Tadashi Okazaki等2023年8月在JHEP期刊发表论文.pdf 
Tadashi Okazaki等2023年8月在JHEP期刊发表论文.pdf
Published for SISSA by Springer Received: May 15, 2023 Revised: July 16, 2023 Accepted: July 29, 2023 Published: August 10, 2023 Tadashi Okazakia and Douglas J. Smithb a Shing-Tung Yau Center of Southeast University, Yifu Architecture Building, No. 2 Sipailou, Xuanwu district, Nanjing, Jiangsu, 210096, China b Department of Mathematical Sciences, Durham University, Upper Mountjoy, Stockton Road, Durham DH1 3LE, U.K. E-mail: tokazaki@seu.edu.cn, douglas.smith@durham.ac.uk Abstract: We propose confining dualities of N = (0, 2) half-BPS boundary conditions in 3d N = 2 supersymmetric SU(N ), USp(2n) and SO(N ) gauge theories. Some of these dualities have the novel feature that one (anti)fundamental chiral has Dirichlet boundary condition while the rest have Neumann boundary conditions. While some of the dualities can be extended to 3d bulk dualities, others should be understood intrinsically as 2d dualities as they seem to hold only at the boundary. The gauge theory Neumann halfindices are well-defined even for theories which contain monopole operators with nonpositive scaling dimensions and they are given by Askey-Wilson type q-beta integrals. As a consequence of the confining dualities, new conjectural identities of such integrals are found. Keywords: Duality in Gauge Field Theories, Extended Supersymmetry, Supersymmetry and Duality ArXiv ePrint: 2305.00247 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP08(2023)048 JHEP08(2023)048 Boundary confining dualities and Askey-Wilson type q-beta integrals Contents 1 3 3 2 Askey-Wilson integral 2.1 SU(2) with Nf = 4 flavors 4 4 3 Nassrallah-Rahman integral 3.1 SU(2) with Nf = 3, Na = 2(+1) flavors 6 6 4 Gustafson integrals 4.1 SU(N ) with Nf = Na = N flavors 4.2 SU(N ) with Nf = N + 1, Na = N (+1) 4.3 USp(2n) with 2n + 2 fundamental chirals 4.4 USp(2n) with 2n + 3(+1) fundamental chirals 4.5 USp(4) with 2 fundamental and 2 antisymmetric chirals 4.6 USp(4) with 3 antisymmetric chirals 4.7 USp(6) with 2 antisymmetric chirals 4.8 SO(N ) with N − 2 fundamental chirals 9 10 13 16 18 20 23 25 26 5 Gustafson-Rakha integrals 5.1 USp(2n) with rank-2 antisymmetric and 5(+1) fundamental chirals 5.2 SU(N ) with rank-2 antisymmetric, 3(+1) fundamental and N antifundamental chirals 5.3 SU(N ) with 2 rank-2 antisymmetric, 3 fundamental and 2(+1) antifundamental chirals 28 28 6 New integrals 6.1 SO(N ) with N − 2 fundamental chirals and Z2 fugacities 6.2 SO(N ) with N − 1(+1) fundamental chirals 6.3 SU(N ) with rank-2 antisymmetric, 4 fundamental and N − 1(+1) antifundamental chirals 44 44 46 1 31 37 49 Introduction and conclusion Confining duality is a phenomenon where the infrared physics of a gauge theory can be described in terms of gauge invariant composites and their interactions. Typically in a confining duality the matter fields of the dual theory transforming in rank-2 tensor representations of the flavor symmetry group can be viewed as mesons of a gauge theory, while –1– JHEP08(2023)048 1 Introduction and conclusion 1.1 Structure 1.2 Future works higher rank representations correspond to baryons. Confining dualities are found in several supersymmetric gauge theories including 4d N = 1 supersymmetric gauge theories [1–13], 3d N = 2 supersymmetric gauge theories [11, 14–19] and 2d N = (0, 2) supersymmetric gauge theories [20]. Confining dualities are useful to construct a sequence of dual theories and to find an essential or basic part of dualities. Our results demonstrate the interplay between physics and mathematics. On the one hand we rephrase mathematically proven identities as half-index identities, giving an interpretation as known or conjectured new dualities of 3d N = 2 theories with boundaries. On the other hand, similar dualities are proposed and the corresponding matching of halfindices leads to new mathematical identities which can be proven using similar techniques to the previously known generalized Askey-Wilson identities. Therefore, as a result of the expected confining dualities, we also find new conjectural identities of Askey-Wilson type q-beta integrals. –2– JHEP08(2023)048 In this paper we propose new confining dualities for 3d N = 2 gauge theories with N = (0, 2) boundary conditions. While several confining dualities can be obtained from Seiberg-like dualities [21–52, 52, 53] and tested by computing the IR protected data, e.g. superconformal index (or full-index) [54–58], our strategy is to consider the theories in the presence of a boundary with boundary conditions preserving the gauge group and to study the half-indices which enumerate the gauge invariant BPS local operators obeying the halfBPS N = (0, 2) boundary conditions [59–68]. These half-indices [59, 61–63] are powerful tools to test the dualities of N = (0, 2) supersymmetric boundary conditions. While the full-indices are not well-behaved for the theories involving monopole operators with nonpositive dimensions, the half-indices encoding the Neumann boundary conditions for gauge fields are well-defined even for such theories. We propose confining dualities of N = (0, 2) boundary conditions for 3d N = 2 SU(N ), USp(2n) and SO(N ) gauge theories from several identities of half-indices as well as the precise matching of the boundary ’t Hooft anomalies. We refer to them as boundary confining dualities. It turns out that the Neumann halfindices of 3d N = 2 gauge theories which have confining descriptions take the form of Askey-Wilson type q-beta integrals [69]. From these integrals we conjecture new boundary confining dualities by checking the agreement of the corresponding half- and full-indices. Some of these cases are standard examples of 3d Seiberg-like dualities in the bulk while other cases correspond to 4d Seiberg-like dualities compactified to 3d, specifically those discussed in subsections 4.1, 4.7, 5.2, 5.3 and 6.3. They include examples of Seiberg-like boundary dualities with different boundary conditions for chiral multiplets transforming in the same representation of the gauge group — as far as we are aware the only similar cases of mixed boundary conditions have been considered for SQED [63] — as well as dualities of theories with chiral multiplets in the rank-2 antisymmetric representation of the gauge group. However, in the cases considered in sections 4.6, 4.7, 4.8 and 6.1 the bulk theories have dimension-zero monopoles and chiral multiplets of non-positive dimensions, so we have examples where there is only a boundary duality. Similarly, in the case of section 6.2, although the bulk monopoles would have positive dimension it is not possible to choose the R-charges so that all chirals have positive dimension. 1.1 Structure 1.2 Future works • It would be interesting to generalize the boundary confining dualities to exceptional gauge groups and explore their Askey-Wilson type q-beta integrals associated with the exceptional Lie algebra. We also expect further boundary confining dualities based on bulk dualities for theories with classical Lie algebras. We hope to report results in our upcoming work [74]. • The Seiberg-like duality of 3d N = 2 SU(N ) gauge theory can be derived from U(N ) Seiberg-like duality by gauging manipulation [29, 30]. While the boundary dualities work in the case with Nf = Na = N (where there is no dual gauge group) as we discussed in this paper, more general cases seem to need more delicate treatment of gauging the Dirichlet boundary conditions by introducing the 2d vector multiplet. It would be nice to figure out the gauging and ungauging manipulations for the case with boundary and we hope to report on this in future work. • While the full-indices of 3d N = 4 ugly or bad theories [75] are not well-defined due to the non-positive dimensions of monopoles, the Neumann half-indices can be computed even for such theories. It would be tempting to explore boundary confining dualities for N ≥ 4 theories.1 • While some of the dualities of N = (0, 2) boundary conditions discussed in this paper can be extended to the bulk, others may not originate from the bulk 3d N = 2 1 See [76] for the boundary confining duality for N = (0, 4) boundary conditions. –3– JHEP08(2023)048 The paper is organized as follows. In section 2 we discuss the boundary confining duality of Neumann boundary conditions for the SU(2) gauge theory with four flavors. The half-index realizes the Askey-Wilson q-beta integral [69] and the duality stems from the Seiberg-like dualities [29, 30]. In section 3 we propose the boundary confining duality for the SU(2) gauge theory with six fundamental chirals where one of the fundamental chirals has Dirichlet boundary condition while the others have Neumann boundary condition. The half-index is identified with the Nassrallah-Rahman integral [70, 71] and as we discuss, the duality also holds in the bulk. In section 4 we discuss the higher-rank generalization of sections 2 and 3. The half-indices are identified with integrals studied by Gustafson [72]. Consequently, we propose new confining dualities for SU(N ), USp(2n) and SO(N ) gauge theories, including cases with rank-2 antisymmetric chirals. In addition, we propose boundary confining dualities for specific USp(4) and USp(6) gauge theories with one or more rank-2 antisymmetric chiral multiplets. Except for the boundary confining dualities in subsections 4.6, 4.7, 4.8 and z 6.1, they can be generalized to bulk confining dualities. In section 5 we find the confining dualities of more general SU(N ) and USp(2n) gauge theories with fundamental and rank-2 antisymmetric chirals. The Neumann halfindices can be identified with the Gustafson-Rakha integrals [73]. In section 6 we find new Askey-Wilson type q-beta integrals which arise from boundary confining dualities. theories so that they should be interpreted as “N = (0, 2) boundary dualities”. Such dualities potentially enlarge the web of known dualities. It would be intriguing to explore the “boundary dualities” in other setups. • Brane construction of the N = (0, 2) boundary conditions can be useful to find more boundary confining dualities as well as the holographic dual descriptions. It should be possible to address this from the T-dual configuration in [78]. 2 Askey-Wilson integral Askey and Wilson evaluated an integral which can be understood as a q-analog of the classical beta integral [69]. We argue that it can be physically understood as a half-index of the 3d N = 2 SU(2) gauge theory with four flavors and that it demonstrates the confining duality. 2.1 SU(2) with Nf = 4 flavors Consider theory A as a 3d N = 2 SU(2) gauge theory with Nf = 4 fundamental chirals QI , I = 1, 2, 3, 4 with R-charge ra . When we later consider SU(N ) gauge group, this corresponds to Nf = N = 2 fundamental and Na = N = 2 antifundamental chirals. Unlike the case of U(2) gauge theory, the SU(2) gauge theory with 4 fundamental flavors is free from gauge anomaly, without a 2d Fermi multiplet in the determinant representation of gauge group, when we impose the N = (0, 2) Neumann boundary conditions for the SU(N ) vector multiplet and the chiral multiplets.2 The half-index is evaluated as3 (q)∞ IIA (N ,N,N ) = 2 I 4 Y ds ±2 1 (s ; q)∞ ra ± 2πis 2 α=1 (q s axα ; q)∞ (2.1) where 4α=1 xα = 1. Turning off the fugacities xα and setting ra = 12 , the half-index (2.1) has the q-series expansion Q 2 1/2 IIA + 20a4 q + (6a2 + 50a6 )q 3/2 + 35(a4 + 3a8 )q 2 (N ,N,N ) = 1 + 6a q + 2a2 (3 + 57a4 + 98a8 )q 5/2 + · · · 2 (2.2) See [63] for full details of Seiberg-like boundary dualities of U(N ) gauge theories with fundamental and antifundamental chirals. 3 See [79, 80] for the notations and conventions of half-indices. –4– JHEP08(2023)048 • The Neumann half-indices can be generalized by introducing the Chern-Simons (CS) coupling and Wilson line operators. It would be interesting to generalize the AskeyWilson type q-beta integrals by introducing additional functions of gauge fugacities in the integrand to figure out more dualities with CS terms and line defects. The duality appetizer [19, 24, 77] with boundary conditions would translate into the Askey-Wilson type q-beta integrals. Also we should be able to examine such decorated half-indices by studying the difference equations which encode the algebra of line operators. The integral (2.1) is identified with the Askey-Wilson integral4 [69] which is a qextension of the classical beta integral. It is equal to 4 Y (q 2ra a4 ; q)∞ . (q ra a2 xα xβ ; q)∞ α<β (2.3) bc SU(2) SU(4) U(1)a U(1)R VM N Adj 1 0 0 Qα N 2 4 1 ra Mαβ N 1 6 2 2ra V D 1 1 −4 2 − 4ra (2.4) Note here that the parameter ra corresponds to shifting the R-charge by ra times the U(1)a charge. In our conventions the scaling dimension is half the R-charge so we require all chirals in theory B to have positive R-charge for Neumann boundary conditions and R-charge less than 2 for Dirichlet boundary conditions. In the case here this means ra > 0. This ensures the unitarity bound is not violated5 and the half-indices do not contain negative powers of q. For a 3d bulk duality the corresponding requirement would be that all R-charges lie in the interval (0, 2). The ’t Hooft anomalies match for these theories and boundary conditions since 3 A = 2 Tr(s2 ) + r2 − 2 Tr(s2 ) + Tr(x2 ) + 4(a − r)2 {z } | {z 2 } | Qα , N VM, N 1 = − Tr(x2 ) + 3(2a − r)2 + (−4a + r)2 | {z } |2 {z } Mαβ , N V, D 5 = − Tr(x2 ) − 4a2 + 8ar − r2 2 (2.5) where s is the field strength for the SU(2) gauge group, x for the SU(4) flavor symmetry group, a for the U(1)a axial symmetry group, r for the U(1)R R-symmetry group.6 Note that we have taken ra for simplicity here since matching of anomalies for any value of ra guarantees matching for all values of ra . 4 In the mathematical literature the expressions are usually given in terms of unconstrained fugacities Xα = axα and with ra = 0. In our notation the parameter ra is introduced by the replacement a → q ra /2 a. 5 Note that, assuming the duality, requiring this for theory B is equivalent to ensuring that no gaugeinvariant operators in theory A violate the unitarity bound. 6 See [63] for the calculation of the boundary anomaly polynomial. –5– JHEP08(2023)048 We observe that the expression (2.3) can be interpreted as the half-index of theory B which consists of free chiral multiplets, specifically a chiral multiplet Mαβ with Neumann boundary conditions transforming as the rank-2 antisymmetric representation of the SU(4) flavor symmetry and a chiral V with U(1)a charge 4 with Dirichlet boundary conditions. The content of the two theories is summarized as To summarize, we propose the following confining duality of N = (0, 2) boundary conditions: SU(2) + 4 fund. chirals Qα with b.c. (N , N ) ⇔ SU(4) antisym. chiral Mαβ + a single chiral V with b.c. (N, D). (2.6) The boundary confining duality (2.6) has a counterpart in the 3d bulk. In fact, the full-index of theory A is calculated as I ra |m| 4 Y ds (q 1− 2 + 2 s∓ a−1 x−1 α ; q)∞ (1−2ra )|m| −4|m| |m| ±2 (1 − q s ) q a ra +|m| 2πis (q 2 s± ax ; q) α=1 α (2.7) ∞ and that of theory B is I B 1−r −2 −1 −1 (q 2ra a4 ; q)∞ (q a a xα xβ ; q)∞ = . (q 1−2ra a−4 ; q)∞ (q ra a2 xα xβ ; q)∞ α<β Y (2.8) The full-indices (2.7) and (2.8) precisely agree with each other as a special case of the SU(N ) Seiberg-like duality [30, 31]. The operators can be identified as Mαβ ∼ Qα Qβ (mesons in theory A) and V corresponds to the minimal monopole in theory A. The superpotentials are WA =0 (2.9) WB =V Pf M (2.10) where here and throughout this article the theory B superpotential is easily interpreted in terms of a Lagrange multiplier (or in other examples several) imposing a constraint which through the operator mapping is seen to be an identity in theory A. E.g. in this case the antisymmetric product of the 4 Qα chirals must be antisymmetric in the 4 SU(2) gauge indices which is obviously not possible, hence it must vanish. 3 Nassrallah-Rahman integral We propose a new type of the Seiberg-like duality of SU(2) gauge theory with fundamental and antifuindamental chiral multiplets where one of the chirals has a different boundary condition. We argue that the half-index of the Neumann boundary condition for the vector multiplet is identified with the Nassrallah-Rahman integral [70, 71] which generalizes the Askey-Wilson q-integral with an additional parameter. In addition, we discuss that the duality can be generalized to bulk theories. 3.1 SU(2) with Nf = 3, Na = 2(+1) flavors Next consider theory A as a 3d N = 2 SU(2) gauge theory with 3 fundamental chirals QI , I = 1, 2, 3 of R-charge 0, 2 antifundamental chirals Qα , α = 1, 2 of R-charge 0 and an antifundamental chiral Q̃ of R-charge 2. This can be interpreted as a special case of SU(N ) gauge theory with Nf = N + 1 and Na = N plus another antifundamental chiral discussed later. Since we have gauge group SU(2), we could also describe this as SU(2) with 5 + 1 –6– JHEP08(2023)048 1 X I = 2 m∈Z A fundamental chirals, but here we describe independently fundamental and antifundamental chirals for easier comparison with later generalizations. We consider the Neumann boundary condition for the SU(2) vector multiplet while Q̃ obeys the Dirichlet boundary condition, and the other chirals satisfy Neumann boundary conditions. The half-index of Neumann boundary conditions for Theory A is calculated as IIA (N ,N ;N,D) I (q (3ra +2rb )/2 a3 b2 s± ; q)∞ ds ±2 (s ; q)∞ Q 3 2πis ra /2 as± x ; q) ∞ I I=1 (q Q 2 rb /2 bs∓ x̃ ; q) α ∞ α=1 (q (3.1) with 3I=1 xI = 2α=1 x̃α = 1. When we choose ra = 12 and rb = 12 and turn off the fugacities xI and x̃α , the half-index (3.1) can be evaluated as Q Q 2 2 1/2 IIA + (6a4 + 16a3 b + 21a2 b2 + 6ab3 + b4 )q (N ,N ;N,D) = 1 + (3a + 6ab + b )q h + 10a6 + 30a5 b + b2 + 48a4 b2 + 54a3 b3 + b6 + 3ab(2 + 2b4 ) i + a2 1 + 13b4 + 2(1 + 4b4 ) q 3/2 + · · · . (3.2) The half-index (3.1) is known as the Nassrallah-Rahman integral [70, 71]. It is shown to be equal to Q Q 3 I=1 3 I=1 q (2ra +2rb )/2 a2 b2 x−1 I ;q Q 2 Q3 q ra a2 x−1 I ;q ∞ α=1 (q ∞ (q rb b2 ; q)∞ I=1 (3ra +rb )/2 a3 bx̃−1 ; q) ∞ α (ra +rb )/2 abx x̃ ; q) ∞ I α α=1 (q Q2 (3.3) We see that the expression (3.3) can be interpreted as the half-index of the theory consisting eα and M fI obeying Neumann, Neumann, of five kinds of chiral multiplets MIα , B I , B, B Neumann, Dirichlet and Dirichlet boundary conditions. The field content and boundary conditions are summarized as bc SU(2) SU(3) SU(2) U(1)a U(1)b U(1)R VM N Adj 1 1 0 0 0 QI N 2 3 1 1 0 ra Qα N 2 1 2 0 1 rb e Q D 2 1 1 −3 −2 2 − 3ra − 2rb MIα N 1 3 2 1 1 ra + r b I B N 1 3 1 2 0 2ra B N 1 1 1 0 2 2rb e Bα D 1 1 2 −3 −1 2 − 3ra − rb f MI D 1 3 1 −2 −2 2 − 2(ra + rb ) (3.4) Here we have explicitly written the shifted R-charges by shifting the R-charge by ra times the U(1)a charge and by rb times the U(1)b charge. Suitable choices of ra and rb (in this case simply rA > 0 and rb > 0) are required to ensure the indices are convergent. However, to save space we will not explicitly present the shifted R-charges in later tables. –7– JHEP08(2023)048 (q)∞ = 2! fI ∼ QI Q; e The operators can be identified as: mesons in theory A, MIα ∼ QI Qα and M eα ∼ Q Q e where without baryons in theory A, B I ∼ QJ QK IJK , B ∼ Q1 Q2 and B α indices indicates antisymmetric contraction of the gauge indices. The anomalies match for these theories and boundary conditions since 3 3 2 A = 2 Tr(s ) + r2 − Tr(s2 ) + Tr(x2 ) + 3(a − r)2 2 2 | {z 2 } 2 | VM, N {z QI , N − Tr(s2 ) + Tr(x̃2 ) + 2(b − r)2 + {z } Qα , N } 2 1 Tr(s2 ) + − 3a − 2b + r |2 {z } e D Q, 3 1 3 1 = − Tr(x ) + Tr(x̃2 ) + 3(a + b − r)2 − Tr(x2 ) + (2a − r)2 − (2b − r)2 2 2 2 |2 {z } 2 | + {z MIα , N 1 Tr(x̃2 ) + − 3a − b + r 2 | {z 2 } + } | {z BI , N } 1 3 Tr(x2 ) + (−2a − 2b + r)2 2 2 | {z eI , D M } Bα, D e 5 = − Tr(x2 ) − Tr(x̃2 ) + 6a2 + 2b2 + 12ab − r2 . 2 B, N (3.5) Therefore we conjecture that the N = (0, 2) Neumann boundary condition in SU(2) gauge theory with 6 flavors has the following confinement: SU(2) + 3 fund. +2 antifund. +1 antifund. with b.c. (N , N ; N, D) ⇔ SU(3) × SU(2) bifund. chiral MIα + an SU(3) antifund. chiral B I eα + an SU(3) fund. chiral M fI + a singlet B + an SU(2) antifund. chiral B with b.c. (N, N, N, D, D). (3.6) Moreover, there is a corresponding confining duality in the bulk by generalizing the boundary confining duality (3.6) SU(2) + 3 fund. +2 antifund. +1 antifund. ⇔ SU(3) × SU(2) bifund. chiral MIα + an SU(3) antifund. chiral B I eα + an SU(3) fund. chiral M fI . + a singlet B + an SU(2) antifund. chiral B (3.7) In particular the theory A index is well-defined as all monopoles have positive dimension. In fact, given the charges of Q̃, the monopole dimensions (R-charges) are given by 2|m| ≥ 2 (3.8) with m ∈ Z, m 6= 0 and there is no dependence on the values of ra and rb indicating that the monopoles have no other charges. Both theories have non-zero superpotentials WA =V (3.9) e α + BM fI B I WB =B I MIα B (3.10) –8– JHEP08(2023)048 | 1 X I = 2 m∈Z A × ra 2 Y (q |m| 3 Y (q 1− 2 + 2 a−1 s∓ x−1 ds |m| ±2 I ; q)∞ (1 − q s ) ra +|m| 2πis (q 2 as± x ; q) I=1 I ∞ I r |m| 1− 2b + 2 α=1 (q rb +|m| 2 b−1 s± x̃−1 α ; q)∞ (q bs∓ x̃α ; q)∞ (q 2 Y 3ra +2rb +|m| 2 3r +2r |m| 1− a 2 b + 2 a3 b2 s± ; q)∞ a−3 b−2 s∓ ; q)∞ q |m| (3.11) and I B = 3 Y (q I=1 (q × 2ra +2rb 2 a2 b2 x−1 I ; q)∞ (q 3ra +rb 2 a3 bx̃−1 α ; q)∞ 2ra +2rb 2 a−2 b−2 xI ; q)∞ α=1 (q 1− 3ra +rb 2 a−3 b−1 x̃α ; q)∞ 1− 3 3 Y 2 Y (q 1− (q 1−ra a−2 xI ; q)∞ (q 1−rb b−2 ; q)∞ Y I=1 (q ra a2 x−1 I ; q)∞ (q rb b2 ; q)∞ I=1 α=1 ra +rb 2 (q −1 a−1 b−1 x−1 I x̃α ; q)∞ ra +rb 2 abxI x̃α ; q)∞ . (3.12) In fact, we have found the precise matching of full-indices (3.11) and (3.12) as strong evidence of the duality (3.7). For example, when we set ra = rb = 1/4 and switch off the flavored fugacities, we find IA = IB = 1 + (3a2 + 6ab + b2 )q 1/4 + (6a4 + 3a−2 b−2 + 2a−3 b−1 + 16a3 b + 21a2 b2 + 6ab3 + b4 )q 1/2 + (12a−2 + 10a6 + 8b−2 + 18a−1 b−1 + 2a−3 b + 30a5 b + 48a4 b2 + 54a3 b3 + 21a2 b4 + 6ab5 + b6 )q 3/4 + · · · . 4 (3.13) Gustafson integrals Gustafson derived several higher-rank generalization of the Askey-Wilson and NassrallahRahman integral identities including for various SU(N ) [72, 73], USp(2n) [72, 73, 81] and SO(N ) [72] groups. Here we give an interpretation of these results as half-index identities arising from boundary dual theories. Several of these boundary dualities correspond to bulk dualities and we provide some checks of matching indices. However, others do not seem to arise from bulk dualities since the natural bulk theories have non-positive dimension chirals or monopoles. –9– JHEP08(2023)048 where V is the minimal monopole operator in theory A having R-charge 2 and no other charges. The theory A superpotential explains the global charges in theory A which do e would have charge 1. Including not include a potential U(1)c symmetry under which Q such a symmetry would result in V having non-zero U(1)c charge, hence the monopole superpotential excludes such a U(1)c . As explained in [29], the linear monopole superpotential arises from compactification of a 4d theory to 3d. Hence, with this superpotential for theory A, this is the 3d realization of the Nf = NA = N case of the 4d SU(N ) ↔ SU(Nf − N ) Seiberg duality, which in this case is a confining duality as the theory B gauge group is the trivial SU(1). The full-indices of theory A and B are evaluated as 4.1 SU(N ) with Nf = Na = N flavors We consider a 3d N = 2 gauge theory with gauge group SU(N ), Nf = N fundamental chirals QI and Na = N antifundamental chirals Qα . We choose the N = (0, 2) half-BPS Neumann boundary condition for the SU(N ) vector multiplet and Neumann boundary conditions for the chiral multiplets. This can be viewed as a higher-rank SU(N ) generalization of the theory discussed in section 2. bc SU(N ) SU(Nf = N ) SU(Na = N ) U(1)a U(1)b U(1)R VM N Adj 1 1 0 0 0 QI N N Nf 1 1 0 0 Qα N N 1 Na 0 1 0 MIα N 1 Nf Na 1 1 0 B N 1 1 1 N 0 0 B N 1 1 1 0 N 0 V D 1 1 1 −N −N 2 (4.1) The anomalies match for these theories and boundary conditions since N2 − 1 2 N N N2 A = N Tr(s2 ) + r − Tr(s2 ) + Tr(x2 ) + (a − r)2 2 2 2 2 | {z } VM, N − | {z QI , N N N N2 Tr(s2 ) + Tr(x̃2 ) + (b − r)2 2 2 2 ! {z } | ! } Qα , N N N N2 1 1 Tr(x2 ) + Tr(x̃2 ) + (a + b − r)2 − (N a − r)2 − (N b − r)2 2 2 2 |2 {z } |2 {z } ! =− | {z } MIα , N B, N B, N 1 + (−N a − N b + r)2 |2 {z } V, D N N N2 2 N2 2 N2 + 1 2 2 2 2 2 =− Tr(x ) − Tr(x̃ ) − a − b + N ar + N br − r . 2 2 2 2 2 7 (4.2) From now on we save space by not explicitly parameterising the R-charge by the possible shifts proportional to other global U(1) charges, although suitable shifts are required to ensure the unitarity bound is satisfied. – 10 – JHEP08(2023)048 We propose that theory A is dual to theory B which has Nf ×Na bifundamental chirals MIα of the non-Abelian flavor symmetry group SU(Nf ) × SU(Na ) and two singlet chirals B, B with Neumann boundary conditions. We also have a singlet chiral V with Dirichlet boundary conditions. In fact, this is an example of dual boundary conditions for a known bulk duality [82]. This is summarized7 as The theory A half-index is N I −1 Y (q)N ∞ A II(N ,N,N ) = N ! i=1 ×Q N dsi Y (si s−1 j ; q)∞ 2πisi i6=j 1 N i=1 Q N ra /2 as x ; q) i I ∞ i=1 (q Q N rb /2 bs−1 x̃ ; q) α ∞ α=1 (q i . (4.3) For example, setting ra = rb = 12 , xI = 1 and x̃α = 1 we have 1/2 IIA + (a3 + b3 )q 3/4 + 45a2 b2 q + 8ab(a3 + b3 )q 5/4 (N ,N,N ) = 1 + 9abq + (a6 + 9ab + 165a3 b3 + b6 )q 3/2 + (a3 + 45a5 b2 + b3 + 45a2 b5 )q 7/4 + · · · , (4.4) N =4: 1/2 IIA + (a4 + 136a2 b2 + b4 )q + 16ab(1 + a4 + 51a2 b2 + b4 )q 3/2 (N ,N,N ) = 1 + 16abq + (a8 + 136a6 b2 + b4 + b8 + 8a2 b2 (32 + 17b4 ) + a4 (1 + 3876b4 ))q 2 + · · · . (4.5) The half-index (4.3) can be viewed as a higher-rank Askey-Wilson integral that generalizes (2.1). Gustafson [72] showed that it is given by q q N ra /2 aN ; q ∞ N ra +N rb 2 q N rb /2 bN ; q ∞ a N bN ; q QN I=1 ∞ (ra +rb )/2 abx x̃ ; q) ∞ I α α=1 (q QN . (4.6) We can exactly identify equation (4.6) with the half-index of theory B with the specific boundary conditions described above. Note that the bulk theory A also has a Seiberg-like dual description [29, 30]. The mapping of operators between theory A and theory B is given by: MIα ∼ QI Qα , B ∼ Q1 · · · QN = det Q and B ∼ Q1 · · · QN = det Q where the antisymmetric tensors are used to contact the gauge indices. Theory A has zero superpotential while theory B has a superpotential WB = −V det M − BB (4.7) which imposes the chiral ring relation det M = BB in theory B which corresponds to det(QQ) = det Q det Q in theory A. It can easily be seen that the contribution of V with Dirichlet boundary condition does exactly this in the theory B half-index. Through two dualities this gives an alternative derivation of the above duality. The first is the Seiberg dual of a SU(N ) theory with Nf = NF fundamental and Na = NF antifundamental chirals which is a U(NF − N ) × U(1)y theory with Nf = NF fundamental and Na = NF antifundamental chirals, two chirals v± with charges ±1 under U(1)y and chirals MIα in the bifundamental representation of the flavor symmetry group SU(Nf ) × SU(Na ). In the case here with NF = N the dual gauge group is just U(1)y and there are hence no fundamental or antifundamental chirals (of U(0)). As the chirals MIα are not charged under U(1)y we can consider them as spectators and dualize the U(1)y gauge – 11 – JHEP08(2023)048 N =3: theory with the two chirals v± which is dual to the XY Z model, where in our notation we identify V , B and B with X, Y and Z. We can extend the above discussion to include boundary conditions as shown in (4.8). Here U(1)A is the axial symmetry in theory A and U(1)G is the global U(1) arising from the gauging/ungauging process to derive the SU(N ) Seiberg duality from the U(N ) duality [29, 30]. (4.8) With these boundary conditions the anomalies match for all three theories provided we include a background mixed CS coupling for U(1)G × U(1)y in theory B, contributing 2N Gy to the boundary ’t Hooft anomaly. In particular the anomalies match for these theories and boundary conditions since N2 − 1 2 N N N2 2 2 A = N Tr(s ) + r − Tr(s ) + Tr(x ) + (A + G − r)2 2 2 2 2 | {z } ! 2 SU(N ) VM, N | {z QI , N } ! N N N2 2 2 Tr(s ) + Tr(x̃ ) + (A − G − r)2 + (N G + y)2 {z } | 2 2 2 − | {z η } Qα , N 1 2 r 2 |{z} =− U(1) VM, D 1 1 + (−N A + y)2 + (−N A − y)2 |2 {z } |2 {z } v+ , D v− , D ! N N N2 Tr(x2 ) + Tr(x̃2 ) + (2A − r)2 + 2N Gy | {z } 2 2 2 − | {z MIα , N } FI N N N2 1 1 2 2 Tr(x ) + Tr(x̃ ) + (2A − r)2 − (N A + N G − r)2 − (N A − N G − r)2 2 2 2 |2 {z } |2 {z } ! =− | {z MIα , N } B, N B, N 1 + (−2N A + r)2 + (N G + y)2 {z } |2 {z } | V, D =− η N N N2 + 1 2 Tr(x2 ) − Tr(x̃2 ) − N 2 A2 + 2N 2 Ar + y 2 + 2N Gy − r . 2 2 2 – 12 – (4.9) JHEP08(2023)048 bc SU(N ) SU(Nf = N ) SU(Na = N ) U(1)A U(1)G U(1)y U(1)R VM N Adj 1 1 0 0 0 0 QI N N Nf 1 1 1 0 0 Qα N N 1 Na 1 −1 0 0 η 1 1 1 0 N 1 0 VM D 1 1 1 0 0 Adj 0 v± D 1 1 1 −N 0 ±1 1 MIα N 1 Nf Na 2 0 0 0 MIα N 1 Nf Na 2 0 0 0 B N 1 1 1 N N 0 0 B N 1 1 1 N −N 0 0 V D 1 1 1 −2N 0 0 2 η 1 1 1 0 N 1 0 To compare to the previous table (4.1), note that we can map U(1)A ×U(1)G to U(1)a ×U(1)b with the following linear combinations of charges Qa = 12 (QA +QG ) and Qb = 12 (QA −QG ). To summarize, we propose the following confining duality of N = (0, 2) boundary conditions: SU(N ) + N fund. chirals QI and N antifund. chirals Qα with b.c. (N , N, N ) ⇔ SU(N ) × SU(N ) bifundamental chiral MIα + singlet chirals B and B with Neumann b.c. and a singlet chiral V with b.c. (N, N, N, D). (4.10) N N N X r a + rb X N (ra + rb ) X N 1− |mi | − |mi − mj | ≥ 1 − |mi | ≥ 2 2 2 i=1 i