From N $$ \mathcal{N} $$ = 4 Super-Yang-Mills on ℝℙ 4 to bosonic Yang-Mills on ℝℙ 2

Published for SISSA by Springer Received: November 27, 2020 Accepted: February 17, 2021 Published: March 22, 2021 Yifan Wang Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, U.S.A. Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. E-mail: Abstract: We study the four-dimensional N = 4 super-Yang-Mills (SYM) theory on the unorientable spacetime manifold RP4 . Using supersymmetric localization, we find that for a large class of local and extended SYM observables preserving a common supercharge Q, their expectation values are captured by an effective two-dimensional bosonic YangMills (YM) theory on an RP2 submanifold. This paves the way for understanding N = 4 SYM on RP4 using known results of YM on RP2 . As an illustration, we derive a matrix integral form of the SYM partition function on RP4 which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial η-invariant of the Dirac operator on RP4 . We also comment on potential applications of our setup for AGT correspondence, integrability and bulk-reconstruction in AdS/CFT that involve cross-cap states on the boundary. Keywords: Conformal Field Theory, Supersymmetric Gauge Theory ArXiv ePrint: 2005.07197 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP03(2021)203 JHEP03(2021)203 From N = 4 Super-Yang-Mills on RP4 to bosonic Yang-Mills on RP2 Contents 1 2 Localization of the N = 4 SYM on RP4 2.1 The SYM on RP4 from involution 2.2 The supersymmetric action for N = 4 SYM on RP4 2.3 Localization to 2d YM on RP2 3 3 6 7 3 2d YM on RP2 and a new matrix model 3.1 Partition function 3.2 Discrete holonomy sectors and phase factors 9 10 13 4 Conclusion and discussion 15 1 Introduction Quantum field theories with time reversal symmetry can be formulated on unorientable spacetime manifolds. This simple idea has led to exciting developments in the classification of topological phases and detection of subtle anomalies that involve time reversal symmetry [1–8], as well as a refinement of the electric-magnetic duality in abelian gauge theories [9]. However relatively little is known about the dynamics of strongly interacting theories on unorientable spacetimes beyond two dimensions. Luckily the bootstrap approach to conformal field theories (CFT) is well suited for this task (see [10] for a recent review). A familiar family of unorientable manifolds are the real projective spaces RPd of even dimensions, which are realized by a free orientation-reversing Z2 quotient of S d (or equivalently Rd ∪ {∞} by a Weyl transformation) and preserve a large residual (Euclidean) conformal subalgebra so(d + 1) similar to the case of co-dimension one defects (planar or spherical) in flat space.1 Consequently one can formulate a bootstrap program for the basic observables in the CFT on RPd , namely the correlation function of local operators, similar to the case with a domain wall or boundary defect [11, 12]. Putting the CFT on RPd introduces new observables beyond those on the flat space, given by the one-point functions of normalized scalar primary operators O(x) [13], hO(x)i = hO (1 + x2 )∆O (1.1) where the position dependence is fixed by the conformal dimension ∆O due to the residual symmetry (which also requires the one-point function of spinning primaries to vanish). 1 The boundary CFT can be thought of as defined by a Z2 quotient of the flat space with a co-dimension one fixed loci. –1– JHEP03(2021)203 1 Introduction The coefficients hO furnish the basic structure constants for the CFT on RPd . Along with the OPE of local operators, they determine general correlation functions on RPd . Solving the CFT on RPd amounts to fixing these coefficients hO in terms of the ordinary OPE data of the CFT on Rd by exploring constraints from the (residual) conformal symmetry, crossing symmetry and unitarity,2 possibly supplemented by additional dynamical inputs from other methods. This program has been pursued for the Ising model in two and three dimensions via numerical techniques [13], and for Lee-Yang theory in 6 −  dimensions [14] and Wilson-Fisher theory in 4 −  dimensions [15] to leading orders in the -expansion.3 In this note, we initiate the study of the four-dimensional N = 4 super-Yang-Mills theory on RP4 using an extension of the localization setup of [35, 36]. There, a particular supercharge Q in the superconformal algebra psl(4|4)4 was used to a localize the theory on S 4 (resp. HS 4 ) to two-dimensional (constrained) Yang-Mills theory on a great S 2 (resp. HS 2 ). Since the antipodal quotient of S 4 gives the RP4 (with round metric), one naturally expects that by implementing a supersymmetric Z2 identification, the SYM on RP4 = S 4 /Z2 should lead to the bosonic YM on RP2 = S 2 /Z2 upon localization. Indeed as will see, such an identification exists and the N = 4 SYM can be defined on RP4 preserving a half-BPS subalgebra osp(4|4) which contains the supercharge Q. The partition function of the YM theory on RP2 has a simple combinatorial formula in terms of the representation data of the gauge group [37], which in turn determines the partition function of the N = 4 SYM on RP4 , which can be re-expressed into a single matrix model. Thanks to the general 2 Note that unlike the more familiar four-point function bootstrap, here the combinations of OPE coefficients that appear in the conformal block decomposition (e.g. of two-point functions on RPd ) have no obvious positivity properties. 3 See [16] for a reformulation of the bootstrap equations on RPd . 4 The Lorentzian N = 4 superconformal algebra is psu(2, 2|4) which is a real form of the complex Lie superalgebra psl(4|4). Here we study the CFT in the Euclidean signature obtained from a Wick rotation. As usual, one then loses the reality condition on the fermionic generators of the superalgebra. This is not a problem for our analysis if the theory is invariant under the supersymmetry transformations regardless any reality conditions, which is the case here [23]. For the same reason, we will also not impose reality conditions on the fermionic generators in the Euclidean superconformal subalgebra osp(4|4) on RP4 in this paper. –2– JHEP03(2021)203 Gauge theories in four-dimensions offer a rich playground to advance this program. On one hand, a large class of CFTs are produced by renormalization group (RG) flows from four-dimensional Yang-Mills theories coupled to matter. On the other hand, on a topologically nontrivial manifold such as RP4 , the gauge theory observables become sensitive to fine details of the theory, such as global structures of the gauge group, topological couplings, and the spectrum of extended defects [9, 17–21]. Due to strong coupling effects, few observables in general four-dimensional gauge theories can be obtained analytically beyond perturbation theory. Fortunately in supersymmetric gauge theories, (...truncated)