JHEP09(2018)076 Springer July 14, 2018 : vacua, and that September 7, 2018 September 13, 2018 q : : = 1 super-Yang-Mills Received N Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP09(2018)076 b [email protected] , and Erich Poppitz . 3 a 1807.00093 = 2 model appears in the worldvolume theory on domain walls between The Authors. two-dimensional Schwinger model. We show that the algebra of the discrete q c q Anomalies in Field and String Theories, Field Theories in Lower Dimensions,

We study the discrete chiral- and center-symmetry ’t Hooft anomaly matching , [email protected] Department of Physics, UniversityToronto, of ON Toronto, M5S 1A7, Canada E-mail: Department of Physics, LewisPortland, & OR Clark 97219, College, U.S.A. b a Open Access Article funded by SCOAP ArXiv ePrint: multiple adjoint fermions and possible lattice tests. Keywords: Supersymmetric Gauge Theory theory and that itThe inherits Schwinger the model discrete resultssurprisingly ’t rich suggest Hooft structure: that anomalies it the oflaw supports high-temperature the a for non-vanishing domain four-dimensional spacelike fermion wall bulk. Wilson condensatefour-dimensional exhibits and low-temperature loops, perimeter theory. a thus We also mirroring discuss many generalizations properties to theories of with the strongly coupled in the charge- symmetry operators involves a central extension, implyingthe the chiral existence of and centerversion symmetries of are the spontaneously broken.center-symmetry We breaking then vacua argue in that the an high-temperature axial SU(2) Abstract: Mohamed M. Anber Anomaly matching, (axial) Schwingerhigh-T models, super and Yang-Mills domain walls JHEP09(2018)076 ]. 1 and 1 G ]. A famous 4 3 ]. In particular, non 10 – 7 , center symmetry that acts , one may try to introduce 5 G , C N 5 4 Z -form symmetry [ is anomalous. In this case, we q 2 G is a 9 Schwinger model , then it might happen that 14 × 2 q G 1 G G 4 × 1 G , then q = – 1 – G ]. If 3 , 2 . If the theory doesn’t maintain its gauge invariance, we say G is said to be a 0-form symmetry if it acts on local operators. If G 1 ) Yang-Mills theory, which enjoys a 1-form in the UV, where the theory is amenable to perturbative analysis. Then, N G With the help of the recently discovered mixed ’t Hooft anomaly, we scrutinize the A global symmetry have no ’t Hooft anomalies, but the product 2.1 Symmetries and2.2 mixed ’t Hooft The anomaly realization of the symmetries and their algebra Schwinger model and symmetry realizations acts on operators of spacetime dimension 2 provide more handles to studytrivial the constraints phases can of be gauge imposedthat theories on enjoy [ the both vacua 0- of and gauge 1-form theories (including discrete their symmetries number) upondomain gauging walls the latter. in hot super Yang-Mills theory and its multi-adjoint nonsupersymmetric say that the theory has a mixed ’t HooftG anomaly. example is SU( on Wilson line operators.symmetries Recently, can it also has be been obstructed realized due that to gauging the the existence 1-form of discrete ’t Hooft anomalies, which can especially powerful in asymptoticallyupon free gauging theories: onethis computes coefficient the has anomaly to be coefficient IR matched in spectrum the of IR, the whichG puts theory, constraints see on [ the strongly coupled sheds light on theGiven nonperturbative a structure QFT of with QFT aa continuous background is or gauge ’t discrete field Hooft’s of globalthat anomaly symmetry it matching has [ a ’tbe Hooft matched anomaly. between The the anomaly infrared is (IR) renormalization and group invariant ultraviolet and (UV) must dynamics. This matching is 1 Introduction Quantum field theory (QFT)laws of is nature. a In universal manytheir situations, paradigm however, nonperturbative QFTs for are behavior writing strongly becomes coupled down and a the learning daunting about fundamental task. One of the powerful tools that 4 Outlook: generalizations and lattice studies 3 The high-T domain wall in SU(2) super-Yang-Mills: the axial Contents 1 Introduction 2 Discrete ’t Hooft anomalies in the charge- JHEP09(2018)076 q T dχ 2 Z phase (i.e. ]. Owing to T 13 center symme- , C 2 5 , Z 4 DW implies a perimeter ]. Further, we find that 5 T anomaly [ C 2 ]. Z - 15 ’t Hooft anomaly. A consequence chirally restored phase, something dχ 4 q Z are broken in the IR. dχ 2 T Z C q - Z C q Z dynamics on the DW resembles the low- 2) center symmetry along with a 0-form T ], see also [ -) symmetry and a 1-form and discrete symmetries is modified by a central ≥ R q 14 – 2 – dχ 2 q =2 Schwinger model results suggest that the 2d q dχ 2 q Z Z (for C q 2. We show that, in the quantum theory, the algebra and Z ≥ symmetry. C q = 1 super Yang-Mills (SYM) theory. The 4d theory has an q Z dχ 4 distinct vacua, which saturate the mixed ’t Hooft anomaly. bulk away from the DW exhibit area law (or unbroken 1-form ’t Hooft anomaly. Anomaly-inflow arguments imply that this Z N q 3 2 ] to ] -) R discrete chiral (or C 2 R 2-form gauge field multiplies the partition function by a non-trivial 12 Z , dχ 4 one-form center symmetry on the high- chiral symmetry on the wall implies that the fermion bilinear conden- -[ C q DW mirrors many of the properties of the strongly-coupled 4d theory Z 11 Z C 2 dχ 4 dχ 4 T ], implying that both Z Z Z 12 model admits phase: the different behavior of the Wilson loop in the bulk and on the DW q , indicating the existence of a mixed T π two-dimensional (2d) vector Schwinger model and uncover the rich structure of its q : 2 i q T e center symmetry). Here, welow- see again that themirrors DW the theory deconfinement reflects of propertiesloop quarks of along on the the the DW) DWs betweenarea (i.e. chiral-breaking law perimeter vacua in law in the for the bulk) the confined as low- Wilson observed in [ dynamics of the bulk, wherediscrete SYM chiral theory (or is known to have two vacua with a broken law for a fundamental WilsonWilson loop loops taken in to the lie in the DW worldvolume. In contrast, sate should be nonzerothat on should the DW be in inchiral the principle symmetry high- measurable is concerned, on the the high- lattice. In some sense, as far as the The broken The broken In the second part of the paper, we argue that the axial version of the charge-2 2. 1. model studied in thethe first bulk part anomaly, of which the is paper.chiral saturated and by The the center results presence symmetries. there oftheory Thus, show two on the degenerate that the vacua the high- with DW broken at inherits low- high temperature 4d SU(2) anomaly-free 0-form try, with a mixed anomaly appears on the DWthe worldvolume as 2d a nature mixed of Schwinger model, its axial version can be mapped to the charge-2 vector extension, signaling thethe presence charge- of aThe mixed theory anomaly, has a as nonelectric in vanishing fermion charges [ bilinear [ condensate and screens arbitrary strength Schwinger model appears in the worldvolume theory of the domain wall (DW) in the phase of this anomalythe is fact that that the thespace, ground Schwinger generalizing model state [ is of exactlyof the solvable operators system and representing is explicitly the construct far its from Hilbert trivial. We exploit domain wall anomaly inflow.charge- We begin the papervacuum. with This a theory warm-up has a exercise:anomaly 1-form free we discrete study chiral symmetry. the Performingthe a global background discrete of chiral a transformation in generalizations, describe their worldvolume effective field theory, and examine the bulk- JHEP09(2018)076 , ) 1 3 2.1 , by (2.1) q > 4 are the (see also x L ) are the left is the gauge − and Understanding e ψ , t ( 4 Schwinger model, 2 − , + x q ψ ]. ψ ) DWs of pure Yang-Mills A bulk and DW physics 5 + , T  4 T t iqA ]—where the model ( that the axial version of A 1 ) is an interesting example + 3 12 ≡ , 1. The theory can be probed with + 2.1 ∂ Schwinger model  ( 11 . The fields − q > A L q ¯ are the hermitean conjugate fields. ψ , i + is irrelevant: we are considering a compact x 2 is an integer and  ) also appear within the framework 3 x q ∂ ¯ + ψ ), and we further assume that space is ≥ 2.1 ≡  + ], but with no reference to anomalies. , we study the charge- − t ψ 6 q x , 2 ) ∂ 1 limit)—is in the assumption that − DW physics and low- ) is of interest from multiple points of view: ≡  T 2.1 iqA – 3 –  , with ∂ eL L + = diag(+ − ] and, as in that reference, we impose antiperiodic ∂ ( kl domain walls on the lattice. 12 ] and take the fermions periodic, with no change in the results g + 11 ¯ T ψ i vectorlike Schwinger model ( + q kl around the spatial circle. f  kl ψ f vector massless Schwinger model with Lagrangian 2 e 1 q 4 − ] which emphasizes the 1 are spacetime indices, = ] but the dynamics here appears richer. Schwinger model was also discussed in [ , 16 5 = 1 charges. One can think of the latter as of very (infinitely) massive dynamical charges. [ q L q π = 0 = θ = 1 was solved exactly in Hamiltonian language for arbitrary values of k, l of four-dimensional gauge theories. We show in section that provides an exactlydiscovered solvable mixed setting discrete to 0-form/1-form study ’t the Hooft manifestation anomalies [ of the recently q Two-dimensional models closely related to ( On its own, the charge- The major difference of our discussion from that in [ This paper is organized as follows. In section The charge- We note that we could also follow [ We caution the reader against concluding that the value of The spirit of the correspondences outlined resembles those found in the high- 2. 1. 3 4 1 2 regarding symmetry realizations and anomalies;further also, below. the utility of Weyl fermion notation will becomeU(1) clear theory with (light)nondynamical dynamical charges with quantized charge theory at and in the correspondingthe global symmetry issues structure and and discrete anomalies anomalies of that ( arise. Our notation follows fromboundary that conditions of on [ with the textbook [ where two-dimensional Minkowski space coordinates, coupling. The spacetime metriccompactified is on a circle of(right) circumference moving components of the Dirac fermion and 2 Discrete ’t HooftConsider anomalies the in charge- the charge- worldvolume of the DW isinflow a and charge-2 axial the Schwinger manifestation model.a of We discussion also the of discuss the anomaly theand generalizations anomaly on a to the proposal QCD(adj) DW. to with study We a the conclude, larger high- in number of section adjoint fermions quite fascinating. The matchingmake of these various properties anomalies worth and pointing the out rich and DW pursuing physics further. its uncovered discrete symmetries, itswe ’t review Hooft the anomalies, DW and solution in the the anomaly high saturation. temperature In SU(2) SYM section theory and show that the We find these correspondences between high- JHEP09(2018)076 ] 20 , (2.2) (2.3) ). In 19 , , where 4 2.1 . phase factor, qχT  ), are exact , the model 2 q i ) is a 0-form ψ  Z e q π 2.3 ψ i transform with 2.2  ), we proceed to e iqα  center symmetry. e ψ 2.1 → C q ) and ( → Z  -dependent  ψ 2.2 µ ψ . ) appear as worldvolume : = 1 there is only a fermion π symmetry ( q 2 q 1 adjoint Weyl fermions. = 1 adjoint Weyl fermions. q 2.1 dχ 2 q i dχ 2 e Z f > Z center symmetry acts by multiplying n f ≡ valued link, or a 1-form; see [ C q n q q Z Z -direction by a ˆ , ω µ , under which dx axial transformations, the anomaly free x V A A H phase factor i 2 theory is the 1-form q e ). Notice that for Z q – 4 – ≥ ω = 2 arises as a worldvolume theory on domain 2.2 pure Yang-Mills theories, where related anomalies q q π → symmetry: A = dx θ x ]), at least for real values of the fermion mass (of course, A ) with 18 with anomaly free subgroup H i 2.1 , e  . : subgroup of the U(1) ψ 4 q C iχ q 2 dχ Z  is the integer topological charge of the gauge field; recall that we e Z = 1 and note that we temporarily adopted Euclidean notation. Thus, Z ], the sign problem does not hinder the lattice studies of these theories , is to multiply it by a 1-form center symmetry action on the Wilson loop around the spatial → q dx ∈ C q ). Thus, the symmetry parameter itself is a 17 x  , transformation, the fermion measure changes by a factor of Z A x ψ 2.3 2 H A 13 i d : , e 5 12 The A f ≡ 5 R 2, a discrete , as also indicated on the r.h.s. of ( W π U(1) 1 q discussed in section lead to a richAs structure opposed of to the the DWs studyarise that [ of is in principle(in amenable the to continuum limit lattice [ here studies. the chiral limit will have to be approached). A proposal for such studies will be walls (DWs) between center-symmetrysuper-Yang-Mills theory, breaking i.e. states Yang-Mills theory in with high-temperatureSimilarly, related SU(2) multi-flavor axialtheories generalizations on of hot DWs ( in SU(2) gauge theories with the Schwinger model ( 2 π ≥ 2 2 It turns out that, in all cases mentioned above, the 0-form/1-form ’t Hooft anomalies Both the chiral 0-form and center 1-form discrete symmetries, ( A further global symmetry of the q This is easiest to understand on the lattice, where the global 3. = 5 = symmetries of the quantumone of theory. the symmetries However, explicitly they breaks suffer the a other ’t sothe unitary that Hooft links representing they the anomaly: gauge canvery field component much not gauging in as the be in ˆ ( simultaneously for a variety of perspectives. circle, number symmetry and no nontrivialsymmetry chiral as symmetry. it The acts on the local degrees ofIt freedom. does not actsuggests. on any local degrees of freedom, but only on line operators, as its name T allow probes with for discrete chiral symmetry, survives. Underχ the discrete chiral symmetry addition to the gauged vectorlikehas symmetry an U(1) anomalous global axial U(1) Under a U(1) 2.1 Symmetries andThus mixed armed ’t with Hooft reasons anomaly the to salient study points. the symmetries We and begin dynamics with of a ( discussion of the symmetries of the model ( JHEP09(2018)076 . ) C q + G Z x - (2.4) 2.17 q A ]. As due to dχ 2 ( ) π Z x 12 ( , = gauge field, ig θ e 11 C q background transforma- Z q x → dχ 2 A x Z A 2 Schwinger model : ≥ G q gauge field background C q Z is transformations, see eq. ( cL . This factor is unity for integer C q Z ]. πT ], dimensionally reduced). 2 . i 24 e under large gauge transformations 22 π , q 2 π ), under a discrete chiral ]. 21 + 20 ), by borrowing the results of [ 2.2 cL ], the holonomy of the gauge field around the and center 2.1 ) (see [ – 5 – q 12 → gauge field to make the 1-form symmetry (acting dχ 2 Z Z C q ) on the holonomy cL ∈ shifted by 2 Z : k mixed ’t Hooft anomaly in the 2.3 ( cL C q C q q k Z ], to turning on nontrivial ’t Hooft fluxes, known to carry Z - is the unit winding number large gauge transformation. center symmetry is most straightforward on the lattice: one = 19 q dχ 2 with C q L πx T Z 2 Z i e cL, = 1) is introduced. The phase in the chiral transformation of the Schwinger model ( when a fractional topological charge (a nontrivial 2-form center ≡ k ≡ q ] argued that this anomaly has to be matched by the infrared (IR) π q ) 2 x i 5 ( , e dx 4 ig 2 Schwinger model appears as a “central extension” of the algebra of the x 7 e = A In continuum language, introducing a 2-form ≥ ]. Enhancement of the discrete symmetry group in Yang-Mills theory at q H q 6 = 0 gauge is to explicitly solve the Weyl equation in the ω 23 , t A ]. Refs. [ 17 , where , ) 5 x ( 17 , ) in ig 5 − To introduce some of the notation of [ We show below that the Now, as argued in the paragraph after eq. ( e 2.1 In two spacetime dimensions, there is no 3-form fieldThis is strength similar associated to to the the appearance 2-form of the CP/center anomaly in the quantum mechanical and field theory ) 6 7 x , but equals thus any background is necessarily topological, see e.g.models [ of [ discrete ’t Hooft anomaly considerations was also discussed in [ spatial circle is The action of the center symmetry ( (this is possibleconstruct in Dirac one sea space states.law, dimension) i.e., and To invariance find under use infinitesimal thethe its gauge vacuum physical states transformations. eigenfunctions ground be and Finally, state, eigenstatesi∂ one eigenvalues of one demands to the that then large imposes gauge Gauss’ transformations In this section,anomaly we in study the the charge- our realization focus of is on the theof symmetry discrete the realization, ground symmetries we state. shall and Briefly, beof the their mostly ( strategy concerned ’t behind with the the Hooft first properties steps of the Hamiltonian solution anomaly in the operators generating the discretein chiral the next section. 2.2 The realization of the symmetries and their algebra the volume of theterm spacetime [ torus anddynamics can of also the theory be and viewed outlined as various options the for variation theis of way reproduced the a matching by can bulk the happen. IR 3d and theory center in symmetries the are “Goldstone” mode spontaneously such broken. that We both also the explicitly discrete show chiral that the mixed tion, the fermion measure changesT by a phase factor gauge background with partition function in the ’t Hooft’t flux Hooft background is anomaly. the manifestation This of phase the mixed is renormalization group invariant — it is independent of introduces a 2-form (plaquette-based) on links) local. is equivalent, see discussion infractional [ topological charge gauged. Gauging the 1-form JHEP09(2018)076 , x 5 A Q ] for (2.6) (2.7) (2.8) (2.9) (2.5) (2.10) 12 in the F anomaly free )) charge H q 2.2 dχ 2 ]. The Dirac sea . Z 12 # states as i 1 12 n | − , recall ( ]. 2 A .  12 ) , π q . 2 π that matter to us. See [ 11 i qcL e i  and we shall simply denote them n − | is a reflection of the chiral anomaly. ≡ 1 is easily incorporated and is seen n , q . π n below: 5 q qcL 2 5 ω 6 Q π ˜ ( with integer eigenvalues  Q q > + 2 qcL − q π 5 1 4 2 are occupied. This left vs. right moving 5 2 ) and acts on the n i ˜ + ˜ " Q Q 2 e , and 5 ; the explicit form is in [ L πn 2.2 π  n q 2 L 5 ≡ Q 2 → i , ω cL q n 5 i 3 – 6 – | 2 ≥ ≡ = ˜ n Q 5 | X  = : ˜ Q i 1 = 12 G n i | 5 n − | Q q 2 5 2 4 Q are eigenstates of the fermion Hamiltonian X  i n π | L 2 states are orthogonal; their norm can be defined as unity, = i F n n | are occupied and the rest are empty, and, simultaneously, all states E charge vanishes, but the chiral (or axial U(1) . 1) − V L n mn ( δ π 2 = i ≤ is the one where the states of all left moving particles of (gauge non-invariant) m | i n n , not displaying their dependence on | (Here and elsewhere we takesame the liberty letter, to hoping denote that operators this and does eigenvalues with not the cause undue confusion.) background and their energies are is invariant under largesubgroup gauge of transformations. the chiral transformations It ( generates the but this operator shifts under large gauge transformations The gauge field-dependence of the axialOne charge can define a gauge-field independent h is nonzero and depends on the holonomy of the gauge field i The Dirac sea states It is clear, however, that the operator Their U(1) The different We now list the properties of the Dirac sea states n | 4. 3. 2. 1. “Fermi level” matching ensures validity of the Gauss’ lawprecise [ definitions and derivations. Weto notice lead that to important new points, see items end result is thatby the states are labeledstate by an integer momenta of the right moving particles of momenta The Dirac sea states obeying Gauss’ law can be found as was briefly outlined above. The JHEP09(2018)076 i ]) P | 11 )) = . For . We (2.15) (2.14) (2.11) (2.12) (2.13) π . Fur- q G 2 the iqθ iqθ − q − dχ 2 e e 1 theory we Z (mod 0 Fourier trans- θ q q > Z − states are not in- θ i states as ( n , (to not confuse them δ | i . 1 i n . | 1 − ), in the states, supplemented by a −  P, θ | , the with eigenvalue i π q π 2.11 2 n | G )), with , , . . . , q π i q , . . . , q by 2 2 mod 1 , θ = 0 , )  , . 0 cL q i states that are eigenstates of θ i ), acts on the with the same eigenvalue q (mod = 0 i ,P 0 − i n θ + 1 2.4 + | G θ ( n , k n  | − i θ, k | | δ 8 cL θ = ) qn = ( + 1(mod q ], but the details will not be relevant for us. kP , shifting q i δ i – 7 – ω + P n G 12 n | | of the Dirac sea states as ) 1 | obtain the same “mass” from the fermion vacuum k q states transform as ( q | =0 − i (mod G Y = θ 0 q k X i ], are degenerate; one way to see this is by noting ) shift of i cL n q | qn θ, k 1 P,P (mod π | operator, representing the center-symmetry action on q 12 √ 0 + δ 2 , states are eigenstates of ], to construct states that are eigenstates of the large k P, θ q ( | i k,k i = Y 11 q δ e parameter (it is unobservable in the massless theory [ , a 2 12 i ≡ i Z , C q θ Dirac sea states. θ, k ∈ X | = ) imply that under the discrete chiral symmetry Z X n i 11 P, θ P, θ n | | i | 0 2.9 . Since the i ≡ , θ G 0 ). We denote the states of this basis by θ, k ), all | 0 P θ, k h | states are also eigenstates of , k 2.13 0 2.11 ) in all ) and ( i θ h . ) states): 0 2.10 2.13 θ P, θ i different linear combinations of the | − θ q ( ), ( θ, k | iqm e 2.14 where we introduced the the gauge field holonomy. holonomy wave function [ that the holonomy fluctuations energy ( variant but transform into each other as As eigenstates of the total Hamiltonian, however, the Z The center symmetry Under large gauge transformations For further use (cluster decomposition, see below), we also define the ∈ Diagonalizing the full Hamiltonian, including fermion excitations above the Dirac sea, in the language 6. 5. 8 m states transform cyclically into each other used here involves a Bogolyubov transformation [ Clearly, the ther, ( form of the basis ( with the As follows from ( note alsoP that gauge transformations can define convenience, we introduce a and define the linear combinations We are now ready, as in [ JHEP09(2018)076 , ) ) i i q 9 q q 2 dχ 2 Y Z X 2.17 θ, k θ, k | | 2.18 (2.19) (2.16) (2.17) (2.18) (2.20) do not states, ) is con- and was C q i ) to show , Z L i 2.18 P, θ states: 2.19 | i P, θ and | n 2 under the | . q ) dχ 2 − (0) x Z φ ( | − 10 ψ , as required by cluster ) P, θ x ih ] shows that the algebra ( ( + 12 → ∞ ¯ , ψ 0 P, θ . | x ) C ) ≡ 0 π q x 2 ) . ( i center symmetry states are degenerate. The action basis ). Their noncommutativity in the x † . It is also clear that ( P,P -dimensional representation on the e ( e φ iθ i i δ q C q | φ − = Z P ∼ e 2.17 . q 0 − q q P, θ P, θ P P, θ | | ω Y ω h C − q ( 0 ] (the ω iθ where ) a “central extension” of the algebra of symmetry i 25 and − q P,P , 2 e q ), the vacua vanish as 2 2.17 X P, θ L πx . δ – 8 – = i X | 2 q , where i i ), implies that, when acting on the in ( Y 2.10 − = q ] in the Hamiltonian formalism for any q P, θ e i | ω const 0 P, θ ω ) has nonzero matrix elements between different 2.16 | 12 → ∞ ). The fermion bilinear has charge ) x C ( = x x → P, θ L , is not a Wilson loop but a generator of discrete chiral ) implies that both symmetries are spontaneously broken. ( | ( φ ), ( q states explained earlier. Using the matrix elements ( q +1 − φ q 2 Y i | ,n ψ Y 0 ) implies that even though the symmetries generated by q n X →∞ ) 2.17 | , θ 2.15 2 ) and nonzero matrix elements between the n x ) gauge theories [ 0 limit on the r.h.s. Furthermore, one can use eq. ( commutes with x

δ q ( i X q P L 2.2 + 2.17 h ω = ), was also found there). Here, however, one of the operators ¯ ψ i P, θ | = 0, but is diagonal in the n | ≡ i 6 ) 2.16 ) (0) x ( x φ ( ), ( θ, k ) φ was computed in [ | | φ x 0 0 φ ( | n † found above, ( h C 2.15 ], we call the appearance of φ q | 5 + 1 Y ) implies that they are eigenstates of the , θ ground states obey the cluster decomposition principle, as opposed to the 0 i P θ, k commute classically, the discrete chiral and center symmetries h and h 2.12 states, ( q q P, θ | 2 Y i Finally, let us argue that ( The ’t Hooft algebra ( A slightly more careful study of the definitions ofFollowing the [ operators from [ X 9 10 P, θ and that connected correlators indecomposition. the holds in the entire Hilbert space. operators, as the new element where we took the infinite- that correlation functions factorize shown to not vanish, includingsistent as with the nature ofit the is straightforward to show that states, discrete chiral symmetry ( The constant commute in the quantum theoryquantum but theory signals instead the obey presence ( of the mixed ’tThe Hooft anomaly. ground states. This isfermion because the bilinear latter are mixed by local operators, the gauge invariant being a center-symmetry generator,loop is operator, indeed but a the (lowertransformations. other, dimensional version of a) ’t Hooft and This algebra is familiarHooft loop from operators the in ’t SU( | Hooft commutation relation between Wilson and ’t Further, following the discussion afterof ( they do not commute but obey the algebra while ( JHEP09(2018)076 ] ]. 1). 12 4 , and > f exhibits f adjoint n 3 n f R n ]. Our goal ], much like = 1 charges ]. For exam- limit). Hence 13 q ground states, , limit, one can 15 ]. Furthermore, 5 , 13 q , symmetry, which L 5 9 31 0-form chiral sym- , ) flavor symmetry. q 4 f f → ∞ dχ 2 n n Z dχ 4 L Z in 4d SU(2) gauge theories 11 (in the ]. /e 15 , 4 is called a 0-form center symmetry, from the Schwinger model has = 1) or its generalizations (for ], see the remarks in [ T q f 30 n (a fundamental Wilson loop in the spatial mixed ’t Hooft anomaly [ T 2 – 9 – ].  C 2 29 ). – Z  26 - f = 1 Wilson loop by taking two static 2.19 n dχ 4 q symmetry is also broken in the large- Z C q , see ( Z 2 Z ] dictated by the anomaly [ ] the symmetry broken at high- 1, cyclically permuted by the discrete chiral 14 center symmetry associated with the Euclidean time direction is 4 − C 2 Z 1-form center symmetry, as well as a continuous SU( , . . . q C 2 1 , Z 5 massless adjoint Weyl fermions, QCD(adj). Here, anomaly inflow arguments also suggest that the DW between the center = 0 ≤ 12 P f , n i Here, we focus on the high-temperature regime of SU(2) QCD(adj), in the deconfined QCD(adj) with SU(2) gauge group has a discrete anomaly free To see that the 1-form To conclude, we have shown that the charge- Schwinger model and symmetry realizations These are really EuclideanIn objects, the see terminology [ of [ 11 12 P, θ point of view of thedimensionally dimensionally reduced reduced 3d 3d theory, unbroken theory. atarea In high- law addition, in there the is deconfined a phase). 1-form center symmetry in the broken vacua inherits the bulk symmetriesis and to discrete study ’t this Hooft in anomaliesworldvolume. some [ detail We and uncover see a howthe rich the Schwinger ’t structure model Hooft of of anomalies the the are previousWe DWs saturated section begin and on (for with the find the DW an Euclidean explicit action connection of to SU(2) Yang-Mills theory endowed with DWs connecting different vacua are foundquark to deconfinement) exhibit [ rich worldvolume physics (for example phase, where the broken. matched by the IR dynamics.bined with At zero continuous temperature, andthe the mixed possible discrete discrete-continuous existence anomaly of anomalies, matching, interestinganomaly can com- new inflow be phases arguments for [ used the to discretephysics ’t on suggest Hooft the anomalies DWs imply separatingple, that at vacua there zero with is temperature, nontrivial broken the discrete discrete symmetries chiral [ symmetry is broken, at least for small with metry and a The discrete symmetries havethe a one in the Schwinger model of the previous section. All ’t Hooft anomalies have to be 3 The high-T domain wall in SU(2) super-Yang-Mills: theIn axial this section, weappear show as that the worldvolume the theory Schwinger on model high-temperature studied DWs above and its generalizations showed that arbitrary charges are screenedpolarization in the effects, massless Schwinger with model, the duethe to screening Wilson vacuum length loop of obeys order perimeter 1 law, signaling the breaking of the 1-form symmetry [ | is spontaneously broken to study the expectation valuesome of distance a apart and studying the potential between them. The calculation of [ JHEP09(2018)076 a ]. ). 2 τ . 3.3 = 0, 27 (3.2) (3.3) (3.1) 4 = , center i a and for 4 26 in ( t [ C 2 A 3 4 3 4 Z ]. The two 1 β A S gT H βA , the 27 i , , h − ]  1, see section , the theory pre- 26
π ) is the one-loop 4 , π 4 2 > = 1 for the rest of such that exp QCD g [0 f A f Λ F ( n QCD → n 2 ∈ a,b O V 3 4 δ ,  3 4
spontaneously breaks the T = βA ) + , simply increasing the number λ βA 4 f ). A DW perpendicular to µ
π n A b , and D t ( for 3.2 is SU(2) Cartan generator and a T µ -component of the gauge field. t V 4 ¯ σ 3 2 1. 2 x τ ¯ λ , , the two 0-form center-symmetry g  π i + 3 = 0) = tr =
2 i , so that at are the spacetime Pauli matrices. In i ) 3 4 3 π 4 i . The covariant derivative is given by x where 2 4 σ A , βA , i A ) + 2 /T is the thermal circle, which is taken along 3 2 D 1 β τ µν S 1 β = 0 ) H S F and the trace of the Polyakov loop vanishes, 3 + 4 = 1 3 4 i x 2 h – 10 – µν π ] is given by replacing
β . To one-loop order, the resulting bosonic part of F βA , many aspects of the theory become amenable to 3 4 π Φ( ), where ) + tr ( β 1 2 : T exp ij S σ tr ( 4 βA T F π, F i, µ QCD 2 δ ij − g 1 π Tr F 2 h 6 = 0) = 1 2 1 β . The inverse width of the DW is of order ) = − 3 = ( tr ( S 3 h x carries an implicit flavor index; we set 1 2 × x 4 σ 3 ]), up to a constant, by (  λ R = 0, restoring the 0-form center symmetry on the DW. Further- 32 3 1 Z πβ i µ DW ) interpolates between 0 and 2 R direction is broken and the theory admits DWs [ → ∓∞ 3 12 Z = is the gauge coupling at the scale A x 4 ] and ¯ ) to 3d after integrating out a tower of heavy Matsubara excitations 2 3 − S g β x 4 and the trace is normalized as tr g = 0) x , , λ 3.1 3 3 µ 3, = , x ) = , ( A 2 4 [ 2 4 , i , A symmetry, the trace of the Polyakov loop vanishes Tr A ( D − boson 3 1 β 13 V C 2 = 1 S S = 1 λ H Z µ i ∂ h are the color-space Pauli matrices. i, j µ, ν -direction and has circumference a 4 = τ The DW is a solution of the equations of motion of ( At temperatures much smaller than the strong coupling scale Λ x exp The analysis of the zero modes in this section goes verbatim for any = 1, is given (see e.g. [ λ is parameterized as F µ 13 f 3 At the DW core,Tr we find Φ( more, the center-symmetric expectation value Φ( of zero modes. The analysis of the DW world-volume theory, however, differs for the trace of the Polyakov loop, x the profile function Φ( breaking vacua, as where the extension to theThe interval two [ minima of thesymmetry potential along are the at center-symmetry breaking vacua are characterized by nonvanishing expectation values of effective potential for theThe Matsubara potential, zero written mode belown of in the terms of the Cartan subalgebra component of the gauge andthe fermion action fields reads along where serves its static charges are confined, and alaw. Wilson At loop temperatures wrapped in larger thesemiclassical than time treatment Λ direction owing to obeys asymptotic the freedom. area reduce the In action this regime, ( we can dimensionally the D addition, the fermion field this section. where and Weyl fermions at finite temperature JHEP09(2018)076 ) ). as has 3.4 3.5 (3.5) (3.6) (3.4) 0 p  µ, p λ -bosons W we obtain W ), are nor- . Thus, the 3 ¯ , λ x T ( 1 ∼ , , = 0 = 0 and bearing 1 4 i β − x + − a , = τ τ -dependent solution p − πp . 3 + p − p 2 λ i (0) x λ λ e 1   (0) ) )   p, 2 i i λ −  p, ia ia τ , the  ) and λ bulk, on the other hand, the 3 # ) with respect to T  − + − x µ, p 1 2 3 i i ( T  is the only massless mode on the 2 R  p,  p, ∂ ∂ 3.1 , ) for the Matsubara zero mode of W , while on the DW the , ( (  λ λ z 4 i i ) T β x -bosons have mass + =0 =0 " 2 z σ σ 0 =0 p + Φ( : g + W λ πp = 1 β µ ,p + + Φ( i, p τ 2 dz a S i a  p 3 T T e dz + x µ, p λ

3 , where it was shown to have interesting ≡  ) ) x π W Z i 3 3 − a x x Z τ +  = = 1). We expand the gauge fluctuations and − p 3 3 Φ( ∓ θ 2 λ τ f x – 11 – 3 0 − + Φ( n x + 0 0 0 µ, p πp + a 2 πp πp πp τ 2. The photon  2 2 − 2 / + p Z ) λ   ∈ 2   p X reads + + + iτ and also have opposite 2d chirality, as can be seen from ( p  3 3  2 2 i ) + τ λ τ 1 , due to the adjoint Higgsing. Thus, in the presence of a DW, 3 a 0 p 0 p τ x — are massive and we neglect them in our treatment. The DW- T λ λ is also expected to be massive as there is no symmetry protecting (
) = exp ) = exp  i = ( 3 3 6=0 =0 ∂ Z ]. Having a 2d abelian gauge field on the worldvolume is not very DW µ x x i  ∈ ,p ( ( p σ X A µ, p 3 1 2 τ 33 a a  p,  p, + = = λ λ 3 and we used the notation is a two-component spinor, i.e., ) has a nonperturbative gap, of order T and , µ λ 0 ]. Here, we show that in theories with adjoints the dynamics is even richer. 2  p 1 2 A 5 , 3.2 λ πp + 2 . A careful examination of the (charged under the unbroken U(1)) components p = 1 T = 0 ∼ i, j p To this end, we study the fermions in the DW background and show that the DW , however, reveals that there are two zero modes on the DW. Setting 3d theory dynamically compactifies to an abelian theory on the 2d worldvolume, in a  p It is easy tomalizable. check that It only is two crucialcharges of under for the these our U(1) solutions, field purposes to note that these two zero modes have opposite in mind that of the equation of motion of where the Cartan gauge field (the U(1)a photon). mass One can immediatelyλ see that the lightest it, is uncharged underinto the the 2d covariant derivative U(1), and varyingthe and the equation we Lagrangian of ignore ( motion it in what follows. Substituting ( where DW. All other gaugewell modes as — the the photons W-bosonsworldvolume and scalar their Kaluza-Klein excitations periodic (anti-periodic) boundary conditions along manner resembling [ interesting in pure YMconsequences theory, [ except at supports two fermionic zero-modesfermion (for fields in Fourier modes, taking into account that the gauge field (fermions) satisfy DW worldvolume supports massless abeliangauge fields. sector ( In the have a larger gap ofwe order expect that at sufficientlyT low energy scales (presumably below the bulk gap) the high- SU(2) gauge symmetry to U(1) and the off-diagonal JHEP09(2018)076 3   λ R λ are The (3.8) (3.7) λ center 14 C 2 Z ). There is apply to the 4 for the axial 2 x transform with 2.1 , under which  . λ + − V + − ψ ψ λ λ )] )] 2 2 )] )] ) inherits the symmetries 2 2 ia ia ia ia discrete “chiral” (from the 3.7 − + The U(1) − + 1 1 dχ 4 ), after integration by parts in 1 1 a a on the DW, while the bulk confined Z a a 15 2( 2( T in the + sector and a relabeling 3.7 i i 2( 2( i i , under which + + + A ¯ λ − + 2 2 2 2 i∂ i∂ = i∂ i∂ − + ) to the Euclidean version of the vector + − + 1 1 ψ 1 1 ∂ ∂ 3.7 [ [ , ∂ ∂ is the bulk confining scale, much larger than the DW is possible in 2d, due to the vector-axial duality is the two dimensional gauge coupling. [ [ + + − ) can be brought to the vector form of Schwinger T bulk). Thus, we can only estimate the value of the 2d 2 ¯ ¯ + − λ – 12 – A ψ ψ 2 e ¯ ¯ i i λ λ 3 i i 3.7 /g = R 1 + + + + + , respectively. Also, to emphasize the fact that ∼ ¯ kl ψ − δ kl F 2, and λ F , kl kl F F 2 and = 1 sector. Performing this in ( 2 e 1 e , where 1 4 + 4 − ]. In what follows, we assume that the results from section λ k, l = T/δ , = . Then, the effective 2d Lagrangian on the DW worldvolume is 13 DW worldvolume theory in SU(2) super-Yang-Mills (SYM) theory . This estimate may raise the issue of scale separation between DW and bulk 2 , symmetry originates from the 1-form center symmetry in the k 2 2 g , 5 a 2 1 T T -bosons and fermions have mass of order l ), the Lagrangian ( C 2 a 4 =  = 1). ∂ in the g axial DW vector DW Z 2 f e 2.1 L W L − − = ∼ n l ¯ λ 2 ) describes the Euclidean axial Schwinger model of charge 2 and, from a 4d 2 a e , mixed ’t Hooft anomaly on the DW worldvolume, as predicted by anomaly 1 k = ∂ 3.7 A C 2 ). ], and as follows directly by repeating the arguments of section − YM case of [ 5 Z ¯ ). = ψ - γ π : we take µ 13 , 2 , kl dχ 4 γ e 3.7 = − which is parametrically the same as the nonperturbative bulk gap. These estimates equally apply 5 F Z λ = θ T 2 ν To more explicitly see the relation of ( It is interesting to note that the DW worldvolume theory ( In what follows, when writing the DW-volume theory of the zero modes, we drop the = We remind the reader that gauging U(1) The localization of the abelian fields on the DW is due to nonperturbative effects in the bulk that g γ 15 14 µν − ∼  e to the DW theory and offerzero the modes, heuristic justification charged thatstates the are only uncharged. light charged states near the DW( are the generate a mass gapthe for DW the background gauge wouldcoupling fluctuations propagate (in in the the width, absence leading of to adynamics: bulk gap, from the the abelian above gauge estimate, field nonperturbative in effects in the 2d Schwinger model occur at scales ψ the + sector we find inflow [ model ( Schwinger model ( model via the change of variables bulk point of view)symmetry symmetry due remaining to anomaly the(this free. fact worldvolume In that addition thebulk there adjoint and is fermions should a carry not twicealso be a the confused with fundamental the charge zero-form center symmetry along perspective, the high- (QCD(adj) with and anomalies ofopposite the charges, bulk is SYM gaugedhave theory. in the the same The axial charge is U(1) charge-2 model. anomalous, instead. There is a where Lagrangian ( Matsubara and 4d spinorthe indices, above and zero denote modes theadjoint by two fermions, dimensional and fields therefore, correspondingof carry to twice variables the fundamentalgiven charge, by we make the change JHEP09(2018)076 )) . We 3.7 (3.9) limit, 13 T subgroup dχ 4 is a subgroup Z symmetry, and dχ 4 Z A in addition to the ), bulk nonperturba- 16 -bulk center symme- 3 3.9 R DW implies a perimeter ). We see that at the level T 2.1 are interchanged w.r.t. ( dynamics on the DW mirrors 0. = 1 SYM case, see Footnote A T f . n → ). − β ψ 2.1 − ¯ ψ and U(1) + phase. This follows already from the fact ψ V + T ¯ = 2. Now the anomaly free ψ , we conclude that the DW theory breaks both q 2 g symmetry. – 13 – 2 (chirally restored!) phase, something that should bulk away from the DW exhibits area law. ∼ 3 T dχ 4 R Z ) with -) Fermi 2.1 bulk coupling in its definition. Thus, in the high- with opposite charges, as in ( R − 4 DW 2 − g L ψ and 1 case in the future. -preserving perturbative effects leading to ( + A > one-form center symmetry on the high- ψ ) (where the roles of U(1) chiral symmetry implies that the fermion bilinear condensate should f C 2 n dχ 4 3.8 ] for the gauged single-flavor Thirring model, with Lagrangian given Z Z ). ), show that it has the spectrum and chiral condensate of the massless 34 dynamics of the bulk, where SYM theory is known to have two vacua with 3.8 3.9 T ), allows for a single classically (and quantum mechanically, as the results acting on A 3.8 . However, when projected on the DW worldvolume, they only induce higher- ) is the only gauge invariant and 2d Euclidean invariant local four-fermi coupling A ) and ( 3.9 law for a fundamental Wilsonrecall loop that taken a to Wilson lie loop in the in DW the worldvolume. In contrast, be in principle measurable on theAs lattice. far as the chiralthe symmetry low- is concerned, thea high- broken discrete chiral (or be nonzero on the DW in the high- ] imply) marginal coupling of the form The broken The broken Thus, borrowing the results of section In addition to U(1) For completeness, we should note that the worldvolume theory, 3.8 We stress again that the present discussion applies only to the 34 2. 1. 16 shall come back to the the worldvolume center symmetrytry) (originating and from the the discreteimplications: 1-form chiral symmetry. This symmetry realization has some interesting dimension non-renormalizable terms thatto are being irrelevant exponentially from suppressedthat in 2d ( the perspective, high- in addition in the theory ( and condensate. Theseassume effects that, are in small our in DW the theory, they limit are of negligible smalltive as effects four-fermi are coupling known and toof we generate U(1) ’t Hooft vertices. These preserve only the indicated this by including the we expect that theThe coefficient results of of thisby [ term ( is small, butSchwinger model, have not with calculated the it 4-fermi precisely. coupling inducing renormalization of the boson mass of [ This gauge invariant four-fermiis coupling hence preserves expected the to anomalous be U(1) induced by perturbative loop effects in the bulk theory; we have of the U(1) of the Lagrangian ( the symmetries are realized exactly as in the modelterms ( in ( which is the Euclidean version of ( JHEP09(2018)076 ), ), 3.9 3.9 (4.1) ) chiral general- ]) and a f denoting f n n 36 phases from j, i , T 35 , and 18 , the same comment phase: the different a, b f . n 4 T = 2.  Z b i , and hence are expected f ] shows that the massive spec- = 1 four-fermi term ( δ However, similar to ( in the chiral limit has not n A and stress that we have neither f , with 37 j a f DW physics are quite tanta- δ n ,j ai 17 n 2 −  T α ψ bj  DW theories requires more work , global symmetry, while the bulk we discuss the higher- + ,a j i T 3 R + δ ) ψ f b a δ n but preserving phase (i.e. area law in the bulk), as found 1 A α 1 high- T SU(  > ,j × − f L – 14 – ψ ) n f i − to the diagonal subgroup, the bulk SU( ¯ DW properties. n ψ ) global chiral symmetry. In the absence of interac- R f ,b T Thus, as opposed to the ) bulk physics and high- n + f n T ψ 18 a + ), which we label as ¯ ψ SU( ), the ’t Hooft anomalies of the additional nonabelian global 3.8 ∼ × 3.8 L ) f Fermi n − ) now has a SU( =2 one can also write an invariant using 4 DW f L ) will affect the IR spectrum and possibly the phase structure and sym- n 3.8 4.1 ) chiral symmetry. motivated by the last remark in section f n = 1 case applies — no such local terms relevant in 2d sense are allowed). The : indices, respectively. The multi flavor generalization of the minimal coupling ]. f f n 14 n term reduces the enhanced global symmetry of the free-fermion DW Lagrangian to L,R ) multiples of those in ( 2 ] pointed out subtleties related to additional ’t Hooft anomalies for behavior of the Wilson loopquarks in on the bulk the and DWs onchiral-breaking (i.e. the vacua perimeter in DW the mirrors law confined the for low- deconfinementin the of [ Wilson loop along the DW) between Here, the DW theory again reflects properties of the low- f α As already alluded to, study of the f 31 n In fact, the solution of the multiflavor massless Schwinger models in [ We note that for n 17 18 ref. [ as in the trum of the multiflavorflavor ’t model Hooft is anomalies. universal while a nonuniversalcalculated massless nor studied sector the matches effect the of nonabelian any of the bulk-induced four-fermi terms. We also note that recently The the bulk SU( we expect that ( metry realization (as far as terms breaking U(1) volume, reducing SU( symmetry. These perturbatively induced classically marginalclidean terms rotations, should preserve gauge the Eu- symmetry,to and have anomalous the chiral form U(1) QCD(adj) theory only hastions the other SU( than those insymmetries ( on the worldvolumedegrees have of to freedom be onfour-fermi matched the terms and DW are one worldvolume allowed expects would and that appear. will new be massless induced by bulk loop effects on the DW world- and we only givein a brief qualitativeSU( discussion. The DWDW fermion Lagrangian zero modes ( now come Higher ization of our study.on Theories the with lattice different number withconclusive of picture various adjoint of motivations flavors their (a haveemerged phase been few yet. structure studied recent as references a function are of [ lizing. One can not helputility in but constraining wonder the about features theirthe of generality often difficult and more to speculate study easily on strongly-coupled tractable their low- high- possible 4 Outlook: generalizations and lattice studies The above relations between low- JHEP09(2018)076 θ T = 1) f n ) in the ) has no , where q π 4.1 2.14 = θ ’t Hooft anomaly, C 2 vacua ( Z i - f n 4 P, θ | Z ), except for q πP 2 domain wall worldvolume theory + T θ dependence of the DW worldvolume ) added will break the discrete chiral , would, in the center-broken phase, two-index symmetric (or antisymmet- 3 f = 1, a mixed bulk theory. cos( x n 4.1 f | f n T n m | ) global symmetry preserved by ( 0 DWs — all of this in the deconfined, chirally and our final remark concerns the effects of mixed f C 1-form center symmetry and an anomaly free T n 2 phase in the action of a single gauge plaquette x – 15 – ∼ C 2 , the degeneracy of the ) Z C 2 P m Z ( E ] and references therein). Other theories with center 38 ), for all β ]. The two-index symmetric (antisymmetric) tensor theories, worldvolume (such configurations are also known as high- one of the more surprising indications of our analysis is that . + pure gauge theories) for real values of the fermion mass. DW 3 32 phase can be induced by imposing appropriate twisted boundary m 4 x - π x 2 T ≡ x = 4 ) 0-form discrete chiral symmetry with a mixed ’t Hooft anomaly. It x θ 4) − ]; in this language, our studies amount to saying that they have a rather N bulk theory, it would be worthwhile to study this further. It would be 19 (2 f plane ( T dχ n 4 Z ( x DW worldvolume theories have, as for - Schwinger model is lifted, 1 f x q n +4) N These effects should be, at least in principle, observable in lattice studies. As lattice An extension of our studies to higher numbers of colors would also be of interest. We expect that the multi flavor model with ( (2 f dχ n theories. there has been recentric) interest tensor in Dirac theories fermionssymmetry with are (see discussed [ in [ with an even numberZ of colors, have a would be interesting to study the matching of this anomaly in the various phases of these Other theories with center symmetry: discrete chiral/center symmetry anomalieslight in or massless other fundamental gauge fermions theories. have no While (not even theories approximate) with center symmetry, regarding the feasibility of suchstudies studies can to be lattice experts.to generalized leading We to only order account note in for thatcharge- the our a fermion analytic small mass nonzeronow includes fermion the mass. phase of For example, conditions. For example,in inserting a the induce a DW withcenter an vortices [ rich structure in theories with adjoint fermions). Needless to say, we leave the judgment simulations are always performedapproached. at While finite this is fermion a mass, difficultlimit, task, a as at least chiral opposed there limit is to nobackgrounds would sign in problem have the (in high- to the continuum be Possible lattice studies: there should be a nonzero fermionWilson condensate and loop perimeter in law the for worldvolume asymmetric of (necessarily spacelike) the phase. high- and the low- especially interesting to seesymmetry whether realization there mirroring is that any of the low- exactly as in the bulk(mixed) theory, anomaly while in the 2d. SU( and center symmetries to matchtantalizing similarities the between anomaly. the In behavior of light the of high- the observed (so far, for higher- JHEP09(2018)076 ]. JHEP JHEP SPIRE ]. , , ]. -angles, IN θ ][ ]. (1985) 220 , walls and (1998) 114 SPIRE (1995) 757] 4 SPIRE IN IN [ ]. 159 SPIRE ][ ]. (2015) 021701 IN arXiv:1805.11423 B 515 B 456 , ][ SPIRE SPIRE IN D 92 ]. 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