JHEP03(2021)203 = 4 N Springer March 22, 2021 : February 17, 2021 November 27, 2020 -invariant of the : : η Published to bosonic Accepted Received 4 RP which, when decomposed into discrete Published for SISSA by super-Yang-Mills (SYM) theory on the https://doi.org/10.1007/JHEP03(2021)203 4 . As an illustration, we derive a matrix 2 RP = 4 RP N . Using supersymmetric localization, we find that for 4 RP submanifold. This paves the way for understanding 2 2 RP RP . . We also comment on potential applications of our setup for AGT Super-Yang-Mills on 3 4 RP 2005.07197 The Authors. = 4 using known results of YM on c Conformal Field Theory, Supersymmetric Gauge Theory

We study the four-dimensional 4 , [email protected] N RP , their expectation values are captured by an effective two-dimensional bosonic Yang- Cambridge, MA 02138, U.S.A. Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. E-mail: Center of Mathematical Sciences and Applications, Harvard University, Open Access Article funded by SCOAP states on the boundary. Keywords: ArXiv ePrint: SYM on integral form of the SYMholonomy partition sectors, function on contains subtleDirac operator phase on factors duecorrespondence, to integrability and the bulk-reconstruction nontrivial in AdS/CFT that involve cross-cap Abstract: unorientable spacetime manifold a large class ofQ local and extendedMills (YM) SYM theory observables on preserving an a common supercharge Yifan Wang From Yang-Mills on JHEP03(2021)203 6 7 3 (1.1) of even d (or equiv- ] for a re- d RP 10 S due to the residual O ∆ ]. Putting the CFT on quotient of 12 2 , Z 11 4 ], RP 13 [ O ) 13 x ∆ quotient of the flat space with a co-dimension ( ) 2 2 O O 9 x Z h SYM on 4 RP (1 + = 4 3 – 1 – = N i ) 2 x , namely the correlation function of local operators, ( d RP hO SYM on RP 15 similar to the case of co-dimension one defects (planar or = 4 from involution + 1) Consequently one can formulate a bootstrap program for the 10 N 4 d ( scalar primary operators 1 RP so and a new matrix model 2 by a Weyl transformation) and preserve a large residual (Euclidean) 1 RP ], as well as a refinement of the electric-magnetic duality in abelian gauge 8 – normalized 1 ∪ {∞} ]. However relatively little is known about the dynamics of strongly interact- d 9 R introduces new observables beyond those on the flat space, given by the one-point A familiar family of unorientable manifolds are the real projective spaces The boundary CFT can be thought of as defined by a 3.2 Discrete holonomy sectors and phase factors 2.1 The SYM2.2 on The supersymmetric2.3 action for Localization to 2d YM on 3.1 Partition function d 1 where the position dependence issymmetry fixed (which by the also conformal requires dimension the one-point functionone of fixed loci. spinning primaries to vanish). basic observables in thesimilar CFT to on the caseRP with a domain wallfunctions or of boundary defect [ dimensions, which are realized byalently a free orientation-reversing conformal subalgebra spherical) in flat space. symmetry [ theories [ ing theories on unorientableapproach to spacetimes conformal beyond field twocent theories review). dimensions. (CFT) is Luckily well the suited bootstrap for this task (see [ 1 Introduction Quantum field theories withspacetime time manifolds. reversal symmetry Thiscation can simple of be idea topological formulated has phases on and led unorientable detection to of exciting subtle developments anomalies in that the involve classifi- time reversal 4 Conclusion and discussion 3 2d YM on Contents 1 Introduction 2 Localization of the JHEP03(2021)203 ] 2 S 3 14 = 4 in this N 4 ) have no d . Solving RP d preserving RP 4 on . Along with RP d -expansion. 4) | RP  dimensions [ (4 RP  super-Yang-Mills − osp 6 ]. There, a particular = 4 . The partition function 36 , N (with round metric), one Q identification, the SYM on 4 35 upon localization. Indeed as 2 2 RP Z in terms of the ordinary OPE Z / 2 O . was used to a localize the theory which is a real form of the complex Lie d h S 4 ]. Due to strong coupling effects, few 4) RP | = 4) 2 | SYM can be defined on 21 2 , gives the – (4 ] to leading orders in the (2 4 RP = 4 17 15 psl S , psu 9 N – 2 – ]. For the same reason, we will also not impose reality ]). , the gauge theory observables become sensitive 4 23 ], and for Lee-Yang theory in 34 – 13 RP 25 possibly supplemented by additional dynamical inputs dimensions [ 2  which contains the supercharge − 4) 4 | ], which in turn determines the partition function of the ] allows for extractions of exact results for a large subset of the (4 has a simple combinatorial formula in terms of the representation 37 24 by exploring constraints from the (residual) conformal symmetry, superconformal algebra is 2 – osp d 22 RP R super-Yang-Mills (SYM) theory on flat space, with possibly extended = 4 furnish the basic structure constants for the CFT on ) to two-dimensional (constrained) Yang-Mills theory on a great amounts to fixing these coefficients 4 . Here we study the CFT in the Euclidean signature obtained from a Wick rotation. O N d h 4) = 4 in the superconformal algebra | using an extension of the localization setup of [ HS should lead to the bosonic YM on (4 4 RP , which can be re-expressed into a single matrix model. Thanks to the general Q N 2 4 ). Since the antipodal quotient of psl Z 2 RP / ] for a reformulation of the bootstrap equations on RP 4 16 (resp. S HS 4 = In this note, we initiate the study of the four-dimensional Gauge theories in four-dimensions offer a rich playground to advance this program. On S The Lorentzian Note that unlike the more familiar four-point function bootstrap, hereSee the [ combinations of OPE co- 4 2 3 4 superalgebra As usual, one thena loses problem the for reality condition ourany on analysis reality the if conditions, fermionic the generators whichconditions theory of is on is the the the invariant superalgebra. fermionic under case generatorspaper. the This here in is supersymmetry [ the not transformations Euclidean superconformal regardless subalgebra SYM on efficients that appear inobvious the positivity conformal properties. block decomposition (e.g. of two-point functions on RP will see, such an identification existsa and half-BPS the subalgebra of the YM theory on data of the gauge group [ theory on supercharge on (resp. naturally expects that by implementing a supersymmetric correlation functions. When combined withway the to bootstrap solve program, the itdimensional provides CFT. a powerful Thisconformal has defects (see been for particularly example successful [ in the study of the four- to fine details of theplings, theory, and such the as spectrum of globalobservables extended structures in defects of general [ four-dimensional the gauge gauge theoriesperturbation group, can theory. topological be cou- obtained Fortunately analytically in beyond localization supersymmetric method [ gauge theories, the supersymmetric dimensions via numerical techniques [ and Wilson-Fisher theory in one hand, a largefour-dimensional class Yang-Mills of theories CFTs coupled arelogically to produced nontrivial matter. by manifold renormalization such On group as the (RG) flows other from hand, on a topo- the OPE of localthe operators, they CFT determine on generaldata correlation of functions the on CFTcrossing on symmetry and unitarity, from other methods. This program has been pursued for the Ising model in two and three The coefficients JHEP03(2021)203 ]. 40 (2.6) (2.1) (2.2) (2.3) (2.4) (2.5) , 39 subalgebra on flat space and the com- 4) | 4) µν (4 | . Together they µ M (4 osp K psl − is defined by identifying µ 4 P , we identify the supersym- . RP , we derive a matrix integral 2 µρ 3 δ 8 , | 1 , involving local operators as well 1 R = x 4 | − preserving the . νρ − (6) 4 = RP I , 2 µ . ) ν 4 so = x x z µ RP µν 1) ( 8 I , translate to (defect) observables in the × | , 0 . In section RP µν Q 2 x (5 ≡ − I 1) is generated by rotations det 2 , ,I 0 0 µ so −| 4 RP SYM on ν x using two dimensional gauge theory techniques x (5 x – 3 – ⊂ 2 = 4 − µ RP so SYM on x → x  = 4 = 2 RP ⊃ µ (5) µ ν 0 x µ ν N 0 SYM on = 4 so − 4) : , the real projective space ∂x | ∂x µ ι ∂x ∂x µν N  x (4 δ is unorientable. = 4 psl = ι 2 det ]. We leave the detailed investigation of such observables to Z . N µν / from involution 38 I 4 4 , R 4 36 = RP 4 , which can be further reduced to computations in the relevant matrix 2 RP partition function. We end by a brief summary and discuss a number of ], observables of the SYM theory enjoys the superconformal symmetry with coordinates 4 4 RP 36 R RP = 4 involution that defines N 2 Z The The residual conformal symmetry on The rest of the paper is organized as follows. In section that leads to the two dimensional YM on which includes the conformal symmetry and R-symmetry and consequently bination of translationdefines and the special subalgebra conformal transformations The Jacobian has negative determinant since The Jacobian of the transformation is where 2.1 The SYM on On flat space points related by a fixed-point-free involution mula for the future directions in section 2 Localization of the metric and carry out the localizationQ computation with respect to theform aforementioned supercharge for the SYM partition functionand on compare with resultsWe from also an comment alternative on localization subtle procedure phase discussed factors in that [ have been missing thus far in a gluing for- as defects preserving theYM common theory supercharge on model as illustrated ina [ future publication. discussions in [ JHEP03(2021)203 . 16 ] for 1 2 , the (2.7) (2.8) (2.9) . (2.10) (2.11) (2.12) (2.13) (2.14) MN δ SYM ι 5 . as a vector. ) = 2 . ι x Spin(10) s . ( }  ι 2 ψ N Z are grouped into µ 790 / ˜ Γ . x 0 2 , Γ 1 2 µ , i S γ 9 − M 1 − Γ = − ] (we follow the conven- { = 2 2∆ . c 35 ,..., | acting on the fermions in the symmetry. The preserved 6 x RP such that a half-BPS subal- | , R 790 i of [ in the SYM which is specified ,  6 = 1 has weight 4) s 2 | Q (6) (3) = 5  ε ) 2 0 ) = S to transform under ι (4 0 I so so x = x ε ( . ( s µ ε ⊕ psl )  Γ . related by a sign flip in the identification of R x ε ), we conclude c ( µ on 4 s 568  , ε  1238 , ψ on the covering space (see appendix A of [ ˜ Γ ι µ , ) 2.9 ι RP µ Γ ˜ structure ( (3) Γ x 790 790 ). Note that 4) ( µ − ) = | ix on the supercharge takes the form + so 0 Γ ν Γ x i ι x V (4 = − 2.14 ( ⊕ → ν pin + − s – 4 – ε µ ) s  = psl I 1)  = R x , is given by the YM on ( µ c R 2∆ ε =  ˜ | 2390 (4 4 Γ x µ with a ε structures on : Γ so 4 RP + ix −| = ⊃ symmetry vector index ] for the spinors and gamma matrices. The 4d spacetime s RP SYM ] for relevant discussions). . The 10d flat space chiral and anti-chiral gamma matrices  ι 4) R 0 ) = 35 | 0 , 41 , x 9 (4 1370 (6) ( related by the involution 9 µ , Γ 0 so osp x 5 , = 1 ,..., s 2 ,V  and and , ) on general spinor fields see ( x x 4 ( ι respectively which satisfy the Clifford algebra 1279 are constant chiral and antichiral 16-component spinors of rotation. The preserved 16 supercharges are parametrized by constant = 1 φ Γ c , such identifications take the following form M R  2∆ M | ,..., ˜ = Γ ∆ 2 x | s (6) ,  and ] here) that defines the 2d Yang-Mills sector on so is induced by an (outer)automorphism of the and s = 1 ) =  36 0 1890 R x µ Γ M ( on the primary operators on the flat space. For scalar, vector and fermion operators In general, CFTs on real projective spaces are defined via the identification induced We note that this subalgebra includes the supercharge We would like an extension of the involution Γ φ For the action of The spinors here are defined on ι 5 6 Euclidean signature). There arethe fermions two at pin points a detailed review and also [ In the next section, weresulting will 2d exploit sector this of fact the to SYM show on that uponby identification by of dimension tion of [ by additional projectors (only three are independent) as below, spinors satisfying which generate a half-BPS subalgebra of Consistency of the superconformal algebraCombined requires with the requirement of solutions to ( up to an where supercharges are those satisfying are Finally gebra is preserved. In general, the action of niently packaged into a conformal Killing spinor We adopt the conventions ofindex [ the 10d index as bosonic subalgebras, as well as Poincaré and conformal supercharges that are conve- JHEP03(2021)203 5 S × 5 (2.23) (2.18) (2.19) (2.20) (2.21) (2.22) (2.15) (2.16) (2.17) limit is , ) N x Ψ( . In particular, R 4 µ S x µ ) that we want to ˜ Γ 2 | 2.11 x ( | i Ω , ε . 3 2 − ) ) c − SYM in the large 0 ). The antipodal map clearly  e ) is simply given by the Weyl ) Φ ) = . I Ψ 0 , 9 Γ 2 . x 2.11 ) on the boundary by taking the = 4 , I 2.17 x Φ → ) y are identified point-wise, and the R I , 5 Ψ( 2.7 N 8 2 S Ψ + 1 dy Φ 1 + SYM I , µ ε . ι ) − and this is identified with the residual , < Γ ≡ , ε dy x = µ 7 Ω 4 + vol ( 2 | Ω | Ω x ν − 1 2 + 5 S Φ x y | e , | e A background with metric µ I AdS 6 ν + ( ε AdS 5 Φ µ = | , e Φ dx s – 5 – I 2 y S (a special case of the lens space). The residual  4 µ − | and 2 ( | S → , 0 3 | (vol × dx 5 ε x dx 1 preserving the half-BPS supersymmetry, we require 1 y I 5 | |→ 4 Φ L 2Ω 4 Φ lim RP y −| with round metric ( = | e > − = 2 4 RP | = ) = , 5 ) = x = ( ds 0 2 µ | F RP x I A x, y ( ds ). More explicitly, ( (with round metric). The SYM fields are also related by µ symmetry of the Φ) for → 4 ) simply amounts to the antipodal map on the R AdS 2.10 4 µ ± ( subject to the same constraint ( rot SYM on ε RP A ,A 2.1 ) is reduced to s,c (5) ]. The holographic dual of the HS  x = 4 ( so in ( 44 I ι – ) in the next section). N = 1 ) by compatibility with the supersymmetry Φ) | 42 µ x R | A ( 2.27 2 | at x or superconformal symmetry is realized in the bulk by Killing spinors on AdS | 3 I is (induced) by ( S Φ = 4 ) = R 0 x N Before we end this section, let us make some comments in relation to the AdS/CFT By stereographic map from the flat space, we put the (Euclidean) CFT on the sphere ( with metric I Φ 4 and they are relatedasymptotic to limit the conformal Killing spinor ( and self-dual five-form flux The correspondence [ given by the IIB string theory on AdS commutes with the conformal symmetry on thanks to the Weyl symmetry. (superconformal) symmetry on transformation of the flatbecome space counterparts. In particular the conformal Killing spinors with constant spinors S Then the inversion the two hemispheres equator Note that the identification(either for the fermionspreserve are (see completely ( fixed by that of the bosons the following supersymmetric identification due to where Here to define the JHEP03(2021)203 . ) a ], · I in T on , · Φ 45 a M , , and (2.27) (2.24) (2.25) (2.26) A = 0 tr ( 23 SYM , where [ θ ι , this is ˆ ≡ µ = 0 quotient is ) . 4 Γ 2 µ S  N M Z ( x background. ˆ µµ m A e on 5 su K S = G m = × µ K . The trace preserving off-shell , g Γ 5 g − 4 given by  isometry in the bulk , I of I and adjoint scalars 5 as reviewed in the last ) 2 Φ . Finally the generators RP m a y I µ Z ν F 0 4 T components, . For Φ , A R m ( . 9 2 λ g RP 2 2 K | R y | + = 1 . Note that ,..., tr = tr ε 2 0 1 + SYM on 2 µ Ψ + y , 2 | ∇ M + x generators I | in the IIB string theory needs to be 2 9 D Φ , = 1 = 4 SYM on follows from that on the SYM scalars y SYM with gauge group 2 M | µI Spin(10) I + y Γ N M y | SYM 2 factors are odd. Thus to preserve the five- 7 ΨΓ identification of the SYM fields by 1 2 = 4 ι µ = 4 . In terms of its + 5 , x 2 − A . In other words, the IIB background dual to , y + 2 N S Z | N – 6 – Ψ ε ∧ ws x for Gamma matrices in the internal directions). | M ] with = 0 Ω MN A . We set the four-dimensional theta angle I − and D MN 8 F 35 ab Γ + anti-hermitian y  Γ , δ 5 M 1 2 Γ = Ψ MN contains the 4d gauge fields dA → − MN m 6 F and preserves the orientation of the 10d spacetime while the and M ) theta angle compatible with the orientation-reversing y ν = F I M 1 2 Γ corresponds to an orbifold by an ˆ denotes the Dynkin index of I 2 1 2 ε − A a M  = ) = , y F Γ Z given by λ b bare µ A 5 ), the involution = = tr T 5 T x y denotes flat space 10d Gamma matrices in the chiral basis (we g SYM a S is given by an orientifold of the usual AdS ι m Ψ = M and is related to the usual trace in a particular representation T √ ˆ ε :( 2.21 M 4 A δ ⊂ K x g ε ε Γ tr ( 4 are auxiliary fields which serve to give an off-shell realization of the 2 δ δ . RP d 4 S 7 SYM 4 S , where ι S λ Z has a fixed locus located at the center of the AdS transforms as a chiral spinor of ,... tr 2 4 λ g 1 with curvature 1 Ψ T 2 2 = 1 A SYM ). Note that this − ι 7 + m = d 2.16 tr = ≡ . with The superconformal transformation of the SYM fields are We start by reviewing the action for 4d SYM in ( π Another possible value of the are normalized by D is the vielbein and S 7 by m a I = ˆ µ µ θ T this paper. by comes with real coefficients is the Killing formλ of identical to the trace in the fundamental representation The gaugino e will not distinguish between K supercharge that we will use to localize the theory. The gauge covariant derivative is defined Here we follow thesection. 10d The notation 10d gauge in fields [ 2.2 The supersymmetricIn action this for section,supersymmetry, we obtained present by performing the a the action covering of space the form flux backgroundsupplemented ( by the worldsheetthe parity supersymmetric Moreover, wrapping a internal where the actionΦ on the internalindividual coordinates volume forms on the AdS The boundary involution JHEP03(2021)203 ) = ). to ρ ε Γ 2.17 δ 2.13 (2.30) (2.31) (2.32) (2.33) (2.28) (2.29) µ in the SO(7) -exact ), and Γ 2 Q ν S Γ 2.26 ρ x , ) are chosen . ν 4 x ]). S M αβ , . 23 2.27 ˜ Γ ) satisfying ( m ε x ε ( K . M ν in ( ) Γ A on the 7 ε x ν 0 ( ). 1 2 µ 6 x , m I , 5 = − K Φ ,..., 9 ) but we have dropped the β Ψ) and ). ε , ε δ α ) = ) = = 1 x 0 0  ¯ 2.18 Ψ ) (with the Weyl factor in ( , x x ( 2.28 = 1 ( ( + ) m ε µ 2 3 δ x 2.9 m in ( ), provided that we implement the ) m ( β x tr ν 4 56 m g + ] (see also appendix C of [ S ν m α with R 2.27 ε 2 2 √ 45 R after taking into account the Weyl factors x x − m µ 4 4 | ν ˜ + d ⊥ Γ ε , ν x R | µ 4 2 1 M M S x i Z Γ – 7 – 2( , x t ε defined by the Killing spinor ) on − ). − 2 + mn Q = 0 is given by = δ ,K 2.15 ) = 4 2.26 RP 0 2 ε ) ] for a streamlined version), upon turning on a x = x x SYM Q ( 10 36 n : S m Ψ( ν ν ,A ) → R M 2 YM x rotation which only acts on the auxiliary scalars µ Γ ( S | x I m x µ | generates rotation transverse to a distinguished great 2 YM SYM ˜ Φ) Γ S S ) with the round metric is defined by implementing the following i SO(7) , ν subalgebra 4 R ] (see also [ − 4 P 35 1) | = 0 − RP ) = ( ) = (1 4 0 0 m gives rise to a combination of the bosonic symmetries (including an x x ν K su ( 2 ε ( ) of the SYM action ( I M δ 1 2 Ψ( m Γ Φ ε K ≡ . 8 ⊥ is an R-symmetry rotation (e.g. it acts on the scalars µ ). We emphasize that this identification is a symmetry of the action ( Γ -closed) deformation (the localizing term) as M 56 µν Q I R As explained in [ The SYM on 2.19 2 Note that the lastIn equation follows checking from these the first one two may [ findWe have useful suppressed an the following identity for gamma matrices | 8 9 , which we denote by x 10 4 −| and (and where S which is compatible with the pure spinor constraints2.3 in ( Localization toWe 2d now YM specialize on toIt the generates supercharge an respects the off-shell SUSYsame identification transformation for laws the ( auxiliarytaken pure into spinors account), as in ( which follows from the identification ( in ( Consequently rotation of identifications between the SYM fields at antipodal points subscript to avoid clutterclose in off-shell, the notations. auxiliaryto 10d In satisfy chiral order spinors for the subalgebra generated by where the conformal Killing spinor JHEP03(2021)203 ) by . For , such (2.34) (2.35) (2.36) (2.37) (2.38) (2.39) 4 2.29 S → ∞ t parametrized ) at the level of 2 YM = 0 2 YM S RP F 2.29 ]) . The 2d YM action is 35 D? by in the limit 4 2 YM A g S , YM on cYM on .  , ]. For this reason, we refer to G ε G 2 Ψ = 0 2 ) 90 ε ) . 35 δ , Γ 2d 11 F ) 2d follows verbatim from the deriva- F ) 0 ? ε, ? 2 ~x , Φ ( 70 i Z , j gauge field tr ( . Γ tr ( 9 / 2 2 A 4 dx 2 ] ε, S Φ G S k ij S , 2 4 79 φ 7 35 g P dV πR ). Γ j dV = 2 2 − x 2 ε, (Φ localization 4 localization S − RP 2.30 68 ijk Z ≡ Z Γ – 8 – Q− ) = Q− = RP i ) ε, ~x 3 1 a 2 YM 1 + 2 ] are excluded in [ YM − g 4 Γ 2 , φ 4 YM g ) and ( ( µ 2 A j g 49 S | ˜ Γ – RP x = A | µ , φ 2.28 ≡ − 1 ij x ≡ − 46 A φ P  ( , )[ related to the 4d SYM coupling YM localizes to the BPS locus YM = G S S 2 YM 4 m SYM on YM S ν S SYM on g G ], this was named “almost Yang-Mills” or aYM. G is the projector to the tangent space of 35 j , which satisfy ( = 4 x = 4 i = 0 x N N 4 The end result simply amounts to the identification ( − x ] since all BPS equations are compatible with the identification ( 4d ij 4d at 12 δ with 36 0 , , ≡ 9 35 , 2 YM ij S P = 7 a To summarize, we have the following from [ Now the localization of the SYM on In the original paperThe [ seven auxiliary pure spinors here are given by (compared to (3.17) in [ -closed observables in the SYM, including the partition function of the SYM on 11 12 An important distinction betweenthat this the effective (unstable) 2d instanton contributions theory (nontrivialby and solutions the to the cocharacter usual lattice 2dthis of Yang-Mills theory is as the constrained-Yang-Mills theory (cYM). with the 2d YM coupling on the Furthermore, they are weighted by an effective 2d Yang-Mills action Q a deformation doesn’t affectlimit. their A expectation careful analysis values,uniquely of determined therefore the by we BPS the are locus value of free shows an that to emergent the take smooth this BPS configurations are the SYM path integral on Therefore we conclude the 2d cYM theory on the where simply tions in [ construction. with JHEP03(2021)203 ) 3.1 (3.2) (3.3) (3.1) . For (2.40) λ we comment ]. 23 3.2 . This makes the Σ . is given by a weighted computed for gauge group ) 2 i 2 x S λ RP since they are incompatible ∼ − , i 2 is however unknown. Here in ) , x , supercharge of [ λ ( ( ) 4 / RP 2 λ . c } ( 2 enjoys a large spacetime symmetry RP c = 2 2 YM g = 1 Σ 2 2 YM , N g 2 3 2 x πR ]. 1 2 πR + − ] which agrees with the 2d YM answer ( − 51 2 2 e G is real is pseudo-real is complex , e SYM on λ x 23 2 λ f λ λ λ – 9 – d of λ + 50 , d λ 2 1 λ ] using the 1 = 4 X 48 λ – 1 − 0 X , x = 40 , N          = 46 , 2 39 = 0 = 2 [ YM S 4 37 λ 4 Z YM [ RP x f { Z respectively where the 2d and 4d gauge couplings are related G RP partition function of SYM is given by a Gaussian matrix model ). Consequently there is no difference between constrained and : 4 4 S RP 2 YM 2.38 and a new matrix model 13 2 RP denote the dimension and second Casimir of the representation and , namely the area-preserving diffeomorphisms on ) RP 4 λ S ( 2 c partition function with interesting features. Then in section 4 we use the effective 2d YM description to determine a matrix integral expression has a simple combinatorial description in terms of representation theory data of ). Indeed, the gauge theories on SDiff(Σ) and is the Frobenius-Schur indicator for the representation ] for a different representation of the YM partition function on using the Reidemeister-Ray-Singer torsion. ]. ) Σ λ λ RP 52 SYM on 36 3.1 R f d 2.37 , = 2 The partition function of the For example, the partition function of the standard 2d YM on , we have instead See [ 2 = 4 N 13 for the on the relation between ourof matrix integral to important partial results for the localization PGL(2 via a different localization computationafter in projection [ to theterms zero-instanton [ sector and taking into account appropriate counter- section Via our localization argumentN in the last section,as they in give ( the partition functions of the where where RP general the underlying gauge group sum over all irreducible representations 3 2d YM on The 2d Yang-Mills theorygiven on by a Riemanntheory surface rather rigid (almost like a 2d TQFT) and consequently its partition function on The nontrivial (unstable)with instantons the are identification ( forbiddenordinary on 2d YM in this case. with JHEP03(2021)203 ) ) 0) N N (3.7) (3.8) (3.9) (3.4) (3.5) (3.6) , mul- If we U( (3.10) G SU( ,..., as, 14 3 . i N x − > ` , πi , . ,N and the representation. 2 2 2 ) 2 ) 4 ··· 1 N  ], − -tuples − 2 N = 0 1 > N + 46 N etc). Together, they i 2 − Yang-Mills is simply − 2 λ i  SU( ]. The dimension and N ` ) ( 2 4 N,N π λ N =1 > ` g 36 4 N i N =1 N i − 1 . −  ` P ). Furthermore, the 2d YM 1 2 i U( i P a . ` − ) = ( log  3.1 N N =1 2 YM as in [ N i 2 λ g (Σ) . charge of the representation and 2 2 N =1 ) P i ) X N χ = j 2 N 2 πR 2 4 , there are no pseudo-real representations. 2 π x + k g − 1 2 ) 8 , . . . , ` , λ 1 − labeled by U(1) N − − ` 1) . We label the representations of e − e i ( ) ) ) 1 x , . . . , `  i = 0 − + 1) case to the expected partition function of ( N 1 N λ ) a i = U( 2 ` i charge vanishes ( N 2 12 N ( − 2 ) G U( ≤ ( N S i − ( Vol(Σ) + ` ∆ +1 G Y = U( i i

1 i ` i ij e , πiz 2 YM z ) 0 2 ! =1 m 2 g N i ij ∆( )∆( i N + e i 2 + ∈{ X det − i 2 ∆( i i z ! P dy z ` R i (1 det ( ∆( m ! 2 πi N i

∆( 0 πi N 2 dz i Z e 2 e 1) dz ! ) ) = i ) = N i =1 − i 2 i YM Y i N 2 =1 is ∆( z Y g N i N =1 iz dz ( Y 2 i

} πiz e λ

N 1 2 ( ∆(

, holonomy (up to conjugation) as ! i partition function of the 2d YM into a matrix model we use the N e =1 πR Z 0 + 1) , λ Y i ( ) ! 2 Z 1) Z 2 χ N λ i =

∈{ X 1 ∈ − N i N Z YM χ i N RP 2 ( real X ` dz ∈ 12 m 2 Z RP i ), we have the renormalized action, which by abusing the notation we N Z real X ) G ( U( ` λ = real X N =1 N N λ Y i 2 real × X ) = e 2 − i

) (1 ) = 1 = N 2 πiz 2 N Z 2 2 2 i − e (1 ( RP = YM RP ( λ N 2 Z ) becomes i χ χ YM RP = 3.9 λ Z 2 We will use the following identity This can be derived as follows To rewrite the real X YM RP Z We can thus write after some algebra Parametrizing the ter for the representation Taking into account the 2dteristic counter-terms ( will still denote by and ( following Fourier transformation formula [ or equivalently JHEP03(2021)203 - ). . G 4 ) = W ]. In , G 2.37 (3.23) (3.24) (3.25) (3.26) (3.19) (3.20) (3.21) (3.22) RP ( 2 48 1 . π  2 j  im , and further m 2 2 i i G + z 2 . j + =1 a N i )) a  a P  ( is the standard ∆ are given by G which depends on the . } 2 YM π πα a g 1 ) 2 , G ∆ i 2 r partition function of the 0 R d ) ], πiz 4 ∈{ X − i 2 j G cochar e y S G cochar i − m Λ 47 z . 2 sinh ( e , 2 i + i 2Λ + a / i )) ) 37 P j z G )∆( ( =1 X a N i i 1) p Y , ∆( δ − G cochar − ! P ) ∈ πiz ) Λ i N 2 2 2 α ) ( 2 4 N ∈ a i e π . The generalized Vandermonde factors g U ( 4 15 πi g using the Weyl integral formula [ m ≡ m ( . π is clear which we state without proof (we − − λ ∆( 2 + j ) e i e ⊂ a G z ). Recall the a ) ) iz ) G ( t i ( i i tr a 2 a πi G ) = 2 ) 4 dUχ = ( π i i g πiz − 4 – 12 – 2 sinh( j 2 z ∆ e ∆( G cochar Z ∆ a − − πiy e Λ 2 is the Weyl group and e G ∆( Y i

a 2 ∆ G πi r e ), we obtain N =1 e Z d Y i Y ( ∆( t λ ∈

Z ∆( ! χ α ) = Λ 3.16 | +2) } 2 Z 1 N N G G , λ ≡ ( (e.g. the fundamental group and center of ) ( 0 X . N ) N ! 1 W 4 N for the effective 2d YM from the localization of the 4d SYM ( Z 2 | 2 ∈{ X a − G r ) to ( i N 2 ( − i 0 RP is determined by the cocharacter lattice 2 m (1 N G = 2 N < and ∆ = × 3.17 2 i m 2 YM RP = 2 YM YM RP Z 2 g Z g coroot denotes the rank of YM = ] does not depend on such global structures, which sets apart the SYM on RP Λ = Z r / 23 4 4 Consequently we have Applying ( SYM RP See next section for additional evidence for this expression from a gluing construction of the SYM SYM RP Z 15 G cochar Z global structure of Λ SYM [ partition function on invariant measure on the Cartanare subalgebra products over the positive roots of The sum over have also dropped an overall phase factor below), where where we have defined The generalization to arbitrary gauge group Note that Thus the appropriate integral contour requires a Wick rotation indicator in terms of an integralreduced of the the character integral with respect tothe to the second the maximal Haar step measure torus we on have of used the completeness relation [ where in the first equality we have used the following expression for the Frobenius-Schur JHEP03(2021)203 4 ⊂ S For 4) | (3.30) (3.27) (3.28) (3.29) 16 (4 osp ) restricted , vector- and ) } ⊂ 2 4 ± 2.29 ] with half-BPS , g = 2 2) | m 58 , that defines the 2d , gauge theory on (4 N {z ] of the 3d boundary a ( 57 Q with the supercharge [ wavefunction 4 63 osp , , 4 4 = 2 – 4 Dir HS case is an additional dis- 70 70 59 ˆ with Dirichlet b.c. HS N Z | RP were studied in relation to HS 4 R R 2 partition function by gluing S Z 4 − + . 4 / ) } 3 s S  56 56 RP m RP were computed explicitly. Mean- , R R 790 a subalgebras 4 , ( + + Γ {z 2 i theory on =2 Z RP 13 13 ] to localize − / = 2 one-loop − N 3 M M 23 S = Q N Z | c = 2 − + + partition function [ ⊥ ⊥ determinants on N 2 =2 ,  M M , s – 13 – Z + N  − − / (a pair of antipodal fixed points on the covering G cochar 3 Q ], the localizing supercharge ± gauge theories on theory on a hemisphere 4 = 0 = = with the round metric, the partition function takes S 2Λ . Importantly, this supercharge, in contrary to the = 36 / = } } } 4 ], the one-loop determinants for RP s X =2 Q = 2 =2 =2 =2 = 2  39 RP in isomorphic multiplets thanks to the residual supersymmetry. N + N − N − N G cochar ] where the dual Toda theory observables involve certain N SYM), one expects to arrive directly at a matrix model Λ Q N Q Q Q 5670 =2 56 ∈ computation compared to the , , , Γ – = 2 ± N m SYM decomposes as 4 . In [ =2 =2 =2 Q 2 53 a = 4 N + N − N + N r Z RP d = 4 t ) where the precise form of the integrand will depend on the mat- N {Q {Q {Q Z | N ) = G 3.23 Z 1 , W ), and the lens space 4 | 3 gauge theories on for the S RP supercharges satisfy ( ) = 1 =2 2 4 = 2 g ] which we will call H ], the localization of ± N = 2 ( ] for general discussions on gluing constructions of supersymmetric partition functions on oriented ] provided a complementary perspective on the ). 4 defined by the projections 23 N Q 4 64 ( 40 N 40 RP S , 4) | Z Consequently, by localizing a general More explicitly, as explained in [ 39 =2 See [ (4 we have been using, squares to a combination of isometry and R-symmetry generators N 16 spacetime manifolds. the following form the partition function of theDirichlet 4d boundary conditions (that areto compatible the with equator the identification ( modes which fall intogeneral 3d locus. A noveltycrete of label the for the BPSone-cycle solutions, by the holonomyhyper-multiplets of in the the gauge trivial fieldswhile along holonomy [ the sector nontrivial on to a zero-dimensional matrix model. Q similar to that ofter ( content of the 4d gauge theory through their one-loop determinants around the BPS and they are precisely of the type considered in [ into a pair of supercharges psl These that admit a singlespace fixed point on YM sector of the 4d In [ the AGT correspondence [ boundary or cross-cap states.one The localizing in supercharge [ inQ these studies follows from the 3.2 Discrete holonomy sectors and phase factors JHEP03(2021)203 ) 18 . 3 . 3.23 S ] and (3.32) (3.33) (3.34) ( (3.35) (3.31) vector  i on the 2 j on m 58 ) , 2 4 t = 2 im RP 57 2 [ ) is given by ∈ − N with opposite j . , A, g 2 a A 4 a 3.31 Z ( = 0  conformal SQCD by / 4 3 ) HS S ∆ N Dir m HS ,  ˆ Z j ,  chiral multiplet is in the 2 i j a im are specified by a constant 2 from 2d YM on , SYM with Dirichlet bound- = 2 SU( im =1 + ) = 2 N i 4 4 2 j 2 4 − N a P N j , g = 4 in our convention here. RP 2 HS  2 a 4 . a π 8 g , thus contributes trivially to the 4 ( 2  N Φ 4 ∆ subalgebra, they correspond to one ) i Toda theory. − j 2 . This is closely related to the last e ) i 1 ∆ = 1 + m j Dir ) HS gauge group, the boundary modes of ) − and a flat connection i i 9 m −  m r Z = 2 N i ) a a Φ j ) independent factor) and trivial − ( A m i 17 gauge fields (up to the Weyl group). The N ( m i , N ( ∆( t m ∆ ]. Here the im a 2 ( πi 2 j G iϕ U( e 63 + ∈ e 6= ! i j +2) – ) agree if the phase factor in ( charge j 2 6= N N a a Y i – 14 – ( P r 59 N ) = gauge theory on N  2 πi 2 2 4 20 = 3.30 = − e i ) ∆ , g i } φ U(1) 1 = 2 m , holonomies between antipodal points on the boundary ( m 0 SYM with , ) = ) = 2 iϕ N relevant for the gluing construction organize into one 3d i 2 4 ∈{ X a e j Z ( 4 4 , g m , m i = 4 i a holonomy labelled by and has Dir a HS ] (up to an overall = ( HS ( ˆ 4 ) ]. In terms of the 3d Z 2 N 2 G Z 64 ]. The nontrivial contribution is entirely due to the  N 64 Dir , / HS j 3 corresponds to the combination Z 60 S U( 58 im ϕ Z , 2 is the usual hemisphere partition function with + ) 57 j , 2 4 a SYM, which admits no nontrivial flat connections. Focusing on the case when the 36 multiplet on , g ) by  -valued 4 a 2 ( S = 4 ∆ Z 4 } multiplets were determined in [ = 4 3.23 labels the discrete holonomy of the 1 N , Dir HS ], this phase factor was treated separately and fixed for the 0 N vector multiplet [ Z is an undetermined phase factor. m 40 ∈{ = 2 X we have . ) j 3 Focusing now on the m m In the With the Dirichlet boundary condition, theSuch gauge connections transformations are picked approachIn out identity by [ at the localization the of bound- the boundary modes on S 19 N = 4 ( , 17 18 19 20 3 iϕ ary explicitly comparing to the cross-cap wavefunction in the we see the two expressionsand for the the gluing SYM (factorization) matrix formula model ( on Together with the hemisphere partitionary function condition of [ the for a fixed factor in ( adjoint representation of one loop determinant [ multiplet, which gives e the 4d N vector multiplet and one chiral3d multiplet. The lens space one-loop determinants of general function on flat connections are labelledS by where BPS boundary conditions ofvalue the of the vectorWe multiplet expect scalar theflat dependence connections of to the beorientations hemisphere a (the partition phase. partition function functions This related is by because complex gluing conjugation) two gives copies the of partition where JHEP03(2021)203 ) ) ) ]. ], 2 2 4 , g 36 m 64 , 3.36 , (3.37) (3.38) (3.39) (3.36) m 35 , )tr ( 58 a , ( (dropping G 4 ( 36 4 [ ∨ Dir h HS in the adjoint 3 RP ˆ Z S ψ = 2 contributes a non- 2 ) ) , it follows that the 3 . We also commented m m factors. 4 S ( / D structures associated to α . Invoking known results RP 2 )) ∆ + ∈ U(1) m det( of the hemisphere partition ( α RP α 0 ( P . 2 πiη → super-Yang-Mills on unorientable ) 2 SYM on . 2 4 m − which takes the following values (in 2 ( g e ) -invariant can be written as, contains α along the one-cycle of ) = 4 4 η m = 4 G ) is effectively the determinant of the πi 4 ) ( G 2 α N m RP e ∆( m N Y ( . Now the pin 2 πi ) on the boundary ∈ 4 1 8 πi 8 e α 2 ]. Here we offer some observations for why πiα are interchanged depending on whether the | , we obtain precisely the phase factor ( ∆ ) − e RP 2.29 e ∈ G ], the determinant α Y m 40 ≡ ± α , – 15 – 9 / SYM to zero on the boundary = D , of η = 39 ) 5 localizes to 2d YM on α ) , m ( 2 4 m det( = 4 ( iϕ RP for a single fluctuating Dirac fermion πiηα -holonomy ∝ | e N 2 2 ) 4 − Z and only fails when e structures on m RP / D G + is even or odd. Thus its 1 on SYM on ), this phase factor is trivial since ) det( . In this case, we expect the nontrivial phase factor to come from ± ) ) N 2 4 m = along the root vector = 4 / ) , g D ]) . As explained in [ ψ . The supersymmetric boundary condition that defines 9 SU( N m m ( G G -invariant of a Dirac fermion on , det( η a using supersymmetric localization. By extending the previous works [ πiα ( , a similar manipulation gives a product over its roots, e is the dual Coxeter number. The triviality of the phase factor extends in an (e.g. 4 4 G ) vector-multiplet contains two 4d Dirac fermions in the adjoint representation of G Dir HS RP G ˆ ( Z ∨ = 4 h N It would be interesting to derive this phase factor from first principle by a careful anal- independent factors below), -holonomy 2 In this note, we havespacetime initiated the study of thewe 4d found that the about the partition functionmodel of expression 2d for YM the partition on function general of Riemann the surfaces, we derived a matrix Taking products over the rootfound vectors earlier. 4 Conclusion and discussion respectively for the twothe pin Dirac fermion Z in terms of the the convention of [ Dirac operator representation of trivial phase depending on the m The the gauge group sets half of theTogether with fermions the in antipodal the relevant identification fermion ( determinant (that depends on ysis following the localizationthis setup is of natural. [ function We consider the weakthe coupling fermion limit one-loop determinant since the bosonic contributions are manifestly positive. For simple where obvious way to semi-simple For general JHEP03(2021)203 and (4.1) ] and SYM -BPS ], one 2 4 1 2 ] for a πi will be g 4 40 40 4 36 , ) is then = 4 = ] that have 39 RP here can be ) is precisely N = 0 τ ] between ob- (see [ 66 4 and certain 4d , z 56 2.13 in the sense that Σ RP – 65 with , = 0 preserves the same 53 τ 38 SYM on z 4 gives the + RP z Σ unusual J = 4 ) SYM on 8 ∼ N Φ -BPS defect observables were i . 1 16 and a Toda degenerate primary quotient to act as the antipodal + 4 = 4 + 1 . SYM on unorientable spacetime. here defined by ( 0 2 SYM on τ z ], the RP Φ Z N Q = 0 3 ∼ = 4 Im x 1 40 ] and can be computed using matrix , = 4 x z ), possibly in the presence of a + N , based on the two fundamental domains 36 39 2 9 using a different supercharge [ at N Z Φ 4 4 2.40 / 2 reduced on such a SYM on ( x Σ RP RP 4 G theory on supersymmetric quotient geometries + related by an S-transformation). Consequently (that preserves the puncture at – 16 – ]. In particular, this will allow us to determine = 4 ⊂ 7 RP 2 0) . 2 Σ Φ , z 4 N 38 1 , Z ⊂ (2 x / of the Toda theory inserted at on RP 3 36 α S → − 2 tr ( 2 YM V properties of the ], the vanilla AGT correspondence [ Z 2 ≡ z ) RP ) Z 40 i to be a (punctured) square torus with holomorphic coordi- , , x theory of type (or The localizing supercharge ( Σ 39 J 1 -BPS local operators ], with respect to which general 0) z 1 8 , O SL(2 36 SYM to also apply here. It would be interesting to carry out this (2 can be naturally extended to include boundaries and cross-caps on -preserving defect networks, which leads to particular insertions in following the dictionary in [ . Furthermore if we choose the 6d 4 (up to a Wick rotation) as in the case with a half-BPS boundary or → − Q subjected to the identification gives an annulus with a modulus 2 S G 1 = 4 2 2 As previously mentioned, the 4) z | RP Z . The action of iz / 2 N (4 Σ + Z . If we take / 2 1 , according the general discussion in [ osp z Z 4 Σ) / As discussed in [ S theory on = × -invariant of the fermions on Σ) η z 4 There are a number of interesting directions to be explored further and we discuss × S = 2 ( 4 S symmetry interface defect. Consequently webeen expect successful the in integrability methods bootstrappingin [ the the interface planar one-pointanalysis functions for of the one-point non-BPS functions operators of expects the relevant Toda boundarytheir states wavefunctions on are the given annulus byhave to a a be cross-cap more state complete inalso understanding the teach of us CFT. the about It the AGT would correspondence beIntegrability. in interesting this to case which will related to the corresponding Todaof theory on restricted to be the quotient operator inserted on one of the two boundary circles. Note that as discussed in [ nate a particular degenerate primary short review), the 6d with gauge group map on AGT. servables of the TodaN theory on a closedthe and Toda oriented side Riemann by surface ( considering the 6d correlation functions of restricted to the submanifold interface defect along the equator the one considered inclassified. [ This meansdecorated our by such localization setupthe 2d of YM the on model techniques as illustrated in [ pointed out subtle phasefrom factors in a gluing formula for the partitionsome function of them that below. come Correlation functions. on relations to other localization results on JHEP03(2021)203 , 1 5 N , ]. (2016) SPIRE (2016) 075135 (2016) 035001 94 operator (with IN 93 88 ][ bulk topological phases ]. ], since the bulk exten- d ) and bulk operators are d 44 Phys. Rev. B , , 2 + 1 RP SPIRE Phys. Rev. B 42 , IN ]. Rev. Mod. Phys. ][ arXiv:1406.7329 , manifold fixes the center of AdS [ Fermionic symmetry protected 4 involves a local 4 SPIRE RP IN SYM. RP dimensions ][ for certain supersymmetric observables. It (2015) 052 = 4 2 ]. In particular, correlation functions involving N 12 RP (3 + 1) – 17 – 75 arXiv:1602.04251 – are reproduced by CFT correlators with cross-cap ]. [ 3 70 Global anomalies on the surface of fermionic ] which relies on a semi-classical gravity dual and [ JHEP 2 , ] for reviews). But in contrary to the Hamilton-Kabat- 69 On time-reversal anomaly of SPIRE ]. ]. 68 ), which permits any use, distribution and reproduction in IN , arXiv:1610.07010 CFT [ ][ / Gapped boundary phases of topological insulators via weak 67 to 2d YM on 3 4 SPIRE SPIRE (2016) 12C101 IN IN RP gravity on AdS ][ ][ ]. For higher dimensional CFTs, it is generally less clear whether CC-BY 4.0 75 2016 – This article is distributed under the terms of the Creative Commons pure 71 (2017) 033B04 Fermion path integrals and topological phases The “parity” anomaly on an unorientable manifold SYM on In the context of AdS/CFT correspondence [ PTEP arXiv:1605.02391 , 2017 [ = 4 ) of the boundary involution defining the N 2.24 arXiv:1508.04715 arXiv:1503.01411 coupling 195150 PTEP [ symmetry-protected topological phases in [ topological phases and cobordisms E. Witten, Y. 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