Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT

Journal of High Energy Physics, Feb 2016

We study superconformal indices of 4d \( \mathcal{N}=2 \) class S theories with certain irregular punctures called type I k,N . This class of theories include generalized Argyres-Douglas theories of type (A k−1 , A N −1) and more. We conjecture the superconformal indices in certain simplified limits based on the TQFT structure of the class S theories by writing an expression for the wave function corresponding to the puncture I k,N . We write the Schur limit of the wave function when k and N are coprime. When k = 2, we also conjecture a closed-form expression for the Hall-Littlewood index and the Macdonald index for odd N. Fromtheindex,wearguethatcertainshort-multipletwhichcanappearintheOPEof the stress-energy tensor is absent in the (A 1 , A 2n ) theory. We also discuss the mixed Schur indices for the \( \mathcal{N}=1 \) class \( \mathcal{S} \) theories with irregular punctures.

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Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT

JHE Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT Jaewon Song 0 1 0 9500 Gilman Dr , La Jolla, CA 92093 , U.S.A 1 Department of Physics, University of California , San Diego We study superconformal indices of 4d N = 2 class S theories with certain irregular punctures called type Ik;N . This class of theories include generalized ArgyresDouglas theories of type (Ak 1; AN 1) and more. We conjecture the superconformal indices in certain simpli ed limits based on the TQFT structure of the class S theories by writing an expression for the wave function corresponding to the puncture Ik;N . We write the Schur limit of the wave function when k and N are coprime. When k = 2, we also conjecture a closed-form expression for the Hall-Littlewood index and the Macdonald index for odd N . From the index, we argue that certain short-multiplet which can appear in the OPE of the stress-energy tensor is absent in the (A1; A2n) theory. We also discuss the mixed Schur indices for the N = 1 class S theories with irregular punctures. Supersymmetric gauge theory; Supersymmetry and Duality; Topological Field - HJEP02(16)45 Theories 1 Introduction 2 Schur index 2.1 2.2 3.1 3.2 4.1 4.2 3 Hall-Littlewood index Index from the 3d mirror Wave function for I2;N 4 Macdonald index Examples 4.2.1 4.2.2 Wave function for the puncture of type Ik;N Examples 2.2.1 2.2.2 Lagrangian theories Argyres-Douglas theories Wave function for the puncture of type I2;n Consistency checks Conjecture for the Macdonald indices of Argyres-Douglas theories can be realized by wrapping 6d N = (2; 0) theory on a Riemann surface C [1, 2]. This description enables us to understand dynamics of the 4d theory in terms of geometry of the Riemann surface. One of the most interesting connection is between the superconformal index [3, 4] of the 4d theory and the 2d topological eld theory [5{9]. It says that the superconformal index of a given theory labelled by C (called the UV curve) is given by a correlation function of the 2d topological eld theory on C, which is a deformed version of Yang-Mills theory. Especially, in the Schur limit of the index, the TQFT is identi ed as the q-deformed Yang-Mills theory [10]. It has been shown via localization of 5d Maximal SYM on S3, that the 2d theory is indeed given by the q-deformed Yang-Mills theory [11{13]. This relation is also extended to the Lens space index [14{16], to the outer-automorphsim twisted index [17], and to other gauge groups [18{22]. A class S theory is not just labelled by the UV curve but also its local data on the punctures. There are regular and irregular punctures depending on the boundary condition { 1 { we impose. The regular punctures are labelled by an SU( 2 ) embedding into that labels the 6d N = (2; 0) theory. The irregular punctures require more elaborate classi cations, and generally lead to non-conformal theories. But when the UV curve is a sphere, we can get a SCFT with one irregular puncture and also with or without one regular puncture [23, 24]. Theories realized in this way includes Argyres-Douglas theory [25, 26] and its generalizations. The (generalized) Argyres-Douglas theories are inherently strongly-coupled and have no weak-coupling limit. Therefore there has been no direct way of computing the superconformal indices or S1 S3 partition functions. Recently, a progress is made in [27], where they obtained an ansatz for the TQFT description of the Schur index for some of the Argyres-Douglas type theories. They were able to verify their result against S-duality [28] and also by studying dimensional reduction to 3d [29]. Their result has been recently extended to the Macdonald index [30]. The result of [27] agrees with the general prediction made in [31], and studied further in [32, 33], that for any N = 2 d = 4 SCFT, there is a protected sector with in nite dimensional chiral algebra acting on it. This implies that the Schur indices have to be given by the vacuum character of the corresponding chiral algebra. Another progress is made in [34]. They observed that the trace of (inverse) monodromy operator appears in the BPS degeneracy counting [35] agrees with the Schur index (of a non-conformal theory). From this observation, they were able to predict the Schur indices of various generalized Argyres-Douglas theories and conjectured that they are given by the vacuum character of certain non-unitary W -minimal models. This agrees with the result of [31], once applied to the Argyres-Douglas theories with no avor symmetry, yield chiral algebra given by Virasoro/W-algebra with central charges of the minimal model series. In this paper, we propose the Schur, Hall-Littlewood (HL) and Macdonald limit of the superconformal index for a class S theory containing certain irregular puncture called (a subset of) type Ik;N . We conjecture the wave function for irregular puncture in these limits, which enables us to use the TQFT description to compute the indices of AD theories. Our strategy is very similar to [27, 30], but we are mainly interested in the theories with no avor symmetry. It turns out the corresponding wave functions are much simpler than the ones with avor symmetries. We rst start with the irregular punctures that appear in the Lagrangian theories and derive their wave function as an integral transformation of regular punctures. From here, we extrapolate the expression to nd a well-behaved wave function corresponding to other irregular punctures. For the Schur index, we are able to nd the wave function for arbitrary coprime k; N . For the HL index and Macdonald index, we nd the wave functions for the k = 2 cases only. Curiously, we nd that the indices for non-conformal theories can also be written in terms of the TQFT. This is the index of the theory at the UV xed point (zero coupling) with Gauss law constraint. Even though the theory is non-conformal for non-zero gauge coupling, the UV index is nevertheless well-de ned. The TQFT description for the \superconformal index" of a non-conformal N = 2 theory enables us to write the index for the conformal 4d N = 1 class S theory [36{39]. The outline of this paper is as follows. In section 2, we study Schur index for the theories with Ik;N punctures. We are able to give a expression when k and N are coprime, { 2 { and it agrees with other proposals. In section 3, we study Hall-Littlewood index with I2;N punctures. We obtain the wave function for I2;N and compare with the direct computation of the index from the 3d mirror theory. In section 4, we conjecture a closed-form formula for I2;N with N odd and perform a number of consistency checks. In section 5, we compute mixed Schur limit of the index for N = 1 class S theories using the result of 2. 2 Schur index Superconformal index of an N = 2 d = 4 SCFT is de ned as I(xi; p; q; t) = Tr( 1 )F pj2+j1 rqj2 j1 rtR+r Y xiFi ; (2.1) symmetries. The trace is taken over the 18 -BPS states that are annihilated by a supercharge Q. The index can be simpli ed by taking certain limits [7]. When p ! 0, it is called the Macdonald index and gets contributions from 14 -BPS states. The Macdonald index can be further simpli ed to the Schur limit upon taking q = t limit. The other limit is to take q ! 0, and this is called the Hall-Littlewood (HL) index. For any class S theories coming from 6d (2; 0) theory of type wrapped on a Riemann surface Cg;n, the superconformal index can be written in terms of a correlation function of a topological eld theory I(ai; p; q; t) = X C2g 2+n Y (i)(ai; p; q; t) ; n i=1 each puncture. constant is xed to be where the sum is over all irreducible representations of and g and n is the genus and the number of punctures respectively. The function C (p; q; t) is sometimes called the structure constant of the theory, and (a; p; q; t) is called the wave function we assign to The Schur index is obtained by specialization p = 0; q = t. In this limit, the structure C 1 = ?(a = q ; q) = Qir=1(qdi; q) ; (q ) where is the Weyl vector of the short-hand notation za de ned as (z; q) wave function is given by Qi1=0(1 zqi). For the case of a full regular puncture, the corresponding = ADE and di are the degrees of the Casmirs. We use Qi ziai and q Q q i. Here the q-Pochhammer symbol is i (z; q) = PE 1 q = Ak 1 and denote the local singularity around the puncture at z = 1 is given as (2.8) (2.9) (2.10) (2.11) (2.12) or equivalently around z0 = 0 via z0 = 1=z as xk = zN (dz)k ; xk = 1 z0N+2k (dz0)k ; to be Ik;N , following the notation of [23, 40]. One can further deform the singularity by adding less singular terms. They serve as deformation parameters of the theory. The regular punctures correspond to N = k. For the case of k = 2 and N = 2n even, the wave function is written by [27] as where R is the spin- 2 representation of SU( 2 ), and j3 = 2 ; 2 + 1; ; . 2 Our goal in this section is to nd a wave function for the punctures of type Ik;N . For example, we conjecture the wave function for the I2;2n 1 puncture to be I2;2n 1 (q) = 0 ( ( 1 ) 2 q 2 ( 2 +1)(n+ 12 ) even; as we will provide evidences in this section. 2.1 Wave function for the puncture of type Ik;N Puncture of type IN; N+1. Let us consider SU(N ) theory with N avors. This theory can be realized by a 3-punctured sphere with one maximal, one minimal and one irregular of type IN; N+1. If we write the index of this theory in terms of a TQFT, we get IN; N+1 denote the wave function corresponding to the minimal and IN; N+1 puncture respectively. Since it is the same as gauging one of the avors of bifundamental hypermutliplets, it can be also written as (2.13) (2.15) (2.17) (2.18) (2.19) Y (1 2 + P 12 n (n +1) I dz z )zP n (z) ; (2.16) where + is the set of positive roots of SU(N ) and W is the Weyl group. Here the index runs from = 1; j +j and the integral measure is given by dz = QiN=11 2dziizi . The last integral can be rewritten upon applying the Weyl character formula as I (w) dzz +w( + ) P n = X w2W (w) w(N =1n) ; where = 12 P the determinant of w and w w( + ) is the shifted Weyl re ection. For the case of N = 2, it is simple to evaluate the above integral. We get is the Weyl vector, (w) is the signature of w which is the same as I2; 1 (q) = 1 X( 1 )nq 21 n(n+1) I 2n z 2n 2 ) 1 jWj(q; q) 21 (N 1)(N 2) X ( 1 )P n q n 2Z X w2W 2 + = 2 ( n2Z 0 ( 1 ) 2 q 21 2 ( 2 +1) dz 2 iz (z even; odd: For N = 3, we nd the expression as below: (I3;1; 22)(q) = < > >8qk(k+1)+`(`+1)+k` > >:0 if 1 = 3k; 2 = 3`; otherwise; qk2+`2 1+(k 1)(` 1) if 1 = 3k 2; 2 = 3` 2; 1The author would like to thank Yuji Tachikawa for the discussions lead to this observation. where k; ` 2 Z 0. We do not have an analytic proof of the formula, but we have checked this expression up to ( 1; 2) = (12; 12). This wave function is an analog of Gaiotto-Whittaker vector [41{43] in the AGT correspondence [44, 45], which realize pure YM theory when we have two punctures of this type. In our case, we expect the two point function of Gaiotto-Whittaker state of the q-deformed Yang-Mills gives us the `Schur index' of the pure YM theory2 IYM(q) = X IN; N+1 (q) IN; N+1 (q) : We will prove this relation for the case of SU( 2 ) in section 2.2.1. Now, we conjecture that the wave function for the irregular puncture of type Ik;N when k and N are relative primes, is simply given by rescaling the q parameter of the wave function for the Gaiotto-Whittaker state (2.14). More precisely, we nd Ik;N (q) = Ik;1 k (qN+k) : This can be thought of as an analog of the coherent state considered by [46{48] in the context of AGT correspondence. We give a number of evidences for (2.21) in later sections. It would be desirable to give a direct proof of this proposal. Puncture of type Ik;kn. We can also consider a irregular puncture of type Ik;N with N being a multiple of k. In this case we have k 1 mass parameters associated to the U( 1 )k 1 avor symmetry. As before, let us rst consider a Lagrangian example. Consider SU(N ) gauge theory with 2N 1 fundamentals. It is realized by a sphere with a maximal, minimal and IN;0 type irregular punctures. Therefore, we write the wave function for the (2.20) (2.21) (2.23) (2.24) 2For a non-conformal theory, the superconformal index really means that of the free UV xed point with the Gauss-law constraint. { 6 { IN;0 puncture as IN;0 (a; q) = = = I I I [dz](q; q)r Y(qzi=zj ; q) Y q i6=j For the case of N = 2, the wave function is given by [27] as where the trace is over the spin- =2 representation R . We nd that the wave function for the irregular puncture of type I2;2n can be written almost as a simple rescaling of the q as We have checked this expression indeed gives us the same wave function as (2.23) found in [27] to high orders in q. We were not able to nd a prescription for N 3. It may be possible to nd a correct prescription by using isomorphism of (A2; A2) = (A1; D4), which has the chiral algebra scu(3) 3 . 2 tures with x2 / (dz0)2=z03, which means I2; 1. From the TQFT, we get ITQFT( 1; 1)(q) = X X m2Z 0 It agrees with the integral expression obtained from blindly applying the index formula for the vector multiplets and then integrating over the gauge group I dz 2 iz ISYM(q) = (q; q)2 (z)(qz 2; q)2 = qm(m+1) ; where (z) = 12 (1 z 2) is the Haar measure of SU( 2 ). This integral can be easily evaluated by using the Jacobi triple identity (2.15). It can be considered as the index at the UV xed point with Gauss law constraint. SU( 2 ) with Nf = 1. The SU( 2 ) gauge theory with 1 avor can be realized by a sphere with 2 irregular punctures I2; 1 and I2;0. From the TQFT, we get ITQFT( 1;0)(q; a) = X I2; 1 I2;0 (a) = 1 + q + + 2a2 a 2 2 1 a2 + 2 q2 + a2 + 4 q4 + a4 + a 2 1 a4 1 a2 + 2 q 3 3 a2 3a2 +5 q5 +O q6 ; { 7 { (2.25) (2.26) (2.27) (2.28) (2.29) which agrees with the one computed from the integral INf =1 = (q; q)2 I dz 2 iz (z) [S3O;0(]4) + [S0O;3(]4) +2 [S2O;0(]4) +2 [S0O;2(]4) +4 [S1O;0(]4) +4 [S0O;1(]4) +6 [S0O;0(]4) +O(q4); This agrees with the integral formula INf =3(ai; q) = (q; q)2 I dz 2 iz (z) upon identifying a1 = bc; a2 = b=c; a3 = a. SU(3) SYM. The Schur index of the SU(3) pure YM can be written as ISU(3) = = 1 3! (q; q)4 I 1 dz1 X dz2 2 iz1 2 iz2 6(q; q)2 n1;2;3;m1;2;32Z Y (1 2 + z )(qz ; q)2 ( 1 )n1+m1 q 21 Pi(ni(ni+1)+mi(mi+1)) n1+n2;m1+m2 n1+n3;m1+m3 : which agrees with the computation from the integral formula. Here we used the Dynkin label to denote the characters. and one I2; 1 puncture. This gives We can also realize the same theory via 3-punctured sphere with 2 regular punctures ITQFT(R;R; 1)(x; y; q) = X C (x) (y) I2; 1 ; which gives the same answer upon matching the fugacities via a ! pxy and b ! px=y. SU( 2 ) with Nf = 3. The SU( 2 ) gauge theory with 3 avors can be realized by a sphere with 2 regular punctures and 1 irregular puncture I2;0. This gives us the index to be ITQFT(R;R;0) = X C (a) (b) I2;0 (c) = 1 + q [S0O;1(;61)] + q2( [S0O;2(;62)] + [S0O;1(;61)] + [0;0;0]) SO(6) + q3( [S0O;3(;63)] + [S0O;2(;62)] + [S1O;2(;60)] + [S1O;0(;62)] + [S0O;1(;61)] + [S0O;0(;60)])+O(q4) : (2.30) (2.31) (2.32) (2.33) (2.35) (2.36) We veri ed this to be the same as the one given by (2.20) with N = 3 to high orders in q. The rst few terms are ISU(3) = 1 + q2 + 2q4 + 2q8 + q10 + 2q12 + q16 + 2q18 + 2q20 + O(q24) : (2.34) 2.2.2 Argyres-Douglas theories (A1; AN 1) theories. The Argyres-Douglas theories of type (A1; AN 1) can be realized by a sphere with single irregular puncture of type I2;N [23, 40]. When N = 2n + 1 is odd, we have where IA2n (q) = X C 1 I2;2n+1 = ( 1 )j qj(j+1)(n+ 12 ) ; X model (2; 2n + 3) to high orders in q, which is consistent with the relation given in [31], and also computation based on the conjectural relation from the BPS degeneracy [34]. We also nd that when n = 0 or N = 1, the index becomes 1, which agrees with the expectation that there is no massless degrees of freedom and vanishing central charges [40] for the (A1; A0) theory. HJEP02(16)45 When N = 2n is even, we get [27] IA2n 1 (x; q) = X C 1 I2;2n (x) = 0 X [ + 1]q TrR with one I2;N type irregular puncture and a regular puncture. When N = 2n This expression exactly agrees with the vacuum character of the a ne Lie algebra 4n . When N = 1, we nd the index of (A1; A3) agrees with that of (A1; D3) I(A1;A3) = up on identifying x = a2. When N = 2n, the index can be written as that the Schur indices for the Argyres-Douglas theories of type (Ak 1; AN 1) as I(Ak 1;AN 1)(q) = 1 Qik=2(qi; q) X (q ) Ik;N (q) : It was conjectured by [34] that for the coprime k; N , the Schur index of (Ak 1; AN 1) theory is given by the vacuum character of the (k; k + N ) Wk-minimal model, which is given by [49] 0 W (k;k+N)(q) = (qk+N ; qk+N ) k 1 k 1 W (k;k+N)(q) to high orders in q for a number of cases. { 9 { For Argyres-Douglas theories realized in class S with a non-trivial Higgs branch,3 once dimensionally reduced to 3d, their mirror theories [51] are known [23, 52]. The Higgs branch of the dimensionally reduced 3d N = 4 theory is the same as the original 4d N = 2 theory, which is the same as the Coulomb branch of the 3d mirror MHiggs = MHiggs = M3Cdoumloirmrobr : 4d 3d Since the Hall-Littlewood (HL) index is the same as the Hilbert series of the Higgs branch [7], it should be the same as the Hilbert series of the Coulomb branch [53] of the 3d mirror theory. This can be also computed using the 3d superconformal index by taking the `Coulomb limit' as discussed in [54]. Therefore we can write IH4dL(t) = IH3diggs(t) = IC3doumloimrrbor(t) : In this section, we review the computation of HL indices from the 3d mirrors [55]. The Coulomb branch index for the 3d N = 4 theory is de ned as IC (t) = Tr( 1 )F tE RH = Tr( 1 )F tE+j ; (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) Index from the 3d mirror where the trace is over the states with E = RC in addition to E RH EC j = 0. Here RH ; RC are the Cartans of SO(4)R = SU( 2 )RH SU( 2 )RC and E is the scaling dimension and j is the Cartan of the rotation group SO(3). The Coulomb branch index contribution for a hypermultiplet is simply IhCyp(z; m; t) = t 21 jmj ; which does not depend on any avor fugacities. Here m is the charge for the background topological U( 1 ) associated to the gauge group. (A1; A2n 1) theories. The 3d mirror theory is given by a quiver gauge theory with k U( 1 ) nodes where all the nodes are connected by n edges. One of the U( 1 ) node has to be ungauged as usual. See the gure 1. For the U( 1 ) gauge theory with n fundamental hypers, the index is given as C I(A1;A2n 1)(w; t) = 1 1 t X m2Z wmt n2 jmj ; where w is the fugacity for the topological U( 1 )J associated to the gauge group. One can sum up the formula to get InC (w; t) = 1 1 t n t 1 (1 n t 2 w)(1 n t 2 w 1) = 1 1 t n t 1 X k=0 [k] SU( 2 )(w)tkn=2 ; where [k] denotes the characters of SU( 2 ) with Dynkin label [k]. Even though we see the index is written in terms of characters SU( 2 ), it is wrong to say that the global symmetry is actually enhanced to SU( 2 ). Note that only when n = 2, we have the extra conserved current coming from the monopoles [56]. 3Even though the theory in the UV might not have a Higgs branch, AD points can sometimes have a quantum Higgs branch [50]. of bifundamentals between a pair of nodes. One of the U( 1 ) has to be decoupled. The Coulomb branch of U( 1 ) gauge theory with n electrons or the Higgs branch of its mirror theory A^n 1 quiver is known to be given by C2=Zn. One can directly see that the Hilbert series of this space is the same as above. For n = 2, the 3d theory is well-known T [SU( 2 )] theory which is self-mirror. Here the SU( 2 ) enhancement of topological symmetry is evident from the mirror perspective. There SU( 2 ) symmetry is nothing but the avor symmetry of two electrons. In this case, the 4d theory is the same as the AD theory found from SU( 2 ) gauge theory with two avors. (A2; A3n 1) theories. The 3d mirror of this theory is given by U( 1 ) U( 1 ) gauge theory with n bifundamentals and n electrons for each U( 1 ). The Coulomb branch index is given by (3.7) (3.8) 1 (1 t)2 X m1;m22Z C I(A2;A3n 1)(w1; w2; t) = w1m1 w2m2 t 21 (jm1j+jm2j+jm1 m2j)n ; which can be written in terms of SU(3) characters as C I(A2;A3n 1)(a; t) = 1 1 t t n 2 1 X k=0 [SkU;k(]3)(a)tnk where [k; k] is the Dynkin label for the k-th powers of adjoint representation of SU(3) and the fugacities are mapped to w1 = a1=a22; w2 = a2=a12. From above, we see that the global symmetry is enhanced to SU(3). This avor symmetry can be understood quite easily when n = 1. In this case, the quiver gauge theory we obtain is nothing but A^2 quiver theory with U( 1 ) nodes, which is mirror to the U( 1 ) gauge theory with 3 electrons. As in the previous case, we have SU(3) avor symmetry for n = 1, which gives us extra conserved current from the monopole operators. For n > 1, even though we get SU(3) character representations, it does not mean that our theory has extra conserved currents. Curiously, we observe that the 3d mirror of (A2; A2) theory is simply given by A^2 quiver theory, which is mirror to U( 1 ) with 3 electrons, which is the 3d mirror of (A1; A3) theory. This is not a surprise, in the sense that generally in class S, there can be many di erent ways to realize the same 4d SCFT. See [57] for a study of such examples among the AD theories. (A1; D2n+2) theories. This theory is obtained by putting extra U( 1 ) punctures to (A1; A2n 1) theory. The 3d mirror is obtained by gluing the 3d mirror theory corresponding to the minimal puncture to the 3d mirror of (Ak 1; Akn 1) theory. See the gure 2. The Coulomb index for the mirror theory is given as C I(A1;D2n+2) = 1 (1 t)2 X m1;m2 w1m1 w2m2 t 21 (njm1 m2j+jm1j+jm2j) : When n = 1, the above sum can be written as X k C I(A1;D4) = [SkU;k(]3)(a)tk ; which is the same as that of (A2; A2) theory as expected. 3.2 Wave function for I2;N N = 2n even. We want to write the index (3.5) using the TQFT as I(A1;A2n 1)(a) = X C 1 I2;2n (a) ; C = Qir=1(1 1 P HL(t 2 ) tdi ) ; with where di are the degrees of the Casmirs of . For = SU(N ), they are given by di = 2; 3; ; N . Here the function P HL(a) is the Hall-Littlewood polynomial labelled by a Dynkin label in general, which we normalize so that h P HL(z); P HL(z)i = Z [dz] (z)PE [t adj(z)] P HL(z)P HL(z) = : (3.13) (3.9) (3.10) (3.11) (3.12) For the case of = SU(2), we get The wave function for the I2;2n puncture can be read o from the expression (3.5) to be P HL(z) = (p p 1 1 t 2 t ( (z) t if = 0, Note that there are many di erent ways decompose the sum (3.5) in the form of (3.11). But we nd our particular form given by (3.15) is the right choice to maintain the consistent TQFT description of the index. We nd that this is indeed consistent with the accidental isomorphism of the AD theories. One can obtain the wave function for the I2;0 puncture as in the previous section by integrating the wave function for the regular puncture with the integration kernel given by vector and hypermultiplets I I I2;0 = where the wave function for the regular puncture is given as We nd it agrees with the expression (3.15) with n = 0. written as From the TQFT structure of the index, the index for the (A1; D2n+2) theory can be We indeed nd this expression agrees with the result (3.9) from the 3d mirror. N = 2n 1 odd. Again, one can obtain the wave function for I2; 1 by integrating the regular puncture wave function via vector multiplet measure. We get I2; 1 = [dz]Ivec(z) (z) = [dz]PE [t adj(z)] P HL(z) > > = < I 8p > >:0 1 p t 1 t 2 t 2 = 0 = 2 otherwise : (z) 1 I From this, we obtain the HL index for the pure SU( 2 ) YM to be ISYM(t) = X I2; 1 I2; 1 = 1 t3 ; which agrees with the direct computation. (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) We can obtain the wave function for I2;1 from using the isomorphism (A1; A3) = (A1; D3). Note that it implies where we identify x = z2 as we learned from the Schur index. Now, we can use the orthonormality of the regular puncture wave functions. Multiply (z) on both sides and then integrate with the vector multiplet measure to obtain I2;1 = X I [dz]Ivec(z)C 1 I2;4 (z2) (z) = p 1 t We conjecture the Hall-Littlewood wave function for I2;2n 1 is the same as that of I2;1, so that for n > 0. This gives the HL index for the (A1; A2n) theory to be 1, which is consistent with the absence of the Higgs branch. (3.21) In this section, we discuss the Macdonald index, carrying two fugacities (q; t). The Macdonald index reduces to the Schur index when q = t and to the Hall-Littlewood index when q = 0. We rst construct the wave function for the puncture of type I2;n and then use it to write the Macdonald index for the (A1; A2n) theory and discuss its implications. 4.1 Wave function for the puncture of type I2;n Odd n. As in the previous sections, we start with the I2; 1 puncture. It can be obtained by a `integral transformation' of a regular puncture via vector multiplet measure. The wave function for the regular puncture is given by q adj(z) P (z; q; t) = 1 (t; q)r Y 2 1 (tz ; q) P (z; q; t) ; (4.1) where P (z; q; t) = N P (z; q; t) is the normalized Macdonald polynomial4 such that (4.2) (4.3) t h ; i = I N (q; t) = [dz]Ivec(z) (z) (z) = ; (q; q)(t2q ; q) (t; q)(q +1; q) (1 tq ) 1 2 : where the normalization factor for the = A1 is given by 4Our normalization is slightly di erent from the usual one. We include the `Cartan piece' ((qt;;qq))rr to the measure. From now on we suppress the dependence on q; t. When = A1, we can explicitly write down the polynomial as P (a) = N i=0 X (t; q)i (t; q) i a2i (q; q)i (q; q) i ; where we de ne the q-Pochhammer symbol as (x; q)n = Qin=01(1 xqi). Now, let us compute the wave function for the Gaiotto-Whittaker state for the theory. We evaluate the integral to get I2; 1 (q; t) = [dz]Ivec(z) (z; q; t) = [dz]PE adj(z) PE t 1 q adj(z) P (z) HJEP02(16)45 I 8 :0 = N i=0 X (t; q)i(t; q) i I (q; q)i(q; q) i = <( 1 ) 2 t 2 q 21 2 ( 2 1)N ((qt;;qq)) ==22 I dz (1 2 iz q t 1 q z 2) 2 even; odd: (qz 2;0; q)z2i wave functions. odd is given by We see that Schur and Hall-Littlewood limits of (4.5) indeed reproduces the corresponding From (4.5), we want to nd a similar rescaling of the fugacities (q; t) as done in the Schur and Hall-Littlewood case. Now, we conjecture that the wave function for I2;n for n 8 :0 I2;n (q; t) = <( 1 ) 2 t 2 (n+2) We will provide some evidences of this proposal in the following subsection. Even n. The wave function for the I2;2n is proposed by [30] as (up to normalization) I2;2n 2 (x) = N tn =2qn 2=4 (t; q) X (t; q)m(t; q) m=0 (q; q)m(q; q) m q n( 2 m)2 x2m : m As in the previous sections, we can obtain I2;0 by integrating the wave function for the regular puncture with vector and hypermultiplet kernel I2;0 (a; q; t) = I (z) = I dz (1 z 2) (qz 2;0; q) 2 iz 2 1 (t 2 z a ; q) P (z; q; t): (4.8) We have veri ed that this expression agrees with (4.7) for n = 1 up to high orders in q. 4.2 and provide consistency checks. In this section, we use (4.6) to compute the Macdonald index for a number of examples (4.4) = A1 (4.5) (4.6) (4.7) Let us rst consider a number of examples where we can cross-check the conjectured formula (4.6) against independent computations. SU( 2 ) SYM. The pure YM theory can be realized by a pair of I2; 1 punctures on a sphere. From the TQFT structure of the index, we write ISYM = X where t = qT . This result indeed agrees with the direct computation. (A1; A0) theory. This should describe a theory with no massless degrees of freedom. We indeed nd I(A1;A0) = X C 1 I2;1 (q; t) = 1 ; (4.10) to high orders in q. This is rather a non-trivial check of the proposal (4.6), since each term in the sum has to cancel exactly up on summing over all terms. (A1; A3) = (A1; D3) theory. The (A1; A3) theory is isomorphic to (A1; D3) theory. The former description can be obtained from a single I2;4 puncture and the latter description can be obtained from I2;1 and a regular puncture. We nd that these two descriptions indeed give us the same index I(A1;A3)(x) = upon identifying x = a2. This result provides a consistency check between (4.6) and (4.7). We nd that the Schur limit of this index can be written in a very simple form I(A1;D3)(a) = PE (1 q q)(1 q 3 q3) adj(a) : The rst term inside the PE is coming from the conserved current multiplet. 4.2.2 Conjecture for the Macdonald indices of Argyres-Douglas theories (A1; A2) theory. It can be obtained from I2;3 punctured sphere. We conjecture its Macdonald index is given as I(A1;A2) = X C 1 I2;3 = 1 + q2T + q3T + q4T + q5T + q6(T 2 + T ) + q7(T 2 + T ) (4.12) (4.13) + q8(2T 2 + T ) + q9(2T 2 + T ) + q10(3T 2 + T ) + O(q11) ; where t = qT . This theory does not have a Higgs branch. This can be seen from triviality of the Hall-Littlewood limit of the index q ! 0. We nd that this expression can be also written as I(A1;A2) = PE q2T q4T 2 (1 q)(1 q5T 2) + O(q11) ; (4.14) where the O(q11) terms vanish in the limit T coming from the short multiplet C^0(0;0) (and their powers) using the notation of [58]. See also appendix B of [7]. This is the multiplet containing the stress-energy tensor. The Macdonald index for the short multiplet C^R(j1;j2) is given as ! 1. The rst term inside the PE is We see that the stress-energy tensor multiplet contributes 1q2Tq to the Macdonald index. Since any SCFT has a stress-energy tensor multiplet, the operator appear in the OPE of it should be also present in the theory. The OPE of the stress-energy tensor multiplet which is the value of (A1; A2) theory [61]. See also [62]. other than ` = 0, which contains the stress-energy tensor. where we have only written short multiplets appear in the OPE that contributes to the Macdonald index. The C^0( 2` ; 2` ) multiplets are higher-spin conserved currents which have to be absent unless the theory is free or has a decoupled sector [60]. This multiplet contributes to the index by q`1+2qT . Indeed, we see from the index (4.13) that there is no C^0( 2` ; 2` ) multiplet Among the terms appear on the r.h.s. of the OPE (4.16), C^1( 2` ; 2` ) multiplet contributes q`+3T 2 1 q to the index. Since the index has coe cient 0 for the q4T 2 term, C^1( 12 ; 12 ) cannot be present. We also see C^1( 2` ; 2` ) with even ` is absent. Our result agrees with the analysis ^ of [59] where they show that C1( 12 ; 1 ) is absent for the theory with central charge c = 1310 2 (A1; A4) theory. It can be obtained from I2;5 punctured sphere. Our conjectured Mac X C 1 I2;5 = 1+q2T +q3T +q4(T 2 +T )+q5(T 2 +T )+q6(2T 2 +T ) donald index is I(A1;A4) = + q7(2T 2 + T ) + q8(T 3 + 3T 2 + T ) + O(q9) : It also reduces to 1 in the Hall-Littlewood limit q ! 0 as expected. We nd the index can be written in terms of a Plethystic exponential as I(A1;A4) = PE q2T q6T 3 (1 q)(1 q7T 3) + O(q15) ; where O(q15) term vanishes as T ! 1. 5The author would like to thank Wenbin Yan for discussions on this point. (4.18) Here we see that some of the short-multiplets appear in the OPE of the 3 stress-energy tensor multiplets should be absent, because there is no term of the form q6T 3 in the index. Among the operators appear in the OPE of 3 stress-energy tensors, the multiplet contributing q16Tq3 has to be absent. The natural candidate would be C^2(1;1), but we cannot rule out other possibilities from the index before working out the selection rule, because any C^R(1;j2) with R + j2 = 3 for an integer j2 will give the same index. (A1; A2n) theory. We put I2;2n+1 puncture on a sphere. We conjecture the Macdonald q)(1 (q2T )n+1 q2n+3T n+1) + ; (4.20) where omitted piece vanishes in the Schur limit T ! 1. There is no (q2T )n+1 term in the index. Therefore the short multiplet that appear in the OPE of (C^0(0;0))n+1 that contributes to the index as (q2T )n+1 1 q multiplet C^n( n2 ; n2 ) contributes the same amount so that it might be absent. is absent. The short (A1; D5) theory. It can be obtained from a sphere with a I2;3 puncture and a regular puncture. We get I(A1;D5)(a) = X I2;3 (a) (4.21) (4.22) (4.23) = 1+qT 3 +q2 T ( 3 + 1)+T 2 5 +q3 T ( 3 + 1)+T 2( 5 +2 3)+T 3 7 + q4 T ( 3 + 1) + T 2(2 5 + 3 3 + 2 1) + T 3( 7 + 2 5) + T 4 9 + O(q5) ; where n denotes the character for the n-dimensional representation of SU( 2 ). When we take the Hall-Littlewood limit q ! 0 with t xed, we get I(A1;D5)(a) = 2n+1(a)tn = X n 0 1 t 2 (1 ta2)(1 ta 2)(1 t) ; which is the same as the HL index of the (A1; A3) theory given in (3.6) with n = 2 and w = a2. This is nothing but the Hilbert series of C2=Z2. The rst term of the index (4.21) comes from the conserved current of the SU( 2 ) avor symmetry. We nd that the index has the form I(A1;D5) = PE qT 1 q adj(a) + : : : ; where the omitted term vanishes in the Schur limit. 5 N = 1 class S theories For every theories in N = 1 class S, the superconformal indices can be written in terms of the correlation functions of a (generalized) topological eld theory on the UV curve [38]. See also [20, 63{65]. In this section, we generalize our discussions to the N = 1 case. For the chiral multiplets with (J+; J ) = (0; 2) in the adjoint representation of G, we get i i # # (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) Ichi(p; q; z) = PE " J p 2 q 2 J+ (z) (1 p 1 J2 q1 J2+ p)(1 q) (z) # ; where is the character of the representation of the gauge group. In the mixed Schur limit, the hypermultiplets charged with (J+; J ) = (1; 0) gives the index purely a function and the hypermultiplets with (J+; J ) = (0; 1) gives The N = 1 index for the class S is de ned as I(p; q; ; x) = Tr( 1 )F pj1+j2+ R20 qj2 j1+ R20 F Y xiFi ; where R0 is the UV R-charge and F is the global U( 1 ) charge conserved for a generic class S theory. One of the simpli cation limit of the above index is to take = pq=p, called the mixed Schur limit [38]. It is given as where we used J F ). In this limit, the chiral multiplet contribution of the index where R is the character of the representation R of G. bundle degrees (p; q) is given by The index of the theory in class S corresponding to the UV curve Cg;n with normal I(ai; p; q; ) = X(C+)p(C ) i (ai) ; n q Y i=1 where labels the representations of and (z) is the wave function associated to the puncture of color . The wave function in the mixed Schur limit becomes = 1 q +(z) = PE IYM = (p; p)(q; q) = X m2Z 0 from the TQFT. (5.9) (5.10) (z) = Ic(h0i;2)(z) +(z) ; +(z) = Ic(h2i;0)(z) (z) : This explains why we attach avor adjoint chiral multiplets to the oppositely colored punctures. conformal, N Equipped with the wave functions for the irregular punctures, we can easily compute the index in the mixed Schur limit. Note that even when the N = 2 counterpart is non= 1 version can actually ow to a SCFT in certain cases. For example, SU(3) theory with Nf = 5 is non-conformal for N = 2, but it is in the conformal window for N = 1 theory. We can indeed compute the indices for these cases using the result of section 2. In section 5.2, we consider simplest examples. 5.2 SU( 2 ) SYM with Nf = 0; 1; 2; 3 In this section, we verify that the mixed Schur index is indeed reproduced by the generalized TQFT we discussed. Pure YM. Let us compute the index naively by using the UV matter content. Here we only have a vector multiplet. We get [66] I dz 2 iz (z)(qz 2; q)(pz 2; q) = 1 2 X m;n2Z p 21 m(m+1)q 12 n(n+1) I dz 2 iz (pq) 21 m(m+1) = X I2; 1;+ I2; 1; ; z2(m n) (5.11) The wave functions for the irregular punctures with color choosing the N = 2 wave function with di erent arguments Ik;n (q). In this limit, the structure constant can be also simply written as = Ik;n;+ = are given by simply Ik;n (p) and Ik;n; = C+ = C (q) C = C (p) ; where C (q) is the structure constant for N = 2 theory. transforming under the adjoint of the avor symmetry: Note that one can ip the color of the wave function by attaching a chiral multiplet HJEP02(16)45 where we used the Jacobi triple product identity. We indeed get the mixed Schur index SQCD with Nf = 1. This theory has a dynamically generated runaway superpotential, therefore we cannot de ne proper superconformal index. Nevertheless, we compute the index at the UV xed point with incorrect R-charges for the chiral multiplets. Namely, we pick R = 12 for the chiral multiplets. This value is the correct R-charge for the massdeformed N = 2 SCFTs such as SU(N ) theory with 2N avors. I 2 iz a 2 2 = 1 + + t4 t y + t2 2a2 y2 1 a2 + y2 2 5 a2y2 + y4 + t3 1 y2 + O(t5) ; Now, we apply the integral formula for the index to compute INf =1(p; q; a) = (p; p)(q; q) (z) INf =1+1(p; q; a; b) = (p; p)(q; q) which agrees with the TQFT expression dz 2 iz (z) 1 (q 2 z a ; q)(p 2 z b ; p) dz 2 iz (z) 1 (q 2 z a ; q)(q 2 z b ; q) ITQFT(p; q; x; y) = X C I2; 1; +(x) +(y) ; where p = ty; q = t=y. This result agrees with the TQFT on a sphere with one irregular puncture I2: 1 and one regular puncture with (p; q) = (0; 0), which is given by HJEP02(16)45 ITQFT(p; q; a) = X I2; 1;+ (a) = X ( 1 )mp 12 m(m+1) Rm(a) : (5.13) 1 (qa 2;0; q) m2Z 0 Note that we have only kept the diagonal subgroup of the full avor symmetry U( 1 )L U( 1 )R. SQCD with Nf = 2. This theory without superpotential con nes with a deformed moduli space, but we perform the computation with the same philosophy as before. Let us take the (wrong) R-charge 1=2 to compute the index. We have two di erent ways to construct the theory, as we have discussed in the case of N = 2 counterpart. Here depending on the choice of the colors on the punctures we actually get di erent indices because these choices determine the superpotential that are allowed [67]. Let us rst consider the case with two irregular punctures I2;0 of each color. This con guration realizes the theory with a quartic superpotential between two quarks. We write the index as INf =2+0(p; q; a; b) = (p; p)(q; q) It agrees with the TQFT expression upon identifying a = xy; b = x=y. ITQFT(p; q; a; b) = X I2;0;+(a) I2;0; (b) : Now, let us consider a 3-punctured sphere realization of Nf = 2 theory. We have two regular + punctures, and one irregular I2; 1 with color. We pick the normal bundle degrees to be (1; 0). This realizes the Nf = 2 theory without quartic superpotential, which gives the index to be 1 1 ; : I I (5.14) (5.15) (5.16) (5.17) SQCD with Nf = 3. This theory can be realized by a sphere with two regular punctures of + color and one irregular puncture I2;0 with color, and normal bundle degrees (p; q) = (1; 0). It splits 3 avors into 2 + 1 with a quartic superpotential interaction. The index INf =2+1(p; q; a; b; c) = (p; p)(q; q) I 2 iz (z) 1 (q 2 z a ; q)(q 2 z b ; q)(p 2 z c ; p) 1 1 : (5.18) We nd that it agrees with the TQFT expression ITQFT(p; q; x; y) = X C +(x) +(y) I2;0; (c) ; (5.19) upon identifying a = xy; b = x=y. Acknowledgments The author would like to thank Abhijit Gadde, Ken Intriligator, Yuji Tachikawa and Wenbin Yan for useful discussions and correspondence. The author is grateful for the hospitality of the Simons Center for Geometry and Physics during the 2015 Simons Workshop in Mathematics and Physics and also Korea Institute for Advanced Study. This work is supported by the US Department of Energy under UCSD's contract de-sc0009919. Open Access. 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Jaewon Song. Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT, Journal of High Energy Physics, 2016, 45, DOI: 10.1007/JHEP02(2016)045