needed, soat most two flavor chemical potentials are allowed, say, x and ˜ x associated with F and ˜ F . We can for example assign R = (2 , , F = (2 , − , − F = (0 , , − N = 16. In actual N = 16formula below x F should be understood as the product, x F ˜ x ˜ F , over the two flavorchemical potentials.One thing special about the pure gauge theories is that we are instructed toignore the poles located at the boundary of the zero mode space ( C ∗ ) r [5]. This isa property which holds generally for theories with the total matter content in a realrepresentation under the gauge group. N = 4 , This gives us an unambiguous procedure of computing the twisted partition functionsΩ G N for all possible G and N . There are some further computational issues, such ashow to deal with the degenerate poles, which complicates the task but still allows usto go forward. We will not give too much details here and instead refer the readersto Ref. [5] for pure Yang-Mills cases, and to Ref. [4] for general gauged quantummechanics.It turns out that, after a long and arduous computer-assisted computation ofJK residues, the twisted partition functions for pure N = 4 , G -gauged quantum7echanics, can be organized into purely algebraic quantities. For N = 4, one findsΩ G N =4 ( y ) = 1 | W G | (cid:48) (cid:88) w y − − y · w ) . (3.3)The sum is only over the elliptic Weyl elements and | W G | is the cardinality of theWeyl group itself. An elliptic Weyl element w is defined by absence of eigenvalue 1;In other words, in the canonical r -dimensional representation of the Weyl group onthe weight lattice, det (1 − w ) (cid:54) = 0 . Some simple examples with N = 4 areΩ SU ( N ) N =4 ( y ) = 1 N · y − N +1 + y − N +3 + · · · + y N − + y N − , Ω SO (4) N =4 ( y ) = 14 · y − + y ) , Ω SO (5) N =4 ( y ) = Ω Sp (2) N =4 ( y ) = 18 · (cid:20) y − + y + 1( y − + y ) (cid:21) , Ω SO (7) N =4 ( y ) = Ω Sp (3) N =4 ( y ) = 148 · (cid:20) y − + y + 6( y − + y )( y − + y ) + 1( y − + y ) (cid:21) , where each term can be associated with a sum over conjugacy classes of the samecyclic decompositions.For pure N = 8 G -gauged quantum mechanics, obtained by adding to the N = 4theory an adjoint chiral, we can include a flavor chemical potential x of the adjointafter assigning a unit flavor charge without loss of generality. With R = 0 for theadjoint chiral, we also have the universal formula,Ω G N =8 ( y , x ) = 1 | W G | (cid:48) (cid:88) w y − − y · w ) · det (cid:0) y − x / − y x − / · w (cid:1) det ( x / − x − / · w ) , (3.4)where again the sum is over the elliptic Weyl elements of G . For example we haveΩ SO (4) N =8 ( y , x ) = 14 · y − + y ) · ( y − x / + y x − / ) ( x / + x − / ) , and the pattern generalizes to higher rank cases in an obvious manner.8he reason why the result can be repackaged into such a simple algebraic formulaehas been explained both for nonequivariant form [2, 10, 16, 17] and for equivariantform [5]. Consider − δ I . This part of Ω has to arise from the continuum and, becauseof this, depends only on the asymptotic dynamics. The latter becomes a nonlinearsigma model on an orbifold O ( G ) N =4 , = R r /W or R r /W , (3.5)so that the δ I 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 (cid:16) I O ( G ) N =4 , (cid:17) bulk + δ I O ( G ) N =4 , = 0which implies [2] − δ I G N =4 , = (cid:16) I O ( G ) N =4 , (cid:17) bulk . (3.6)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. [10,16],with the result 1 | W | (cid:48) (cid:88) w − w ) . (3.7)What we described above in (3.3) and in (3.4), individually confirmed by directlocalization computation, are the equivariant uplifts of this expression for N = 4 , G N =4 , is abundantly clear. They come entirely fromthe asymptotic continuum states spanned by the free Cartan dynamics, modulo theorbifolding by the Weyl group; The path-integral-computed Ω G N =4 , has no room fora contribution from threshold bound states. Therefore, the true enumerative part I inside Ω has to be null, I G N =4 = 0 = I G N =8 , (3.8)for any simple group G . Recall that, for classical groups G , N = 4 , S and S in K3 and Calabi-Yau three-fold, possibly togetherwith Orientifold planes, and the Witten index of these theories must vanish. Thisphysical expectation dovetails with the above structure nicely.The same principle generalizes to N = 16 cases. However, their asymptotic dy-namics will no longer be captured by analog of O ( G ) alone; The presence of thresholdbound states implies that the continuum sectors Ω G N =16 will no longer be that sim-ple. There could be additional sectors involving partial bound states tensored withcontinuum of remaining asymptotic directions. We turn to this next. N = 16 Continuum Sectors
The same kind of continuum sectors as the above N = 4 , N = 16, with the asymptotic dynamics of the form, O ( G ) N =16 = R r /W G , (3.9)and we can easily guess the contribution to Ω G N =16 from this sector to take the form,∆ G N =16 ≡ (3.10)1 | W G | (cid:48) (cid:88) w y − − y · w ) · (cid:89) a =1 det (cid:0) x F a / y R a / − − x − F a / y − R a / · w (cid:1) det ( x F a / y R a / − x − F a / y − R a / · w ) , as a straightforward generalization of N = 4 , a labels the threeadjoint chirals. Indeed, as we will see below, each Ω G N =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 expectedin general. For all SU ( N ), e.g., a single threshold bound state must exist for M-theory/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 morecomplex network of continuum sectors exist. Generally a product of subgroups ⊗ A G A < G correspond to a collection of one-particle-like states, each labeled by A . When thissubgroup equals the Cartan subgroup of G , the corresponding continuum sector con-10ributes the universal ∆ G N =16 to Ω G N =16 . When at least one of G A is a simple group,the corresponding partial bound state(s) can contribute a new fractional piece toΩ G N =16 . The relevant continuum sector is the asymptotic Coulombic directions wherethe “particles” forming the bound state associated with G A moves together. In otherwords, the asymptotic Coulombic directions are parameterized by a subalgebra h [ ⊗ A G A ]of the Cartan of G , where ⊗ A G A is the centralizer of h [ ⊗ A G A ].Then, the argument leading to (3.6) can be adapted to this slightly more involvedcase; A continuum contribution from this sector would be associated with a subgroup W (cid:48) of W G that leaves h [ ⊗ A G A ] invariant yet act faithfully. Contribution to Ω would arisefrom generalized elliptic Weyl elements of W (cid:48) ,det (1 − w (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) h [ ⊗ A G A ] (cid:54) = 0 , where the determinant is now taken in the smaller representation over h [ ⊗ A G A ]. Ina slight abuse of notation, it turns out that the continuum contribution from W (cid:48) toΩ G N =16 can be expressed as a product of the form, (cid:89) I ∆ H I N =16 where ∆ H I N =16 are defined for some subgroups H I of G in the same manner as (3.10).Each H I is a simple subgroup of G whose Weyl group is a subgroup factor of W (cid:48) . N = 16 Ω G N =16 can also be directly computed using the HKY procedure [4]. One then searchesfor a unique decomposition as sum over such continuum pieces asΩ G N =16 = I G N =16 + (cid:88) ⊗ G A 4. For Ω O − (2 N ) N =4 , , , we made an explicit JK-residueevaluation as in the previous section. The insertion of P can be represented by a Z holonomy along the Euclidean time circle,diag N × N (1 , , . . . , , − , (4.3)whereby the zero mode space shrinks by one dimension, so r = N − O − (2 N ).The reduced zero modes, t , ,...,N − = e πiu , ,...,N − , parameterize O − (2 N ) holonomy14s e πiσ u · · · e πiσ u · · · · · · · · · · · · e πiσ u N − 00 0 0 0 σ , which sets t N = 1 in g ( t ). The N -th Cartan elements in all multiplets becomemassive, instead, and now contribute factors with the signs flipped, e.g., one of the N overall y − y − factors in the denominator for the Cartan is flipped to y + y − .See Appendix B. However, we must caution against viewing this as a spontaneoussymmetry breaking of the dynamics. Consider very long (Euclidean) time β . The“symmetry breaking effect” becomes diluted arbitrarily, as the size of the time-likegauge field scales with 1 /β . Moreover, at each time slice, this A can be gauged away,locally, and thus will not alter the dynamics. It is only when we are instructed toperform the trace, this P makes a difference.Finally, one needs to be careful about the usual division by the Weyl group whencomputing O − (2 N ) contributions. Recall that the Weyl group of O + (2 N ) = SO (2 N )is W SO (2 N ) = S N (cid:110) ( Z ) N − , (4.4)with the latter factor representing the even number of sign flips. For the O − (2 N )sector of the path integral, the N -th zero mode is turned off and hence, the nontrivialpermutation reduces to S N − while the effective number of sign-flips remains the same.We thus need to divide by | S N − (cid:110) ( Z ) N − | = 2 N − · ( N − , (4.5)instead of dividing by | W SO (2 N ) | = 2 N − · N !. We warn the readers not to confusethese groups with the Weyl group of O (2 N ) W O (2 N ) = S N (cid:110) ( Z ) N , (4.6)which will enter the continuum interpretation of the rational pieces below. Just asin O + (2 N ) = SO (2 N ), the results for the twisted partition function for O (2 N ) canbe organized physically, in terms of plane-wave-like states that explore the classicalvacua. These plane waves will see all N Cartan directions as flat, even though in the15ocalization computations one must regard the N -th as massive. This means thatthe continuum contributions to Ω O (2 N ) will take a similar form as those to Ω SO (2 N ) with W O (2 N ) replacing W SO (2 N ) . However, W O (2 N ) itself does not enter the residuecomputation of Ω O − (2 N ) N directly. N = 4 , N = 4 , N = 16 contin-uum sectors should look like. This will enable us to decompose uniquely N = 16results into the integral part and the rational parts, in much the same way asΩ O + (2 N )= SO (2 N ) N =16 ’s were decomposed. Having computed Ω O − (2 N ) N =4 , by a direct path in-tegral evaluation, we again find the results can be all organized into the followingsimple expressions,Ω O − (2 N ) N =4 ( y ) = 1 | W SO (2 N ) | (cid:48)(cid:48) (cid:88) ˜ w y − − y · ˜ wP ) . (4.7)The sum is now over the Weyl elements of SO (2 N ) such thatdet (1 − ˜ wP ) (cid:54) = 0 , where P inside the determinant P = diag N × N (1 , , . . . , , − P on the weight lattice of SO (2 N ). In this note, we will callthese ˜ w ’s the twisted Elliptic Weyl elements. Why this happens is fairly clear in view of the heuristic arguments in Section 3. As an illustration, we list the first few for Ω O − (2 N ) N =4 ( y ),Ω O − (4) N =4 ( y ) = 12 · y − + y , Ω O − (6) N =4 ( y ) = 124 (cid:20) y − + y + 1( y − + y ) (cid:21) , Ω O − (8) N =4 ( y ) = 116 (cid:20) y − + y + 1( y − + y )( y − + y ) (cid:21) . (4.8) G N =4 , was understood as a result of the orbifolding of the asymptoticCartan dynamics by the Weyl action, or equivalently via the insertion of the Weylprojection operator in the Hilbert space trace for O ( G ),1 | W | (cid:88) σ ∈ W σ . Only the elliptic Weyl elements w with det(1 − w ) (cid:54) = 0 contribute to Ω, and produce1 | W | (cid:48) (cid:88) w y − − y · w ) . For O − ’s, the operator P multiplies on the right, so the only difference is that theWeyl projection for O − (2 N ) is now shifted to1 | W SO (2 N ) | (cid:88) σ ∈ W σP . This leads to the modified sum (4.7), where w is replaced by ˜ w · P . See Appendix Afor more details on Elliptic Weyl elements and twisted Elliptic Weyl elements.Although we computed O ± (2 N ) sector contributions separately, the total partitionfunction Ω O (2 N ) N =4 ( y ) = 12 (cid:16) Ω SO (2 N ) N =4 ( y ) + Ω O − (2 N ) N =4 ( y ) (cid:17) (4.9)can be more succinctly written as Ω O (2 N ) N =4 ( y ) = Ξ ( N ) N =4 with Ξ ( N ) N =4 ≡ | W ( N ) | (cid:48) (cid:88) w y − − y · w ) , (4.10)where the sum is now over elliptic Weyl elements of O (2 N ) and, likewise, W ( N ) = W O (2 N ) . This follows from the fact that P is a Weyl element of O (2 N ) which generates W O (2 N ) /W SO (2 N ) . The universal role played by elliptic Weyl elements is evident hereagain.As in the previous section, N = 8 is a straightforward extension of this, with17dditional factors from the single adjoint chiral multiplet,Ω O − (2 N ) N =8 ( y , x ) = 1 | W SO (2 N ) | (cid:48)(cid:48) (cid:88) ˜ w y − − y · ˜ wP ) · det (cid:0) y − x / − y x − / · ˜ wP (cid:1) det ( x / − x − / · ˜ wP ) , (4.11)the simplest of which isΩ O − (4) N =8 ( y , x ) = 12 · y − + y · y − x + y x − x + x − . (4.12)Again, we can write the total partition function asΩ O (2 N ) N =8 ( y , x ) = Ξ ( N ) N =8 ≡ | W ( N ) | (cid:48) (cid:88) w y − − y · w ) · det (cid:0) y − x / − y x − / · w (cid:1) det ( x / − x − / · w ) , (4.13)where the sum is over elliptic Weyl elements of O (2 N ). N = 16After computing Ω O − (2 N ) N =16 , we again wish to decompose it into the integral part andother rational parts from various continuum sectors. Our findings for N = 4 , O − (2 N ) N =16 , of theform∆ O − (2 N ) N =16 ≡ (4.14)1 | W SO (2 N ) | (cid:48)(cid:48) (cid:88) ˜ w y − − y · ˜ wP ) · (cid:89) a =1 det (cid:0) x F a / y R a / − − x − F a / y − R a / · ˜ wP (cid:1) det ( x F a / y R a / − x − F a / y − R a / · ˜ wP ) , where the sum is over the twisted elliptic Weyl elements of SO (2 N ). For Ω O − (2 N ) N =16 , wecan also have continuum contributions constructed from,∆ O − (2 r +1) N =16 = ∆ SO (2 r +1) N =16 , for r < N . (4.15)The reason for the equality is explained in next subsection.Upon direct computations of the twisted partition functions, the analogs of (3.12)18nd (A.1) are found for O − (2 N ) as followsΩ O − (4) N =16 = 1 + ∆ O − (4) N =16 , Ω O − (6) N =16 = 1 + 3∆ O − (3) N =16 + 2∆ SO (3) N =16 · ∆ O − (3) N =16 + ∆ O − (6) N =16 , Ω O − (8) N =16 = 2 + 2∆ O − (3) N =16 + ∆ O − (5) N =16 + ∆ SO (3) N =16 · ∆ O − (4) N =16 + ∆ O − (8) N =16 . (4.16)Note that the decomposition is unique. The fact that each term on the right handside has only one of the latter type factor is also reasonable, as at most one subgroup H would see the projection operator P .As with N = 4 , 8, the full partition function of O (2 N ) gauge theory can also beexpressed in terms of the elliptic Weyl sums, Ξ ( N ) N =16 ≡ | W ( N ) | (cid:48) (cid:88) w y − − y · w ) · (cid:89) a =1 det (cid:0) x F a / y R a / − − x − F a / y − R a / · w (cid:1) det ( x F a / y R a / − x − F a / y − R a / · w ) , (4.17)as followsΩ O (4) N =16 = 1 + Ξ (1) N =16 + Ξ (2) N =16 , Ω O (6) N =16 = 1 + 2 Ξ (1) N =16 + Ξ (1) N =16 · Ξ (1) N =16 + Ξ (3) N =16 , Ω O (8) N =16 = 2 + 3 Ξ (1) N =16 + Ξ (1) N =16 · Ξ (1) N =16 + 2 Ξ (2) N =16 + Ξ (2) N =16 · Ξ (1) N =16 + Ξ (4) N =16 . The partition functions of SO (2 N ) theories do not equal those of O (2 N ) theories,Ω O (2 N ) N =16 (cid:54) = Ω SO (2 N ) N =16 , (4.18)yet we observe that the integral pieces that enumerate threshold bound states doagree between O (2 N ) and SO (2 N ), I O (2 N ) N =16 = I SO (2 N ) N =16 . (4.19)Explicit computations have shown this latter identity for up to rank 4, and we believe Up to the accidental identity, ∆ O − (2) N =16 = 2∆ O ± (3) N =16 . See the subsection 4.3. N . O (2 N + 1) One can similarly compute Ω O − (2 N +1) N for N ≥ O − (2 N +1) N = Ω O + (2 N +1) N . Perhaps the simplest way to understand this is to usea different form of P , diag (2 N +1) × (2 N +1) ( − , − , . . . , − . (4.20)On representations with an even number of vector-like indices, such as the adjointrepresentation or symmetric 2-tensors, the action of P is trivial. Neither the deter-minants nor the zero modes are affected by P , so we findΩ O (2 N +1) N = Ω SO (2 N +1) N , (4.21)for all N and all N = 4 , , 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 (2 N + 1). This, from the trivial action of P on the Cartan of SO (2 N + 1), impliesthat the decomposition into continuum sectors are also intact under the projection,leading us from (4.21) to I O (2 N +1) N = I SO (2 N +1) N . (4.22) O (2) and O (1) Let us close with two exceptional cases of O (2) and O (1). In the O + (2) = SO (2)sector, the twisted partition function vanishesΩ O + (2) N =4 , , = 0 , (4.23)as all fields are charge-neutral and the determinant g ( t ) is independent of the gaugevariable t ; the relevant JK-residue sum has to vanish identically, since we are sup-posed to pick up residue only from physical poles for these pure Yang-Mills quantummechanics [5]. See section 5 for related discussions. O − (2) sector, however, t no longer appears as a zero mode, so there isno final residue integral to perform. The localization merely reduces to a product ofdeterminants,Ω O − (2) N =4 = 1 y − + y = 2 Ξ (1) N =4 Ω O − (2) N =8 = 1 y − + y · x / y − + x − / y x / + x − / = 2 Ξ (1) N =8 Ω O − (2) N =16 = 1 y − + y · (cid:89) a =1 x F a / y R a / − + x − F a / y − R a / x F a / y R a / + x − F a / y − R a / = 2 Ξ (1) N =16 (4.24)and, in view of (4.23), Ω O (2) N = Ξ (1) N (4.25)for each N = 4 , , 16. Since Ξ ’s are inherently of continuum contributions, thisimplies that not only for N = 4 , N = 16, Ω O (2) N , the integral indexvanishes, I O (2) N =16 = 0 = I SO (2) N =16 . (4.26)Finally, O (1) means a single D0 trapped in O0. As such, even though the theory isempty literally, it still makes sense to assign, I O (1) N =16 = 1 , (4.27)as the counting of a IIA quantum state. This, together with higher rank compu-tations above, completes O ( m ) cases. This result may look a little odd in that, ofall orientifold theories, the O (2) theory proves to be the only case with null Wittenindex. In the next section, we will explain this from a simple and elegant M-theoryreasoning. R / Z (cid:88) n ≥ z n I Sp ( n ) N =16 = (cid:89) k =2 , , ,... (1 + z k ) , (5.1)1 + (cid:88) m ≥ z m I O ( m ) N =16 = (cid:89) k =1 , , ,... (1 + z k ) . (5.2)The two generating functions count the number of partitions of 2 n and m into, re-spectively, distinct even natural numbers and distinct odd natural numbers. Ourpath-integral computation confirmed this formulae up to 2 n = 8 and m = 9, that is,up to nine D-particles in the covering space. Recall that O (2) is the only Orientifoldtheory with no bound states, I O (2) N =16 = 0. We find the manner in which (5.2) realizesthis m = 2 result, quite compelling and elegant: m = 2 is the only positive integerthat 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. [16],which counted classical isolated vacua of mass-deformed theories instead. The massdeformation is easiest to see when N = 16 theory is viewed as N = 4 with threeadjoint chirals and a particular trilinear superpotential W . Adding a quadratic massterm 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 andSmilga proposed the counting of such special subsets of classical vacua equals thetrue Witten index of the undeformed theory. Interestingly, this drastic approach hadpreviously produced the desired results of I SU ( N ) N =16 = 1 [22].Extending this to SO and Sp groups, Kac and Smilga found numbers whichcan 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 I SO ( m ) = I O ( m ) for all m , but this equality follows easily: The classical vacua for the mass-deformed SO ( m ) theory can be thought of as a triplet of m × m matrices forming a su (2)representation [16]. The defining representation of SO ( m ) is real, so only integralspins can enter, while the absence of centralizer demands these spins be distinct.Each partition of m into distinct odd natural numbers, m = (cid:88) k s ; k s + 1 ∈ Z + , k s (cid:54) = k s (cid:48) if s (cid:54) = s (cid:48) then gives a solution where the three adjoints are block-diagonal with k s × k s blocks.The action of P on such solutions is trivial, up to possible shift along SO ( m ) orbits,22egardless of even or odd m , for the same reason as P acts trivially on SO (2 N + 1)pure Yang-Mills theories.It has been observed by Hanany et. al. [21] that spectrum of type (5.1) and (5.2)have a simple explanation in M-theory. For this, we must first go back to the storyof M-theory on T p +1 / Z originally due to Dasgupta and Mukhi [23]. p = 0 is thewell-known Horava-Witten [24], while p = 1 is relevant for D -type (2,0) theories andanomaly inflow thereof [25–27]. The lesser-known case of p = 2 was also discussed,however, where the authors noted that the net anomaly after the projection can becanceled by a single chiral fermion supported at each fixed point. As first proposedin Ref. [21], this implies certain spectrum of D-particle states at the Orientifoldpoint R / Z . Upon a further S compactification, the fixed point will become a IIAorientifold point, and at this point the chiral fermion will generate infinite towers ofharmonic oscillators, with either integral or half-integral KK momenta, depending ona choice of the spin structure.With the anti-periodic spin structure, we have fermionic harmonic oscillators b k/ , b † k/ with odd k ’s. The Hilbert space built out of these, with positive KK mo-menta k/ O (2 N ) cases of O0 − while an odd number ofoscillators corresponds to O (2 N + 1) cases of (cid:103) O0 − . With the periodic boundaryconditions, we have b k/ , b † k/ with even k ’s, instead, so this would lead to partitionfunction of the first type, i.e., (5.1). With periodic spin structure, the zero mode b , b † also appear, meaning that there are actually two towers, built on either the vacuum | (cid:105) or on b † | (cid:105) . It looks reasonable that we associated these two towers with O0 + and (cid:103) O0 + , respectively. The correspondence is complete once we recall that 2 n and m arethe D-particle charges in the covering space and must be divided by 2. These fourtowers also explain neatly the four possible types of O0’s.There are a few noteworthy facts. First, apart from the anti-D0 towers due tooscillators with negative k ’s, there are additional states with positive and negative k oscillators mixed. These correspond to mixture of D0 and anti-D0 from the standardM/IIA duality, and a pair annihilation must occur to reduce them to collection ofeither D0 and anti-D0 only. The relevant coupling involves the closed string multipletin the bulk, as the energy must be radiated away to transverse space. With nothingthat prevents the necessary couplings, the above four towers we reproduced fromD0-O0 perspective are the only stable states from these free fermions.23econd, each of these stable states is, for any such collection of k ’s of the samesign, a single quantum state rather than a supermultiplet. Although this may soundstrange given the extensive supersymmetry, there is really no contradiction as thesestates are strictly one-dimensional. Supersymmetry does not always imply an on-shellsupermultiplet for quantum mechanical degrees of freedom. Recall that the usual D0problem in the flat IIA case is governed by U ( N ) = U (1) × SU ( N ), and U (1) isresponsible for R center of mass degrees of freedom and the BPS multiplet structureof 256. In the orientifold analog, this U (1) is projected out, which is consistent withthe fact that O0 breaks the spatial translational invariance completely.Finally, the number of states at a given large D-particle quantum number k seemsto 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 asymptoticformula [28], 14 · / · n / exp (cid:16) π (cid:112) n/ (cid:17) + · · · . (5.3)This exponential growth is a straightforward consequence of the single chiral fermionalong the M-theory circle at the origin of the IIA theory. Whether this has otherphysical consequences remains to be explored. For N = 4 quiver theories based on U ( N )-type gauge groups, it has been observedthat there is a universal relationship between Ω’s and I ’s of the form,Ω Γ ( y ) = (cid:88) N | Γ N · y − − yy − N − y N · I Γ /N ( y N ) (6.1)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 structureevident in the final answers but also in the computational middle steps as well, andis thus quite ubiquitous in counting problems in the wall-crossing [17, 29, 30]. Theobject of type (6.1), prior to being identified as the twisted partition functions [5], was This has been extensively tested in the class of quivers where 1-cycles and of 2-cycles are absent,meaning absence of adjoint chirals and of complex conjugate pairs. N · y − − yy − N − y N (6.2)in this expression coincides with Ω SU ( N ) N =4 , and carries the continuum contributionfrom a plane-wave sector of N -identical 1-particle-like states. This is because thecontinuum 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 sametype of contributions. Universality of this begs for the question whether there is ananalog of this rational structure for D-brane theories with Orientifolds.Indeed, one of the most tantalizing outcome is the “orientifolded” version of (6.2) Ξ ( N ) N precisely defined in (4.10), (4.13), and (4.17), as building blocks for Ω G N for orthogonaland symplectic groups. These functions Ξ ( N ) N appear universally for these theories,simply because O (2 N ), O (2 N + 1), and Sp ( N ) share a common Weyl group; W O (2 N ) = W O (2 N +1) = W Sp ( N ) = W ( N ) ≡ S N (cid:110) ( Z ) N . One difference of Ξ ( N ) N =4 from the above U ( N ) version (6.2) is that Ξ ( N ) N has increas-ing large number of linearly independent terms, due to large number of contributingconjugacy classes. Another complication is that, as we saw in various N = 16 Ori-entifolded theories, the continuum sectors are no longer constrained to sectors withidentical partial bound states.We note here that at least the first issue has a simple and elegant solution; Ξ ( N ) N ,even though they look individually quite complicated, can be all constructed from asingle function Ξ (1) N . Introducing χ ( n ) N ( y , · · · ) ≡ Ξ (1) N ( y n , · · · ) (6.3)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 n -thpower, Ξ ( N ) N can be seen to be sums of products of χ ( n ) N with contributing n ’s sum to25 . One then finds the generating functions,1 + ∞ (cid:88) N =1 q N · Ξ ( N ) N = Exp (cid:32) ∞ (cid:88) k =1 q k k χ ( k ) N (cid:33) = P . E . (cid:104) q · χ (1) N (cid:105) (6.4)for all N , where P.E. is the Plethystic Exponential [31]. We expect that these quan-tities, term by term in q -expansion, should play a role similar to (6.2), now forOrientifolded quiver theories.We are not aware of a general answer to the second complication, yet. Trivialexamples, in this sense, are N = 4 , ∞ (cid:88) N =1 q N Ω G N N =4 ( y ) = P . E . (cid:20) q y − + y ) (cid:21) , (6.5)and 1 + ∞ (cid:88) N =1 q N Ω G N N =8 ( y , x ) = P . E . (cid:20) q y − + y ) · x / y − + x − / y x / + x − / (cid:21) , (6.6)common for G N = O (2 N ), O (2 N + 1), or Sp ( N ). But the analog of (6.1) for generalOrientifolded quiver theories, which may have nontrivial ground states, is yet anothermatter. Even for N = 16 theories computed in this note, we are yet to find a closedform of generating functions, inclusive of all ranks. We wish to come back to theproblem of finding generic Orientifold version of the rational invariants in near future. Acknowledgement We would like to thank Chiung Hwang and Joonho Kim, for discussions on their workinvolving other types of Orientifold planes, Amihay Hanany for bringing our attentionto his old work on Orientifold points, and Matthew Young for illuminating discussionson the quiver stability. SJL is grateful to Korea Institute for Advanced Study forhospitality. The work of SJL is supported in part by NSF grant PHY-1417316.26 Elliptic Weyl Elements and Rational Invariants An elliptic element w of Weyl group W is defined by absence of eigenvalue 1 in thecanonical representation of W on the weight lattice.For SU ( N ), the Weyl group S N 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 allof these belong to a single conjugacy class. For SO (2 N ), SO (2 N + 1), and Sp ( N )groups, the Weyl groups are S N semi-direct-product with ( Z ) N − , ( Z ) N , and ( Z ) N ,respectively. The elements can be therefore represented as follows σ = ( ab ˙ c ˙ d . . . )( klm ˙ n . . . ) · · · where dots above a number indicate a sign flip. For example (12 ˙3) represents theelement, − · . In this form, the above ( Z ) N − for SO (2 N ) means that the total number of sign fliphas 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 thisrequires 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, forsome low rank cases, from which the pattern should be quite obvious, • SU ( N ) (123 · · · N ) • SO (4) ( ˙1)( ˙2) • SO (5) and Sp (2) (1 ˙2) , ( ˙1)( ˙2) • SO (6) (1 ˙2)( ˙3)27 SO (7) and Sp (3) ( ˙1 ˙2 ˙3) , (12 ˙3) , (1 ˙2)( ˙3) , ( ˙1)( ˙2)( ˙3) • SO (8) ( ˙1 ˙2 ˙3)( ˙4) , (12 ˙3)( ˙4) , (1 ˙2)(3 ˙4) , ( ˙1)( ˙2)( ˙3)( ˙4) • SO (9) and Sp (4)(1 ˙2 ˙3 ˙4) , (123 ˙4) , ( ˙1 ˙2 ˙3)( ˙4) , (12 ˙3)( ˙4) , (1 ˙2)(3 ˙4) , (1 ˙2)( ˙3)( ˙4) , ( ˙1)( ˙2)( ˙3)( ˙4)We may classify the twisted elliptic Weyl elements, ˜ w , for O ( m )’s, similarly. Wetake this to be defined by absence of eigenvalue 1 in ˜ w · P where ˜ w is an element of W SO ( m ) . One immediate fact is that the underlying action of P is trivial on the rootlattice of SO (2 N + 1), so for SO (2 N + 1), the elliptic Weyl elements coincide withthe twisted elliptic Weyl elements. This is, in retrospect, another reason behind whyΩ O − (2 N +1) = Ω O + (2 N +1) and hence Ω O (2 N +1) = Ω SO (2 N +1) . For O (2 N ), 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 as follows, • O − (4) (1 ˙2) • O − (6) ( ˙1 ˙2 ˙3) , (12 ˙3) , ( ˙1)( ˙2)( ˙3) • O − (8) (1 ˙2 ˙3 ˙4) , (123 ˙4) , (1 ˙2)( ˙3)( ˙4) , • O − (10) ( ˙1 ˙2 ˙3 ˙4 ˙5) , (12 ˙3 ˙4 ˙5) , (1234 ˙5) , ( ˙1 ˙2 ˙3)( ˙4)( ˙5) , (12 ˙3)( ˙4)( ˙5) , (1 ˙2)(3 ˙4)( ˙5) , ( ˙1)( ˙2)( ˙3)( ˙4)( ˙5)Note that P is in fact nothing but the generator of W O (2 N ) /W SO (2 N ) = Z . Therefore,one can also think of ˜ w · P as elliptic Weyl elements of O (2 N ) which are not in W SO (2 N ) .28n particular, this means that W O (2 N ) = W O (2 N +1) = W Sp ( N ) and the the respectiveelliptic Weyl elements also coincide. A.1 Ω G N =16 with Simple and Connected G We list results for twisted partition functions with N = 16, from Ref. [5];Ω SO (4) N =16 = 1 + 2∆ SO (3) N =16 + ∆ SO (4) N =16 , (A.1)Ω SO (5) N =16 = 1 + 2∆ SO (3)= Sp (1) N =16 + ∆ SO (5)= Sp (2) N =16 = Ω Sp (2) N =16 , Ω G N =16 = 2 + 2∆ SU (2) N =16 + ∆ G N =16 , Ω SO (6) N =16 = 1 + ∆ SO (3) N =16 + ∆ SO (6) N =16 , Ω SO (7) N =16 = 1 + 3∆ SO (3) N =16 + (cid:16) ∆ SO (3) N =16 (cid:17) + ∆ SO (5) N =16 + ∆ SO (7) N =16 , Ω Sp (3) N =16 = 2 + 3∆ Sp (1) N =16 + (cid:16) ∆ Sp (1) N =16 (cid:17) + ∆ Sp (2) N =16 + ∆ Sp (3) N =16 , Ω SO (8) N =16 = 2 + 4∆ SO (3) N =16 + 2 (cid:16) ∆ SO (3) N =16 (cid:17) + (cid:16) ∆ SO (3) N =16 (cid:17) + 3∆ SO (5) N =16 + ∆ SO (8) N =16 , Ω SO (9) N =16 = 2 + 4∆ SO (3) N =16 + 2 (cid:16) ∆ SO (3) N =16 (cid:17) + 2∆ SO (5) N =16 + ∆ SO (3) N =16 · ∆ SO (5) N =16 + ∆ SO (7) N =16 + ∆ SO (9) N =16 , Ω Sp (4) N =16 = 2 + 5∆ Sp (1) N =16 + 2 (cid:16) ∆ Sp (1) N =16 (cid:17) + 2∆ Sp (2) N =16 + ∆ Sp (1) N =16 · ∆ Sp (2) N =16 + ∆ Sp (3) N =16 + ∆ Sp (4) N =16 , where ∆’s are defined in (3.10). As with SU ( N ) case in (3.12), these decompositionsare unique. A.2 Common Building Blocks for Orthogonal and SympleticGroups Since the Weyl groups of O (2 N ), O (2 N + 1), and Sp ( N ) coincide, the quantitiesdefined in (4.10), (4.13), and (4.17) are common to all three classes of the gaugegroups. These can be classified by the rank alone, without reference to the type oforientifolding projection, suggesting universal building blocks for continuum contri-29utions. Here we list a few low rank examples of Ξ ( N ) N =4 ( y ) of (4.10); • rank 1 12 1 y − + y (A.2) • rank 2 18 (cid:20) y − + y + 1( y − + y ) (cid:21) (A.3) • rank 3 148 (cid:20) y − + y + 6( y − + y )( y − + y ) + 1( y − + y ) (cid:21) (A.4) • rank 4 1384 (cid:20) y − + y + 32( y − + y )( y − + y )+ 12( y − + y ) + 12( y − + y )( y − + y ) + 1( y − + y ) (cid:21) (A.5) • rank 5 13840 (cid:20) y − + y + 240( y − + y )( y − + y ) + 160( y − + y )( y − + y )+ 80( y − + y )( y − + y ) + 60( y − + y ) ( y − + y )+ 20( y − + y )( y − + y ) + 1( y − + y ) (cid:21) (A.6)Elevating these to building blocks of N = 8 , 16 orientifolded theories is a matter ofattaching chiral field contributions to each linearly-independent rational pieces, as in(4.13) and in (4.17). Ω N =4 , and ∆ N =16 ’s are related simply to these as Ξ ( N ) N =4 , = Ω O (2 N ) N =4 , = Ω O (2 N +1) N =4 , = Ω SO (2 N +1) N =4 , = Ω Sp ( N ) N =4 , , (A.7)30nd Ξ ( N ) N =16 = ∆ O (2 N ) N =16 = ∆ O (2 N +1) N =16 = ∆ SO (2 N +1) N =16 = ∆ Sp ( N ) N =16 . (A.8) A.3 Ω G N =16 for D-Particles on an Orientifold Point Although there is a universal form (4.17) of continuum contributions to N = 16theories with an Orientifold point, the actual partition functions and the indicesdiffer among O (2 N ), O (2 N + 1), and Sp ( N ) groups. Here we list all three series, forcomparison, although O (2 N + 1) and Sp ( N ) cases were already shown in Section A.1in a different notation;Ω O (2) N =16 = 0 + Ξ (1) N =16 , (A.9)Ω O (4) N =16 = 1 + Ξ (1) N =16 + Ξ (2) N =16 , Ω O (6) N =16 = 1 + 2 Ξ (1) N =16 + (cid:16) Ξ (1) N =16 (cid:17) + Ξ (3) N =16 , Ω O (8) N =16 = 2 + 3 Ξ (1) N =16 + (cid:16) Ξ (1) N =16 (cid:17) + 2 Ξ (2) N =16 + Ξ (1) N =16 · Ξ (2) N =16 + Ξ (4) N =16 , Ω O (3) N =16 = 1 + Ξ (1) N =16 , (A.10)Ω O (5) N =16 = 1 + 2 Ξ (1) N =16 + Ξ (2) N =16 , Ω O (7) N =16 = 1 + 3 Ξ (1) N =16 + (cid:16) Ξ (1) N =16 (cid:17) + Ξ (2) N =16 + Ξ (3) N =16 , Ω O (9) N =16 = 2 + 4 Ξ (1) N =16 + 2 (cid:16) Ξ (1) N =16 (cid:17) + 2 Ξ (2) N =16 + Ξ (1) N =16 · Ξ (2) N =16 + Ξ (3) N =16 + Ξ (4) N =16 , Ω Sp (1) N =16 = 1 + Ξ (1) N =16 , (A.11)Ω Sp (2) N =16 = 1 + 2 Ξ (1) N =16 + Ξ (2) N =16 , Ω Sp (3) N =16 = 2 + 3 Ξ (1) N =16 + (cid:16) Ξ (1) N =16 (cid:17) + Ξ (2) N =16 + Ξ (3) N =16 , Sp (4) N =16 = 2 + 5 Ξ (1) N =16 + 2 (cid:16) Ξ (1) N =16 (cid:17) + 2 Ξ (2) N =16 + Ξ (1) N =16 · Ξ (2) N =16 + Ξ (3) N =16 + Ξ (4) N =16 . B Integrand for the O − (2 N ) The determinant g O − (2 N ) ( t ) that appears in the localization formula (3.1) for thetwisted partition function of the O − (2 N ) pure Yang-Mills theory can be obtained bymodifying the following O + (2 N ) counterpart, g O + (2 N ) ( t ) = (cid:18) y − y − (cid:19) N · (cid:89) a (cid:18) x − F a / y − ( R a / − − x F a / y R a / − x F a / y R a / − x − F a / y − R a / (cid:19) N × (cid:89) α t − α/ − t α/ t α/ y − − t − α/ y · (cid:89) a (cid:89) α t − α/ x − F a / y − ( R a / − − t α/ x F a / y R a / − t α/ x F a / y R a / − t − α/ x − F a / y − R a / = (cid:18) y − y − (cid:19) N · (cid:89) a (cid:18) y − ( R a / − − x F a y R a / − x F a y R a / − y − R a / (cid:19) N × (cid:89) α − t α t α y − − y · (cid:89) a (cid:89) α y − ( R a / − − t α x F a y R a / − t α x F a y R a / − y − R a / , (B.1)so that the parity action is appropriately taken into account. Here, α ’s are the rootsof SO (2 N ) and a ’s label the 0, 1, and 3 adjoint chiral multiplets for N = 4, 8, and16 theories, respectively. With the parity represented as in Eq. (4.3),diag N × N (1 , , . . . , , − , (B.2)the N -th zero mode is frozen to t N = 1 and some of the one-loop determinantsrelevant to the N -th Cartan U (1) undergo appropriate sign flips as described in theparagraph including Eq. (4.3). The determinant g O − (2 N ) ( t ) is then a function of the N − t = { t , . . . , t N − } , and can be written as g O − (2 N ) ( t ) = g O + (2 N − ( t ) · y + y − · (cid:89) a y − ( R a / − + x F a y R a / − x F a y R a / + y − R a / (B.3) × N − (cid:89) i =1 − t i t i y − − y t i t i y − + y · N − (cid:89) i =1 − t − i t − i y − − y t − i t − i y − + y (cid:89) a N − (cid:89) i =1 y − ( R a / − − t i x F a y R a / − t i x F a y R a / − y − R a / y − ( R a / − + t i x F a y R a / − t i x F a y R a / + y − R a / × (cid:89) a N − (cid:89) i =1 y − ( R a / − − t − i x F a y R a / − t − i x F a y R a / − y − R a / y − ( R a / − + t − i x F a y R a / − t − i x F a y R a / + y − R a / , where the expression for g O + (2 N − ( 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: g N =4 O − (4) ( t ) = 1 y − y − · y + y − × − t t y − − y · t t y − + y · − t − t − y − − y · t − t − y − + y ,g N =8 O − (4) ( t ) = 1 y − y − · y + y − · y − x y − x − · y + x y − x + 1 × − t t y − − y · t t y − + y · − t − t − y − − y · t − t − y − + y × y − t x y − t x − · y + t x y − t x + 1 · y − t − x y − t − x − · y + t − x y − t − x + 1 ,g N =16 O − (4) ( t ) = 1 y − y − · y + y − · − x x y − y − · x x y + y − × y − x − ˜ x y − x − ˜ x − · y + x − ˜ x y − x − ˜ x + 1 · y − x − ˜ x − y − x − ˜ x − − · y + x − ˜ x − y − x − ˜ x − + 1 × − t t y − − y · t t y − + y · − t − t − y − − y · t − t − y − + y × − t x t x y − y − · t x t x y + y − · − t − x t − x y − y − · t − x t − x y + y − × y − t x − ˜ x y − t x − ˜ x − · y + t x − ˜ x y − t x − ˜ x + 1 · y − t − x − ˜ x y − t − x − ˜ x − · y + t − x − ˜ x y − t − x − ˜ x + 1 × y − t x − ˜ x − y − t x − ˜ x − − · y + t x − ˜ x − y − t x − ˜ x − + 1 · y − t − x − ˜ x − y − t − x − ˜ x − − · y + t − x − ˜ x − y − t − x − ˜ x − + 1 , where R-charges and flavor charges have been assigned as R = 0 and F = 1 to the33djoint chiral multiplet of the N = 8 theory and as R = (2 , , F = (2 , − , − F = (0 , , − 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, just like they do in the O + theory, y − ( R a / − − x F a y R a / − x F a y R a / − y − R a / = − x F a / y R a / − − x − F a / y − ( R a / − x F a / y R a / − x − F a / y − R a / . (B.4)If true, the formula would have an incorrect overall sign for N = 8 and 16 casesas there exist one and three such massive Cartan factors, respectively. However, wepropose that they do not come with an expected minus sign and Eq. (B.3) is correctas it is. For a consistency check, let us consider N = 4 O − (2 N ) theory with anadjoint chiral multiplet, to which R = 1 and F = 0 are assigned. Since this theoryadmits a mass term for the chiral field, it should flow to pure N = 4 O − (2 N ) theoryand hence, the twisted partition functions of the two theories must agree, with thesame overall sign. We have indeed confirmed this for N = 2 and 3 based on theone-loop determinants (B.3). References [1] E. Witten, “String theory dynamics in various dimensions,” Nucl. Phys. B (1995) 85 [hep-th/9503124].[2] P. Yi, “Witten index and threshold bound states of D-branes,” Nucl. Phys. B (1997) 307 [hep-th/9704098].[3] S. Sethi and M. Stern, “D-brane bound states redux,” Commun. Math. Phys. (1998) 675 [hep-th/9705046].[4] K. Hori, H. Kim and P. Yi, “Witten Index and Wall Crossing,” JHEP (2015) 124 [arXiv:1407.2567 [hep-th]].[5] S. J. Lee and P. Yi, “Witten Index for Noncompact Dynamics,” JHEP (2016) 089 [arXiv:1602.03530 [hep-th]]. Similar argument applies to all the flipped factors in the other lines of Eq. (B.3), although thetotal number of such factors is always even so that they may never affect the final result. (2000) 77 [hep-th/9803265].[7] M. Staudacher, “Bulk Witten indices and the number of normalizable groundstates in supersymmetric quantum mechanics of orthogonal, symplectic and ex-ceptional groups,” Phys. Lett. B (2000) 194 [hep-th/0006234].[8] V. Pestun, “N=4 SYM matrix integrals for almost all simple gauge groups (ex-cept E(7) and E(8)),” JHEP (2002) 012 [hep-th/0206069].[9] E. Witten, “Constraints on Supersymmetry Breaking,” Nucl. Phys. B (1982)253.[10] M. B. Green and M. Gutperle, “D Particle bound states and the D instantonmeasure,” JHEP (1998) 005 [hep-th/9711107].[11] M. Stern and P. Yi, “Counting Yang-Mills dyons with index theorems,” Phys.Rev. D (2000) 125006 [hep-th/0005275].[12] M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral Asymmetry and Rie-mannian Geometry I” Math. Proc. Cambridge Philosophical Society 77 Issue 01(1975) 43-69; “Spectral Asymmetry and Riemannian Geometry II” Math. Proc.Cambridge Philosophical Society 78 Issue 03 (1975) 405-432; “Spectral Asym-metry and Riemannian Geometry III” Math. Proc. Cambridge PhilosophicalSociety 79 Issue 01 (1976) 71-99.[13] F. Benini, R. Eager, K. Hori and Y. Tachikawa, “Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups,” Lett. Math. Phys. (2014) 465 [arXiv:1305.0533 [hep-th]].[14] F. Benini, R. Eager, K. Hori and Y. Tachikawa, “Elliptic Genera of 2d N = 2Gauge Theories,” Commun. Math. Phys. (2015) 3, 1241 [arXiv:1308.4896[hep-th]].[15] L. C. Jeffrey and F. C. Kirwan, “Localization for nonabelian group actions,”Topology (1995) 291-327, arXiv:alg-geom/9307001.[16] V. G. Kac and A. V. Smilga, “Normalized vacuum states in N=4 supersymmetricYang-Mills quantum mechanics with any gauge group,” Nucl. Phys. B (2000)515 [hep-th/9908096]. 3517] H. Kim, J. Park, Z. Wang and P. Yi, “Ab Initio Wall-Crossing,” JHEP (2011) 079 [arXiv:1107.0723 [hep-th]].[18] S. Kachru and E. Silverstein, “On gauge bosons in the matrix model approachto M theory,” Phys. Lett. B (1997) 70 [hep-th/9612162].[19] C. Hwang, J. Kim, S. Kim and J. Park, “General instanton counting and 5dSCFT,” arXiv:1406.6793 [hep-th].[20] Y. Hwang, J. Kim and S. Kim, “M5-branes, orientifolds, and S-duality,”arXiv:1607.08557 [hep-th].[21] A. Hanany, B. Kol and A. Rajaraman, “Orientifold points in M theory,” JHEP (1999) 027 [hep-th/9909028].[22] M. Porrati and A. Rozenberg, “Bound states at threshold in supersymmetricquantum mechanics,” Nucl. Phys. B (1998) 184 [hep-th/9708119].[23] K. Dasgupta and S. Mukhi, “Orbifolds of M theory,” Nucl. Phys. B (1996)399 [hep-th/9512196].[24] P. Horava and E. Witten, “Heterotic and type I string dynamics from eleven-dimensions,” Nucl. Phys. B (1996) 506 [hep-th/9510209].[25] K. A. Intriligator, “Anomaly matching and a Hopf-Wess-Zumino term in 6d,N=(2,0) field theories,” Nucl. Phys. B (2000) 257 [hep-th/0001205].[26] P. Yi, “Anomaly of (2,0) theories,” Phys. Rev. D (2001) 106006 [hep-th/0106165].[27] K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, “Anomaly polynomialof general 6d SCFTs,” PTEP (2014) no.10, 103B07 [arXiv:1408.5572 [hep-th]].[28] M. Abramowitz and I. A. Stegun, ed., “Handbook of Mathematical Functionswith Formula, Graphs, and Mathematical Table” (1964), National Bureau ofStandards, United States Department of Commerce.[29] J. Manschot, B. Pioline and A. Sen, “Wall Crossing from Boltzmann Black HoleHalos,” JHEP (2011) 059 [arXiv:1011.1258 [hep-th]].3630] M. Kontsevich and Y. Soibelman, “Stability structures, motivic Donaldson-Thomas invariants and cluster transformations,” arXiv:0811.2435 [math.AG].[31] B. Feng, A. Hanany and Y. H. He, “Counting gauge invariants: The Plethysticprogram,” JHEP0703