AGT/ℤ2

Journal of High Energy Physics, Dec 2017

We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional \( \mathcal{N}=2 \) gauge theories. Our construction naturally involves four-dimensional theories with fields defined on different ℤ2 quotients of the sphere (hemisphere and projective space) but nevertheless interacting with each other. The six-dimensional origin is a ℤ2 quotient of the setup giving rise to the usual AGT correspondence. To test the correspondence, we work out the ℝℙ4 partition function of four-dimensional \( \mathcal{N}=2 \) theories by combining a 3d lens space and a 4d hemisphere partition functions. The same technique reproduces known ℝℙ2 partition functions in a form that lets us easily check two-dimensional Seiberg-like dualities on this nonorientable space. As a bonus we work out boundary and cross-cap wavefunctions in Toda CFT.

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AGT/ℤ2

HJE Bruno Le Floch 0 1 Gustavo J. Turiaci 0 0 Princeton University , Princeton NJ 08544 , U.S.A 1 Princeton Center for Theoretical Science We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional N = 2 gauge theories. Our construction naturally involves four-dimensional theories with on di erent Z2 quotients of the sphere (hemisphere and projective space) but nevertheless interacting with each other. The six-dimensional origin is a Z2 quotient of the setup giving rise to the usual AGT correspondence. To test the correspondence, we work out the RP4 partition function of four-dimensional N = 2 theories by combining a 3d lens space and a 4d hemisphere partition functions. The same technique reproduces known RP2 partition functions in a form that lets us easily check two-dimensional Seiberg-like dualities on this nonorientable space. As a bonus we work out boundary and cross-cap wavefunctions in Conformal and W Symmetry; Supersymmetric Gauge Theory; Supersymme- Toda CFT. Keywords: Conformal and W Symmetry, Supersymmetric Gauge Theory, Supersymmetry and Duality, Duality in Gauge Field Theories 1 Introduction 2 Liouville/Toda boundary CFT Boundary wavefunctions Disk two-point function 3 Hemisphere and branes Orbifold identi cation Partition function on hemisphere 2.1 2.2 3.1 3.2 3.4 4.1 4.2 5.1 5.2 5.3 3.3 Identity brane Other branes 3.5 S-duality 4 Projective space and cross-caps Partition function on projective space Cross-cap two-point function 5 Generalizations Quivers Verlinde loops Surface operators 5.4 Boundary operators 6 Conclusion A ADE Toda boundary states A.1 Boundary CFT A.2 Representations and characters A.3 D Conjectural hypermultiplet determinant { 1 { Introduction The AGT correspondence relates correlation functions in Liouville/Toda 2d CFT on a Riemann surface and 4d N = 2 gauge theories on S4 [3]. The interplay between these two completely di erent setups has provided new ways to obtain and motivate new results on both sides, mainly for four dimensional gauge theories. From the two dimensional perspective this correspondence has been studied on Riemann surfaces with punctures and arbitrary genus. Nevertheless, a question originally posed in [3] has not been addressed in the literature: What does CFT on surfaces with boundaries correspond to on the gauge theory side? concurrently exchange the poles of Sb4. In 2d CFT, adding boundaries to the surface in which the theory lives has proven to be very fruitful towards understanding their structure. Important progress in this direction was initiated by Cardy in a seminal paper [19]. Given the importance of his construction in CFT one is led to wonder what it teaches us about four dimensional gauge theories. This is the motivation for this work, and we take a rst step by answering the question raised above. A second motivation is to learn more about the 6d N = (2; 0) theory that gives rise to the AGT correspondence. The 6d theory admits no supersymmetric boundary conditions, because of chirality. Instead, 2d boundaries (and cross-caps, as we will see) arise from a Z2 quotient of the usual AGT setup.1 The 2d surface is a Z2 quotient of a closed Riemann surface b and we consider the 6d N = (2; 0) theory on several quotients (Sb4 b)=Z2. Since the AGT correspondence relates (anti)holomorphic parts of 2d CFT correlators to (anti)instantons the two poles of the ellipsoid Sb4, and since the Z2 reverses the orientation on b it must Discrete quotients of the AGT setup considered in previous works xed the poles of the ellipsoid, where instantons and anti-instantons are located. For example [5, 7, 8, 12] considered Zp subgroups of (x1; x2) and (x3; x4) rotations; instanton contributions matching Virasoro conformal blocks are then changed to those of a di erent chiral algebra. In contrast, our Z2 exchanges poles of the ellipsoid, hence identi es the instanton and antiinstanton sums. Correspondingly, the Z2 action identi es left-moving and right-moving modes of the CFT. An interesting extension would be to combine this Zp with our Z2 quotients to learn about boundary states of the parafermionic CFTs corresponding to Sb4=Zp, including N = 1 super Liouville for p = 2, and N = 2 super Liouville obtained by adding a surface operator on the gauge theory side [18]. Our main tool to probe the proposed correspondence is supersymmetric localization. The Z2 action must thus leave the localizing supercharge invariant, and in particular its square. In terms of embedding coordinates x 2 R5 the ellipsoid is de ned by x20 + b 2(x21 + x22) + b2(x23 + x42) = 1; (1.1) 1A toy example to keep in mind is that the quotient (S1 on S1. This is true for both actions in (1.2). ! and ! R)=Z2 gives rise to a half-line upon reduction + mod 2 on S1, analogous to the two actions { 2 { (a) = b=Z2 b P1 P2 P1 (b) (Sb4 b)=Z2 aries . (b) The lift of this construction to a 6d orbifold. We indicate how it acts on the ellipsoid S4. b S1, where the rst factor corresponds to the equator of the middle ellipsoid and the second factor to the boundary of the Riemann surface. with poles at x0 = 1, and the supercharge squares to rotations in the (x1; x2) and (x3; x4) of Sb4 and to the antipodal map: planes. We thus restrict our attention to the re ection x0 ! x0 around the Sb3 equator x0 ! x0 (re ection) and x ! x (antipodal map): (1.2) The quotient Sb4=Z2 is the (squashed) hemisphere HSb4 and projective space RPb4, respectively. and Sb4. In gure 1 we show a simple example of our setup. We begin with a Riemann surface composed of a torus with a single circular boundary. In the same gure we show the Schottky double b which has no boundary but has genus 2. In the right panel we show how this construction naturally lifts to 6d, with the involution acting simultaneously on b The 4d theory corresponding to the surface with boundaries b=Z2 is then obtained by a Z2 identi cation of elds in the 4d N = 2 theory T [b] associated to b . Recall how the 4d theory T [b] is constructed for a choice of pants decomposition of b. One associates isolated CFTs (matter theories) to three-punctured spheres and one gauges avor symmetries using vector multiplets associated to tubes connecting these spheres. Given two three-punctured spheres or two tubes mapped to each other by the Z2, the corresponding matter theories or vector multiplets are identi ed up to the space-time symmetry (1.2). The vector multiplet corresponding to a tube invariant under Z2, namely to a boundary of , for instance the middle tube in gure 1, is instead identi ed to itself under this space-time symmetry. The theory can be described more parsimoniously by keeping only one of each pair of elds identi ed by the Z2. This associates a vector multiplet to each tube in the bulk of = b=Z2, and to each boundary of it associates a vector multiplet living on Sb4=Z2 with the identi cation (1.2). The recipe in this description is to start from the theory on Sb4 associated to the Riemann surface obtained by replacing all boundaries (and crosscaps) of by punctures, then to gauge each corresponding avor symmetry using a vector multiplet on Sb4=Z2. From this point of view there is no reason for all of these vector { 3 { Ishibashi state ZZ brane (all) FZZT brane Unusual brane on the squashed hemisphere HSb4 and 2d CFT Ishibashi states, branes (Cardy states), and crosscaps. ZZ branes are labeled by a fully-degenerate primary hence by a pair of representations of the Lie algebra. FZZT branes are labeled by semi-degenerate or non-degenerate primaries. The 6d construction also singles out an unusual boundary state, whose wavefunction is that of a cross-cap state, and an unusual cross-cap state, whose wavefunction is that of an identity brane. On the gauge theory side these correspond to turning on a discrete theta angle. multiplets to be subject to the same Z2 identi cation among (1.2). This more general 4 setup is obtained by replacing Sb b by a bration over b. Apart from this description based on the fundamental domain Sb4 there is another description based on the fundamental domain HSb4 of the Z2 action, b where HSb4 is the x0 ! squashed hemisphere x0 > 0. Then the full eld contents of T [b] are kept, but restricted to HSb4, and with a boundary condition implementing the Z2 identi cation. When Z2 acts as x0 the vector multiplet associated to a tube joining to its image is given Neumann boundary conditions. When Z2 acts as the antipodal map the boundary condition is nonlocal; it relates elds at opposite points of the equator so that 4d elds e ectively live on the projective space RPb4. Two-dimensional CFTs have a rich set of boundary conditions. In Cardy's formalism, which we review and apply to Liouville/Toda CFT in appendix A, boundary conditions are labeled by primary operators. So far, in our construction, boundaries only depend on whether Z2 acts on the ellipsoid by the re ection or antipodal map. The re ection case turns out to correspond to identity branes, labeled by the identity operator. For this Z2 action x0 ! x0, there is a xed-point locus2 S 3 b S 1 S 4 b b for each boundary of . Inserting along this singular locus some codimension 2 and 4 operators of the 6d theory has the e ect of changing the boundary condition of the CFT. Strikingly, the basis of boundary conditions in Liouville/Toda CFT obeying the Cardy condition are in one-to-one correspondence with these operators of the 6d theory. The explicit correspondence is listed in table 1. In analogy to branes in Liouville CFT [35, 93] we call ZZ branes the branes labeled by a fully degenerate primary (including the identity) and FZZT those labeled by a semi-degenerate or a non-degenerate primary. The ZZ brane 2A priori it may be problematic to consider the 6d N = (2; 0) theory on an orbifold, but our concrete match of 2d and 4d observables in the main text suggests otherwise. { 4 { is labeled by a pair of representations (R1; R2) of the simply-laced Lie algebra g that de nes the 6d N = (2; 0) theory. It is obtained by inserting a pair of Wilson lines at x3 = x4 = 0 and x1 = x2 = 0 in representations R1 and R2 of the gauge group associated to the boundary. The FZZT branes correspond to symmetry-breaking boundary conditions, essentially identical to symmetry-breaking walls obtained for topological defects [25]: the gauge group is broken to a subgroup H G at the boundary and FI parameters for the remaining U(1)'s are turned on, as well as Wilson lines of H stuck to the boundary by gauge invariance. Each of these 2d CFT boundary states is a linear combinations of Ishibashi states, and we match the latter (up to a scaling) to Dirichlet boundary conditions for the corresponding vector multiplet of the theory T [b]. Our task then boils down to equating the wavefunction of the identity brane to the one-loop determinant of a vector multiplet with x0 ! x0 identi cation. To circumvent the well-known issue of in nite fusion multiplicities in Toda CFT, the concrete examples of correlators that we give are for the 6d N = (2; 0) theory of Atype with \degenerate-enough" insertions (see the main text). They correspond in gauge theory to quivers of SU(N ) gauge groups. Our results however apply to all branes in ADE Toda CFT.3 Besides having boundaries, can also be nonorientable. Such surfaces can also be assembled by inserting cross-caps in an orientable surface, so they t in Cardy's formalism. We work out the ADE Toda cross-cap wavefunction in appendix A, namely what linear combination of cross-cap Ishibashi states is needed to describe the nonorientable surface. The cross-cap turns out to correspond to an antipodal identi cation of the corresponding vector multiplet. To identify in detail the Liouville/Toda cross-cap to a vector multiplet on RPb4 we worked out the partition function of such a multiplet. We obtained it by a gluing prescription that combines ingredients from HSb4 and lens space Sb3=Z2 partition functions. For completeness we extended this to any 4d N = 2 Lagrangian gauge theory (see section 4, that can be read independently). The 6d construction leaves a choice of whether to construct boundaries or cross-caps of the 2d CFT, and of whether Z2 acts by re ection or the antipodal map on the ellipsoid. Before listing the four cases let us make some comments. In 2d CFT, boundary and crosscap Ishibashi states are formally related by an analytic continuation of a real cross-ratio to its opposite. The AGT correspondence relates cross-ratios to complexi ed gauge couplings, and this analytic continuation corresponds to turning on a theta angle # = . Note that the Sb4 integral of the topological term Tr F ^ F vanishes for vector elds that are invariant under the re ection or antipodal maps. The theta term must thus only be integrated over a fundamental domain of the Z2 action. To avoid dependence on the choice of fundamental domain, # must be invariant (up to the 2 periodicity) under # ! #, so # = 0 or . The four cases are as follows. 1. Boundary and re ection: this gives rise to a vector multiplet on HSb4 with Neumann boundary conditions, and on the CFT side to an identity brane. 3We did not nd in the literature a clear dictionary relating D-type and E-type Toda CFT to gauge theory. The main missing ingredient seems to be nding any Toda CFT three-point functions beyond the A-type case. { 5 { 2. Cross-cap and re ection: this gives the same vector multiplet, but with a discrete theta angle # = . On the CFT side it gives an unusual cross-cap state whose wavefunction is that of an identity brane. 3. Boundary and antipodal map: this gives an RPb4 vector multiplet and on the CFT side an unusual boundary state whose wavefunction (given by the RPb4 vector multiplet one-loop determinant) is equal to the cross-cap wavefunction. 4. Cross-cap and antipodal map: this also gives rise to an RPb4 vector multiplet, but the theta angles in the two cases must di er by . On the CFT side it gives a standard locus where codimension 2 or 4 operators are naturally placed; correspondingly they do not have the additional label of a primary operator. It would be very interesting to understand whether the unusual cross-cap and boundary states coming out of the 6d construction are singled out from the 2d CFT perspective. In the main text we only consider cases 1. and 4. All of the usual bells and whistles of the AGT correspondence can be inserted: defects of the 6d theory have the same 2d and 4d interpretations as before provided they are away from the xed point locus. We discuss these and a few other cases in section 5. On the CFT side, besides correlation functions of bulk operators in the presence of a boundary one can also insert boundary operators. Their e ect on correlation functions can be obtained by bootstrap methods and we nd cases that correspond to duality walls put on the equator of the 4d geometry. We were informed by Bawane, Benvenuti, Bonelli, Muteeb and Tanzini of their upcoming work [6] in this direction. The paper is organized as follows. In section 2 we set up Liouville/Toda CFT notations and explain the construction of boundary states. In section 3 we describe how 4d N = 2 SQCD on HSb4 and 2d CFT disk two-point functions correspond for various choices of boundaries. In section 4 we write the RPb4 partition function (4.2) of 4d N = 2 (Lagrangian) gauge theories as a combination of HSb4 and Sb3=Z2 localization results, then detail the correspondence between RPb4 and cross-caps. In section 5 we give generalizations: we explain how arbitrary Riemann surfaces with boundaries give 4d N = 2 quiver gauge theories, how to include the usual loop, surface, and domain wall operators, and discuss boundary-changing operators. We conclude with some outlook in section 6. Appendices collect interesting results we obtained in the course of our investigations. In appendix A we give a streamlined derivation of Liouville/Toda boundary CFT results that exclusively uses modular bootstrap of the annulus. From the Mobius strip bootstrap we derive the ADE Toda cross-cap wavefunction (A.40), which appears to be new. In appendix B we motivate the gluing procedure we use for the RPb4 partition function by checking that the same technique reproduces the RP2 partition function of [63], and we check Seiberg dualities. In appendix C we describe quotients of S b4 consistent with localization; this suggests possible generalizations of our work. In appendix D we consider hypermultiplet determinants on RPb4. { 6 { Liouville/Toda boundary CFT We brie y review here the construction of boundary states for ADE Toda eld theory. Branes in Liouville CFT were classi ed in [35, 93]. For Toda CFT they were classi ed in [34] (see also [33, 83]) and found to be in one-to-one correspondence with representations of the extended algebra de ning the theory, following the intuition of the Cardy construction. We describe the modular bootstrap of these boundary states in appendix A. In this subsection we cherry-pick details that are essential for the main text and describe how the results reduce to more familiar Liouville ones. The Toda eld theory is a nonrational CFT with diagonal spectrum. It depends on a Lie algebra g that we take to be simply-laced, and it reduces to Liouville theory when g = su(2). We let t denote the Cartan algebra of g and r = rank g. The theory has an extended W-algebra symmetry generated by currents (2.1) (2.2) W (si); i = 1; : : : ; r with spins si = li + 1 determined by the set of exponents flig of the Lie algebra. This includes the stress tensor, of spin s = 2. The algebra of these currents has a central charge c = r + 12hQ; Qi, where we de ned the background charge to be Q = (b + 1=b) , where is the Weyl vector of the algebra. In the Lagrangian formulation the theory consists of a scalar eld in the Cartan subalgebra t of g, with exponential potentials. In terms of the rescaled cosmological constant ^ = ( (b2)b2 2b2 )1=b the theory is invariant under (b; ^) ! (1=b; ^). Let us summarize the di erent W-algebra representations that we use in this work,4 classifying the primary operators of the CFT. The W-algebra has non-degenerate, semidegenerate, and fully degenerate representations with various amounts of null vectors. Non-degenerate representations. These representations have no null states. In the Lagrangian formulation they are constructed as eh ; i in terms of a vector = Q + m with m 2 it. Their conformal weight is ( ) = ( ) = h2Q ; i=2. Representations whose momenta m are related by the Weyl group are identi ed up to a re ection amplitude as (2.8). Semi-degenerate representations. They are labeled by a choice of a full-rank (hence simply-laced) subalgebra h g and two representations R1, R2 of h with highest weights 1 , 2 . We denote by I the set of simple roots of the subalgebra h and indicate group theory quantities related to this algebra by an index I. The momentum of these states is m = ( I + 1)b ( I + 2)=b + m~; (2.3) where m~ is a vector with imaginary components that is orthogonal to all roots in I. 4Yuji Tachikawa pointed out to us that the D-type and E-type 6d N = (2; 0) theories have codimension 2 operators that fall outside this classi cation. These should correspond to other W-algebra representations but the dictionary does not seem to be available in the literature. { 7 { Wn(s) ( 1)sW (s) n k ii = 0: jBi = 1 Z jWj = R d(Im ) means an integral over all imaginary Q with standard real measure in the basis of simple roots ej : for any function F , where C is the determinant of the Cartan matrix, since that matrix Cij = hei; ej i is the metric on the Cartan algebra in the coordinates x. While we divide all our integrals by the order jWj of the Weyl group to reduce them to a Weyl chamber (the integrands are always Weyl-invariant), we keep this factor explicit in equations such as (2.6). Let us explain the appearance of = 2Q in (2.6). Vertex operators with momenta related by Weyl symmetry are proportional: V = Rw( )VQ+w( Q) for some re ection amplitude Rw( ) that we do not need explicitly [28]. The two-point function on the plane is normalized as (2.5) (2.6) (2.7) (2.8) Fully degenerate representations, namely semi-degenerate representations with h = g. They are labeled by two highest weights 1 , 2 of representations R1 and R2. The momentum is discrete, m = ( + 1)b ( + 2)=b: (2.4) When R1 and R2 are the trivial representation ( 1 = 2 = 0) this operator is the identity. The Virasoro algebra (corresponding to g = su(2)) only has fully degenerate and nondegenerate representations.5 To de ne boundary states associated to holes in the Riemann surface, we need to namely formal sums of descendants of a primary j i obeying6,7 specify the boundary condition on the currents. We take untwisted Ishibashi states k ii, Branes are linear combinations of Ishibashi states with non-degenerate momenta, hV 1 (z1)V 2 (z2)i = 1 X jz1 z2j4 ( 2) w2W Rw( 2) ( 1 Q) + w( 2 Q) (2.9) instance = L(QL squared he; ei = 2. ( 1)senrW (s) by Wn(s) 2Q ( 1)s ^ W (s) n are replaced by ^(2Q ). 5It is worth noting a discrepancy between standard Liouville momenta L and Toda notations: while = Le where e is the positive root of su(2), the parameters Q are related by Q = QLe=2. Hence for L) = h ; 2Q i=2 where the overall factor of 2 is due to roots having length6The insertion of this Ishibashi state gives a hole of unit size in the plane. A hole of arbitrary size er is obtained as k iir = e2rL0 k ii by applying the dilatation operator L0 + L0. It obeys e nrWn(s) n k iir = 0. The size of the hole simply alters cross-ratios. 7Ishibashi states can be twisted by an automorphism ^ of the W-algebra, replacing the condition (2.5) k ii = 0. In the Schottky double described in section 2.2, the image momenta { 8 { where R d(Im ) ( function on the plane with a unit-sized hole (2.10) is8 0)f ( ) = f ( 0). From (2.6) we then deduce that the one-point h V (z)iB = j1 zzj 2 ( ) B ( ): (2.10) B This justi es the convention of using in (2.6). By a conformal transformation, the disk one-point function is also proportional to ( ). In particular, ( ) must transform like V (2.8) under the Weyl group. It is straightforward to check that the wavefunctions below B have this property. tion result from adding to the identity brane a Verlinde loop operator along the boundary. The wavefunction of the identity brane is (A.25) vertex operator labeled by (R1; R2) we obtain the degenerate brane (R1;R2)( ) = 1( ) R1 (e2 iba) R2 (e2 ia=b) in terms of characters of g. The semi-degenerate brane is given by ;R1;R2 ( ) = 1( ) X Finally, the analogue of the FZZT brane associated to a non-degenerate operator is given by ( ) = 1( ) X w2W Qe>0 e2 ihw(m);ai On a related note, we give the cross-cap wavefunction in (4.9). All of the wavefunctions obey ( ) and are invariant under Weyl group actions on their labels. Like vertex operators, they are multiplied by a re ection amplitude when the Weyl group acts on . For the Liouville theory, these branes are the ZZ branes [93] and FZZT branes [35] found long ago. We parametrize Liouville momenta as the well-known results (with a pleasant normalization thanks to our choice of measure): = 12 (b + 1=b) + iP and reproduce Liouville(P ) = (^b2b 2=b) iP 1 ZLZio(umvi;lnle)(P ) = 1 (b + 1=b) + iP 1 2 2 iP (1 2iP b) (1 2iP=b) sinh(2 mP=b) sinh(2 nbP ) sinh(2 P=b) sinh(2 bP ) ; LFiZoZuTvi(lsle)(P ) = (^b2b 2=b) iP (1 + 2ibP ) (1 + 2iP=b) 1 cos(4 sP ): 2 iP B(0) 6= 1. Another common choice that we will not use is to normalize h1iB = 1. 8We use the natural normalization coming from the modular bootstrap, which is such that h1iB = ; 2 { 9 { (2.11) (2.12) : (2.13) (2.15) (2.16) (2.17) Correlators on a surface with boundaries can be computed using the method of images. The surface is written as a quotient Riemann surface. Each vertex operator on = b=Z2 of its Schottky double, a closed is split into two chiral operators on b in conjugate representations of the W-algebra, labeled by momenta and 2Q . The original correlator is then a chiral correlation function on b, integrated over the internal Liouville/Toda momenta with certain weights. For each brane B, the momentum that ows along the corresponding tube from the Riemann surface to its double is integrated with a weight B(2Q ) given in the last subsection. In the normalization of conformal blocks (chiral correlators) we include the square root of an OPE coe cient at each trivalent vertex. Since each trivalent vertex appears twice in the Schottky double (with all momenta mapped as ! 2Q ), this is equivalent to the single OPE coe cient that one would obtain when computing the correlator on using a bulk OPE. Our formulas thus only involve these normalized conformal blocks and the boundary (and later cross-cap) wavefunctions. The simplest observable with a boundary is the one-point function (2.10). We now focus mainly on the simplest non-trivial observable: the two point function in the region z j j 1 with a single boundary along jzj = 1. This readily generalizes to higher genus or to more boundaries. For example a correlator on a cylinder (genus-zero Riemann surface with two boundaries) has as its Schottky double a chiral correlator on a torus. In this simple case of a single boundary, for each operator V (z), we consider a chiral correlator on the plane including the image insertion V2Q This construction is shown in the left of gure 2. It gives at the image position z? = 1=z. V 1 (z1; z1)V 2 (z2; z2) = V 1 (z1)V 2 (z2)V2Q 1 (1=z1)V2Q B = j 1 z2z2j2 1 2 2 1 Z j 1 z1z2j4 2 jWj 2 (1=z2) chiral B B is the wavefunction of the boundary state. To avoid in nite fusion multiplicities we restrict concrete calculations to Liouville CFT or to A-type Toda CFT with one momentum 2 = !1 proportional to the rst fundamental weight !1, hence in particular semi-degenerate. As summarized in the right panel of gure 2, we are computing the two-point function in the channel that involves the OPE of the two operators and the OPE of their images. The result must be equivalent to the one computed using the OPE of an operator with its image. This constraint is one of the sewing (crossing symmetry) constraints that are known [66] to be su cient to imply consistency of the CFT with boundaries. Cardy's condition, which we use in appendix A to derive wavefunctions, is insu cient to imply full consistency in the most general case. jBi 1 2 2 1 1 jBi HJEP12(07)9 In this section we explain the dictionary between Liouville/Toda boundary states and 4d described in the introduction, this comes from an orbifold (Sb4 N = 2 gauge theories. We rst reproduce the identity brane B1 from gauge theory. As b)=Z2 of the 6d theory, where Z2 acts by a Z2 involution of b such that b=Z2 is the surface with boundaries that we are interested in, and by x0 ! x0 on the ellipsoid (1.1). We analyze in section 3.1 how the latter identi cation a ects the 4d N = 2 vector multiplet: this gives a vector multiplet on the squashed hemisphere HSb4 with Neumann boundary condition. In section 3.2 we write the hemisphere partition function of [43] (generalized from b = 1 to b 6= 1). We match in section 3.3 the HSb4 partition function with Neumann boundary conditions to the wavefunction of the identity brane. This is the main result of this section: the identity brane B1 corresponds to a vector multiplet with a Z2 identi cation that makes it e ectively live on HSb4 with Neumann boundary. All other elds live on the whole ellipsoid. As a concrete example we match the partition function of 4d N = 2 SU(N ) SQCD with Nf = 2N fundamental hypermultiplets of pairwise equal masses mj to a disk two-point function in AN 1 Toda CFT: ZHSQSb4C;NDe;Numf=an2nN m1; m1; : : : ; mN ; mN ; q) = hV (z1)V !1 (z2)iB1 (3.1) up to unimportant factors. The masses mj are encoded in the labels and !1 = 2 1 N (b + 1=b) i P j mj !1 of primary operators on the Toda side, where hj are weights of the fundamental representation of SU(N ). The cross-ratio of their positions z1, z2 and their Z2 images z1? and z2? is the instanton-counting parameter q. = Q + i P j mj hj As we review in appendix A, all boundary states are obtained from the identity brane (also called the ZZ brane) by inserting Verlinde loops along the boundary. The gauge theory interpretation of these is already understood [25] and we recall it in section 3.4. We end with a brief discussion of S-duality of boundary conditions in section 3.5. ( ; ; '; ) $ ( ; ; '; ): where 0 '; 2 are periodic, 0 6d orbifold is located along the equator and 0 =2. The singular locus of the = =2 of Sb4, and the Z2 identi cation acts as Since d ! under this Z2 while other components do not. d and d , d', d are unchanged, A and F for The Z2 changes chirality hence must exchange gauginos and up to some invertible 2 2 matrix. Supersymmetry then constrains this matrix: the supersymmetry variation 2 f ; '; g change sign sphere HSb4 with appropriate boundary conditions at x0 = 0. We The rst step is to rewrite elds on the 4d orbifold Sb4=Z2 as elds on the squashed hemind that the correct prescription is to take Neumann boundary conditions, as de ned for example in [25, 48]. In a 4d N = 2 gauge theory the vector multiplet is composed of vector multiplet : A ; A ; A; = 1 + i 2; 2 to ease comparison with [43]. Following [50] we , normalized by pg . The Killing spinors used in [43, 50] do indeed obey (3.7).9 The supersymmetry variations = A A $ A A = so the Z2 exchanges $ . We get the following Z2 identi cation together with (3.6) up to gauge transformations, of course. This identi cation can be twisted by an outer automorphism ^ of the gauge algebra, changing for instance the last equation above to 9In the 6d setup [20] with partial topological twist along the Riemann surface b, supercharges are constant scalars on b hence are Z2-invariant regardless of the Z2 action on b. ( ; ; '; ) = ^ ( ; ; '; ) . We only consider the untwisted case: the twisted case should correspond to twisted boundary states discussed in footnote 7. The (untwisted) identi cation implies Neumann boundary conditions along the equator, F j = =2 = 0; 1 = =2 = 0; i 3 A A = =2 = 0; (3.9) where j 2 f ; '; g, together with in nitely many conditions on higher derivatives. These match with the Neumann boundary condition (5.9) of [43, v2] when their a is set to zero and the relation between their and 1 is taken into account. The way Neumann boundary conditions are conventionally de ned [25, 48] is (4 =g2)F i + (#=(2 )) 12 ijkF jk = 0 where i, j, k are transverse to the equator and # is the theta angle. We take # = 0, consistent with the fact that for CFT with boundaries the cross-ratio is real and of a de nite sign. In fact, under Z2 the theta angle changes sign, so # = 0 and # = are both sensible choices (integrated over a hemisphere); on the CFT side this changes the sign of cross-ratios, interchanging cross-cap and boundary states. On the hemisphere 0 =2 the variation of the action, both the original and the localization term, vanishes up to boundary terms. As shown in [43], the boundary terms vanish upon imposing the Neumann boundary conditions (3.9), but also upon imposing Dirichlet boundary conditions, namely the supersymmetric completion of i 3 A + A The sign change compared to (3.9) leads to Aj j = =2 = 0 in some gauge and = =2 = 0. Fij = =2 = 0; 1 = =2 = constant; 2 = =2 = 0; i 3 A + A = =2 = 0; (3.10) where i; j 2 f ; '; g. One may wonder whether the Dirichlet boundary conditions can be 4 obtained from a Z2 orbifold of the whole Sb . To obtain Aj j = =2 = 0, the Aj components should change sign under Z2 and A not. Then the rst two terms in F j [A ; Aj ] are invariant under ! ! @ ) but the commutator changes sign. Therefore we cannot get a consistent identi cation of the eld strength unless the gauge group is abelian. (In the abelian case the Dirichlet boundary arises from the orbifold (3.8) twisted by charge conjugation.) From the CFT side this impossibility may be related to Cardy's condition. We check below that a hemisphere with Neumann boundary conditions corresponds to the identity brane, and Dirichlet boundary conditions to an Ishibashi state (up to normalization). As we review in appendix A Ishibashi states do not satisfy Cardy's condition and therefore do not correspond to physical boundary conditions. It would be very interesting to understand Cardy's condition from the 4d point of view in a more general setting. Before moving on we mention what changes for the Z2 action that creates an unoriented surface RPb4, used in section 4. In this case the points that are to be identi ed are HJEP12(07)9 The identi cation is consistent with supersymmetry if the elds transform as10 Partition function on hemisphere The partition function of a 4d N = 2 gauge theory on a squashed hemisphere di ers from the one on the full ellipsoid in two ways. On the one hand, while on an ellipsoid the instanton partition function appears in pairs as Zinst(q)Zinst(q) due to the presence of instantons localized at the north pole and anti-instantons localized at the south pole, for the squashed hemisphere a single instanton contribution Zinst(q) appears, localized at a single pole.11 At the same time, the classical action is half of that on the ellipsoid.12 Both are expected if we want this partition function to match with a CFT correlator on a disk, as that is given by a chiral correlator on the Schottky double and therefore a single conformal block appears for the channel associated to the boundary state. On the other hand, the one-loop determinants are di erent. The one-loop determinants for Dirichlet and Neumann boundary conditions are respectively ZHveScb4t;oDririchlet = Y ZHveScb4t;oNreumann = Y e e 1 b(b + 1=b 1 b(he; i i) he; i i) e>0 2 he; i ib(b 1=b)he;i i = Y Q (1 + b 1he; i i) = Y e>0 Q 2 he; i ib (b 1=b)he;i i (1 b 1he; i i) b(he; i i) ; (3.13) b(he; i i) ; (3.14) where products range over all roots e of G or only positive roots e > 0. In the Neumann case the Coulomb branch parameter is integrated over the Cartan algebra t with the measure (2.7) normalized to match with CFT. The usual Jacobian factor (Vandermonde determinant) converting the integral from g to t is included in the one-loop determinant displayed here. The properties of the functions appearing in these one-loop determinants are as follows. b is the Barnes double gamma function; it has poles at bZ 0 + b 1 under b ! 1=b and obeys b(x+b) = b(x)p2 bxb 1=2= (bx) and b((b+1=b)=2) = 1; Z 0, is invariant b(x) = 1=( b(x) b(b + 1=b x)) has zeros at x 2 (bZ 0 + b 1 Z 0) [ (bZ 1 + b 1 Z 1); 10Here we use [43]'s conventions for the Killing spinors; the identi cation would have opposite sign with conventions in [50]. 11While one may select a given instanton sector (power of q) by imposing the value of the Chern-Simons term integrated along the boundary, this is unnatural since it is a nonlocal boundary condition. 12If one localized on the region < 0 instead of < =2, one-loop determinants would be unchanged and the classical action would be proportional to the volume of the region. HJEP12(07)9 Sb(x) = at x 2 bZ 1 + b 1 Z 1. x) used later has poles at x 2 bZ 0 + b 1 Z 0 and zeros invariance. The b = 1 result from [43] in the Dirichlet case has zeros of order n 1 For b = 1 these one-loop determinants were computed in [43]: these authors used spherical harmonics to nd eigenvalues of the operator associated to quadratic uctuations of the localizing term in the action, and restricted to modes that are consistent with the boundary condition or equivalently with the Z2 identi cation (in the Neumann case). The precise forms of the localizing term and quadratic uctuations are given there. To nd the generalization to b 6= 1 we kept track of zeros and imposed b ! b 1 and ! at he; i i = n for integers n 2 and positive roots e. Given the ellipsoid answer [50] these multiple zeros should split into two lattices he; i i 2 t(b) + bZ 0 + b 1 Z 0 for some function t(b). It is natural to expect points in the lattice and its opposite to di er by multiples of b and b 1, so 2t(b) = kb + `b 1 for some integers k and `. Using t(b) = t(b 1) and t(1) = 2 xes k = ` = 2, telling us the set of zeros of (3.13). The normalization is not xed by these arguments; we choose it to have two properties (besides being nonnegative, reducing to the known b = 1 case and being ant). First, the determinant for the Neumann case is equal to the Dirichlet one times an Sb3 vector multiplet determinant Qe 1=Sb(he; i i). Second, this 3d determinant times the square of the Dirichlet one gives the vector multiplet determinant on the whole ellipsoid S4. As discussed more near (4.5), these two properties are justi ed by gauging along Sb3 the global symmetry of a vector multiplet on one or two squashed hemispheres with Dirichlet b ! invariboundary conditions. Let us quote the result for the one-loop determinant for matter elds, even though we do not use it in our work. Supersymmetry leads to mixed Dirichlet/Neumann boundary conditions. Then the one-loop determinant of a hypermultiplet in the representation R of G is ZHhySpb4er = Y w2weights(R) b 21 (b + 1=b) + hw; i i 1 We now match the gauge theory results to an identity (ZZ) brane. The momentum integral translates through = Q+i to the Coulomb branch integral of the gauge theory partition function with Neumann boundary conditions. In the vector multiplet determinant (3.14) we recognize the wavefunction 1( ) of the identity brane The factor could be absorbed into a normalization of vertex operators that makes them Weyl invariant. In correlators it cancels with parts of the OPE coe cients that depend separately on each momentum, speci cally the numerator of the DOZZ formula or its Toda CFT analogue. For concreteness consider the disk two-point function (3.1) of AN 1 Toda CFT, with a non-degenerate momentum 1 and a semi-degenerate momentum !1. The result of the boundary state formalism translates to gauge theory as (3.17). Besides an instanton partition function and classical contribution, the conformal block contains a normalization since we included in it a single OPE coe cient C 1; !1 . From the AGT correspondence on Sb4 we know that this OPE coe cient is equal to hypermultiplet determinants on Sb4, up to the factor discussed below (3.16). More precisely, this is the one-loop determinant on Sb4 of N 2 hypermultiplets that transform in the bifundamental representation of the gauge group SU(N ) and a avor symmetry group U(N ). They can alternatively be described as pairs of hypermultiplets on Sb4 pairwise-identi ed at the boundary. The real cross-ratio q translates to an instanton counting parameter. Altogether, hV 1 (z1)V !1 (z2)iB1 = = jWj 1 Z jWj d(Im ) 1( )F !1 2Q 1 2Q !1 ; q 1 d ZHveScb4t;oNreu.( )ZSh4yper( )Zinstanton( ; q); b (3.17) HJEP12(07)9 where we omitted some powers of jz2 z2?j and jz1 z2?j given in (2.18). We get the partition function of a theory constructed as follows: start with N 2 hypermultiplets living on Sb4 (this is the theory associated to a three-punctured sphere with momenta 1 , !1, ), then gauge an SU(N ) symmetry using a vector multiplet that has the Z2 identi cation (3.8). In other words, gauge it using a vector multiplet living on HSb4 with Neumann boundary conditions. Alternatively the whole theory can be considered on HSb4 by describing the hypermultiplets as a pair of hypermultiplets on the squashed hemisphere. Finally, consider the gauge theory we just described on the squashed hemisphere, namely a pair of identical-mass hypermultiplets coupled to a vector multiplet, but change the vector multiplet boundary conditions to Dirichlet. Up to a normalization, the partition function is a disk two-point function with Ishibashi boundary state: ZHSb4 = F !1 2Q 1 2Q !1 ; q = hV 1 (z1)V !1 (z2)iIshibashi( ): 1 (3.18) This is a special case of the CFT interpretation of Dirichlet boundary conditions (3.10) as selecting internal momenta of conformal blocks [25]. 3.4 Other branes As we review in appendix A the Cardy construction of boundary CFT implies that a general boundary state labeled by is obtained from the identity brane by acting with a Verlinde loop operator along the boundary. The gauge theory counterpart of Verlinde loops was worked out in [25] and here we present their results for completeness. From the 6d orbifold perspective, Verlinde loops correspond to the insertion of defects on the singular locus Sb3 S1. For fully-degenerate the defects are codimension 4 defects of the 6d theory and span a great circle of Sb3, times the S1 boundary; they reduce to Wilson loops in 4d. Otherwise they are codimension 2 and span the whole singular locus; they reduce to symmetry-breaking boundary conditions in 4d. Writing the squashed sphere in coordinates b2 x20 + x21 + x22 + b2(x23 + x42) = 1; (3.19) the supercharge used for localization squares to Q2 = R12 + R34; where Rij corresponds to a rotation in the (xi; xj ) plane. These rotations admit two invariant circles, and we put there two supersymmetric Wilson lines in representation R1 and R2 of g: R1 R2 along along From [78] we know the e ect of these insertions is to include the character of these representations inside the partition function HJEP12(07)9 R1 (e2 iba) R2 (e 2b i a): Since we know the HSb4 result with Neumann boundary conditions reproduces the identity brane, these insertions add the factors necessary to turn the boundary state into a completely degenerate one, as in equation (2.12). Another type of defect we can insert along the equator is a symmetry breaking wall. This was de ned for N = 4 SYM in [48] and extended to N = 2 in [25]. They have the property of breaking the group G along the equator to a subgroup H by requiring the vector multiplet to take values in h rather than g at the wall. This can be done in a way that preserves the 3d N = 2 supersymmetry at the equator. In [25] it is argued that the action of this symmetry breaking wall is to introduce the following term in the partition function (when H has full rank) X w2W=WH Q e2 + w( +H ) e2 ihw( m~);ai : (3.23) From the gauge theory point of view the parameter m~ consists of Fayet-Iliopoulos terms for the U(1) factors localized at the wall. Semi-degenerate momenta are parametrized by H and m~ but also representations R1 and R2 of H; correspondingly, to reproduce the action of a Verlinde loop operator one needs to insert along the symmetry breaking wall a pair of Wilson lines in representations R1 and R2 of the residual symmetry H. Combining these with the identity brane wavefunction yields the semi-degenerate wavefunction (2.13). 3.5 S-duality To conclude this section, we indicate how to use CFT results to probe how boundaries transform under S-duality for some N = 2 gauge theories, as thoroughly studied in [49] for N = 4. We limit ourselves to SU(2) for this discussion, but keep notations adapted to Toda CFT (see footnote 5). In this case, we have shown above that ZHNSeub4mann = Z 1 d(Im ) 1 p 2 ZZ( ) F 21 21 ; q : (3.24) (3.20) (3.21) (3.22) The 4d N = 2 theory on HSb4 is SU(2) Nf = 4 SQCD where hypermultiplets have pairwise equal masses. Then we can use an identity found in [93] between Virasoro conformal blocks (where = 0 denotes the identity)13 ZHNSeub4mann(q) = Z d(Im ) p 2 ZZ( )F 2 2Q 1 2Q 2 ; q 1 (3.25) = F =0 2Q 1 2Q 1 2 q = ZHDSirb4ichlet(1 q): In terms of the theories living on a squashed hemisphere this identity relates by S-duality q ! 1 these two channels. q a gauge theory with Neumann boundary conditions with the same gauge theory but with Dirichlet boundary conditions. See gure 3 on page 23 for a CFT depiction of The vacuum block is known explicitly in the semiclassical limit [32], and we can deduce from the CFT result (3.25) an explicit expression of the partition function on HSb4 with Neumann boundary condition in that limit. In gauge theory that limit is the NekrasovShatashvili limit in which the Omega deformation parameter 2 ! 0 with 1 xed since c 6 1= 2 ! 1 in the CFT. The analysis in [32] apply when 1 is heavy, meaning that 1=c is nite, and 2 light, meaning that 2 O(1) in the c ! 1 limit. This corresponds to taking all hypermultiplet masses to in nity, keeping their di erences nite. It would be interesting to understand purely from gauge theory why the simpli cation coming from combining (3.25) with the results of [32] occurs in this limit.14 An exhaustive classi cation of N = 2 boundary conditions and how S-duality acts on them is not the purpose of this paper and therefore we leave it for future work. Nevertheless we see that the relation with Liouville theory with boundaries might be helpful as a guiding principle. 4 Projective space and cross-caps fx 2 RPb4 j x0 = 0g where We outline in section 4.1 a method to localize 4d N = 2 gauge theories on RPb4 by splitting that space into a hemisphere HSb4 and a lens space Sb3=Z2. We con rm our procedure in appendix B by reproducing known RP2 localization results from an analogous splitting. For completeness we hazard a conjecture in appendix D for the hypermultiplet one-loop determinant. These parts can be read independently from the rest of the paper. In section 4.2 we discuss the AGT correspondence for a CFT two-point function with a cross-cap; this con rms RPb4 results other than the hypermultiplet one-loop determinant. All of the spaces are of unit size and squashed in a U(1) U(1) invariant way: Sb3=Z2 = RPb4 = x20 + b 2(x21 + x22) + b2(x23 + x42) = 1 =(x x): (4.1) 13These authors also have a similar identity for the degenerate brane (2; 1) instead of the identity brane, which indicates that Wilson lines map to 't Hooft lines, as expected under S-duality. 14Some aspects of Liouville theory has been recently related to quantum chaos in 2d [61, 70, 87, 92]. One might wonder whether the behavior of the out-of-time-ordered four-point function has a gauge theory interpretation. The 4d N tion reads Of course, Sb3=Z2 is neither a boundary nor a singularity of RPb4 and we are simply cutting along Sb3=Z2 to reuse known localization results. In this section we denote by ; ; : : : the 4d indices and i; j; : : : the 3d indices. = 2 theory we consider has gauge group G, and hypermultiplets in a representation R (more generally, half-hypermultiplets). Unsurprisingly its partition funcZRPb4 (q) = 1 X Z jWj y2T;y2=1 t d ZRvePcbt;or(4.6)ZRhyPpb4e;r(mDu.6lt)ipletq 12 Tr 2 4 Zinstanton(q); (4.2) with a sum over the holonomy y along the nontrivial element of 1(RPb4) = Z2, with y in the Cartan torus T , and an integral over the Cartan algebra t whose measure is de ned in (2.7). The classical action is half of the one on the sphere, and there is only a single instanton partition function, normalized as Zinst(q) = 1 + (: : :)q + : : : 15 Contrarily to Sb4, to HSb4, or to at space, R Tr(F ^ F ) vanishes on RPb4 hence there can be no continuous theta term; the instanton counting parameter q = exp( 8 2=g2) for each simple group factor is thus real.16 This is a manifestation of the fact that only parityinvariant theories can be put on nonorientable surfaces, unless one allows nonconstant couplings or domain walls. One can also turn on an FI parameter for each U(1) factor in G but they play no role in our paper so we do not include such terms. Our approach is not sensitive to phases that only depend on the holonomy y, and we add such phases \by hand" when matching with Toda CFT. In that context we also need to couple the RPb4 b elds to hypermultiplets on S4. These can also be described as a pair of hypermultiplets on RPb4 with a non-trivial avor holonomy yF = ( 01 10 ), namely a periodicity that exchanges the two hypermultiplets when going along an antipodal loop. 4.1 Partition function on projective space The locus Sb3=Z2 is preserved by a 3d N = 2 subalgebra of the 4d N = 2 supersymmetry algebra so the restrictions to Sb3=Z2 of 4d elds and (all of) their normal derivatives decompose into (an in nite set of) multiplets of this subalgebra. Most importantly the 4d N = 2 vector multiplet with bosonic components (A ; +i ) includes (see for instance [25, 27, 36]) a 3d N = 2 vector multiplet whose bosonic components are Ai and a real (i.e. Hermitian) scalar ; a 3d N = 2 adjoint chiral multiplet whose bosonic components are the real scalars A4 and ; in nitely more multiplets involving normal derivatives of 4d elds. This decomposition di ers from that of a 3d N = 4 vector multiplet into vector and chiral multiplets in the way fermions are arranged. Let Q be the RPb4 projection of the supercharge used on Sb4 by Pestun [78]. Its restriction to Sb3=Z2 is the supercharge used in localization calculations [1, 14, 38, 59, 60, 72]. Add to the 4d action the (Q-exact and Q-closed) localization term used in the latter works, 15Note that this di ers from the normalization we chose for the conformal blocks in previous sections. 16The discrete theta angle # = ips the sign of q, which on the CFT side exchanges boundary and cross-cap Ishibashi states. As brie y explained in the introduction this corresponds to di erent quotients of the 6d setup. way ZB1B2 ( ) = TrHB1B2 e2 i (L0 2c4 ) = ( ); where the trace is taken over the open-string states propagating between the boundaries. When boundary conditions preserve a diagonal subalgebra of A A this Hilbert space is a direct sum HB1B2 = L N B1B2 V . Each term contributes the corresponding character ( ) with a multiplicity N B1B2 . When the spectrum is discrete (in particular in RCFTs) these multiplicities must be positive integers. When the spectrum is continuous, the sum is replaced by an integral and N becomes a density of states. Alternatively one can quantize the theory along the closed channel, and rewrite ZB1B2 as a cylinder amplitude between two boundary states jB1i; jB2i in Hclosed. This gives hB1je i = (L0+L0 1c2 )jB2i = ZB1B2 ( ) = X N B1B2 ( ): The goal of Cardy's construction is to classify states jBi such that multiplicities of the spectrum between two branes are integers. The rst step consists of nding a proper basis to expand the boundary states, such that they are invariant under conformal transformations that preserve the boundary. If the boundary is placed on the real axis then this condition means classically that W (z) = ^ W (z) ; z = z; where W , W are the currents generating the algebra A A. Here, ^ denotes an automorphism of the algebra that leaves the stress-tensor invariant. In terms of modes in the closed channel this condition translates to Wn(s) ( 1)hs ^ W (s) n jBi = 0; X fkg k ii j ; ki j ?; ki: where hs is the conformal dimension of W and s labels the generators of A. For ^ = id a basis of solutions were found by Ishibashi [58], labeled by representations V are given by formal sums over an orthonormalized basis of descendants in V , of A. They The contribution of this state to the partition function (A.2) is proportional to the character of the corresponding representation V . Namely, it is hh k e2 i( 1= )Hclosed k ii = h j i ( 1= ); (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) R( ) = R( ?). where Hclosed is the closed-channel Hamiltonian. The two-point function is proportional to Kronecker/Dirac delta as appropriate: h j i = hV ? V i = R( ) . We deduce { 32 { Z Boundary states can then be expanded in Ishibashi states as22 jBi = ( ?) 2S d B R( ?) k ii; that is; hBj = Z 2S d B ( ) R( ) hh k ; where the boundary state wavefunction B characterizes the boundary condition in the closed channel. The normalization by R( ) = R( ?) ensures that the disk one-point function and the one-point function with a hole are equal to the wavefunction: hV jBi = hBjV i = B ( ) up to cross-ratios. Then Cardy's condition (A.2) reads 2S d B1 ( ) B2 ( ?) R( ) ( 1= ) = X N B1B2 ( ): (A.8) When A is the Virasoro algebra, ( ) = R 2S d S ( 1= ) so by linear independence B1 ( ?) B2 ( ) R( ) = X N B1B2 S : (A.7) (A.9) (A.10) modes W0(s). condition implies For the W-algebra (for Toda CFT) the S-matrix is far from being xed by how it acts on characters since those are not linearly independent. However, there is a concrete conjecture [25] and we assume that (A.9) holds. Their formula is supported by a relation with gauge theory but it would be valuable to check it for characters re ned by the insertion of We assume that there exists a unique brane such that the open string spectrum between two such brane consists only of the identity. By uniqueness it is invariant under ?. Cardy's 1( ) = pS1 R( ) Consider next a brane jBi such that the spectrum of an open string between that brane and the identity brane is discrete. Cardy's condition in the form (A.9) implies that the wavefunction is an integer linear combination of basic branes B ( ) = X N 1B ( ); ( ) = S R( ) 1( ) = S p p S1 R( ) S S1 = 1( ) : (A.11) These basic branes are labeled by representations V of A. The second expression for is the well-known Cardy ansatz (R( ) is a phase). The third shows that the brane can also be obtained from the identity brane by inserting a Verlinde loop [89] labeled by along the ( ) boundary. Indeed, the ratio of S-matrix elements D = S =S1 is precisely the action of a topological defect or Verlinde loop [25]. This observation makes it easier to compare boundary states in Toda to 4d gauge theory calculations later on. 22In ADE Toda CFT labeled by g, the spectrum is a Weyl chamber S = t=W of the Cartan algebra, and we use the measure (2.7) on that Weyl chamber rather than integrating throughout t with a factor of 1=jWj. The two-point function R( ) given in (A.24) is called a maximal re ection amplitude. In the main text we use transformation properties under the Weyl group to recast B for non-degenerate momenta is 2Q ( ?)=R( ?) = B(2Q ), which Cardy's solution also speci es multiplicities in the Hilbert space of an open string stretching between two branes. For the Toda CFT we check in section A.4 that they are integers when the spectrum is discrete, in other words when one of the branes is labeled by a fully degenerate representation (see de nition in section 2). Using the explicit solution (A.11), equation (A.9) takes the form of the Verlinde formula [89] and therefore the degeneracy of the representation between branes B 1 and B 2 is equal to in cases where Verlinde's formula holds. the fusion rule coe cient, N B 1 B 2 = N 1? 2 , which is automatically a nonnegative integer A.2 Representations and characters First we give a brief summary of highest-weight representations of W-algebras, which classify the primary operators of ADE Toda CFT. We follow the notations of [25]. The theory is labeled by a simply-laced Lie algebra g (for Liouville CFT, sl2). It has central charge c = rank g + 12hQ; Qi, where Q = (b + 1=b) is a multiple of , the half-sum of positive roots. Rather than the cosmological constant we use ^ = (b2)b2 2b2 1=b. The theory has a symmetry (b; ^) ! (1=b; ^). First, we consider the generic (non-degenerate) representation which consists of the full Verma module of the W-algebra and has no null states. It is labeled by a vector (A.12) (A.13) (A.14) (A.15) hQ; Qi=2 q = exp 2 i ) where the momentum m is an imaginary vector in the Cartan algebra of g. The corresponding primary operator of ADE Toda theory has dimension ( ) = h ; 2Q i=2 = hm; mi=2. The character of this representation of the W-algebra is (in terms of ( ) Tr(qL0 c=24) = The W-algebra also has completely degenerate representations denoted = (R1; R2), labeled by two representations R1 and R2 of g with highest weights respectively 1 and 2 . They have one null vector for each positive root of g and their character is ( ) = (w) Q bw( + 1) ( + 2)=b( ) X w2W = q ( ) (c rank g)=24 = Q + m; where the Toda momentum is Q = b( + 1) ( + 2)=b. Finally, there are also semi-degenerate representations for which the null states are associated to a subset I 2 + of the positive roots (I = + for completely degenerate representations and I = ; for non-degenerate ones). Following [25] we only consider the case where I is the set of positive roots of a (simply-laced) Lie subalgebra g I full rank. These representations are labeled by gI , two representations R1 and R2 of gI g with with highest weights 1 and In terms of their Toda momentum 2, and an imaginary vector m~ orthogonal to all roots in I. Q = m = m~ b( I + 1) ( I + 2)=b where I is the half-sum of the roots in I, the character of these representations is m~ ; ;I ( ) = (w) Q+ m~ bw( I+ 1) ( I+ 2)=b( ) (w)qh I+ 1 w( I+ 1); I+ 2i: S = X e2 ihw(m);ai: w2W Between the identity (denoted 1) and non-degenerate: S1 = Y e>0 4 sin( bha; ei) sin( b 1ha; ei) : Between fully degenerate = (R1; R2) and non-degenerate: S = S1 R1 e2 iba R2 e2 ia=b : Here, R(ex) is the character of the representation R of g. It is de ned as a sum over weights of the representation R or, using the Weyl character formula, as a sum over Weyl group: X w2WI = In this subsection we list the elements of the S-matrix found in [25] giving the transformation of the W-algebra characters under a modular transformation, meaning ( ) = d S ( 1= ) with the real measure (2.7). This choice removes from S-matrix elements all powers of the Cartan matrix determinant and of the Weyl group order compared to [25], as well as the factor irank g (which was caused by a sign i ! i in their (3.30)). These matrix elements are building blocks of ADE Toda boundary states (see section A.4). The results can be checked using modularity of and a Gaussian integral. Denoting by jWj the order of the Weyl group and C the determinant of the Cartan matrix Cij , they are as follows. Between non-degenerate representations with momenta m = Q and a = Q: R(ex) = X 2weights(R) eh ;xi = P P w2W w2W (w)ehw( + );xi (w)ehw( );xi : (A.16) (A.17) (A.18) (A.19) (A.20) (A.21) (A.22) Between semi-degenerate (I; m~; ) and non-degenerate (note that Ri are characters of gI ): S(I;m~ ; ) = X From the general formula (A.10) in terms of S-matrix elements (A.20) and the re ection amplitude R( ) = (^b2b 2=b) 2h ;ai Y e>0 (1 + ha; eib) (1 + ha; ei=b) (1 ha; eib) (1 ha; ei=b) where a = Q, one works out the wavefunction for the identity brane, and other branes given below (2.11). This generalizes the su(2) (Liouville) [35, 93] and su(N ) [34, 83] results. While these wavefunctions are in principle only de ned in the fundamental Weyl chamber t=W, their analytic continuation obeys a re ection formula B ( ) = Rw( ) B Q + w( Q) for the same re ection amplitude Rw( ) as (2.10), independent on the brane. That independence is not surprising since the S-matrix elements themselves, analytically continued, are Weyl invariant. We now check Cardy's condition (A.9) when neither brane is the identity. To get a discrete spectrum we consider the case where one brane is labeled by a degenerate primary operator. Cardy's condition is that the multiplicities in this spectrum are integers. Using the explicit solution (A.11), the condition reads where we used S ? = S ? . The following relations are useful: S ? = S ? ; S(I;m~ ; ) ? = S(I?;m~ ?; ?) ; S ? = S ? : Consider two degenerate branes with labels 12 = (R1; R2) and 34 = (R3; R4). We rst work out the Verlinde formula in the form of [62]: R1 (e2 iba) R2 (e2 ia=b) R3 (e2 iba) R4 (e2 ia=b) = X N1k3N2`4S k` : k;` = X N 1 2 S Here we used S 12 =S1 = R1 (e2 iba) R2 (e2 ia=b) then rewrote the product of the g characters as a sum over single characters R1 (e2 iba) R3 (e2 iba) = X N1k3 Ri (e2 iba): (A.30) k This formula comes from the tensor product rules R1 of the Lie algebra g, hence N1k3 are nonnegative integers that can be worked out explicitly for any Lie group. This version of the Verlinde formula, with (R1; R2) ! (R1?; R2?) as per (A.28), expresses the degeneracies for the open string spectrum between branes 12 R3 = P k N1k3Rk for representations and 34, namely N 1?2 34 k` = N1? 3kN2? 4`. Now we repeat this procedure with the partition function in an annulus with boundary conditions associated with a degenerate brane = (R1; R2) and a non-degenerate brane labeled by its momentum m. The relevant Verlinde formula is now 1; 2 X e2 ihw(m);ai R1 (e2 iba) R2 (e2 ia=b); X e2 ihw(m);ai X N R11 N R22 e2 ibh 1;ai+2 ih 2;ai=b; 1; 2 w2W N R11 N R22 X e2 ihw(m+b 1+b 1 2);ai; N R11 N R22 Sm+b 1+b 1 2; : In the second line we used the de nition of the g characters as a sum over weights of the representation. The Cardy condition is obeyed because multiplicities N R are integers. Incidentally, when the degenerate brane is the identity, the annulus partition function reduces to a single character. A very similar story can be done for semi-degenerate representations and we will not present it: the procedure should now be clear. A.5 Cross-cap state We end this section by using the modular bootstrap to nd the wavefunction of ADE Toda describing the boundary state that reproduces the CFT on RP2. The analogue of Cardy's ( ) = ei ( ) P1 p S1 ; ( ) = 1( ) P1 : S1 (A.31) (A.32) (A.33) (A.34) ansatz in this case is where P character Finding the cross-cap state is thus equivalent to nding this element of the P-matrix. is a matrix that generates the S modular transformation on the Mobius strip ^ ( ) = e i ( ( ) c=24) ((1 + )=2) = ei rank g=24 e i ha;ai=2 ((1 + )=2)rank g where the second equality uses (A.14) hence only holds for a non-degenerate . Consistency on the Mobius strip requires that the overall phase ei ( ) has to be the same as the one for boundary states. Therefore we can write the cross-cap state as Using the following Gaussian integral (for p and s real) e ip2( 1= )=2 ((1 1= )=2) = Z ds e isp e i s2=2 ((1 + )=2) and its rank g analogue, we get the P-matrix between non-degenerate states, with normalization as in (A.18): We need to nd P1 . Just like what happens in Liouville, the character of a degenerate state is a sum of non-degenerate ones. For the identity character, HJEP12(07)9 P = X e ihw(m);ai: X 1( ) = (w) Q+bw( )+ =b( ): (A.35) (A.36) (A.37) (A.38) (A.39) (A.40) When we write the Mobius strip character, besides ! (1 + )=2 there is an overall phase which depends on w 2 W through the dimension (Q + bw( ) + =b) = h w( ); i. Explicitly, ^1( ) = X ( 1)h w( ); i (w) ^Q+bw( )+ =b( ): Now we can use the knowledge of the P-matrix between non-degenerate states to deduce ( 1)h ; i hw( );w0( )i (w)e ibhw( );ai (w0)e ihw0( );ai=b Finally, (A.34) tells us to multiply P1 by 1( )=S1 = 1= 1(2Q ) read from (A.25): ( ) = (^b2b 2=b) h ;ai Y e>0 (1 + bhe; ai) (1 + b 1he; ai) ( 1)h ; i hw( );w0( )i (w)e ibhw( );ai (w0)e ihw0( );ai=b: P1 = X w;w02W X w;w02W B RP2 partition function As a consistency check of the gluing prescription, we apply it to 2d N = (2; 2) gauged linear sigma models on RP2. We naturally obtain a \Higgs branch" expression of the RP2 partition function (with a sum over vortices), which must be equal to the \Coulomb branch" integral of [63]. We show the equality for a class of theories by summing up the residues in the Coulomb branch integral. The same two types of expressions have been found in various dimensions: 2d [9, 26], 3d [15, 31], 4d [22, 77, 80], and 5d [76]. The 2d N = (2; 2) gauge theories we consider have a connected gauge group G and are constructed from vector multiplets and from chiral multiplets in a representation R of G. Each U(1) gauge factor admits an FI parameter a. The vortex counting parameters za = exp( 2 a) are real, as there can be no continuous theta angle on RP2. Indeed, R Tr F = 0 on RP2 as the topological term is parity odd. We ignore discrete theta terms, which a ect how di erent topological sectors are summed up, because we neglect sign factors between topological sector. For simplicity we consider theories where FI parameters do not run, namely where the sum of charges under each U(1) vanishes. We make an important technical genericity assumption on the FI parameters, which excludes for instance SU(n) gauge theories for which beautiful dualities are known. Let g be the gauge algebra, t its Cartan subalgebra, a the abelian subalgebra of g, and be the collection of FI parameters. We assume that cannot be written as a positive linear combination of less than dim t weights of R and roots of G. Another technical assumption 2 a_ t_ is that the trial central charge is positive where R denotes the R-charge. In other words we assume that the theory has enough matter with low R-charges. This holds for instance when the theory has a phase in which it reduces to a non-linear sigma model with some nontrivial target space X; indeed c=3 = dimC X > 0. When applying zeta-function regularization, we omit powers of the cuto energy scale (in units of the inverse radius of RP2). They can be reinstated by the replacement c 3 = TrR(1 R) dim G > 0; (B.1) u 2 1 . (u) ! u (u); sin u ! sin u; sin 1=2 sin u 2 ; (B.2) similarly for cos, and multiplying factors linear in u by B.1 Localization by gluing as a boundary condition for the HS2 partition function. We cut RP2 = fx02 + x21 + x 22 = 1g=Z2 along the circle at x0 = 0, which we denote S1=2 = fx12 + x22 = 1g=Z2. The decomposition preserves 1d N = 2 supersymmetry and the 1 1d multiplets we encounter are vector, chiral, and Fermi multiplets. We rst localize using a Q-exact and Q-closed term supported on S11=2, then use the resulting constant 1d elds Supersymmetry equations set the 1d matter multiplets to zero. The gauge eld has a holonomy y around S11=2. Contrarily to higher dimensions, y needs not square to 1 since 1(S11=2) = Z rather than Z2. To simplify notations we assume G is connected (and compact), so y = exp( im) with 12 m 2 t= coweight lattice. The normalization is due to S11=2 having circumference . Supersymmetric con gurations for the S11=2 vector multiplet are parametrized by an additional constant real scalar that commutes with m hence can be diagonalized. Altogether this is (after gauge transformation), where is the u = m + i 2 tC=(2 ): (B.3) The 1d path integral restricts to an integral over this space with real dimension 2 dim t. More precisely one has to omit neighborhoods of singular points u = u at which (enough) chiral multiplets become massless. In the purely 1d partition function [21, 51, 52, 74], the integrand turns out to be an antiholomorphic total derivative @=@u, which by Stokes theorem reduces to an integral over a middle-dimensional cycle surrounding the singular points; the latter integral computes a Je rey-Kirwan residue (JK-res) of the integrand at these u . As explicited below we must assume that is a valid JK parameter. In 1d there is also a contribution from the region j j ! 1. The HS2 partition function will turn out to depend holomorphically on u and to decay at in nity, see (B.13), so the steps above all apply and the RP2 partition function reduces to a sum over the singular points. Localization on HS2 was performed with various combinations of Neumann and Dirichlet boundary conditions for the vector and chiral multiplets in [54, 57, 84]. However, none of these papers gives exactly the data we need. The restriction of one of the 2d vector multiplet scalars ( in the notations of the review [13]) to the equator is part of a 1d adjoint chiral multiplet hence vanishes. The HS2 BPS equations [84, v2, (3.40)] set F12 = = 0, which is incompatible with the nontrivial holonomy m thus would eliminate contributions from most u = u . However, these BPS equations were derived under a speci c reality condition on the auxiliary eld D and one should relax that assumption to make the contour pass through saddle-points. BPS equations on S2 with no reality condition on D are given in [13, v4, (2.9){(2.10)]; the only di erence for HS2 is the range of the latitude 0 radius to 1) that is a constant, = ( ) and that =2. There it is found (we set the HJEP12(07)9 (B.4) (B.5) (B.6) The gauge eld A( =2) = (0) d' at the equator must have holonomy y2 hence A = ( (0) ( ) cos ) d' : (0) = m + k for k 2 . Note that the Wilson line joining antipodal points along the equator is not gauge-invariant on the hemisphere hence needs not give y in the gauge we chose. Reducing the domain of integration in [13, v4, (2.35)] to the hemisphere and using that = 0 on the equator, one nds that the FI term is i Z 2 HS2 (D + ) d2x = i + (0) = u + k: Since other terms in the action are Q-exact, the classical contribution is exp( 2 (u + k)). In principle, one should then compute one-loop determinants in the non-constant background we just found. On manifolds without boundaries, one-loop determinants can be computed by localizing to xed points of Q2. On the other hand, boundary conditions typically a ect one-loop determinants by selecting some modes of the Laplace/Dirac operators without altering the eigenvalues for these modes. It is thus plausible that our non-constant vector multiplet only a ects chiral multiplet one-loop determinants through its value near the poles and the gauge holonomy along the equator (this is con rmed by the fact that we reproduce RP2 results [63]). Both coincide with the constant = (0) background studied in [84]. After zeta-function regularization, the one-loop determinant of chiral multiplets with Dirichlet boundary conditions is ZHchSi2ra;Dl;1ir-loop = Y p (1 + w( R=2 + uF + u + k)) ; (B.7) Here, w denotes a combined weight under all symmetry groups (vector U(1) R-symmetry, avor, gauge), R is the vector R-charge of each chiral multiplet, and uF combines background holonomies and twisted masses (background scalars ). The contribution from o -diagonal components of the vector multiplet must be the inverse of that of an adjoint chiral multiplet of R-charge 0 because in a theory containing both, the super-Higgs mechanism could make both sets of components massive. Thus ZHveSc2t;oDr;i1r-loop = Y 2roots(G) 2 (1 + (u + k)) = : (B.8) Y >0 (u + k) 2 sin (u + k) HJEP12(07)9 This includes the Vandermonde factor that relates integration over the gauge Lie algebra and its Cartan subalgebra. A consistency check is that the theory with Dirichlet boundary condition admits a 1d avor symmetry corresponding to constant gauge transformations at the boundary; gauging it should give Neumann boundary conditions. We nd indeed that our result di ers from [54, 57] precisely by the one-loop determinant Q sin (u) of a vector multiplet on the full boundary S1 (of circumference 2 ). Our result also di er from [84] since their vector multiplet and adjoint chiral multiplet determinants do not cancel, most likely because the super-Higgs mechanism is incompatible with their boundary conditions. In any case, the one-loop determinant (B.8) is appropriate for the gluing procedure: for instance the vector multiplet one-loop determinant on S2 is the one on S1 times the square of (B.8). B.2 Higgs and Coulomb branch results Combining all ingredients, the RP2 partition function reads23 Q 2 sin 2 (u) w 2 sin 2 w( R=2 + uF + u) >0 21 (u + k)= sin (u + k) 1 + w( R=2 + uF + u + k) = 2 where jWj is the order of the Weyl group of G. What singular points u does the sum run over? Recall that this sum (and the JK prescription) comes from cutting out neighborhood of singularities in a higher-dimensional version of R d2u @u(: : : ). Both the singularities due to 1d chiral multiplets (namely denominator sine factors outside the sum over k) and those due to HS2 vector multiplets (namely 1= sin in the last numerator) must be cut out. Each factor has poles along hyperplanes of the form fu j w(u) = : : : g where w is a weight of R or a root of G. The u 2 tC=(2 ) 23As our 1d contribution we included the one-loop determinant of a 1d vector multiplet and a 1d chiral 2d N multiplet, namely the 1d N = 2 multiplets that contain the 1d restriction of bottom components of the = (2; 2) multiplets. These are the ones needed to reproduce the S 2 partition function from two hemispheres. Since this paper is mostly about the AGT correspondence we did not track down why 1d N = 2 multiplets containing other components of the 2d N = (2; 2) multiplets do not contribute. e 2 (u+k) ddim tu ! (B.9) to sum over are intersections of dim t such hyperplanes and the JK prescription selects intersections such that The sum over k 2 2 t_ is a positive linear combination24 of the corresponding w. is in fact truncated. Consider one singular u de ned by weights wi. By construction wi(R=2 uF u) are integers. The zeros of 1= (1+wi( R=2+ uF + u + k)) truncate the sum over k to wi(k) wi(R=2 uF u). Since is a positive linear combination of the wi, the truncation makes the classical action 2 (u + k) bounded below, which is essential for convergence. It is heartening to see considerations of 1d localization (the positivity required by the JK residue) play such an important role in making the 2d sum over vortices converge. We now convert our results to the form (B.12) found in [63]. If the sum ran over k 2 2 , it could be combined with the sum over singular points u 2 tC=(2 ) into a single sum over all singular points in tC. As it stands, the sum over k has many more terms, accounted for by also summing over the class of k in =2 . To write explicit expressions, we pick an arbitrary lift l : =2 . We also change variables to ! e 2 (v) Y ZRvePc2tor >0 R=2); w(k) ddim tv ! dim t=2 Y ZRchPi2ral w(v + uF ZRvePc2tor(v; k) = ( 1)kv tan ZRchPi2ral(v; k) = sin 2 (v p =2 2 1=2+v; (B.10) (B.11) where we reinstated factors of the cuto for completeness (r in the notations of [63]). Two minor sign di erences are worth noting. Our chiral multiplet determinant for sign under k ! k + 2. In the models we consider these signs cancel thanks to P k 2 f0; 1g is equal to (minus) what they get for even/odd holonomy, but ours changes w w = 0: gives a sign Q ( 1)w( ) = 1. Through the relation ZRvePc2tor(v; k)Zchiral(v; k)ZRchPi2ral( v; k) = 1 this rst sign di erence implies a sign di erence RP2 in the vector multiplet determinant. Besides an unimportant ( 1)(dim g dim t)=2 we have an additional sign Q >0( 1) (k). This is equal to the sign that was originally missed in S2 localization calculations [9, 26, 41], pointed out by Hori and collaborators [56, 57], and described in detail in [13]. It would be good to clarify the correct sign on RP2. Interestingly, at least for U(K) quiver gauge theories, contributions from poles of the HS2 vector multiplet determinant seem to cancel out, hence one can simply sum over the set of v used in the purely 1d localization calculation.25 Consider an intersection v of 24To be a valid JK parameter, must be generic in the sense that it is not a positive linear combination of fewer than dim t weights of R or roots of G. We made precisely this genericity assumption. 25It would be valuable to nd a 1d localization term that directly localizes the index to the discrete points v rather than an integral. The calculation of the 2d elliptic genus [10, 11, 39] has a similar structure, dim t hyperplanes de ned by some weights w1; : : : ; wm and n 1 roots 1; : : : ; n, such that = aiwi + bj j (with implicit summation) is a positive linear combination of these weights and roots. Since only U(1) gauge group factors have FI parameters, is orthogonal to all j and in particular to bj j , thus aibj hwi; j i = h b j j ; bj0 j0 i < 0. We deduce that hwI ; J i < 0 for some I and J , thus wI + J is a weight of R.26 Now we use the structure of the integrand: at v , each j (v k) is an odd integer and each wi(v k + uF R=2) is an even integer so (wI + J )(v k + uF R=2) is an odd integer. Let k 2 be such that wI ( k) is odd and all other wi( k) and j ( k) are even (this is possible for quivers). Then shifting k ! k + k, parities of wI (v k + uF R=2) and (wI + J )(v k + uF R=2) are exchanged. This describes another contribution to the partition function and explicit calculations in examples (see next subsection) indicate that the contribution exactly cancels the unshifted one thanks to a sign in the JK prescription. For even k the chiral multiplet determinant has poles at v = 0; 2; 4; : : : while for odd k it has poles at v = 1; 3; 5; : : : In both cases the poles are in the half-space Re w(v) > 0, as long as R-charges are in the standard interval (0; 2) and there are no avor holonomies (Re uF = 0).27 The sum over singular points is then exactly the sum of residues obtained by closing the following contour integral jWj k2l( =2 ) it (2 i)dim t e 2 (v) Y ZRvePc2tor X >0 Y ZRchPi2ral w(v + uF w R=2); w(k) : (B.12) (B.13) The Coulomb branch integral (B.12) is precisely the answer that can be assembled from ingredients given in [63]. From this point of view, the sum over k 2 l( =2 ) is a sum over holonomies describing at connections on RP2. It would be interesting to directly obtain this formula in our approach, presumably by a di erent choice of 1d localizing term such as the one used in [75] to get a Coulomb branch expression for the 1d partition function. It is straightforward to check the following asymptotics as v ! i1 with constant real part, ZRvePc2tor v; k = O(jvj); ZRchPi2ral v R=2; k = O jvj(R 1)=2 Re v : At least when the di erent components of Im v are scaled by the same factor , the vector multiplet contribution combines with the measure to give dim G and the matter contribution compensates for it, provided the trial central charge (B.1) is positive. This supports our earlier claim that the region jvj ! 1 does not contribute in (B.9). It also suggests that the integral (B.12) converges. giving an integral over v that is reduced afterwards to a sum over singular points and it would be interesting to get the result directly from a Higgs branch localization. 26Kenny Wong's answer to the related https://math.stackexchange.com/questions/2356883/ can be adapted to show this. Consider the sl2 subalgebra hH ; E ; F i associated to J . If wI + J were not a weight of R then the raising operator E would act trivially on the wI weight space, in other words this weight space would consist of highest-weight vectors of sl2. We would then have hwI ; J i 27The contour integral (B.12) is analytically continued to R-charges outside (0; 2) and to nonzero avor holonomies (nonimaginary uF ) by deforming the contour so that no pole crosses the contour. Higgs branch expressions are instrumental in proving that Seiberg dual theories have equal S2 partition functions. Using our expressions we perform the same check for RP2 partition functions of one dual pair. The electric theory is a U(K) vector multiplet coupled to N fundamental and N antifundamental chiral multiplets. We denote R-charges by RA and ReA for 1 A N . The magnetic theory is a U(N K) vector multiplet coupled to N fundamental and N antifundamental chiral multiplets, themselves coupled by a cubic superpotential to N 2 neutral chiral multiplets (corresponding to mesons of the original theory). These three sets of chiral multiplets have R-charges (1 RA), (1 ReB) and (RA + ReB) for 1 A; B N , compatible with the fact that the cubic superpotential has total R-charge 2. The RP2 partition function of the electric theory is Y N Y i=1 A=1 e 2 Tr v JK-res v=v K i<j Y ZRvePc2tor(vi ZRchPi2ral(vi RA=2; ki)ZRchPi2ral( vi ReA=2; ki) vj ; ki kj ) ! dK v with ZRvePc2tor and ZRchPi2ral given in (B.11). The poles to consider are at 8 <vj = RA=2 + lj with lj 2 Z 0 and lj kj 2 2Z :vj 2 vi + ki kj + 1 + 2Z (chiral multiplet pole) (vector multiplet pole) (B.14) (B.15) where in the former case 1 A N and in the latter case vi should be another component, itself set to RA=2 + li or to some other vi0 plus an integer, and so on. As argued above, vector multiplet poles do not contribute. Let us make the argument explicit in the case of U(2). If a vector multiplet pole is used then we have (up to exchanging 1 $ 2) v1 = RA=2 + lI and v2 = v1 + = RA=2 + l2 with l2 k1 + (k2 k1 + 1) k2 + 1 mod 2. For l2 < 0 the residue vanishes due to the (2; A) factor in the chiral multiplet determinant, leaving only poles for which l2 0. Denote such a pole by (l1; l2; k1; k2). Since shifting k's by even integers does not change signs the residue there is the same as the residue at (l1; l2; l1; l2 + 1). That residue is antisymmetric in l1 $ l2 because the arguments (v k) in (B.11) are invariant, the 1= factors in (1; B) and (2; B) components of chiral multiplet determinants are interchanged, and the linear factor in the vector multiplet one-loop determinant changes sign. We kept parities of l1 k1 and l2 k2 xed so that the same components of one-loop determinants are singular to avoid having to track a sign due to the JK prescription. In the general argument above, instead of using the pole at (l2; l1; l2; l1 + 1) we had used its Weyl re ection (l1; l2; l1 + 1; l2) because that contribution is easier to locate on general grounds. Altogether, the enclosed poles are at vj = RAj =2 + lj for some choice of distinct avors Aj and for some vorticities lj 2 Z 0. We denote A = fA1; : : : ; AK g and compute ZRchPi2ral RA RB ; 0 Zv(Aor)tex R; Re; x where x = ( 1)N+K 1e 2 and ZvfoArjtegx(R; Re; x) = X flj 0g QiK6=j ( li QjK=1(xlj =lj !) QN B=1 QjK=1( Re2B + RAi + 2 R2Aj )lj QN B62fAg QjK=1(1 R2Aj )l j R2B + R2Aj )l j This vortex partition function is identical to the one appearing in the factorization of the S2 partition function, or in the hemisphere partition function. One di erence is that on RP2, x = e 2 2 is real. This can be traced back to the fact that the rst Chern class RP2 Tr F vanishes so one cannot turn on a continuous theta term. Using known results about vortex partition functions (found when comparing S2 partition functions of Seiberg dual pairs) it is straightforward to get ZRP2;U(K);N (R; Re; ) = x (N K PA RA)=2 1 x N K PA(RA+ReA)=2 (B.16) : (B.17) RA + ReB ; 0 : (B.18) In other words the partition functions of Seiberg dual theories are equal up to powers of x and (1 x). In the S2 case these were understood as regularization ambiguities and background FI parameters [17]. We must mention a subtle point that we haven't elucidated. The BPS equations on RP2 (when the auxiliary eld D is taken to be real) make ?F be covariantly constant. The authors of [63] choose a gauge where ?F is constant and deduce F = 0, but this is only possible locally on RP2. Consider the double-cover of RP2. Since the Hodge star changes sign under the antipodal map, ?F must take opposite values at antipodes, and D(?F ) = 0 simply forces them to be conjugate. For G = U(2), an explicit class of examples for B 2 2Z + 1 (to simplify expressions) is F = B 2 cos e iB' sin eiB' sin cos ! sin d ^ d' : (B.19) In the region 0 < it derives from a potential A = i 2 0 e iB' d + B sin 2 sin e iB' cos eiB' cos ! sin (B.20) which obeys A ( ; + ') = A ( ; ') and A'( ; + ') = A'( ; '). In principle one should take these nonabelian BPS con gurations into account when localizing. However, our check of Seiberg duality forbids them from contributing since we matched the RP2 partition functions of some nonabelian gauge theories to abelian ones. C Other quotients of the ellipsoid In this appendix we discuss what discrete quotients Sb4=G can be probed using available 4d N = 2 localization results. We use embedding coordinates x 2 R5 in which the squashed sphere is x20 + b 2(x21 + x22) + b2(x23 + x24) = 1. The round sphere (b = 1) has more diverse quotients so this is what we focus on. In the following, only the quotients involving products of cyclic groups are allowed for b 6= 1. The relevant 4d theories are in general not conformal so G must act by isometries, namely G O(5). The localizing supercharge Q should be left invariant by the orbifold action, and in particular the poles should be xed or exchanged (since they are xed points of Q2) hence G Z2 O(4) where the Z2 is generated by the re ection x0 ! x0. The two SU(2) factors inside SO(4) = (SU(2)L SU(2)R)=Z2 act on the complex coordinates (x1 + ix2; x3 + ix4) and (x1 + ix2; x3 ix4) respectively and their central elements ( 1)L and ( 1)R are identi ed since they both ip the sign of (x1; x2; x3; x4). Since Q2 generates a U(1)L SU(2)L, its commutant is (U(1)L SU(2)R)=Z2 and we deduce that G Z2 (Zp )= for some discrete subgroups Zp U(1)L and SU(2)R. Here, identi es ( 1)L( 1)R 1 if it belongs to the group. The discrete subgroups SU(2) are well-known, namely the odd cyclic groups Z2n+1 U(1) and binary polyhedral groups. The latter contain Z2 and have an ADE classi cation: =Z2 SO(3) are symmetry groups of pyramids, prisms, the tetrahedron, the octahedron, and the icosahedron. Some of these quotients do not appear to preserve supersymmetry, even after turning on an R-symmetry current. Since our analysis is not meant to be exhaustive let us concentrate on whether the quotient S3=G of the equator preserves supersymmetry. The round S3 is parametrized by unit-norm quaternions on which the two SU(2) act by multiplication on the left/right respectively. In the absence of background R-symmetry current, the round sphere admits two left-invariant and two right-invariant Killing spinors. Thus, when G is a (discrete) subgroup of SU(2)R, the quotient S3=G preserves two Killing spinors. This case includes lens spaces L(p; 1) = S3=Zp, on which localization was performed: the one-loop determinants of 3d N = 2 multiplets on L(p; 1) are obtained by keeping only the Zp-invariant modes in the one-loop determinants on S3. It also includes the quotients of S3 by dicyclic, binary tetrahedral, binary octahedral, and binary icosahedral groups, which have not been studied yet in the supersymmetric localization literature. Through the 3d-3d correspondence such geometries might give rise to new topological theories in 3d.28 For a more general quotient S3=G one must turn on a U(1) R-symmetry current, so that Killing spinors are not constant anymore (neither in the left-invariant nor the rightinvariant frames). Two of the Killing spinors are then only invariant under an O(2) 28We thank Mauricio Romo for this comment. HJEP12(07)9 SU(2)R, which contains dicyclic groups but not other binary polyhedral groups. It should be possible to localize on quotients of S3 by subgroups of (Zp when is cyclic or dicyclic. This includes lens spaces L(p; q) and more exotic manifolds and orbifolds. However, we were unable to write an R-symmetry current such that Killing spinors are invariant both under a tetra-/octa-/icosahedral subgroup of SU(2)R and under a nontrivial subgroup of U(1)L, so even the smooth quotients (spherical 3-manifolds) among these do not seem to preserve supersymmetry. Let us give a very short outline of the localization result on the quotient of S3 by a dicyclic subgroup of SU(2)R. The one-loop determinant of a free 3d N = 2 chiral multiplet of mass is obtained by keeping modes that are invariant under the dicyclic group Y (1 + m + n i ) 0 m n; m n 0 mod p (1 + m + n + i ) In a gauge theory, one must sum over at gauge connections (labeled by conjugacy classes of morphisms from the dicyclic group to the gauge group). The gauge connection a ects which modes to keep in the chiral multiplet one-loop determinant, namely it changes the conditions m n 0 and m n. An easy generalization is to give an R-charge to the chiral multiplet. The one-loop determinant of o -diagonal components of the vector multiplet is then (essentially) one over that of a chiral multiplet of R-charge 0 (because of the superHiggs mechanism) hence equal to that of an adjoint of R-charge 2 (because adjoints of R-charges summing to 2 can be given a superpotential mass and can be integrated out). There are sign ambiguities between di erent holonomy sectors, but they are no worse than with other methods of computing fermion determinants. The common solution is to compare dual gauge/non-gauge theories and nd what choice of signs makes partition functions equal. It would be interesting to see what localizing on the quotients of S3 described here teaches us about global structures in 3d dualities. It would also be interesting to perform the 4d localization on the corresponding quotients of S b4 (with or without an action of functions on orbifolds of C2 such as those studied in [5, 7, 8, 12]. x0 ! x0) and relate the results to various 2d CFTs. This will involve instanton partition D Conjectural hypermultiplet determinant We make here a conjecture for the one-loop determinant of a single 4d N = 2 hypermultiplet on the squashed projective space RPb4. Our justi cation is quite schematic. A proper treatment would require describing precisely how the Z2 action on Sb4 acts on fermions, as we did for the vector multiplet in (3.12). As pointed out to us by Yuji Tachikawa, this issue is rather thornier for hypermultiplets [88] as there are several choices of CP action on fermions in class S theories. It would be interesting to de ne the setup more precisely and give a rst principle derivation of our proposal. This appendix nds its logical place after we give the one-loop determinant (4.7) on Sb4, namely the one-loop determinant of a pair of hypermultiplets subject to holonomies However, to emphasize that the main text does not rely on the conjecture made here, we moved it to the present appendix. While the symmetry is broken by a boundary condition used as an intermediate step, it is restored in the nal result. The hypermultiplet one-loop determinant in [43] is the square root of the sphere determinant (4.7) so no 3d matter is needed in the decomposition analogous to (4.5). This would suggest to use that square root as our hypermultiplet determinant on RPb4. However, the one-loop determinant computed in [43] was for mixed Dirichlet/Neumann boundary conditions, while the 3d deformation term treats all matter scalar elds on a same footing. This suggests looking for a hypermultiplet determinant on HSb4 that would correspond to a di erent boundary condition. The gluing (4.5) involved only a 3d N = 2 vector multiplet, even though the 3d restrictions of elds of the 4d vector multiplet belong to a 3d N = 2 vector and an adjoint chiral multiplet. The restrictions of elds of the hypermultiplet belong to two chiral multiplets q, q~ in conjugate representations, so in analogy we propose to use a hypermultiplet determinant that obeys a gluing similar to (4.5) with one of these chiral multiplets in 3d. Namely, ZHhySpb4e;fromruglltuipinlegt( ) = ZSh4ypermultiplet b ZSch3iral 1=2 b w2weights(R) b( 21 (b + 1=b) hw; i i) where we used ZSch3iral = Q b w Sb( 12 (b + 1=b) hw; i i), the one-loop determinant29 for q or q~. We select the top sign here for de niteness. We remind the reader that Sb(x) = x) and (x) = 1= b(x) b(b + 1=b x) . The Barnes Gamma function obeys a duplication formula of the following form30 Sb b + b 1 b + b 1 2 2 + ix + ix = c(b)2x2=2 Y Y Sb b + b 1 2 b + b 1 2 4 b 1 4 ix 2 proven by checking that both sides obey the same shift relations under x ! x + 2b 1. This does not x the constant c(b) = c(1=b). The chiral multiplet determinants on Sb3=Z2 contain two of these four factors, depending on the holonomy. Combining with (D.1) and using the duplication formula we obtain an RPb4 hypermultiplet determinant Y Y s1= s2= b + b 1 2 + s1 4 + s2 4 b 1 s1s2 hw; i i 2 where the depends on the holonomy, but also on whether we used q or q~ in our procedure. This dependence (equivalently the lack of symmetry ) is inconsistent. It is 29These 3d determinants are inverses of each other for the following reason. Hypermultiplet scalars (denoted q and q~y) transform in a doublet of the SU(2) R-symmetry in 4d hence have R-charges +1 and 1 under its Cartan torus, the U(1) R-symmetry in 3d. The normalization is xed by giving R-charge 1 to supercharges. Thus, the 3d N = 2 chiral multiplets q and q~ (without y) have R-charge 1 and transform in conjugate representations of G and of the avor group. Their 3d one-loop determinants must cancel, because one could turn on a superpotential mass term W = q~q which makes both massive. { 48 { (D.1) (D.2) (D.3) (D.4) would have so little di erence. to keep in the Sb3=Z2 determinant are 8 w >> > ZSchb3=irZa2l = Y <> b( 41 (3b + b 1) + 12 hw; i i) b( 41 (b + 3b 1) + 12 hw; i i) 12 hw; i i) b( 41 (b + 3b 1) 12 hw; i i) > : b( 41 (b + b 1) + 12 hw; i i) b( 41 (3b + 3b 1) + 12 hw; i i) b( 41 (b + b 1) 12 hw; i i) b( 41 (3b + 3b 1) 12 hw; i i) This leads to our conjecture for the RPb4 hypermultiplet one-loop determinant: ZRhyPpb4ermultiplet = Y w2weights(R) b + b 1 2 b + ywb 1 4 hw; i i 2 where yw = +1 for even holonomy and yw = 1 for odd holonomy. The product of determinants of hypermultiplets with even and odd holonomy in the same representation reproduces the determinant of a hypermultiplet on Sb4 (4.7). Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. (even holonomy), (odd holonomy). (D.5) not restored by the sum over holonomies since there can be several hypermultiplets with di erent mass parameters. In addition it seems strange that the even and odd holonomies A somewhat arti cial resolution to both of these issues would be if the correct modes [1] L.F. Alday, M. Fluder and J. Sparks, The large-N limit of M 2-branes on Lens spaces, JHEP 10 (2012) 057 [arXiv:1204.1280] [INSPIRE]. [2] L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. 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Bruno Le Floch, Gustavo J. Turiaci. AGT/ℤ2, Journal of High Energy Physics, 2017, 99, DOI: 10.1007/JHEP12(2017)099