Elliptic genera of pure gauge theories in two dimensions with semisimple non-simply-connected gauge groups
aa r X i v : . [ h e p - t h ] S e p KIAS-P20047
Elliptic genera of pure gauge theories in two dimensionswith semisimple non-simply-connected gauge groups
Richard Eager , Eric Sharpe School of PhysicsKorea Institute for Advanced StudySeoul 02455, Korea Dep’t of PhysicsVirginia Tech850 West Campus Dr.Blacksburg, VA 24061 [email protected] , [email protected] In this paper we describe a systematic method to compute elliptic genera of (2,2) su-persymmetric gauge theories in two dimensions with gauge group G/ Γ (for G semisimpleand simply-connected, Γ a subgroup of the center of G ) with various discrete theta angles.We apply the technique to examples of pure gauge theories with low-rank gauge groups.Our results are consistent with expectations from decomposition of two-dimensional theorieswith finite global one-form symmetries and with computations of supersymmetry breakingfor some discrete theta angles in pure gauge theories. Finally, we make predictions for theelliptic genera of all the other remaining pure gauge theories by applying decomposition andmatching to known supersymmetry breaking patterns.September 2020 1 ontents G . . . . . . . . . . . . . . . . . 52.2 Non-simply-connected G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 SU (2) / Z = SO (3) gauge theories 135 Pure SU (3) / Z gauge theories 176 Pure SO (4) gauge theories 197 Pure Spin (4) / ( Z × Z ) gauge theories 218 Pure SO (5) gauge theories 229 Pure Sp (6) / Z gauge theories 2710 Predictions for general cases 3211 Conclusions 3412 Acknowledgements 35References 36 Introduction
The low energy infrared (IR) limits of gauge theories have been of interest for many years.Pure gauge theories in two dimensions with N = (2 ,
2) supersymmetry have long beenbelieved to be gapless, as a result of the chiral R-symmetry and anomalous two-point func-tions [1, section 3]. The paper [2] made a more refined conjecture: that the IR limit of a(2,2) supersymmetric pure G gauge theory, G semisimple and simply-connected, should bea theory of free twisted chiral multiplets, as many as the rank of G , with R-charges propor-tional to Casimir degrees. Using nonabelian mirrors [3] it was checked in [3–5] that the IRtheory contains as many twisted chirals as the rank, and in pure G/ Γ gauge theories for Γa subgroup of the center of G , that one gets an identical free theory for one value of thediscrete theta angle, and supersymmetry breaking in the IR for other values of the discretetheta angle.All that said, the work [3–5] did not compute elliptic genera, which would provide a veryexplicit concrete check of R-charges of free IR twisted chirals. For a pure (2,2) supersym-metric G gauge theory for G simply-connected, methods to compute elliptic genera exist (seee.g. [6–9]), and it is being checked [10, 11], that those elliptic genera match the expectationsof [2].The purpose of this paper is to develop the technology to compute elliptic genera of pure(2,2) supersymmetric G/ Γ gauge theories for various discrete theta angles. The elliptic genusis given by a sum of Jeffrey–Kirwan residues of a meromorphic form over the moduli spaceof flat G/ Γ-connections the torus using supersymmetric localization [6, 7]. The meromorphicform is obtained by evaluating the one-loop determinants corresponding to G/ Γ-bundleswith non-trivial characteristic classes. We combine the results from different components ofthe moduli space, weighted by phases from the discrete theta angle, to determine the ellipticgenus.In section 2 we review known results for elliptic genera of pure supersymmetric gauge the-ories in two dimensions. In section 3 we describe the procedure we will use to compute ellip-tic genera of pure supersymmetric gauge theories with semisimple but non-simply-connectedgauge groups. The remainder of this paper is spent working out low-rank examples. Webegin in section 4 by discussing pure SO (3) gauge theories. For these, the elliptic genera inquestion were previously derived in [12, appendix A], but this case acts as a test and demon-stration of our strategy. In section 5, we compute elliptic genera of pure SU (3) / Z gaugetheories; in section 6, pure SO (4) gauge theories; in section 7, pure Spin(4) / ( Z × Z ) gaugetheories; in section 8, pure SO (5) gauge theories; and, in section 9, pure Sp (6) / Z gaugetheories. In each case, the elliptic genus vanishes (and supersymmetry is broken) unless thediscrete theta angle takes the value described in [3–5]. We conclude by making predictionsfor elliptic genera of all other pure gauge theories with semisimple non-simply-connectedgauge groups, in section 10. 3e will also note in each case that the results are consistent with decomposition [13–15].(See also e.g. [16, 17] for four-dimensional versions and related analyses.) Briefly, decompo-sition is the statement that a two-dimensional theory with a finite global 1-form symmetry(such as a two-dimensional gauge theory in which a finite center acts trivially) decomposes into a disjoint union of theories which individually do not have a 1-form symmetry. In thecase of a pure G gauge theory for G simply-connected, with Γ a finite subgroup of the cen-ter, the G gauge theory has a global one-form Γ symmetry (sometimes denoted B Γ), and sodecomposes into a disjoint union of G/ Γ gauge theories with various discrete theta angles,which we write schematically as G = ⊕ θ ∈ ˆΓ ( G/ Γ) θ . (1.1)In particular, the elliptic genus of a pure G gauge theory should be the sum of elliptic generaof pure G/ Γ gauge theories with various discrete theta angles. The result that the ellipticgenera of G/ Γ gauge theories vanish except for a single discrete theta angle, for which theelliptic genus matches that of the G gauge theory, is consistent with the decomposition above.Also, although we will not emphasize this perspective in this paper, in principle thesecomputations have a mathematical understanding. Elliptic genera of pure G gauge theoriesshould, in principle, match [18–20] elliptic genera of classifying stacks BG , the G -equivariantelliptic genera of points [21,22], and so we are also making predictions for those elliptic genera. Pure N = (2 ,
2) supersymmetric G gauge theory can be described in terms of vector multipletconsisting of a gauge field A µ , gauginos λ and ¯ λ , scalars σ, ¯ σ, and a real auxiliary scalar D. The gauge field strength is a twisted chiral superfield Σ with lowest component σ. TheEuclidean Yang-Mills Lagrangian is L YM = Tr (cid:16) F + D + D µ ¯ σD µ σ + iD [ σ, ¯ σ ] − i ¯ λγ µ D µ λ − i ¯ λP + [ σ, λ ] − i ¯ λP − [¯ σ, λ ] (cid:17) , (2.1)where P ± = 1 ± γ . (2.2)The classical potential is proportional to Tr (cid:2) σ, σ † (cid:3) . The classical vacua occur at the mini-mum of the potential and satisfy (cid:2) σ, σ † (cid:3) = 0 . Equivalently, the classical Coulomb branch ofvacua can be described by the vacuum expectation values of the gauge invariant polynomialsin σ . It is a classical result that this ring of functions is freely generated by rank( G ) genera-tors. However, the potential receives quantum corrections, so the IR behavior is potentiallymore complex. This is a stronger statement than just superselection. For example, only in infinite volume does one geta selection rule from superselection sectors, whereas decomposition holds at finite volume. This distinctionis discussed in greater detail in [16]. .1 Prediction for simply-connected semisimple G The paper [2] proposed that for G semisimple and simply-connected, the IR theory should bea free theory of twisted chiral multiplets, Y i (Σ), i = 1 , . . . , rank( G ) , built out of the generatorsof the invariant functions on Σ, with axial R-charges r i given by twice the Casimir degrees d i of G computed from and in one-to-one correspondence with the possible Casimirs (of whichthere are as many as the rank). The contribution of a single twisted chiral multiplet Y (Σ)with axial R-charge r to the elliptic genus is [7, equ’n (2.11)]Tr RR ( − F q H L q H R y J = θ ( τ | (1 − r/ z ) θ ( τ | − ( r/ z ) , (2.3)where q = exp(2 πiτ ), y = exp(2 πiz ), J is the left-moving U (1) R charge, and the genus iscomputed for periodic left-moving fermions. Since the low energy theory is a theory of freetwisted chiral multiplets, the elliptic genus is expected to be Y i θ ( τ | (1 − r i / z ) θ ( τ | − ( r i / z ) . (2.4)For simply-connected G , this will be demonstrated by explicit computation in [11].For later use, we collect in table 1 the degrees of Casimirs for simple Lie algebras, each ofwhich is half the R-charge of a corresponding twisted chiral in equation (2.4). For example,the elliptic genus of a pure G gauge theory is predicted to be θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) . (2.5)As a consistency check, the Casimir degrees d i and the dimension of the group G are relatedby dim G = X i (2 d i − . (2.6)In passing, identifying R-charges r i = 2 d i , we can apply the central charge formula [23][equ’n (15)] to see that c X i (1 − r i ) = − dim G. (2.7)It is rather unusual to have a negative central charge, but we can get the same result fromthe modular transformation properties. Applying [7, equ’n (2.7)] Z (cid:18) − τ , zτ (cid:19) = exp (cid:20) c πiτ z (cid:21) Z ( τ, z ) (2.8) This follows from the Harish-Chandra isomorphism that relates Casimirs to symmetric invariants. In conventions in which the superpotential obeys W ( λ r i x i ) = λ W ( x i ). θ (cid:18) − τ (cid:12)(cid:12)(cid:12)(cid:12) zτ (cid:19) = − i √− iτ exp( πiz /τ ) θ ( τ | z ) , (2.9)we see that under τ
7→ − /τ , z z/τ , the elliptic genus of a twisted chiral with R-charge r (equ’n (2.3)) picks up a phase exp (cid:0) πi (1 − r ) z /τ (cid:1) , (2.10)and the elliptic genus of a pure G gauge theory (2.4) picks up a phaseexp πi X i (1 − r i ) z /τ ! = exp (cid:0) − πi (dim G ) z /τ (cid:1) , (2.11)hence we see again that the (left-moving) central charge is given by c − dim G. (2.12)Intuitively, for theories formulated on S , the sign of the central charge above is surely relatedto the fact that for R charge greater than two, the action has a curvature-dependent termof the wrong sign [24, section 3.4].Mathematically, this has a simple understanding. A pure G -gauge theory is a sigma modelon [18–20] the stack BG = [point /G ], and this stack has dimension (see e.g. [25, section7], [26, example 2.44]) dim [point /G ] = − dim G, (2.13)matching c/ G In this paper, we will compute elliptic genera of pure supersymmetric gauge theories withgauge groups G/ Γ, where G is simply-connected and Γ is a subgroup of the center of G .Now, a principal G/ Γ bundle on worldsheet T admits a characteristic class we shall denote w ∈ H ( T , Γ) ∼ = Γ. (For example, for SO ( k ) bundles, w is the Stiefel-Whitney class w .)Such theories admit analogues of theta angles, known as discrete theta angles, in which thepath integral is weighted by phases of the form exp( iθ · w ) for θ a (log of a) character of Γ,the set of which we shall denote ˆΓ.The papers [3–5] have looked at IR behavior of two-dimensional pure (2,2) supersym-metric gauge theories with non-simply-connected gauge groups G/ Γ. (See also [12, 27, 28] forcomputations of elliptic genera in some examples related to Hori’s dualities [29].) Briefly,these papers found 6auge group Dimension Casimir degrees SU ( n + 1)( A n ) ( n + 1) − , , , · · · , n + 1Spin(2 n + 1)( B n ) n (2 n + 1) 2 , , , · · · , nSp (2 n )( C n ) n (2 n + 1) 2 , , , · · · , n Spin(2 n )( D n ) n (2 n − n ; 2 , , , · · · , n − G
14 2 , F
52 2 , , , E
78 2 , , , , , E
133 2 , , , , , , E
248 2 , , , , , , , SU ( k ) / Z k − (1 / k ( k −
1) mod k Spin(2 k + 1) / Z k ) / Z × Z k (2 k −
1) mod 2 , k + 2) / Z k (2 k −
1) mod 4 Sp (2 k ) / Z (1 / k ( k + 1) mod 2 E / Z E / Z • If the gauge group is not simply-connected, then for precisely one value of the discretetheta angle, the IR limit is a theory of free twisted chirals, as many as the rank (and asmany as IR limit of a pure gauge theory with corresponding simply-connected gaugegroup). For other values of the discrete theta angle, there are no supersymmetricvacua, hence supersymmetry is broken in the IR. • For the one nontrivial case, the IR theory is a theory of as many twisted chiral multi-plets as the rank, matching the IR behavior of a pure G gauge theory.This structure is consistent with the predictions of decomposition [13–15] for two-dimensionaltheories with one-form symmetries, as discussed in [3–5].In this paper, we will compute elliptic genera to check these claims for more generaltheories. 7o understand some of the quantum subtleties that will arise when studying pure G/ Γgauge theories, let us briefly review such theories more concretely. The Lagrangian for sucha theory can be written in (2,2) superspace in the form − g Z d θ Tr ΣΣ + (cid:18) − r + i θ π (cid:19) Z dθ + dθ − TrΣ | θ − = θ + =0 + c.c., (2.14)where Σ is a twisted chiral superfield encoding the gauge field strength, r is a Fayet-Iliopoulosparameter, and θ the theta angle. In analyzing the low-energy behavior of such theoriesone often works on the Coulomb branch, along which there is a twisted one-loop effectivesuperpotential which for a pure G/ Γ gauge theory with G simply-connected and Γ a subgroupof the center, takes the form W eff = − X a Σ a " − r a + i θ a π + 1 | Γ | X ˜ µ α a ˜ µ ln X b Σ b α b ˜ µ ! − ! , (2.15)where now r a and θ a are the FI parameters and theta angles for each of the unbroken U (1)’son the Coulomb branch. (No further corrections exist beyond one-loop order.) The firsttwo terms are the ( − r + iθ/ π )TrΣ of the classical action along the Coulomb branch, andthe last is a loop correction, of the same form commonly seen in theories with matter, hereultimately due to W bosons. The α a ˜ µ are the root vectors of the nonzero roots (indexed by˜ µ ) of the Lie algebra of the gauge group. The second term can be simplified, and written as(see e.g. [5, section 2.1])1 | Γ | X ˜ µ α a ˜ µ ln X b Σ b α b ˜ µ ! − ! = X ˜ µ pos ′ iπ | Γ | α a ˜ µ , (2.16)giving what amounts to a gauge-group-dependent shift of the theta angle. (This was firstobserved in [24, equ’n (10.9)].) These additional phases will play an important role in ourcomputations of elliptic genera of pure G/ Γ gauge theories.
The elliptic genus of a pure G/ Γ-gauge theory reduces to a residue integral over the modulispace M of flat G/ Γ-connections on T [6, 7]. Principal G/ Γ bundles have a degree-twocharacteristic class, valued in Γ, which we shall denote w ∈ H ( T , Γ) ∼ = Γ, so the modulispace of flat G/ Γ connections is a disjoint union of moduli spaces M = G w ∈ H ( T , Γ) M G/ Γ ,w . (3.1) See e.g. [31, section 4.1].
8n the sector of bundles with w = 0, any G/ Γ bundle lifts to a G bundle. Essentially as aresult, the elliptic genus of a pure G gauge theory matches that of a pure G/ Γ gauge theoryin the sector w = 0, up to a volume factor 1 / | Γ × Γ | and a Jacobian factor | Γ | : Z ( G/ Γ , w = 0) = | Γ || Γ × Γ | Z ( G ) = 1 | Γ | Z ( G ) . (3.2)Now, we turn to a G/ Γ gauge theory in a sector in which w = 0. Computations in thesesectors will occupy most of the effort in this paper. To describe such bundles, we pick twoholonomies p , q around cycles of the torus, which commute up to an element w ∈ Γ: pq = wqp. (3.3)The matrices p and q are the holonomies of any bundle about two cycles of the torus, liftedfrom G/ Γ to G . Put another way, these almost-commuting holonomies are the result oflifting commuties holonomies in G/ Γ to pairs in G . Next, we simultaneously diagonalize theadjoint action of p and q on the generators of the Lie algebra in the adjoint representation,writing pT α p − = ω αp T α , (3.4) qT α q − = ω αq T α , (3.5)where ω αp,q are phases, which enter into the elliptic genus computation. These phases alsoappeared in the calculation of the four-dimensional Witten index [32, 33] . Note that such adiagonalization is not possible for every possible representation in which the T α may appear;in particular, for the diagonalization above to be possible, one needs for the representationto be acted upon nontrivially by the center detected by p and q . Additionally the phasesfor the adjoint representation are sufficient to determine the phases for all representationswhen the center of G/ Γ is trivial since the adjoint is a tensor generator of the representationcategory [34].If the phases ω p,q are different from one, then, those ‘directions’ in the group are fixed. Ifthey are equal to one, on the other hand, then the group is unconstrained in those directions,and so one must integrate over corresponding Wilson lines, over the corresponding modulispace of flat connections, to get the elliptic genus.To the latter end, it can be shown that [35, 36] M G/ Γ ,w = M ˜ G ( w ) , (3.6) This arises from the different normalization of the root systems. A potentially useful reference is [37], describing representations for which such a diagonalization ispossible. For a representation in which such a diagonalization is not possible, consider the case G = SU (2),Γ = Z , with p and q in the of SU (2). It is easy to check that the resulting 3 × p and q . / Γ w ˜ G ( w ) A n − ∼ SU ( n ) / Z n d SU ( m ), m = gcd ( n, d ) B n ∼ Spin(2 n + 1) / Z Sp (2 n − n − C n ∼ Sp (4 n ) / Z Sp (2 n ), Spin(2 n + 1) C n +1 ∼ Sp (4 n + 2) / Z Sp (2 n ), Spin(2 n + 1) D n +1 ∼ Spin(4 n + 2) / Z Sp (2 n − n − Sp (4 n − n − Sp (2 n − n − D n ∼ Spin(4 n ) / Z × Z (1 , Sp (2 n ), Spin(2 n + 1)(0 , Sp (4 n − n − , Sp (2 n ), Spin(2 n + 1) E / Z G G E / Z F Table 3: List of groups ˜ G ( w ) whose moduli space of flat connections matches that of amoduli space of flat G/ Γ connections with nontrivial characteristic class w ∈ H ( T , Γ). Ineach case, we assume Γ is all of the center of simply-connected G , and not a subgroup. In D n , the (0 ,
1) indicates the Z whose quotient of Spin(4 n ) is SO (4 n ). Note that becausethe ranks and Weyl groups match, the moduli space of flat Spin(2 k + 1) connections matchesthat of flat Sp (2 k ) connections. This table summarizes results in [38, section 5.4], [39, table6], and [40, appendix A].for some other group ˜ G ( w ) that depends upon G/ Γ and w , where M denotes the modulispace of flat connections. Such groups ˜ G ( w ) are listed in [38, section 5.4], [39, table 6],and [40, appendix A], and we summarize their results in table 3. Roughly speaking, we canthink of the groups ˜ G ( w ) as being obtained by folding the affine Dynkin diagram accordingto the action of w ∈ Γ.To describe the moduli spaces M G/ Γ ,w =0 more concretely we recall some notions from thetheory of compact Lie groups. Let T a maximal torus of G/ Γ with corresponding Cartansubalgebra h . Let Q be the root lattice, P be the weight lattice, and Λ char be the characterlattice of G/ Γ. Similarly, let Q ∨ be the coroot lattice, P ∨ be the coweight lattice, and Λ ∨ char be the co-character lattice. Then the Cartan torus of G/ Γ can be identified with h / π Λ ∨ char .The center of and fundamental groups of G/ Γ are Z ( G/ Γ) ∼ = P ∨ / Λ ∨ char ∼ = Λ char /Q, (3.7) π ( G/ Γ) ∼ = Λ ∨ char /Q ∨ ∼ = P/ Λ char . (3.8) In addition, the paper [41] relates the moduli spaces ˜ G ( w ) to ˜ G ( w = 0) by Galois coverings. Not to be confused with the elliptic curve T . M = h C / (Λ ∨ char + τ Λ ∨ char ) , (3.9)then the moduli space of flat G/ Γ-connections on T with w = 0 is M G/ Γ ,w =0 = M /W, (3.10)where W is the Weyl group of G/ Γ.For G simply-connected the cocharacter lattice is equal to the coroot lattice. In theopposite extreme of G/ Γ with trivial center, the cocharacter lattice is equal to the coweightlattice. The relations between the cocharacter lattices mean that the moduli space M G, isan order | Γ × Γ | cover of M G/ Γ ,w =0 . The elliptic genus of a pure G/ Γ theory (with bundles of vanishing characteristic class)is given by [7] Z T ( τ, z, w = 0) = 1 | W | X u ∗ ∈ M ∗ sing JK-Res u = u ∗ (cid:0) Q ( u ∗ ) , η (cid:1) Z ( τ, z, u ) (3.11)where | W | is the order of the Weyl-group of G. The Jeffrey–Kirwan residue operation [42]JK-Res u = u ∗ (cid:0) Q ( u ∗ ) , η (cid:1) assigns a residue to each pole of Z in M ∗ sing depending on thecharge vectors Q ( u ∗ ) responsible for the pole and a covector η. The parameter q = e πiτ in Z specifies the complex structure of the torus T and y = e πiz is the fugacity for theleft-moving U (1) R-symmetry. The coordinates u a on the moduli space M can equivalentlybe described by the coordinates x a = e πiu a . The contribution of a vector multiplet V withgauge group G/ Γ to Z for the w = 0 characteristic class is Z V,G/ Γ ( τ, z, u ) = (cid:18) πη ( q ) θ ( q, y − ) (cid:19) rank G Y α ∈ G θ ( q, x α ) θ ( q, y − x α ) rank G Y a =1 d u a . (3.12)The product is over the roots α of the gauge group and η ( q ) is the Dedekind eta function.For bundles with non-trivial characteristic classes w , the contribution to Z is modi-fied. Using the eigenvalues ω αp,q , one can then construct an elliptic genus for bundles of fixedcharacteristic class w as a product of ratios θ ( τ | v α ) θ ( τ | − z + v α ) , (3.13)for nonzero v α , where v α = ln ω αp πi + τ ln ω αq πi , (3.14) We omit the flavor holonomies ξ since they are absent in pure theories. (cid:18) πη ( q ) θ ( q, y − ) (cid:19) rank ˜ G ( w ) Y α ∈ G θ ( τ | v α ) θ ( τ | − z + v α ) rank ˜ G ( w ) Y a =1 d u a . (3.15)for every vanishing v . The resulting residue integral is computed as a Jeffrey-Kirwan residueover (a cover of) the moduli space of those flat connections preserving the holonomy.This determines the elliptic genus (for fixed bundle characteristic class w ) up to anoverall normalization factor, which reflects residual gauge transformations that preserve theholonomies. For theories of the form SU ( n ) / Z n , that normalization factor is computed ine.g. [43, section 2.2.1].So far we have described how one computes contributions to the elliptic genus frombundles with different characteristic classes w ∈ H ( T , Γ). Finally, we will combine them, toform the elliptic genus as a function of the discrete theta angle. These different contributionsare each weighted with potentially two different phases. First, there is a factor exp( iθ · w ),where θ ∈ ˆΓ is a choice of discrete theta angle. Second, as studied in detail in [5] andreviewed in section 2.2, there is a factor of the form exp( iw · t ), where [5, equ’n (2.7)] t a = − πi | Γ | X ˜ µ pos ′ α a ˜ µ , (3.16)and w is encoded in w a so that t · w = X a t a w a . (3.17)Strictly speaking, the t a are not uniquely defined, as there are e.g. branch cut ambiguities,but the phase factor above is well-defined, as discussed in detail in [5]. Put another way, the t a encode a constant shift, due to quantum corrections, to the discrete theta angle θ .Thus, if we label the contribution to the elliptic genus of a pure G/ Γ gauge theory ina sector with bundles of characteristic class w by Z ( G/ Γ , w ), then the elliptic genus for ageneral characteristic class has the form Z ( G/ Γ , θ ) = X w exp( iw · θ ) exp ( iw · t ) Z ( G/ Γ , w ) . (3.18)In the next several sections we will carry out this program for several low-rank exam-ples. Specifically, we will apply the procedure above to derive elliptic genera for SU (2) / Z , SU (3) / Z , SO (4), Spin(4) / ( Z × Z ), SO (5), and Sp (6) / Z gauge theories with various dis-crete theta angles. The special case of SU (2) / Z was previously discussed in [12, appendixA]; we recover their results through this systematic method. In each case, we will find thatthe elliptic genus vanishes unless the discrete theta angle takes the value listed in table 3, asexpected [3–5]. We will also see that the results are consistent with decomposition [13–15].12urthermore, in each case we discuss, we will also find that the contribution to the ellipticgenus from bundles with characteristic class w = 0 matches (up to a phase) the contributionfrom bundles of characteristic class w = 0. This is reminiscent of the fact that ellipticgenera are independent of deformations, and so one is naturally led to wonder if there isa more elegant approach to these computations that demonstrates that contributions tothe elliptic genus are (modulo an overall phase) independent of w . For example, for sigmamodels on Calabi-Yau manifolds, the scale r of the Calabi-Yau is a marginal parameter,so as the elliptic genus is an index, it is independent of r , and the resulting elliptic generaare necessarily independent of worldsheet instanton corrections [44, 45]. In two-dimensionalgauge theories, on the other hand, the gauge coupling is irrelevant , so this argument doesnot apply. In any event, we leave this question for future work. SU (2) / Z = SO (3) gauge theories The elliptic genus of pure SU (2) gauge theory is [6]12 X u ∗ ∈ M +sing iη ( q ) θ ( τ | − z ) I u ∗ d u θ ( τ | u ) θ ( τ | − z + 2 u ) θ ( τ | − u ) θ ( τ | − z − u ) , (4.1)where the contributing poles are located at M +sing = n z , z + 12 , z + τ , z + τ + 12 o . (4.2)Elliptic genera of pure SO (3) gauge theories were computed in [12, appendix A]. Briefly, theauthors argued that the pure SU (2) and the SO (3) − theories have the same elliptic genus,given by θ ( τ | − z ) θ ( τ | − z ) = 12 θ ( τ | + 1 / θ ( τ | − z + 1 / θ ( τ | + τ / θ ( τ | − z + τ / θ ( τ | − (1 + τ ) / θ ( τ | − z − (1 + τ ) / , (4.3)while the elliptic genus of the pure SO (3) + theory vanishes identically. This is consistentwith the results of [5], which argued that in pure SO (3) gauge theories, only for the nontrivialdiscrete theta angle are there supersymmetric vacua, and supersymmetry is broken in theIR in SO (3) + . It is also consistent with decomposition [13–15], which in this case can beschematically expressed as SU (2) = SO (3) + + SO (3) − . (4.4) We should be careful as terms such as ‘marginal’ and ‘irrelevant’ are not well-defined away from fixedpoints of renormalization group flow, but we are not aware of examples of two-dimensional (2,2) supersym-metric gauge theories in which the gauge coupling flows in the IR to a marginal operator.
13n more detail, [12, appendix A] combined the contributions of the two distinct typesof SO (3) bundles. The contribution to the SO (3) elliptic genus from bundles of vanishingcharacteristic class is obtained from12 X u ∗ ∈ M +sing iη ( q ) θ ( τ | − z ) I u ∗ d u θ ( τ | u ) θ ( τ | − z + u ) θ ( τ | − u ) θ ( τ | − z − u ) , (4.5)with a single contributing pole located at M +sing = z/ . This results in Z ( SO (3) ) = 12 θ ( τ | − z ) θ ( τ | − z ) , (4.6)which is the SU (2) elliptic genus up to a factor of 1 / | Γ | = 1 /
2. As explained in section 3,this factor arises from the differing character lattices of the SU (2) and SO (3) groups. Notethat all four poles in equation 4.2 contribute equally to the SU (2) elliptic genus, but thereis only one pole for the SO (3) elliptic genus. Since there are only 1 / | Γ × Γ | = 1 / | Γ | relative to the SU (2) poles, we arriveat the previously claimed factor of | Γ | / | Γ × Γ | = 1 / | Γ | = 1 / . The contribution from bundles of nonzero characteristic class is Z ( SO (3) ) = − θ ( τ | − z ) θ ( τ | − z ) . (4.7)For a discrete theta angle θ ∈ { , π } , the possible SO (3) elliptic genera are Z ( SO (3)) = Z ( SO (3) ) + exp( iθ ) Z ( SO (3) ) , (4.8)= 12 θ ( τ | − z ) θ ( τ | − z ) (1 − exp( iθ )) . (4.9)When θ = 0, this vanishes, and when θ = π , this is nonzero and matches Z ( SU (2)).For later use, the elliptic genus, given up to numerical factors we will describe later, is θ ( τ | + 1 / θ ( τ | − z + 1 / θ ( τ | + τ / θ ( τ | − z + τ / θ ( τ | − (1 + τ ) / θ ( τ | − z − (1 + τ ) /
2) (4.10)can be derived directly from thinking about the contribution of Z − loop in the sector with w = 0, in the notation of [6]. Briefly, for w = 0, the moduli space of flat connections isa point, so that one does not integrate over a space of u ’s. Instead, the u ’s are fixed, withholonomies about the T which can be taken to bediag( − , − , +1) , diag(+1 , − , − . (4.11)An SO (3) bundle with these holonomies cannot be lifted to an SU (2) bundle. A heuristicway to see this is to observe that the lifts of the holonomies to SU (2) are given in equation14.19 and they anticommute. We can also see this more formally by computing the secondStiefel-Whitney class w , which gives the obstruction to lifting, in this case, an SO (3) bundleto an SU (2) bundle. With the holonomies above, we can describe this bundle as L ⊕ L ⊕ L , (4.12)where L and L each have nontrivial monodromy about a single S on T , and L = L ⊗ L .Thus, for example, w ( L ) = 1 + J , w ( L ) = 1 + J , w ( L ) = 1 + J + J , (4.13)where J , J generate H ( T , Z ) = ( Z ) , and in this case give w of L , L , respectively.Thus, w ( L ⊕ L ⊕ L ) = w ( L ) w ( L ) w ( L ) = 1 + J J + · · · , (4.14)hence w ( L ⊕ L ⊕ L ) = J J , (4.15)and in particular is nonzero. Thus, indeed, this SO (3) bundle has nonzero w , and can notbe lifted to an SU (2) bundle.Returning to the computation of the elliptic genus for the pure SO (3) gauge theory in asector in which w = 0, in terms of holonomies encoded in the parameter u , it can be written Y roots α θ ( τ | α · u ) θ ( τ | − z + α · u ) = θ ( τ | u ) θ ( τ | − z + u ) θ ( τ | θ ( τ | − z ) θ ( τ | − u ) θ ( τ | − z − u ) , (4.16)corresponding to the three generators of the Lie algebra of SO (3). The three boundaryconditions correspond to values of u as follows: u ( U , U )0 (+1 , +1)1 / − , +1) τ / , − τ ) / − , − w = 0 sector of thepure SO (3) gauge theory is proportional to θ ( τ | + 1 / θ ( τ | − z + 1 / θ ( τ | + τ / θ ( τ | − z + τ / θ ( τ | − (1 + τ ) / θ ( τ | − z − (1 + τ ) / , (4.17)confirming the results of [12, appendix A] up to numerical factors we will describe momen-tarily. 15o far, we have discussed known results for SU (2) elliptic genera, and also used a trickto compute the SO (3) elliptic genus in a sector where the characteristic class is nontrivial.Let us now repeat the computation systematically using the method of section 3, which wewill apply to other examples.Following the method of section 3, we compute the contribution to the elliptic genus from SO (3) bundles of vanishing characteristic (Stiefel-Whitney) class w . As discussed there, thecontribution in this sector is the same as that of a pure SU (2) theory, albeit with a constantfactor of 1 / | Γ | = 1 / Z ( SO (3) , w = 0) = 12 θ ( τ | − z ) θ ( τ | − z ) . (4.18)Next, we compute the contribution from SO (3) bundles of nontrivial characteristic class.As in section 3, we define this sector through holonomies lifted to SU (2), where they anti-commute. Specifically, consider the SU (2) matrices p = (cid:20) − (cid:21) , q = (cid:20) i − i (cid:21) . (4.19)It is easy to verify that pq = − qp . Viewing p and q as holonomies, they define a flat SU (2) / Z = SO (3) bundle with nontrivial characteristic class. Under the adjoint action of p and q , the Pauli sigma matrices are diagonal: pσ p − = − σ , pσ p − = + σ , pσ p − = − σ , (4.20) qσ q − = − σ , qσ q − = − σ , qσ q − = + σ . (4.21)From table 3, we see that the moduli space of flat SO (3) connections with nontrivialcharacteristic class is a point. We compute the contribution to the elliptic genus for thisnontrivial characteristic class by applying equation 3.15 with the phases listed above to getthe the product of theta functions in equation 4.17 up to a constant factor.Finally, to derive the elliptic genus for bundles of nonzero second Stiefel-Whitney class,we need to add a suitable numerical factor, corresponding to dividing out by the numberof residual gauge transformations which preserve the holonomies. From [43, section 2.2.1]for this case, we multiply the theta function product (4.17) by a factor of 1 / | W | , where W = Z × Z . Thus, we have that Z ( SO (3) , w = 0) = 14 θ ( τ | + 1 / θ ( τ | − z + 1 / θ ( τ | + τ / θ ( τ | − z + τ / θ ( τ | − (1 + τ ) / θ ( τ | − z − (1 + τ ) / , = 12 θ ( τ | − z ) θ ( τ | − z ) . (4.22)16ow, let us assemble these contributions. For a discrete theta angle θ , Z ( SO (3) , θ ) = Z ( SO (3) , w = 0) + exp( iw · t ) exp( iw · θ ) Z ( SO (3) , w = 0) . (4.23)As computed in [5, section 3.1], t = − πi , henceexp( iw · t ) = − , (4.24)and trivially exp( iw · θ ) = exp( iθ ), hence Z ( SO (3) , θ ) = Z ( SO (3) , w = 0) − exp( iθ ) Z ( SO (3) , w = 0) , (4.25)= 12 θ ( τ | − z ) θ ( τ | − z ) (1 − exp( iθ )) , (4.26)which duplicates the SO (3) elliptic genus as a function of θ computed in [12, appendix A]. SU (3) / Z gauge theories In this section, we apply the method of section 3 to compute the elliptic genus of a puresupersymmetric SU (3) / Z as a function of the discrete theta angle. First, for a vanishingcharacteristic class, from equation (3.2), the elliptic genus of the pure SU (3) / Z gauge theoryis the same as the elliptic genus of the pure SU (3) gauge theory, up to a factor of 1 / | Γ | : Z ( SU (3) / Z , w = 0) = 13 Z ( SU (3)) = 13 θ ( τ | − z ) θ ( τ | − z ) . (5.1)Next, we consider the elliptic genus of a pure SU (3) / Z gauge theory with a nontrivialcharacteristic class. We can describe an SU (3) / Z bundle with nonzero w ∈ H ( T , Z ) astwo holonomies p and q in SU (3) such that pq = wqp (5.2)for w = exp(2 πik/
3) with k = ± . To that end, consider the SU (3) matrices p = w w − , q = , (5.3)then, using w = 1, one can verify that pq = wqp. (5.4)17aking linear combinations of the Lie algebra generators λ a (in the three-dimensionaladjoint representation) to solve pλ a p − = ω ap λ a , qλ a q − = ω aq λ a , (5.5)we find that( ω p , ω q ) ∈ { (1 , w ) , (1 , w ) , ( w , , ( w , w ) , ( w , w ) , ( w, , ( w, w ) , ( w, w ) } . (5.6)In particular, the dimension of this component of the moduli space of flat SU (3) / Z connec-tions is zero, as can be confirmed from table 3.Using equation 3.15 with these phases, we find that the elliptic genus of the pure SU (3) / Z gauge theory with nontrivial bundle is1 | W | θ ( τ | τ k/ θ ( τ | − z + τ k/ θ ( τ | − τ k/ θ ( τ | − z − τ k/ θ ( τ | − k/ θ ( τ | − z − k/ θ ( τ | − k/ − τ k/ θ ( τ | − z − k/ − τ k/ · θ ( τ | − k/ τ k/ θ ( τ | − z − k/ τ k/ θ ( τ | k/ θ ( τ | − z + k/ θ ( τ | k/ − τ k/ θ ( τ | − z + k/ − τ k/ · θ ( τ | k/ τ k/ θ ( τ | − z + k/ τ k/ , (5.7)where W is the unbroken gauge symmetry of the pair ( p, q ), which for this case is [43, section2.2.1] W = Z × Z , hence | W | = 9. Recall that w = exp(2 πik/
3) for k = ± k ↔ − k .For k = 1, the product above can be written more succinctly as19 Y j,ℓ = − (cid:20) θ ( τ | j/ ℓτ / θ ( τ | j/ ℓτ / − z ) θ ( τ | − z ) θ ( τ | (cid:21) , (5.8)where in the product one should omit the case j = k = 0. One can show that Y j,ℓ = − (cid:20) θ ( τ | j/ ℓτ / θ ( τ | j/ ℓτ / − z ) θ ( τ | − z ) θ ( τ | (cid:21) = 3 θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) , (5.9)where y = exp(2 πiz ).Now, let us assemble these pieces to build the elliptic genus of the pure SU (3) / Z theorywith discrete theta angle θ ∈ { , π/ , π/ } . From [5, section 3.2], the quantum correctionis given by t a = 2 πi m a , (5.10) A careful reader will observe that if we had instead chosen k = 1 ,
2, we would have crossed a branchcut, which can generate factors such as y . We note that fact here, but it will not play a role in our furthercomputations. X a m a ≡ . (5.11)Without loss of generality, we can choose m = 0 = m , hence the phase factorexp( iw · t ) = +1 , (5.12)and so the elliptic genus can be written as a function of θ ∈ { , π/ , π/ } as Z ( SU (3) / Z , θ ) = Z ( SU (3) / Z , w = 0) + exp( iθ ) Z ( SU (3) / Z , w = 1)+ exp( − iθ ) Z ( SU (3) / Z , w = 2) , (5.13)= 13 θ ( τ | − z ) θ ( τ | − z ) (1 + exp( iθ ) + exp( − iθ )) . (5.14)As a consistency check, the reader should note that for θ = 0, the expression above forthe elliptic genus vanishes, whereas for θ = 0, it matches that of the pure SU (3) gaugetheory. This is consistent with the computation in [5, section 3.2] that supersymmetry isonly unbroken in a supersymmetric pure SU (3) / Z gauge theory when θ = 0.Furthermore, X θ =0 , ± π/ θ ( τ | − z ) θ ( τ | − z ) (1 + exp( iθ ) + exp( − iθ )) = θ ( τ | − z ) θ ( τ | − z ) = Z ( SU (3)) . (5.15)This matches the prediction of decomposition [13–15], which in this case schematically saysthat SU (3) = ( SU (3) / Z ) θ =0 + ( SU (3) / Z ) θ =2 π/ + ( SU (3) / Z ) θ =4 π/ . (5.16) SO (4) gauge theories Let us now turn to the elliptic genera of pure SO (4) gauge theories. These can be derivedfrom the results above for pure SO (3) gauge theories.First, consider a pure SO (4) theory in the sector in which w vanishes (so that all bundlescan be lifted to Spin(4) bundles). Now, Spin(4) = SU (2) × SU (2), so the elliptic genus inthis sector is the product of elliptic genera corresponding to two pure SU (2) gauge theories.Thus, as explained in section 3, the elliptic genus of a pure SO (4) gauge theory in a sectorwith w = 0 is 12 (cid:18) θ ( τ | − z ) θ ( τ | − z ) (cid:19) , (6.1)19aking into account the constant factor of 1 / | Γ | from section 3. This is consistent with theprediction (2.4) since there are two Casimirs each of the form Tr Σ .Now, let us turn to the sector in which w = 0. Here, we can apply the same analysisas in the case of the analogous SO (3) sectors. A set of holonomies describing such SO (4)bundles are given by diag(+1 , − , − , +1) , diag(+1 , +1 , − , − . (6.2)It is straightforward to check that these holonomies describe an SO (4) bundle with nonzero w , and from table 3, the moduli space of flat SO (4) connections with nonzero w is apoint. These holonomies emerge as a special case of the results in [9, equ’n (3.2)]. Wecan think of these holonomies as describing transformations under one of the two factors in SO (4) = ( SU (2) × SU (2)) / Z . Now, the nonzero roots of SO (4) can be expressed as ± u ± u , (6.3)where u , couple to Cartan holonomies. If one of the two SU (2) factors has trivial holonomy,then we can set u = 0, in which case, these roots become two copies of the roots of SO (3).Using previous results for SO (3) holonomies and elliptic genera, we immediately have thatthe SO (4) elliptic genus for w = 0 is proportional to (cid:20) θ ( τ | + 1 / θ ( τ | − z + 1 / θ ( τ | + τ / θ ( τ | − z + τ / θ ( τ | − (1 + τ ) / θ ( τ | − z − (1 + τ ) / (cid:21) = (cid:20) θ ( τ | − z ) θ ( τ | − z ) (cid:21) , (6.4)which from equation (4.3) is proportional to the elliptic genus for pure SO (4) gauge theorieswith vanishing w .Now, let us assemble these contributions. In principle, for discrete theta angle θ ∈ { , π } , Z ( SO (4) , θ ) = Z ( SO (4) , w = 0) + exp( iw · t ) exp( iw · θ ) Z ( SO (4) , w = 0) . (6.5)As computed in [5], t a = iπm a where X a m a ≡ , (6.6)hence exp( iw · t ) = − . (6.7)Thus, the elliptic genus is given by Z ( SO (4) , θ ) = 12 (cid:18) θ ( τ | − z ) θ ( τ | − z ) (cid:19) (1 − exp( iθ )) . (6.8)As a consistency check, note that Z ( SO (4) , θ ) vanishes for θ = 0, which is consistentwith the result [3, section 13.1] that supersymmetry is broken in this theory for θ = 0.20s another consistency check, note that X θ =0 ,π (cid:18) θ ( τ | − z ) θ ( τ | − z ) (cid:19) (1 − exp( iθ )) = (cid:18) θ ( τ | − z ) θ ( τ | − z ) (cid:19) , (6.9)the elliptic genus of the pure Spin(4) theory. This confirms the prediction of decomposition[13–15] in this case, which schematically saysSpin(4) = SO (4) θ =0 + SO (4) θ = π . (6.10) (4) / ( Z × Z ) gauge theories The group Spin(4) = SU (2) × SU (2), so the analysis of this group will be closely relatedto the analysis of SU (2). We can describe the Lie algebra of Spin(4) in terms of block-diagonal matrices and we can describe sectors with nontrivial characteristic classes by takingholonomies to be of the form diag( p, , diag( q,
1) (7.1)for one Z and diag(1 , p ) , diag(1 , q ) (7.2)for the other Z . Proceeding in a simple generalization of the analysis for a single copy of SU (2), we find results for elliptic genera as follows:1. Vanishing characteristic class. In this case, the elliptic genus is a product of two copiesof the SU (2) elliptic genus (divided by a factor of | Z × Z | = 4):14 ( Z ( SU (2))) = 14 (cid:18) θ ( τ | − z ) θ ( τ | − z ) (cid:19) . (7.3)2. Nontrivial characteristic class in one Z . Here, if we let Z ( SO (3) ) denote the ellipticgenus of a single SO (3) theory with nontrivial characteristic class, then the ellipticgenus is given by 12 Z ( SU (2)) Z ( SO (3) ) = 14 (cid:18) θ ( τ | − z ) θ ( τ | − z ) (cid:19) (7.4)(up to a phase).3. Nontrivial characteristic classes in both Z ’s. Here, the elliptic genus is given by( Z ( SO (3) )) = (cid:18) θ ( τ | − z ) θ ( τ | − z ) (cid:19) (7.5)(up to a phase). 21n the expressions above, we have used that Z ( SU (2)) = θ ( τ | − z ) θ ( τ | − z ) (7.6)and Z ( SO (3) ) = 12 θ ( τ | − z ) θ ( τ | − z ) (7.7)up to a phase, matching [12].Now, let us assemble these results. In principle, a sector of bundles of nontrivial char-acteristic class should be weighted by factors exp( iw · t ) and exp( iθ ), for θ a discrete thetaangle, and using results in [5], one can derive both phases for each sector. However, in thiscase there is a faster way, as the gauge group can equivalently be written as SO (3) × SO (3),so we can reuse the results of [12, appendix A] to immediately write the elliptic genus of apure Spin(4) / Z × Z gauge theory with discrete theta angles ( θ , θ ), θ i ∈ { , π } as (cid:20) θ ( τ | − z ) θ ( τ | − z ) (cid:21) (cid:18) − exp( iθ )2 (cid:19) (cid:18) − exp( iθ )2 (cid:19) . (7.8)In particular, note that X θ ,θ ∈{ ,π } (cid:20) θ ( τ | − z ) θ ( τ | − z ) (cid:21) (cid:18) − exp( iθ )2 (cid:19) (cid:18) − exp( iθ )2 (cid:19) = (cid:20) θ ( τ | − z ) θ ( τ | − z ) (cid:21) , (7.9)and so we see that the elliptic genus of the pure Spin(4) theory matches that of the sumof the elliptic genera of pure Spin(4) / Z × Z theories with the various possible discretetheta angles, as expected from decomposition [13–15] of two-dimensional theories with a B ( Z × Z ) symmetry. SO (5) gauge theories Now, let us turn to elliptic genera for pure SO (5) gauge theories. From equation (2.4) andthe fact that there are two operators, tr Σ and tr Σ , of R-charges 4 and 8, one expectsthat the elliptic genus of the pure Spin(5) theory and that of a pure SO (5) theory for onevalue of the discrete theta angle is θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) , (8.1)as discussed in section 2. This will also be derived by a direct residue computation in [11].22 p ω q θ argument − − − (1 + τ ) / − / − τ / − exp(2 πiλ ) − exp(2 πiλ ) − (1 + τ ) / u + exp( − πiλ ) − exp( − πiλ ) τ / − u − exp( − πiλ ) − exp( − πiλ ) − (1 + τ ) / − u exp(2 πiλ ) − exp(2 πiλ ) τ / u − exp(2 πiλ ) + exp(2 πiλ ) 1 / u − exp( − πiλ ) + exp( − πiλ ) 1 / − u Table 4: List of eigenvalues of SO (5) under the adjoint action of p , q .For bundles with vanishing w , from equation (3.2), the contribution to the elliptic genusof the pure SO (5) gauge theory is 1 / θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) , (8.2)as discussed in section 3.Next, let us consider the case of nonzero w , which we analyze following the pattern ofsection 3. Following [9, equ’n (3.3)], we can express the holonomies p , q in the form p = diag (exp(2 πiλ σ ) , − , − , +1) , q = diag (exp(2 πiλ σ ) , +1 , − , − . (8.3)Since we have already descended to SO (5) matrices, and are not working in Spin(5), thesematrices commute. Then, we diagonalize, finding a basis T α of the Lie algebra such that pT α p − = ω αp T α , qT α q − = ω αq T α . (8.4)Doing so, we find the eigenvalues ω αp,q , which we list in table 4. In each case, the θ argumentis computed as ln ω αp πi + τ ln ω αq πi , (8.5)and u = λ + τ λ . The number of eigenvalues ( ω p , ω q ) = (1 ,
1) gives the dimension of theresidue integral, as it reflects moduli of flat connections that are not constrained by theholonomies p , q .Alternatively, one could think of table 4 in terms of a (maximal-rank) SO (2) × SO (3)subgroup of SO (5). The weights of the nonzero roots of SO (5) are α · u ∈ {± u ± u , ± u , ± u } , (8.6)23n principle, for nonzero holonomies, the product over roots is of the same form as in the case w = 0, except that the values of one of the u i are constrained (to match those of SU (2),while the other is unconstrained. Thinking of the roots above in this fashion can also beused to generate table 4.In any event, from table 4, we read off a one-dimensional residue integral, proportionalto N (cid:18) πη ( q ) θ ( τ | − z ) (cid:19) I du πi · θ ( τ | u + 1 / θ ( τ | − z + u + 1 / θ ( τ | u + τ / θ ( τ | − z + u + τ / θ ( τ | u − (1 + τ ) / θ ( τ | − z + u − (1 + τ ) / · θ ( τ | − u + 1 / θ ( τ | − z − u + 1 / θ ( τ | − u + τ / θ ( τ | − z − u + τ / θ ( τ | − u − (1 + τ ) / θ ( τ | − z − u − (1 + τ ) / , (8.7)where N = θ ( τ | + 1 / θ ( τ | − z + 1 / θ ( τ | + τ / θ ( τ | − z + τ / θ ( τ | − (1 + τ ) / θ ( τ | − z − (1 + τ ) / , (8.8)= 2 θ ( τ | − z ) θ ( τ | − z ) , (8.9)where the second line follows from [12, equ’n (A.6)].From table 3, the moduli space of flat SO (5) connections with nontrivial characteristicclass is the same as the moduli space of flat SU (2) connections, i.e., T / Z , which is theorigin of the integral above. We integrate over the covering space T , and add a factor of1 / u coefficients. (Alternatively, we could sum only over poles with negative u coefficients, butwe pick the former convention in this paper.) These poles are given by u = z − / , z − τ / , z + (1 + τ ) / . (8.10)The fact that the integrand is symmetric under u
7→ − u reflects the Weyl group action on themoduli space of flat SU (2) connections. Also note that the prescription above is summingover distinct residues which are not related by the Weyl group.We will use the identity [6, equ’n (B.6)] θ ′ ( τ |
0) = 2 πη ( q ) , (8.11)24here the derivative is taken with respect to the second variable. As a result, and using thefact that [6, equ’n (B.4)] θ ( τ | z + a + bτ ) = ( − ) a + b exp( − πibz − iπb τ ) θ ( τ | z ) (8.12)for a, b ∈ Z , one has [6, equ’n (B.7)]12 πi Z u = a + bτ duθ ( τ | u ) = ( − ) a + b exp( iπb τ ) θ ′ ( τ |
0) = ( − ) a + b exp( iπb τ )2 πη ( q ) , (8.13)for a, b ∈ Z .From the pole at u = z − /
2, we have a contribution12 Nθ ( τ | − z ) θ ( τ | + z ) θ ( τ | z − / τ / θ ( τ | − / τ / θ ( τ | z − / − (1 + τ ) / θ ( τ | − / − (1 + τ ) / · θ ( τ | − z + 1 / θ ( τ | − z ) θ ( τ | − z + 1 / τ / θ ( τ | − z + 1 / τ / θ ( τ | − z + 1 / − (1 + τ ) / θ ( τ | − z + 1 / − (1 + τ ) / . (8.14)From the pole at u = z − τ /
2, we have a contribution12 Nθ ( τ | − z ) θ ( τ | + z ) θ ( τ | z + 1 / − τ / θ ( τ | + 1 / − τ / θ ( τ | z − τ / − (1 + τ ) / θ ( τ | − τ / − (1 + τ ) / · θ ( τ | − z + τ / / θ ( τ | − z + τ / / θ ( τ | − z + τ ) θ ( − z + τ ) θ ( τ | − z − / θ ( τ | − z − / . (8.15)From the pole at u = z + (1 + τ ) /
2, we have a contribution12 Nθ ( τ | − z ) θ ( τ | + z ) θ ( τ | z + 1 / τ ) / θ (1 / τ ) / θ ( τ | z + 1 / τ ) θ ( τ | / τ ) · θ ( τ | − z − τ / θ ( τ | − z − τ / θ ( τ | − z − / θ ( τ | − z − / θ ( τ | − z − − τ ) θ ( τ | − z − − τ ) . (8.16)One can verify (e.g. numerically) that the sum of these residues is2 θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) . (8.17)To derive Z ( SO (5) , w = 0), we still need a numerical factor, 1 / | W | for some W as in [43].Rather than compute W directly, for the moment, we write Z ( SO (5) , w = 0) = αZ ( SO (5) , w = 0) (8.18)25or some positive real number α , which we will compute by using known results for super-symmetry breaking.Now, let us assemble these results into the elliptic genus for SO (5) with discrete thetaangle θ ∈ { , π } . The contribution from the sector with w = 0 is independent of θ , and isjust a factor of 1 / | Γ | away from the elliptic genus of Spin(5): Z ( SO (5) , w = 0) = 12 θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) . (8.19)Next, we consider the contribution from the sector with w = 0. There is a factor ofexp( iθ ) from the discrete theta angle θ ∈ { , π } . In addition, there is also a phase exp( iw · t )where, from the analysis of [5], t a = iπm a , (8.20)where X a m a ≡ . (8.21)As a result, exp( iw · t ) = − SO (5) gaugetheory as a function of discrete theta angle θ ∈ { , π } : Z ( SO (5) , θ ) = Z ( SO (5) , w = 0) − exp( iθ ) Z ( SO (5) , w = 0) , (8.22)= 12 θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) (1 − α exp( iθ )) . (8.23)From [3, section 13.2], we know that supersymmetry is broken in pure SO (5) theorieswith θ = 0, hence we must require that α = 1, hence the elliptic genus of the pure SO (5)theory with discrete theta angle θ is Z ( SO (5) , θ ) = 12 θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) (1 − exp( iθ )) . (8.24)As a consistency check, note that α is a positive real number, as expected – phasefactors have already been accounted for. As another consistency check, note that for θ = π ,the elliptic genus of the pure SO (5) gauge theory matches that of the Spin(5) theory, inagreement with expectations from [3, section 13.2].As another consistency check, note that this implies that the elliptic genus of the pureSpin(5) theory is the sum of the elliptic genera of the pure SO (5) theories with either valueof θ : Spin(5) = SO (5) θ =0 + SO (5) θ = π , (8.25)which is consistent with decomposition of two-dimensional theories with a B Z symmetry[13–15]. 26 Pure Sp (6) / Z gauge theories We now turn to pure Sp (6) / Z gauge theories (in conventions in which Sp (2) = SU (2)).Since Sp (2) = SU (2) and Sp (4) = Spin(5), the first interesting case amongst Sp (2 k ) / Z is Sp (6) / Z .As before, for bundles of vanishing characteristic class, from equation (3.2), the ellipticgenus matches that of the pure Sp (6) gauge theory, up to the factor 1 / | Γ | : Z ( Sp (6) / Z , w = 0) = 12 Z ( Sp (6)) = 12 θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) , (9.1)as discussed in section 3.To describe a nontrivial bundle, we give two anticommuting holonomies in Sp (2 k ), whichfollowing [46, section 4.1], [47, equ’n (8)] we can take to be p = diag (cid:0) λ , − λ , i, − i, − λ − , λ − (cid:1) , (9.2) q = diag (cid:18)(cid:20) − λ − λ (cid:21) , (cid:20) − i − i (cid:21) , (cid:20) − λ − − λ − (cid:21)(cid:19) , (9.3)and where we take the symplectic form to beΩ = −
10 0 0 0 − − , (9.4)so that p T Ω p = Ω , q T Ω q = Ω . (9.5)Following the procedure of section 3, we diagonalize a basis of the Lie algebra withrespect to the diagonal action of p , q above. The eigenvalues ω αp,q of the adjoint action aregiven in table 5.In table 5, u = ln λ πi + τ ln λ πi , (9.6)and the θ coefficient is ln ω αp πi + τ ln ω αq πi . (9.7) In case the reader finds it helpful, the Lie algebra with the symplectic form given in Equation (9.4) isdescribed in detail in [48, Chapter 30]. p ω q θ argument λ − − λ − τ / − uλ − λ − − u − λ − λ − / − u − iλ − − iλ − / τ / − uiλ − − iλ − / τ / − u − iλ − iλ − / τ / − uiλ − iλ − / τ / − u − − − (1 + τ ) / − − − (1 + τ ) / − τ / − τ / − / − /
21 1 0 − iλ − iλ / τ / uiλ − iλ / τ / u − iλ iλ / τ / uiλ iλ / τ / uλ − λ τ / uλ λ u − λ λ / u Table 5: Table of eigenvalues of the adjoint action of the holonomy matrices.28s a simple consistency check, note that the number of entries, 21, is the same as thedimension of Sp (6).The fact that there is only one entry in table 5 with p , q eigenvalues (1 ,
1) means thatthe elliptic genus will be computed by a one-dimensional residue integral. From table 3, wesee that the moduli space of flat Sp (6) / Z connections with nontrivial characteristic class isthe same as the moduli space of SU (2) connections – indeed, one-dimensional. The modulispace is T / Z , but we will integrate over the T cover, quotienting by a factor of 2 to reflectthat fact.Putting this together, the elliptic genus of a pure Sp (6) / Z gauge theory with bundles ofnontrivial characteristic class is proportional to N (cid:18) πη ( q ) θ ( τ | − z ) (cid:19) I du πi (9.8) · θ ( τ | / τ / u ) θ ( − z + 3 / τ / u ) θ ( τ | / τ / u ) θ ( τ | − z + 1 / τ / u ) θ ( τ | / τ / u ) θ ( τ | − z + 3 / τ / u ) · θ ( τ | / τ / u ) θ ( τ | − z + 1 / τ / u ) θ ( τ | τ / u ) θ ( τ | − z + τ / u ) θ ( τ | u ) θ ( τ | − z + 2 u ) θ ( τ | / u ) θ ( − z + 1 / u ) · θ ( τ | / τ / − u ) θ ( − z + 3 / τ / − u ) θ ( τ | / τ / − u ) θ ( τ | − z + 1 / τ / − u ) θ ( τ | / τ / − u ) θ ( τ | − z + 3 / τ / − u ) · θ ( τ | / τ / − u ) θ ( τ | − z + 1 / τ / − u ) θ ( τ | τ / − u ) θ ( τ | − z + τ / − u ) θ ( τ | − u ) θ ( τ | − z − u ) θ ( τ | / − u ) θ ( − z + 1 / − u ) , where N = (cid:20) θ ( τ | − (1 + τ ) / θ ( τ | − z − (1 + τ ) / θ ( τ | τ / θ ( τ | − z + τ / θ ( τ | / θ ( τ | − z + 1 / (cid:21) , (9.9)= (cid:20) θ ( τ | − z ) θ ( τ | − z ) (cid:21) , (9.10)using [12, appendix A]. The overall factor of 1 / T of the moduli space of flat connections.The reader will note that the expression above is symmetric under u ↔ − u . This reflectsthe Weyl group action on the moduli space of flat SU (2) connections, whose double-coverwe are integrating over in the expression above.Following the Jeffrey-Kirwan residue prescription, we will take poles of denominatorsin which u appears with a positive coefficient. (Alternatively, we could equivalently takepoles in which u appears with negative coefficient, but we will use the positive coefficientprescription in this paper.) In passing, note that none of these poles are related by the Weylgroup action to one another. 29our of the poles are at u = z − / − τ / , z − / − τ / , z − / − τ / , z − / − τ / . (9.11)To find all of the remaining poles, one must take into account the periodicities of the thetafunction. Taking those into account, we find four poles at2 u = z − τ / { , , τ, τ } , or u = z/ − τ / { , / , τ / , (1 + τ ) / } , (9.12)another four poles at u = z/ { , / , τ / , (1 + τ ) / } , (9.13)and another four at u = z/ − / { , / , τ / , (1 + τ ) / } , (9.14)for a total of 16 residues that must be summed over.We illustrate a few examples of these residues here, to illustrate the complexity of thecomputation. The residue at u = z − / − τ / Nθ ( τ | − z ) θ ( τ | + z ) θ ( τ | z − / θ ( τ | − / θ ( τ | z − τ / θ ( τ | − τ / θ ( τ | z − (1 + τ ) / θ ( τ | − (1 + τ ) / · θ ( τ | z − / − τ ) θ ( τ | z − / − τ ) θ ( τ | z − / − τ / θ ( τ | z − / − τ / θ ( τ | z − − τ / θ ( τ | z − τ / · θ ( τ | − z + 3 / τ ) θ ( τ | − z + 3 / τ ) θ ( τ | − z + 3 / τ / θ ( τ | − z + 3 / τ / θ ( τ | − z + 2 + 3 τ / θ ( τ | − z + 2 + 3 τ / · θ ( τ | − z + 3 / τ / θ ( τ | − z + 3 / τ / θ ( τ | − z + 1 + 3 τ / θ ( τ | − z + 1 + 3 τ / θ ( τ | − z + 3 / τ ) θ ( τ | − z + 3 / τ ) · θ ( τ | − z + 1 + τ ) θ ( τ | − z + 1 + τ ) . (9.15)Similarly, the residue at u = z/ − τ / Nθ ( τ | − z ) θ ( τ | z/ / τ / θ ( τ | − z/ / τ / θ ( τ | z/ / τ / θ ( τ | − z/ / τ / θ ( τ | z/ / θ ( τ | − z/ / · θ ( τ | z/ / θ ( τ | − z/ / θ (+ z ) θ ( τ | z/ − τ / θ ( τ | − τ / θ ( τ | z + 1 / − τ / θ ( τ | / − τ / · θ ( τ | − z + τ ) θ ( τ | − z + τ ) θ ( τ | − z + τ / θ ( τ | − z + τ / θ ( τ | − z + 1 / τ / θ ( τ | − z + 1 / τ / θ ( τ | − z/ / τ ) θ ( τ | − z/ / τ ) · θ ( τ | − z/ / τ ) θ ( τ | − z/ / τ ) θ ( τ | − z/ / τ / θ ( τ | − z/ / τ / θ ( τ | − z/ / τ / θ ( τ | − z/ / τ / . (9.16)A leading factor of 1 / u not u . An overall factor of 1 / T and not T / Z . For reasons ofbrevity, we do not list the other fourteen residues here, though they are straightforward tocompute.One can verify numerically that the sum of the residues above, the integral (9.8) equals8 θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) . (9.17)The product of theta functions above should be proportional to the elliptic genus of thepure Sp (6) / Z theory with nonzero characteristic class. The proportionality factor should bea real number of the form 1 / | W | for W a finite subgroup of the gauge group that preservesthe holonomies. For the moment, we will write Z ( Sp (6) / Z , w = 0) = αZ ( Sp (6) / Z , w = 0) , (9.18)for some positive real number α . We will compute this factor indirectly, using known resultsfor supersymmetry breaking for various discrete theta angles.Finally, we need to weight the w = 0 contribution with relevant phases. There is a factorexp( iθ ) arising from the discrete theta angle θ ∈ { , π } . In addition, there is potentially afactor of exp( iw · t ). From [5, section 5], t a = πim a (9.19)where X a m a ≡ Sp (6) / Z ), so without loss of generality we can take all m a = 0, hence exp( iw · t ) = +1.Now, putting this together, combining the result for the elliptic genus in the sector with w = 0 with the result above, determined up to a proportionality factor, for w = 0, we havethat the elliptic genus of a pure Sp (6) / Z gauge theory with discrete theta angle θ ∈ { , π } is given by Z ( Sp (6) / Z , θ ) = Z ( Sp (6) / Z , w = 0) + α exp( iθ ) Z ( Sp (6) / Z , w = 0) , (9.21)= 12 θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) (1 + α exp( iθ )) . (9.22)It was argued in [5, section 5] that a pure Sp (6) / Z gauge theory has supersymmetricvacua only if the discrete theta angle θ = 0, hence for θ = π , supersymmetry is broken, andthe elliptic genus should vanish. Imposing this as a constraint, we find that α = +1, hencethe elliptic genus of a pure Sp (6) / Z gauge theory as a function of discrete theta angle θ is Z ( Sp (6) / Z , θ ) = 12 θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) (1 + exp( iθ )) . (9.23)31s a consistency check, note that α is real and positive, as expected – phase factors havealready been accounted for. As another consistency check, note that for θ = 0, the ellipticgenus of the pure Sp (6) / Z gauge theory matches that of the pure Sp (6) gauge theory, inagreement with expectations from [5, section 5].As a further consistency check, it is straightforward to see that this result is consistentwith decomposition [13–15]: X θ =0 ,π θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) θ ( τ | − z ) (1 + exp( iθ )) = Z ( Sp (6)) , (9.24)consistent with the expectation Sp (6) = ( Sp (6) / Z ) θ =0 + ( Sp (6) / Z ) θ = π (9.25)(expressed schematically).
10 Predictions for general cases
So far, we have performed direct computations to compute elliptic genera of pure gaugetheories with semisimple, non-simply-connected gauge groups in some low rank cases. Next,we are going to make a proposal for all cases, utilizing (a) our knowledge of the contributionfrom w = 0, (b) supersymmetry breaking for most discrete theta angles, and (c) decompo-sition. These three constraints form sufficiently many algebraic equations to enable us tosolve algebraically for the elliptic genera.We illustrate the method using the pure SU (4) / Z gauge theory as an example. First,we know that Z ( SU (4) / Z , w = 0) = 14 Z ( SU (4)) . (10.1)Given the results for low-rank cases, let us assume that Z ( SU (4) / Z , w = 0) ∝ Z ( SU (4) / Z , w = 0) , (10.2)so we can write Z ( SU (4) / Z , θ ) = 14 Z ( SU (4)) (1 + α exp( iθ ) + α exp(2 iθ ) + α exp(3 iθ )) , (10.3)for θ ∈ { , π/ , π, π/ } . From table 2,we see that supersymmetry is broken unless θ = π ,which gives the constraints 1 + α + α + α = 0 , (10.4)1 + iα − α − iα = 0 , (10.5)1 − iα − α + iα = 0 , (10.6)32or θ = 0 , π/ , π/
2, respectively, and from decomposition, since the elliptic genera vanishfor θ = π , the elliptic genus at θ = π must match that of SU (4), hence1 − α + α − α = 4 . (10.7)These are four linear algebraic equations in three unknowns, which happen to admit a uniquesolution: α = α = − , α = +1 . (10.8)Putting this together, we have that Z ( SU (4) / Z , θ ) = 14 θ ( τ | − z ) θ ( τ | − z ) (1 − exp( iθ ) + exp(2 iθ ) − exp(3 iθ )) . (10.9)We have used our knowledge of supersymmetry breaking and decomposition, and only as-sumed that the contributions from sectors of various characteristic classes are proportionalto one another. One can check that the resulting phase factors, derived algebraically, areconsistent with those described in section 3.Proceeding in this fashion, using our knowledge of supersymmetry breaking and decom-position, elliptic genera are straightforward to predict for all other cases. We summarize theresults below.First, for SU ( k ) / Z k , for k odd, supersymmetry is unbroken only for θ = 0 (from table 2),and we predict the elliptic genus Z ( SU ( k ) / Z k , θ ) = 1 k θ ( τ | − z ) θ ( τ | − kz ) k − X m =0 exp( imθ ) , (10.10)for θ ∈ { , π/k, π/k, · · · , k − π/k } . For k even, supersymmetry is unbroken only for θ = π and we predict the elliptic genus Z ( SU ( k ) / Z k , θ ) = 1 k θ ( τ | − z ) θ ( τ | − kz ) k − X m =0 ( − ) m exp( imθ ) . (10.11)Proceeding similarly, for Spin(2 k + 1) / Z , we predict the elliptic genus Z (Spin(2 k + 1) / Z , θ ) = 12 Z (Spin(2 k + 1)) (1 − exp( iθ )) , (10.12)where Z (Spin(2 k + 1)) denotes the elliptic genus of the pure Spin(2 k + 1) gauge theory, asgiven in section 2, and for θ ∈ { , π } .For Spin(4 k ) / Z × Z , we predict the elliptic genus Z (Spin(4 k ) / Z × Z , θ , θ ) = 14 Z (Spin(4 k )) (cid:0) − ) k exp( iθ ) (cid:1) (cid:0) − ) k exp( iθ ) (cid:1) , (10.13)33or θ , ∈ { , π } .For Spin(4 k + 2) / Z , we predict the elliptic genus Z (Spin(4 k + 2) / Z , θ ) = 14 Z (Spin(4 k + 2)) X m =0 ( − ) km exp( imθ ) , (10.14)for θ ∈ { , π } .For Sp (2 k ) / Z , we predict the elliptic genus Z ( Sp (2 k ) / Z , θ ) = 12 Z ( Sp (2 k )) (1 + ( − ) m exp( iθ )) , (10.15)for θ ∈ { , π } , where m = (cid:26) k/ k even , ( k + 1) / k odd . (10.16)For E / Z , we predict the elliptic genus Z ( E / Z , θ ) = 13 Z ( E ) (1 + exp( iθ ) + exp(2 iθ )) , (10.17)for θ ∈ { , π/ , π/ } .For E / Z , we predict the elliptic genus Z ( E / Z , θ ) = 12 Z ( E ) (1 − exp( iθ )) , (10.18)for θ ∈ { , π } .As a consistency check, note that the elliptic genus of SU (2) / Z matches that of Spin(3) / Z ,the elliptic genus of SU (4) / Z matches that of Spin(6) / Z , and the elliptic genus of Sp (4) / Z matches that of Spin(5) / Z , as expected since the Lie groups are the same.In each case, the elliptic genus vanishes for discrete theta angles θ for which supersym-metry is broken in the IR (from table 2), and decomposition [13–15] is obeyed: Z ( G ) = X θ Z ( G/ Γ , θ ) . (10.19)
11 Conclusions
In this paper we have described a systematic method to compute elliptic genera of puretwo-dimensional (2,2) supersymmetric G/ Γ gauge theories with various discrete theta angles.34ur results agree with previous computations of elliptic genera of pure SO (3) gauge theories,and we also derived the elliptic genera of pure SU (3) / Z , SO (4), Spin(4) / Z × Z , SO (5)and Sp (6) / Z gauge theories. In each case, the results are consistent with predictions ofsupersymmetry breaking for certain discrete theta angles in [3–5], and the resulting ellipticgenera are also consistent with expectations from decomposition [13–15] of two-dimensionalgauge theories with finite global one-form symmetries. Finally, we applied these two criteriato make predictions for elliptic genera of higher-rank cases.Pure two-dimensional (2,2) supersymmetric gauge theories have also been extensivelystudied by lattice simulations [49–53]. Our results also provide new analytic results that canbe used to test and callibrate future lattice studies of pure two-dimensional supersymmetricgauge theories. They also suggest new avenues for research such as varying the globalstructure of the gauge group and including discrete theta angles.Gauge theories correspond to sigma models on stacks [18–20], and the elliptic genera wehave computed in this paper should correspond to elliptic genera of the classifying stacks BG [21, 22].The sensitivity of the elliptic genus to the global structure of the gauge group makes ita powerful tool to investigate of two-dimensional dualities. The elliptic genus has alreadybeen used to test several of Hori’s proposed dualities [29] in [12, 27, 28]. Looking forward, weexpect the elliptic genus of G/ Γ gauge theories will be useful to establish new dualities andwill help with exploring the dynamics of two-dimensional supersymmetric gauge theories .
12 Acknowledgements
We would like to thank D. Berwick-Evans, C. Closset, M. Hanada, T. Johnson-Freyd,K. Hori, E. Poppitz, S. Razamat, Y. Tachikawa, A. Tripathy, and Piljin Yi for useful dis-cussions. We would especially like to thank Y. Tachikawa for his careful reading of themanuscript. R.E. would like to thank Kavli IPMU for hospitality while this work was beingcompleted and the World Premier International Research Center Initiative (WPI), MEXT,Japan. R.E. is supported in part by KIAS Individual Grant PG075901. E.S. was partiallysupported by NSF grants PHY-1720321 and PHY-2014086. That said, elliptic genera should be applied with care. For example, we have seen earlier in this paperthat the SU (2) elliptic genus matches that of SO (3) − . However, these two theories are not dual to oneanother. Instead, the SU (2) theory is a sum of the two SO (3) theories, with each value of the discrete thetaangle. Because supersymmetry is broken in the SO (3) + theory, the elliptic genus only receives contributionsfrom the SO (3) − theory. We see that relying solely upon the equality of elliptic genera can be misleadingin trying to find dualities. eferences [1] E. Witten, “Bound states of strings and p-branes,” Nucl. Phys. B (1996) 335–350, arXiv:hep-th/9510135 .[2] O. Aharony, S. S. Razamat, N. Seiberg and B. Willett, “The long flow to freedom,”
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