Index theorem for topological excitations on R^3 * S^1 and Chern-Simons theory
aa r X i v : . [ h e p - t h ] M a r Preprint typeset in JHEP style - HYPER VERSION
Index theorem for topological excitations on R × S and Chern-Simons theory Erich Poppitz ∗ and Mithat ¨Unsal † Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada SLAC and Physics Department, Stanford University, Stanford, CA 94025/94305, USA
Abstract:
We derive an index theorem for the Dirac operator in the background of various topological excitationson an R × S geometry. The index theorem provides more refined data than the APS index for aninstanton on R and reproduces it in decompactification limit. In the R limit, it reduces to theCallias index theorem. The index is expressed in terms of topological charge and the η -invariantassociated with the boundary Dirac operator. Neither topological charge nor η -invariant is typicallyan integer, however, the non-integer parts cancel to give an integer-valued index. Our derivationis based on axial current non-conservation—an exact operator identity valid on any four-manifold—and on the existence of a center symmetric, or approximately center symmetric, boundary holonomy(Wilson line). We expect the index theorem to usefully apply to many physical systems of interest,such as low temperature (large S , confined) phases of gauge theories, center stabilized Yang-Millstheories with vector-like or chiral matter (at S of any size), and supersymmetric gauge theorieswith supersymmetry-preserving boundary conditions (also at any S ). In QCD-like and chiral gaugetheories, the index theorem should shed light into the nature of topological excitations responsiblefor chiral symmetry breaking and the generation of mass gap in the gauge sector. We also show thatimposing chirally-twisted boundary condition in gauge theories with fermions induces a Chern-Simonsterm in the infrared. This suggests that some QCD-like gauge theories should possess componentswith a topological Chern-Simons phase in the small S regime. ∗ [email protected] † [email protected] ontents
1. Introduction 1
2. The index for Dirac operator on R × S
3. SU ( ) with arbitrary representation fermions 164. Interpolating from Callias to APS index 185. Remarks on anomalies and induced Chern-Simons terms on R × S R × S A. Another calculation of the η -invariant 25B. Index for higher representation fermions 26
1. Introduction
A method to study non-perturbative aspects of an asymptotically free gauge theory on R is to begin with a compactification on R × S . S may be either a spatial or temporalcircle, determined according to the spin connection of fermions. In asymptotically free gaugetheories, the size of S is a control parameter for the strength of the running coupling at thescale of compactification. At radius much smaller than the strong length scale, the theoryis weakly coupled and at large radius, it is strongly coupled. It is well-known that certainaspects of weakly coupled gauge theories are amenable to perturbative treatment. It is lessknown that such gauge theories also admit a semi-classical non-perturbative treatment if theboundary Wilson line (Polyakov loop) satisfies certain conditions.– 1 –et U ( x ) denote the holonomy of Wilson line wrapping the S circle: U ( x ) = P exp i I A dy, x ∈ R , y ∈ S , (1.1)where A ( x, y ) is the component of the gauge field along the compactified direction. If theeigenvalues of the A field (which are gauge invariant) repel each other in the weak couplingregime, for a dynamical reason or due to a deformation described below, then the boundaryvalue of the A field as | x | → ∞ takes the form: A (cid:12)(cid:12) ∞ = diag(ˆ v , ˆ v , . . . ˆ v N ) , ˆ v < ˆ v < . . . < ˆ v N . (1.2)In the weak coupling regime, A behaves as a compact adjoint Higgs field and the vacuumconfiguration (1.2) induces gauge symmetry breaking, G → Ab ( G ), to the maximal abeliansubgroup Ab ( G ). This means that there exist a plethora of stable topological excitations insuch four dimensional gauge theories, such as magnetic monopoles, magnetic bions, instantonsand other interesting (stable) composites.It is well-known that in a thermal set-up at sufficiently high temperatures (and weakcoupling), dynamics disfavors configurations for Wilson lines such as (1.2) [1], and favorsconfigurations for which A (cid:12)(cid:12) ∞ = diag(ˆ v, ˆ v, . . . ˆ v ) = (0 , , . . . , a .) center-stabilizing double-trace deformations, b .)adjoint fermions with periodic boundary conditions, or mixed representations of adjointsand a few complex representation fermions all with periodic boundary conditions, and c .)supersymmetry and supersymmetry preserving boundary conditions. In this sense, the caseof non-trivial holonomy (1.2) at weak coupling is as generic as the high-temperature trivialholonomy. In particular, with the center stabilizing double-trace deformations, certain gaugetheories on R , such as Yang-Mills theory and vector-like and even chiral theories can besmoothly connected to small S × R [3, 4, 5]. There already exist evidence from latticegauge theory (where deformations were also suggested independently to explore phases ofpartial center symmetry breaking) that the conjecture of smoothness holds for Yang-Millstheory [6]. Therefore, there is currently a strong incentive to study in detail the topologicalexcitations on R × S and the index theorems associated with these excitations. The center-stabilized small- S regime of gauge theories is amenable to both numerical lattice simulationsand non-perturbative semi-classical techniques. In this sense, this regime provides a first example in whichwe can confront a controlled approximation, including non-perturbative effects, with the lattice, and is, in ouropinion, an important opportunity for both lattice and continuum gauge field theory. – 2 –ur interest is in the index of the Dirac (or Dirac-Weyl) operator ˆ D (1.5) in the back-ground of topological excitations pertinent to the gauge theory on R × S . The reason thatthis is interesting for non-perturbative physics is two-fold: • The topological excitations with non-vanishing index will carry compulsory fermion zeromodes attached to them, and may induce chiral symmetry breaking. Generically, thefermionic index of a monopole operator on R × S is (much) smaller than the APSindex for the BPST instanton. Thus, they are in principle more relevant for low energyphenomena. • The generation of mass gap (and confinement) for gauge fluctuations, in the weakcoupling regime, requires the existence of topological excitations with vanishing index.In typical QCD-like and chiral gauge theories, most of the leading topological excitations(monopoles) carry fermionic zero modes, hence cannot contribute to the mass gap.Therefore, the index theorem can be used to identify composite topological excitations(such as magnetic bions) for which the sum of individual indices add up to zero.A simple example which illustrates both issues is Yang-Mills theory with adjoint fermions(QCD(adj)) and N = 1 SYM. In both cases, magnetic monopole operators (which appearat order e − S = e − π / ( g N ) in the semi-classical e − S expansion) induce a certain chiralcondensate. However, the topological excitations responsible for the existence of mass gap ofthe dual photon and thus for confinement are the magnetic bions with vanishing index, whichappear at order e − S . The index theorem on R × S should help us identify both classes ofnon-perturbative topological excitations for any gauge theory.The non-perturbative semi-classical analysis provides reliable information about the gaugetheory in the weak coupling regime. However, the semi-classical treatment does not ex-tend over to the large radius, strong coupling regime, where the eigenvalues of A fluc-tuate rapidly, and there is no “Higgs regime” where the long-distance theory abelianizes.In the partition function, we need to sum over all gauge inequivalent configurations. At x = ∞ , A field can acquire a profile consistent with the unbroken center symmetry, suchas (1.2). In fact, the boundary Wilson line (1.2) defines an isotropy group at infinity [1]: G A | ∞ = { g ∈ G | gU ( ∞ ) g † = U ( ∞ ) } . For example, in low temperature pure Yang-Millstheory, the isometry group is isomorphic to the maximal abelian subgroup, G A | ∞ ∼ Ab ( G ).This does not mean that a dynamical abelianization takes place in this regime, nor semi-classical techniques apply. However, the index theorem for the Dirac operator can be inter-polated from R to R . The index theorem is valid at any radius, regardless of the value ofthe coupling constant.Although the index theorem and topological excitations consistent with the isotropygroup G A | ∞ continue to exist in the large S strong-coupling domain, the semi-classicaltechniques no longer usefully apply. Nonetheless, we believe that there is value in studyingthe form of the topological operators, dictated by the appropriate index theorem, and at– 3 –east qualitatively study their dynamical effects. This is the goal of the recent “deformationprogram.” We introduce our notation in Section 1.3 below. In Section 2, we begin the calculation of theindex. Our calculation can be thought of as a generalization of that of [7, 8], see also [9]. Weshow that the index on R × S has two contributions—a topological charge and surface termcontribution.In Section 2.1, we first calculate the index for static monopole backgrounds. The surfaceterm contribution, Section 2.1.1, is expressed in terms of the η -invariant of the boundaryDirac operator, while the topological charge contribution is given in Section 2.1.2. The finalformula for the index in the “static” background is eqn. (2.32) for the fundamental of SU ( N )and eqns. (B.1–B.4) from Appendix B for other representations. The calculation of the indexin a Kaluza-Klein (“winding”) monopole background is given in Section 2.2, with the resultfor the fundamental of SU ( N ) in (2.44), and in (B.7) for general representations.We note that an expression for the index on R × S similar to ours—given in terms of thetopological charge and the η -invariant—can be extracted from the appendix of ref. [12]. Thecontribution of this paper consists of: a .) a derivation of the index accessible to physicistsalong the lines given in the physics literature for R and by using exact operator identitiesvalid on any four-manifold and b .) a calculation of the index in specific backgrounds and adiscussion of its jumps—properties which are of interest for concrete quantum field theoryapplications.In Section 3, we discuss in some more detail the index for the three lowest representationsof SU (2) and the fundamental of SU ( N ). We explain the jumps of the index which occur asthe ratio of boundary holonomy to the size of S is varied.In Section 4, we explain the relation to the Callias [10] and APS [11] indices.In Section 5, we consider the generation of fermion-loop induced Chern-Simons termson R × S . We show when Chern-Simons terms are induced and how their coefficients arequantized. We consider the effect of turning on of discrete Wilson lines for background fieldsgauging anomalous flavor symmetries (similar effects are known from the string literature).The resulting Chern-Simons terms have a profound effect on the phase structure of the theoryon R × S .Finally, Appendix A contains another calculation of the η -invariant. In Appendix B, wegive formulae for the index for general representations. We take the four-dimensional Euclidean Dirac operator of a vector-like fermion in the repre-sentation R to be: ˆ D ≡ γ µ D µ , D µ ≡ ∂ µ + iA aµ T a . (1.3)We use hermitean T a ’s, obeying Tr T a T b = T ( R ) δ ab , taking T(fund.)= 1 / SU ( N ).To further set and check notation, note that we use, in a given representation, ψ → U ψ ,– 4 – µ → U A µ U † − iU ∂ µ U † under gauge transformations, hence F µν = ∂ µ A ν − ∂ ν A µ + i [ A µ , A ν ].Roman indices run from 1 . . .
3, while Greek indices span 1 . . .
4; the x ≡ y direction isperiodic, y ≡ y + L . The hermitean Euclidean γ -matrix basis we use is: γ k = σ ⊗ σ k , γ = − σ ⊗ σ , γ = σ ⊗ σ , (1.4)where σ k are Pauli matrices and σ is the unit matrix. The vector-like Dirac operator (1.3)is: ˆ D = σ k D k + iσ D σ k D k − iσ D ! ≡ − D † D ! . (1.5)In the above equation, we defined the 2 × D , obeying: D † D = − D µ D µ + σ m ǫ mkl F kl + σ k F k = − D µ D µ + 2 σ m B m (1.6) DD † = − D µ D µ + σ m ǫ mkl F kl − σ k F k = − D µ D µ . (1.7)Here, D µ is as defined in (1.3), and we assumed, without loss of generality, that the backgroundof interest is anti self-dual, namely that F k = ǫ kpq F pq ≡ B k (all expressions can be easilygeneralized for self-dual backgrounds). In this paper, we will use “Tr” to denote traces ofoperators over spacetime as well as spinor indices, while “tr” will refer to traces over spinorindices only.We are interested in computing the index of the Dirac operator in topologically nontrivialbackgrounds on R × S , generalizing the R result of [10]. The simplest example of anontrivial background is given by the three-dimensional SU (2) Prasad-Sommerfield (PS)solution of unit magnetic charge, embedded in R × S . The other backgrounds on interest canbe constructed by taking superpositions of the fundamental monopoles and other solutions,obtained by non-periodic “gauge” transformations.The PS solution is “static” (i.e. y -independent) and the A -component of the gauge fieldplays the role of the Higgs field. For example, consider the anti self-dual solution, which obeys F k = − ǫ kpq F pq = ǫ kpq F pq ≡ B k . In our conventions and in regular (“hedgehog”) gauge,the SU (2) solution reads: A = A a ( r, v ) T a = ˆ r a f ( r, v ) T a , A j = A aj ( r, v ) T a = ǫ jba ˆ r b g ( r, v ) T a , (1.8)where ˆ r a = r a r and: f ( r, v ) = 1 r − v coth vr , g ( r, v ) = − r + v sinh vr . (1.9)The asymptotics of the B k , A fields of the PS solution at infinity are: A (cid:12)(cid:12) ∞ = − v ˆ r a T a (cid:18) − vr + . . . (cid:19) B k (cid:12)(cid:12) ∞ = ˆ r k r ˆ r a T a + . . . , (1.10)– 5 –here dots denote terms that vanish as e − vr . To cast the solution in string gauge, we needto gauge transform ˆ r a T a → T . The asymptotics of A and B k in string gauge are obtainedfrom (1.10) by replacing ˆ r a T a with T , for example the asymptotics of SU (2)-holonomy is A | ∞ = diag( − v, v ).For the applications we have in mind, we want to also consider monopole solutions of SU ( N ). The SU (2) PS solution considered above can be embedded in SU ( N ) as describedin [8]. For simplicity, we will use the fundamental generators of SU ( N ) to describe theembedding (a description of the embedding using roots and weights can be also given, see[8]; however, we find that for our purposes using N × N matrices is both sufficient andilluminating). The general form of the asymptotics of the Higgs field A is: A (cid:12)(cid:12) ∞ = diag(ˆ v , ˆ v , . . . ˆ v N ) , ˆ v < ˆ v < . . . < ˆ v N , N X i =1 ˆ v i = 0 (1.11)where, without loss of generality, we have ordered the eigenvalues as in our SU (2) example (forexample, eqn. (1.10) corresponds to taking ˆ v = − ˆ v = − v ). A background with an additionaloverall U (1) “Wilson line” a , often also called “real mass” term (when fermions are included),allows the holonomies ˆ v j to be more general:ˆ v j → ˆ v j + 1 √ N a , (1.12)where we also normalized the overall U (1) generator multiplying a to Tr T = 1 /
2. Includinga non-vanishing a can be used to incorporate different boundary conditions for the fermionsin all our formulae.The asymptotic form of the U ( N ) holonomy (1.11, 1.12) admits N types of elementarymonopoles; N -1 of these are associated with the positive simple roots α i of the SU ( N ) Liealgebra, for which: α i · H = 12 diag(0 , . . . , |{z} i , − |{z} i +1 , . . . , , i = 1 , . . . N − , (1.13)where H = ( H , . . . , H N − ), where H a denote the Cartan generators of SU ( N ). The Cartangenerators and simple roots are normalized as Tr H a H b = δ ab , α i · α j = δ i,j − δ i,j ± . The N th type of fundamental monopole arises due to compactness of the “Higgs” field A , and isassociated with the “affine” root: α N · H ≡ − N − X j =1 α j · H = 12 diag( − , , , . . . , . (1.14)– 6 – monopole solution corresponding to the i th simple root (1.13) of SU ( N ) can be con-structed from (1.10) as follows. First, rewrite the holonomy (1.11): A (cid:12)(cid:12) ∞ = diag(ˆ v , ˆ v , . . . , ˆ v i − , V |{z} i , V |{z} i +1 , ˆ v i +2 , . . . , ˆ v N ) + 12 diag(0 , , . . . , , − ˜ V |{z} i , ˜ V |{z} i +1 , , . . . , , (1.15)where V = (ˆ v i +1 + ˆ v i ) and ˜ V = ˆ v i +1 − ˆ v i . We now diagonally embed the SU (2) generators τ a into SU ( N ), such that their only nonzero elements are equal to one-half the Pauli matricesembedded in a 2 × N × N matrices (thus, their diagonalelements are the i th and i +1 th ones singled out in (1.15)). With this embedding it is easy toexplicitly verify that: A = ˆ r a f ( r, ˜ V ) τ a + diag(ˆ v , ˆ v , . . . , ˆ v i − , V |{z} i , V |{z} i +1 , ˆ v i +2 , . . . , ˆ v N ) ,A m = ǫ mba ˆ r b g ( r, ˜ V ) τ a , (1.16)with f ( r, ˜ V ) and g ( r, ˜ V ) defined in (1.9) solves the anti-self-duality condition F k = B k inside SU ( N ). The large-radius asymptotics can be immediately read off (1.10) by replacing T a with τ a and inserting in (1.16).Finally, a collection of n , n , ..., n N − fundamental monopoles of the type correspondingto the 1 st , 2 nd , ..., N − th , respectively, simple root of SU ( N ), has an asymptotic magneticfield which is the natural generalization of (1.10) and is given, in the string gauge, by: B m (cid:12)(cid:12) ∞ = ˆ x m | x | N − X i =1 n i ( α i · H )= 12 ˆ x m | x | diag( n , n − n , . . . , n j − n j − , . . . , − n N − ) . (1.17)The asymptotic form of the SU ( N ) holonomy is as in (1.11) and the string gauge asymp-totics of the gauge field is best described in polar coordinates, with A φ its only nonvanishingcomponent: A φ (cid:12)(cid:12) ∞ = 1 − cos θ n , n − n , . . . , n j − n j − , . . . , − n N − ) . (1.18)The Kaluza-Klein monopole solution corresponding to the affine root (1.14) will be con-structed in Section 2.2.
2. The index for Dirac operator on R × S We define the Callias index of a Weyl fermion (with equation of motion Dψ = 0 and D defined in (1.5)) in the representation R on S × R as in [10, 7]: I R = lim M → Tr M D † D + M − Tr M DD † + M . (2.1)– 7 –his is the definition most convenient for explicit calculations, despite the fact that in thelocally four dimensional case of interest an additional regularization will be required, havingto do with the need to perform the sum over the Kaluza-Klein tower implicit in (2.1). Nonzerodiscrete eigenvalues do not contribute to the formal expression (2.1)—if ψ is an eigenfunctionof D † D with a nonzero eigenvalue, Dψ is an eigenfunction of DD † with the same eigenvalueand so their contributions to the trace cancel—hence I R counts the number of zero modes of D minus the number of zero modes of D † ; the continuous spectrum also does not contribute to(2.1) if all ˆ v j are different, see the discussion in the appendix of ref. [7]. The arguments giventhere continue to hold on R × S and we will not repeat them here—as will become clearfrom our results, I R of eqn. (2.1) always yields an integer value for finite action backgroundson R × S . Furthermore, we will show that the index reduces to the Callias index in theappropriate limit and experiences discontinuous jumps, which can also be explained physically,upon changing the ratio of the circumference of S ( L ) to the holonomies at infinity (ˆ v j ) .Using the operator ˆ D from (1.5) and our notation for γ (1.4), we find that: I R ( M ) = Tr γ M − ˆ D + M = M Tr γ ˆ D + M − ˆ D + M , (2.2)where the second identity is true because of cyclicity of trace and γ ˆ D = − ˆ Dγ . Finally wecan cancel the ˆ D + M factor between numerator and denominator and arrive at the expressionwe will actually use: I R ( M ) = M Tr γ − ˆ D + M . (2.3)In our study, we closely follow the derivation of the Callias index on R of ref. [7], payingrespect to the differences due to the locally four-dimensional nature of spacetime. The maindifference—apart from the already mentioned sum over Kaluza-Klein modes—occurs in thevery first step below and has to do with the fact that anomalies occur in a locally fourdimensional spacetime. To elucidate, we note that: h x | D − M | y i = h ψ ( x ) ψ ( y ) i , (2.4)where h . . . i denotes an expectation value in a Euclidean quantum field theory of a Diracfermion ψ, ψ with action − S = ψ ( − ˆ D + M ) ψ . For such theories in a locally four dimensionalbackground the following operator identity holds: ∂ µ J µ ≡ ∂ µ ( ψγ µ γ ψ ) = − M ψγ ψ − T ( R )8 π G aµν ˜ G aµν . (2.5)The index (2.3), via (2.4, 2.5), can be rewritten as: I R ( M ) = − M Tr γ h ψψ i = M Z d x L Z dy h ψγ ψ i = − Z S ∞ d σ k L Z dy h J k i − T ( R )16 π Z d x L Z dy G aµν ˜ G aµν , (2.6)– 8 –here we used periodicity of the current on S to argue that the integral of ∂ y h J i vanishes.Eqn. (2.6) is our main tool, allowing us to smoothly interpolate the index from R to R byvarying the size of the circle and the appropriate background. As a first simple check, takethe limit of an infinite L , i.e. R , where eqn. (2.6) becomes: I R ( M ) = − Z S ∞ d σ µ h J µ i − T ( R )16 π Z d x G aµν ˜ G aµν . (2.7)This is the index theorem appropriate for a BPST instanton background (provided that thesurface term vanishes: that this is so follows from the fact that the surface contribution in (2.7)could only be due to BPST fermion zero modes, as the nonzero modes vanish exponentiallyat S ∞ and so does their current; the fermion zero modes in an instanton fall off as a powerlaw ψ | x →∞ ∼ ρ | x | , in nonsingular gauge [13]).Going back to R × S , consider now the integral over S ∞ × S in (2.6). We rewrite thesurface term in (2.6) as follows: I R ( M ) ≡ − Z S ∞ d σ k L Z dy h J k i = − Z S ∞ d σ k L Z dy tr h x | γ k γ − ˆ D + M | x i (2.8)= − Z S ∞ d σ k L Z dy tr h x | (cid:18) γ k γ ˆ D − ˆ D + M (cid:19) | x i , (2.9)where we performed the operations that led to eqn. (2.3) in reverse. Further, from (1.5), theexpressions (1.6) for D † D and DD † , and the explicit form (1.4) of the γ -matrices, we have: I R ( M ) = 12 Z S ∞ d σ k L Z dy tr h x | σ k σ l D l (cid:18) − D ν + M + 2 σ m B m − − D ν + M (cid:19) | x i− Z S ∞ d σ k L Z dy tr h x | iσ k D (cid:18) − D ν + M + 2 σ m B m + 1 − D ν + M (cid:19) | x i , (2.10)and we recall that (2.10) is written for an anti self-dual background. The final formula forthe index which will be used in our further computations is: I R ( M ) = I R ( M ) − T ( R )16 π Z d x G aµν ˜ G aµν ≡ I R + I R , (2.11)with I R defined in (2.10) and I R , the topological charge contribution to the index, in (2.11). To avoid (or add) confusion, recall that the index (2.1) for a fundamental Weyl fermion in an anti-selfdualinstanton should be +1, as it is D , in the notation of Section 1.3, that has a normalizable zero mode. – 9 – .1 The index in a “static” BPS monopole background and Callias index Consider first a 3d BPS “static” monopole background, independent on the S coordinate.Physical intuition tells us that if we consider a small S , hence weak coupling, we expect theindex on R × S to be the same as that on R provided Lv ≪
1, such that KK modes do notinfluence physics at scales of order the size of the monopole. On the other hand, one expectsthat when Lv ≫
1, the index can differ from the one on R . To study how this expectationplays out in detail and under what conditions the index can jump, in the following Sectionswe successively evaluate the two contributions to the index (2.11). To evaluate the contribution of the surface term (2.10), we note that at infinity the dominantterms in the expansion of the operators appearing in I R ( M ) in the static BPS backgroundare: − D ν + M ≃ − ∂ m + M − D , with − iD → πnL + A , (2.12)where we used the string-gauge asymptotics of A (1.11) and A m (1.18). We now expand thesurface term contribution, recalling that B m ∼ r − , observing that only the second term in(2.10) contributes after the Pauli matrix traces are taken, and using (2.12): I R ( M ) = 2 L Z dy Z S ∞ d σ k tr h x ; y | iD − ∂ m + M − D B k − ∂ m + M − D | x ; y i . (2.13)Next, we substitute the asymptotic form for a “static” BPS solution, eqn. (1.17), to obtain using (2.12) to replace iD : I R ( M ) = − Z S ∞ d σ k ˆ x k | x | ∞ X p = −∞ N X j =1 (ˆ v j + 2 πpL )( n j − n j − ) Z d k (2 π ) h k + M + (ˆ v j + πpL ) i . (2.14)After taking the three dimensional momentum and surface integrals ( d σ k ≡ | x | ˆ x k d Ω S ),as well as the M → I R (0) = − N X j =1 ( n j − n j − ) ∞ X p = −∞ ˆ v j + πpL | ˆ v j + πpL | . (2.15)The Kaluza-Klein (KK) mode sum in (2.15) is a periodic generalization of the sign function,which appears in the Callias index for gauge theories on R (upon taking L → p = 0 term contributes in the sum and so (2.15) reproduces the Callias index result, seeAppendix B). Such a generalization is necessary, since on R × S the eigenvalues of the“Higgs” field A are compact and the index should be a periodic function of the expectationvalues of A , with periodicity determined by the representation R . Recall that n = n N = 0 is understood for the static solution. – 10 –he KK sum (2.15) can also be thought of as a sum of the indices of a KK tower ofthree-dimensional Dirac operators, each that of a KK fermion of mass πpL . The Callias indextheorem shows that for a given Higgs vev only a finite number of massive operators in the KKtower have a nonvanishing index (essentially, those with | m | < O ( | v | )), thus only a few termsin the sum over indices of KK Dirac operators can contribute to the index. While followingthis logic is a quick way to find our formula for the index for static backgrounds, recall thatthere is also a non-integer topological charge contribution given by the second term in (2.10),which should be cancelled by a corresponding non-integer contribution to (2.15) to yield aninteger value. Thus, to obtain a formula for the index that works for general backgrounds[1], specified by the holonomy at infinity, magnetic charge, and topological charge, we mustregulate the sum over KK modes in (2.15).For a given j , the KK sum is equal to η j [0], the spectral asymmetry of the differentialoperator h j = i ddy + ˆ v j acting on the space of periodic functions f ( y ) = f ( y + L ). The η -invariant is defined by analytic continuation from sufficiently large Re( s ) > η [ v j , s ] ≡ η j [ s ] ≡ X λ =0 sign λ | λ | s , (2.16)where λ are the eigenvalues of h j . Thus the surface term contribution to the index is: I R (0) = − N X j =1 ( n j − n j − ) η j [0] . (2.19)To calculate η j [0], begin with its definition (2.16), rescaling both numerator and denominatorby πL : η j [ s ] = ∞ X p = −∞ sign (cid:16) ˆ v j L π + p (cid:17) | ˆ v j L π + p | s = ∞ X p = −∞ sign (ˆ a j + p ) | ˆ a j + p | s . (2.20)We defined: ˆ a j ≡ ˆ v j L π − (cid:22) ˆ v j L π (cid:23) ⊂ (0 , , (2.21)having noted that since η j is a periodic function of ˆ a j of unit period, by relabeling the KKmodes, we can take the argument to lie in the fundamental interval (0 , ⌊ x ⌋ is thefloor function: ⌊ x ⌋ = max { n ∈ Z | n ≤ x } , (2.22) An equivalent way to to define the η -invariant is via its integral representation. Let H = i ddy + A . Then, η [ H, s ] ≡ tr H ( H ) ( s +1) / ≡ s +12 ) Z ∞ dt t ( s − / tr[ He − H t ] . (2.17)This representation makes sense for large Re( s ) > ζ function regularization, for which: ζ [ H, s ] ≡ tr[ H − s ] ≡ s ) Z ∞ dt t ( s − tr[ e − Ht ] . (2.18) – 11 –hich denotes the largest integer smaller than x , and ˆ x = x − ⌊ x ⌋ is the fractional part of x .It then follows that all terms in the sum (2.20) with p ≥ p < η j [ s ] = X p ≥ a j + p ) s − X p ≥ p + 1 − ˆ a j ) s = ζ ( s, ˆ a j ) − ζ ( s, − ˆ a j ) , (2.23)where ζ ( s, x ) is the incomplete zeta-function. Finally [14], since ζ (0 , x ) = − x , we find ourfinal expression for η j [0]: η j [0] = 12 − ˆ a j − (cid:18) − (1 − ˆ a j ) (cid:19) = 1 − a j = 1 − v j L π + 2 (cid:22) ˆ v j L π (cid:23) . (2.24)For another calculation of the η -invariant, see Appendix A.From (2.24), the surface term contribution (2.15) to the index for the fundamental rep-resentation of SU ( N ) becomes: I fund. (0) = − N X j =1 ( n j − n j − ) (cid:18) − ˆ v j L π + (cid:22) ˆ v j L π (cid:23)(cid:19) . (2.25) Consider now the second term in (2.11)—the topological charge contribution to the index,which is well-known to be a surface term: I R (0) = − T ( R ) Q = − T ( R )16 π Z d x L Z dy G aµν ˜ G aµν = − T ( R )16 π L Z dy Z S ∞ d σ m K m , (2.26)The topological current is: K µ = 4 ǫ µνλκ tr (cid:18) A ν ∂ λ A κ + 2 i A ν A λ A κ (cid:19) . (2.27)In writing the surface integral in (2.26), we used the fact that for the static BPS background K µ is a periodic function of y . To evaluate (2.26) we note that the spatial component of K µ can be rewritten as: K m = 4 ǫ mij tr ( A F ij − A i ∂ A j − ∂ i ( A A j )) . (2.28)Now we use ǫ ijk F jk = 2 B i and the fact that in the static anti self-dual BPS background(1.8-1.10), assuming SU (2) for now, 8tr A B m (cid:12)(cid:12) ∞ = − v ˆ r m r ˆ r b ˆ r c tr T b T c = − v ˆ r m r . Thus, theonly contribution to the surface integral (2.26) comes from the first term in K m , yielding, for T ( R ) = 1 / I fund.,SU (2) (0) = 132 π πL v = Lv π . (2.29)– 12 –his is, of course, the known result for the negative of the topological charge of an antiself-dual BPS monopole.To obtain the SU ( N ) result in the multimonopole background, it is best to transformthe surface integral (2.26) to string gauge and use (1.11, 1.17). The singular nature of thestatic gauge transformation does not change the periodicity of K µ used in (2.26) and does notaffect the surface integral. Thus, for an arbitrary representation of SU ( N ) the topologicalcharge contribution to the index is: I R (0) = − T ( R )16 π L Z dy Z S ∞ d σ m A B m ]= − T ( R ) N X j =1 ( n j − n j − ) L ˆ v j π . (2.30) Combining the two contributions to the index, eqns. (2.30) and (2.25), gives our final formulafor the index. Note that neither the topological charge contribution (2.30), nor the surfaceterm (2.25) is an integer. However, in the combined result, the non-integer parts comingfrom the two cancel neatly. With some work, our expression can also be extracted fromthe formulae in the Appendix of [12]; it was derived here in a physicists’ manner by usingeqn. (2.5), the axial-current non-conservation which is an exact operator identity valid on any4-manifold. In this respect, our derivation is a natural generalization of [7].For the fundamental representation of SU ( N ), adding (2.30) to (2.25), the index is: I fund. ( n , n , . . . , n N − ) = − N X j =1 ( n j − n j − ) (cid:18)
12 + (cid:22) L ˆ v j π (cid:23)(cid:19) , = − N − X j =1 n j (cid:18)(cid:22) L ˆ v j π (cid:23) − (cid:22) L ˆ v j +1 π (cid:23)(cid:19) , (2.31)where in the first line, as usual n N = n = 0.It is fairly easy to extract the Callias index theorem from (2.31). Let us restrict − π 1, elements, explicitly: U ≡ . . . − . (2.39)The point of (2.38) is that transforming A µ ( ˜ V ′ ) with U leads to a twisted (i.e. y -dependent)solution in the vacuum with asymptotics given by (2.33) with ˜ V replaced by ˜ V ′ − πL = − ˜ V .The role of the U transformation acting on A is to flip the sign of ˜ V and thus generate asolution in the desired vacuum (2.33). The A asymptotics of the KK monopole solution A KKµ is thus the desired (2.33), while the B -field flips sign at infinity due to the U conjugation.Thus the KK monopole solution has magnetic charge opposite that of the corresponding antiself dual solution—its magnetic charge given by the affine root (1.14) and asymptotics (for n N copies of the solution): B mKK (cid:12)(cid:12) ∞ = − n N ˆ x m | x | N X i =1 ( α i · H )= n N x m | x | diag( − , , . . . , , . . . , . (2.40)To find the topological charge of the KK monopole, eqn. (2.38) plus gauge covariance ofthe field strength allow us to argue that: Q = 132 π Z d xdyG aµν ˜ G aµν h A KK ( ˜ V ) i = 132 π Z d xdyG aµν ˜ G aµν h A P S ( ˜ V ′ ) i = L π Z S ∞ d σ m tr A ( ˜ V ′ ) B mP S = − n N ˜ V ′ L π = − n N − ˜ V L π ! , (2.41)the calculation in complete analogy with (2.30), using the asymptotics of A , eqn. (2.33)with ˜ V → ˜ V ′ , and of B mP S = − B mKK of (2.40). Thus, remembering from eqn. (2.26) that I R = − T ( R ) Q , we obtain that for n N KK monopoles, the topological charge contributionto the index is: I ,KK R (0) = 2 T ( R ) n N − ˜ V L π ! . (2.42)– 15 –he computation of the surface term I R (0) is also simplified by the fact that the asymp-totics of the KK monopole solution at infinity are x independent and are, as explained above,the same as those for the PS monopole, except for a switch in the sign of the magnetic field.Thus, despite the fact that in the “bulk” the solution is twisted around S , we can still use(2.13) to calculate the surface term contribution. Substituting eqns. (2.40) and (1.11) into(2.13), we obtain for the fundamental representation of SU ( N ), instead of (2.25): I ,KKfund. (0) = n N ˜ V L π − (cid:22) ˆ v N L π (cid:23) + (cid:22) ˆ v L π (cid:23)! . (2.43)Combined with (2.42), this gives for the total index of the KK monopole: I KKfund. (0) = n N (cid:18) − (cid:22) ˆ v N L π (cid:23) + (cid:22) ˆ v L π (cid:23)(cid:19) . (2.44)In the case where for all j the holonomies obey | ˆ v j | < πL , taking into account our orderingof the holonomy (1.11) (ˆ v < 0, ˆ v N > n N = 1, that I KKfund = 0. Recallfrom the discussion around eqn. (2.32) that in the background of n , n , ..., n N − monopolescorresponding to the 1 st , 2 nd , etc., simple roots there are n j ∗ fermionic zero modes, where j ∗ is the position of the last negative ˆ v j from (1.11). Thus, the combination of a n j ∗ = 1monopole and an n N = 1 KK monopole have a combined number of zero modes equal to thatof a four-dimensional BPST (anti) instanton (one for the fundamental of SU ( N )); the sumof their topological charges also adds to minus one.At this stage, we can also combine (2.31) and (2.44) into a single formula: I fund. [ n , n , . . . , n N − , n N ] = I fund. ( n , n , . . . , n N − ) + I KKfund. ( n N )= n N − N X j =1 n j (cid:18)(cid:22) L ˆ v j π (cid:23) − (cid:22) L ˆ v j +1 π (cid:23)(cid:19) . (2.45)where L ˆ v N +1 ≡ L ˆ v . 3. SU ( ) with arbitrary representation fermions The calculation of the index is particularly simple for arbitrary representations of SU (2).Consider, for example, a Weyl fermion in the spin- j representation of SU (2) in the staticBPS background. The asymptotic form of the A and magnetic fields are: A | ∞ = − v ( T ) j = − v diag ( j, j − , . . . , − j ) , B m (cid:12)(cid:12) ∞ = ˆ x m | x | ( T ) j , (3.1)where we set n = 1 for simplicity. The index receives contribution from the surface term(2.15) and topological charge (2.26). Instead of (2.15), we now have: I j (0) = − j X m = − j m ∞ X p = −∞ sign (cid:18) − vm + 2 πpL (cid:19) , (3.2)– 16 –here the minus sign in the sign-function is because in our convention the holonomy at infinityis A ≃ − vT . We perform the KK sum in a way similar to (2.24) to obtain: I j (0) = j X m = − j − m vLπ − m (cid:22) − vmL π (cid:23) . (3.3)For the topological charge contribution, we can use the first line of (2.30) and following thesteps that led to (2.29), we obtain: I j (0) = 2 T ( j ) Lv π . (3.4)Recall that for the spin- j representation of SU (2), the Casimir is given by T ( j ) = j P m = − j m = j ( j + 1)(2 j + 1). Therefore, summing over the two contributions (3.3) and (3.4) to the index,we find: I j (0) = j X m = − j m (cid:18) − mvL π − (cid:22) − mvL π (cid:23)(cid:19) + 2 T ( j ) Lv π = − j X m = − j m (cid:22) − mvL π (cid:23) . (3.5)The relation between the index for the BPS monopole and KK monopole is also especiallysimple in SU (2), where there are only two kinds of monopoles; in the spin- j representationthe index in the KK monopole background can be obtained by using techniques of the section(2.2), with the result: I KKj = 2 T ( j ) − I j (3.6)where I j is the index of the j -representation in the monopole field and 2 T ( j ) is the numberof zero modes in a BPST instanton background.Let the number of monopoles and KK monopoles in a given background be, respectively, n and n . The main result of this section is captured in the index and the topological chargeformulae: I j [ n , n ] = n I j + n I KKj = n T ( j ) − ( n − n ) j X m = − j m (cid:22) − mvL π (cid:23) ,Q [ n , n ] = n Q BP S + n Q KK = − n + ( n − n ) vL π . (3.7)We consider now as an example the three lowest representations of SU (2). We alreadydiscussed the fundamental representation of SU ( N ). In the Appendix, we give expressionsfor other SU ( N ) representations of interest. Index for the fundamental ( j = 1 / ): We have, from (3.5): I / (0) = − (cid:22) − vL π (cid:23) + (cid:22) vL π (cid:23) . (3.8)– 17 –egin with the case 0 < v < πL , when we obtain I / = 1. That this is so can be easily verifiedby explicitly solving the zero mode equation for the Weyl operator D in the PS background[18]. This is also the result of the Callias index theorem on R , as expected on physicalgrounds when L is small and the scale v of SU (2)-breaking is below the KK scale.Upon increasing v , taking πL < v < πL , we have I / = 3. More generally, eqn. (3.8)implies that the index jumps by two every time v crosses another πL threshold. This jumpof the index occurs because every time v increases by πL , two zero-mode solutions withnonvanishing KK number become normalizable. This jump of the index can be easily seenexplicitly by considering the normalizability of the zero-mode solutions of the D ( A ) ψ = 0Weyl equation in the static PS background on S × R , along the lines of the Appendix ofref. [8]. Index for the adjoint ( j = 1 ): Now we have from (3.5): I (0) = − (cid:22) − vL π (cid:23) + 2 (cid:22) vL π (cid:23) . (3.9)Begin with 0 < v < πL , where I (0) = 2, the well-known value in three dimensions. As weincrease πL < v < πL , we obtain I (0) = 6. Thus, the index jumps by 4 every time v crossesa KK threshold. Again, this is because as v passes beyond πL every L = 0 normalizable zeromode acquires two more normalizable KK partners. Index for three-index symmetric tensor ( j = 3 / ): Our final example is the three-index symmetric tensor ( j = 3 / SU (2)). This representationalone is free of a Witten anomaly and gives an example of a chiral four-dimensional theorywith interesting non-perturbative dynamics. The index of the representation is T (3 / 2) = 5.For this case (3.5) implies that the index is: I / (0) = − (cid:22) − vL π (cid:23) − (cid:22) − vL π (cid:23) + 3 (cid:22) vL π (cid:23) + (cid:22) vL π (cid:23) . (3.10)For 0 < v < π L , where I / (0) = 4, as on R . As v increases across the first KK threshold to π L < v < π L , we have I / = 10—a jump of the index by 6. As v crosses the next threshold π L < v < πL , we similarly find that the index jumps by 6, giving I / (0) = 16. Similarly to theprevious cases, the jumps are interpreted as due to more KK-fermion zero modes becomingnormalizable as v increases through each threshold. 4. Interpolating from Callias to APS index It is useful to put together the results for the index theorem on R × S and see how itinterpolates between the Callias index theorem on R and the APS index theorem on R .This will also provide a crisp notion of an elementary versus composite topological excitationon R × S . In order to study these excitations, it is useful to recall some basic facts about– 18 –he root system of a Lie algebra and the distinction between the simple root system and affineroot system.For a given Lie algebra, we can construct all roots ∆, positive roots ∆ + , and simplepositive roots ∆ , satisfying ∆ ⊃ ∆ + ⊃ ∆ . For example, all roots in ∆ + can be written aspositive linear combinations of simple roots which constitute ∆ :∆ = { α , . . . , α N − } , (4.1)where α i are N − R .On R × S , there is an extra monopole, the KK-monopole, which is on the same footingwith the monopoles. The existence of this extra topological excitation is significant in multipleways. For example, as it will be seen below, one can only construct the four dimensional BPSTinstanton out of the “constituent monopoles” due to the existence of the KK monopole.Incorporating the KK-monopole into the set of “elementary” monopoles also has a simplerealization in terms of Lie algebra. There is a unique extended root system (or extendedDynkin diagram) for each ∆ , which is obtained by adding the lowest root to the system ∆ :∆ = ∆ ∪ { α N } ≡ { α , . . . , α N − , α N } (4.2)Let n , . . . , n N denote the number of elementary monopoles whose charges are propor-tional to α , . . . , α N ∈ ∆ , respectively. The Callias index on R , for sufficiently small | ˆ v j L | ,is equal to the index of the Dirac operator on R × S for elementary monopoles with chargestaking values in the simple root system ∆ , i.e.: I R [ n , . . . , n N − , 0] = I R × S [ n , . . . , n N − , . (4.3)This is already demonstrated in obtaining (2.32) from (2.31) by using | ˆ v j L | ≤ π .We now discuss the relation between the APS index for the BPST instanton and theindex theorem on R × S . The result is: I instanton = I R × S [1 , , . . . , , 1] = N X i =1 I R × S [0 , . . . , |{z} i th , . . . , 0] (4.4)The proof of this statement necessitates a convenient rewriting of the index for the “static”(2.31) and “winding” (2.43) solutions. The important technical detail to keep in mind is thatfor static solutions (2.31), we set n = n N = 0. The index formula for [ n , . . . , n N ] monopolestakes the simple form: I fund. [ n , . . . , n N ] = n N − N X j =1 n j (cid:18)(cid:22) L ˆ v j π (cid:23) − (cid:22) L ˆ v j +1 π (cid:23)(cid:19) . (4.5)We also need to show that the topological excitation for which [ n , . . . , n N ] = [1 , . . . , P Ni =1 α i = 0. We also need to show that the topological charge– 19 –dds up to the one of a BPST instanton. Using formula (2.30) for static BPS monopoles (andsetting n = n N = 0 therein) and (2.41) for the KK monopole, we obtain the topologicalcharge of the excitation: Q [ n , . . . , n N ] = − n N + N X i =1 n i (cid:18) L ˆ v i π − L ˆ v i +1 π (cid:19) . (4.6)For [ n , . . . , n N ] = k [1 , . . . , v i . This is indeed the instanton with winding number k . For indextheorem aficionados, eqn.(4.4) can also be expressed as:dim ker / D inst − dim ker / D † inst = X α i ∈ ∆ (cid:16) dim ker / D α i − dim ker / D † α i (cid:17) . (4.7)It is evident that the index for Dirac operators on S × R has more refined data than thefamiliar APS index theorem for instantons on R . Remark on some special cases: In the derivation of the index theorem for the Diracoperator in the background of a monopole, we used the local axial-current non-conservation(2.5), which is an exact operator identity valid on any four-manifold, and a certain boundaryWilson line A | ∞ (1.11). In fact, the index is only well-defined for invertible A | ∞ . Inthis case, the corresponding Dirac operator is called a Fredholm operator. An eigenvalue of A | ∞ can always be rotated to zero by turning on an over-all Wilson line as in (1.12), whichcorresponds to a non-Fredholm operator. In those cases, the index for the monopole as wellas the η -invariant are not well-defined.What happens physically as the overall U (1) Wilson line is dialed? In that case, ineqn. (4.5), we replace ˆ v j → ˆ v j + √ N a following eqn. (1.12). As a is dialed smoothly, thefermionic zero mode will jump from the monopole it is localized into (say, with charge α j ∗ )to a monopole which is nearest neighbor, α j ∗± , depending on the sign of the a . In themean time, note that the index for the BPST instanton I instanton = I R × S [1 , , . . . , , 1] ineqn. (4.4) should remain invariant. As the normalizable zero mode jumps from α j ∗ to α j ∗± ,exactly at the value of a where one of the eigenvalues becomes zero, a non-normalizable zeromode appears and the exponential decay of the zero mode wave function is replaced by apower law decay of the three dimensional massless fermion propagator. 5. Remarks on anomalies and induced Chern-Simons terms on R × S Consider a chiral four-dimensional gauge theory compactified on R × S . In the limit ofzero radius, one expects that a generic theory on R with complex-representation fermionswill violate three dimensional parity. This is because the R theory can not be regulated byPauli-Villars (PV) fields in a simultaneously parity- and gauge-invariant manner.For example, in the SU (5) theory with left-handed Weyl fermions in the and ∗ ,compactified to R , four-dimensional Lorentz and gauge invariance would forbid mass terms– 20 –or the fermions, but on R real mass terms are allowed. Real mass terms in three dimensionscan be thought of as expectation values of the A (Wilson line) components of background U (1) gauge fields gauging global chiral symmetries. These mass terms are gauge invariantbut break three dimensional parity. On R , one can regulate the theory in a gauge invariantmanner via real-mass Pauli-Villars fields in the and ∗ . It is a well-known result [19]that every PV regulator gives rise to a Chern-Simons (CS) term, proportional to the indexof the representation (1 / and 3 / ∗ ) and to the sign of its mass. Thus, at oneloop, the fermion effective action has a parity-violating CS term, whose coefficient is 1 or 2,depending on the chosen relative sign of the two PV mass terms. This CS term does notgive rise to a “parity anomaly,” which would require the addition of a gauge-noninvariantbare half-integer coefficient CS term, since the integer coefficient assures its invariance undergauge transformations with nontrivial π ( G ) (for a brief reminder of the quantization of theCS coefficient, see the footnote in the beginning of Section 5.1). However, it gives a topologicalmass term to the gauge boson. If a bare CS term with integer coefficient is added, the CScoefficient becomes a free parameter of the three dimensional “chiral” gauge theory. Whenthe gauge group is broken to its maximal Abelian subgroup (by an adjoint Higgs field, as inthe applications we have in mind) this will give rise to CS terms with quantized coefficientsfor the various U (1). × S Now, consider the same theory on the locally four-dimensional background R × S . PVregulators with complex masses are not allowed by gauge invariance, while a real mass dueto a Wilson-line expectation value is neither local nor Lorentz invariant. Hence, we are ledto reconsider the calculation of the CS term, this time on R × S . We would like to knowwhether such a term is generated and what the freedom in the CS coefficient found in the R case corresponds to on R × S . Our main interest is in the case when the gauge group isbroken to its maximal Abelian subgroup by the nontrivial holonomy on S . Thus, considerthe loop-induced CS coefficient k ab : S CS = Z d x k ab π ǫ lim A al ∂ i A bm , (5.1)where a and b run over the Cartan generators of the gauge group. A straightforward loop Recall that in the nonabelian case, S CS = R d x k π ǫ lim tr( A l ∂ i A m + i A l A i A m ), where the trace is in thefundamental, and that k is quantized. To see this, let U ( x ) denote a gauge rotation for which π ( G ) is non-trivial, i.e, R π ǫ νλκ tr[ U∂ ν U † U∂ λ U † U∂ ν U † ] ≡ R ω ( x ) ∈ Z . Under a gauge transformation, the variation ofthe action is given in footnote (5) and yields S CS ( A U ) = S CS ( A ) + i (2 πk ) R ω ( x ), in Euclidean space, showingthat gauge invariance of the partition function demands quantization of k . – 21 –alculation of k ab in the background holonomy A gives: k ab = ∞ X n = −∞ Z d kπ tr T a k + ( A + πnL ) T b ( A + πnL ) k + ( A + πnL ) = tr T a T b ∞ X n = −∞ sign (cid:18) A + 2 πnL (cid:19) , (5.2)where a sum over all fermion matter representations is understood in the trace. To obtain thesecond equality we noted that all generators above are in the Cartan and took the momentumintegral, leading to a KK sum identical to the one appearing in (2.15). Finally, we regulatethe sum via ζ -function as in the calculation of the η -invariant, and obtain: k ab = tr T a T b η [ A , 0] = tr T a T b (cid:18) − LA π + 2 (cid:22) LA π (cid:23)(cid:19) , (5.3)where the function ⌊ ... ⌋ is applied to each element of the diagonal matrix A . To furtherunderstand (5.3), note that if | A | < πL , we have 1 − L A π + 2 (cid:22) LA π (cid:23) = − LA π + sign A ,and that after inserting this in (5.3) and using k ab = k ba , we find: k ab = − tr( { T a , T b } A ) L π + tr( T a T b sign A ) . (5.4)To understand the meaning of the two terms in (5.4), we now use the decomposition ofthe sign matrix sign( A ) in each representation R in terms of the unit matrix and Cartangenerators:sign( A R ) = s r X c =1 s c T c , s = 1dim( R ) tr R [sign( A )] , s a = 1 T ( R ) tr R [sign( A ) T a ] , (5.5)and a similar decomposition for the holonomy A itself: A L π (cid:12)(cid:12)(cid:12)(cid:12) R = a r X c =1 a c T c , a = L π dim( R ) tr R [ A ] , a c = L πT ( R ) tr R [ A T c ] . (5.6)After inserting these in (5.4), we find: k ab = X R h tr R (cid:16) { T a , T b } T c (cid:17) ( s c − a c ) + T ( R ) δ ab ( s − a ) i . (5.7)If A is entirely in the Cartan subalgebra of the gauge group, then a = 0. Furthermore, ifthe sign matrix is traceless ( s = 0)—which is the case for SU (2 N ) theories with a centersymmetric background—we find that the CS coefficient on R × S is proportional to the If this condition is not obeyed, the following equations have to be modified accordingly, as was done inthe computation of the index. – 22 –oefficient of the gauge anomaly in four dimensions (recall that the anomaly coefficient fora representation R is tr R (cid:0) { T a , T b } T c (cid:1) ). In this case, we find that for anomaly-free chiralgauge theories in four dimensions there is no loop induced CS term in three dimensions.It can happen that the sign matrix is not traceless, in which case the only contributionto the CS term is from the second term in (5.4), proportional to tr sign A . For example, inanomaly-free SU (2 N + 1) gauge theories with an almost center symmetric holonomy, whilethe first term in (5.4) vanishes, the second term in (5.4) may still be non-zero. In such cases,one can tune a background Wilson line associated with an axial, non-anomalous U (1) toisolate a point where CS-term vanishes (an example of this kind is SU (5) theory with and ∗ ).In conclusion, we find that the CS coefficient on R × S receives two contributions. Thefirst is a “four-dimensional” one and is given by the first term in (5.4). If the only Wilson linesthat are turned on correspond to anomaly-free gauge and global symmetries, the contributionof this term vanishes. On the other hand, turning on Wilson lines corresponding to anomaloussymmetries leads to a nonvanishing first term in (5.4)—its origin is in the four-dimensionalWess-Zumino term induced when anomalous background fields are included (the reason thecalculation in the nontrivial holonomy phase is so simple is that breaking the gauge symmetryand having massive fermions propagate in the loop allowed us to turn the four dimensionalWess-Zumino term into a local three dimensional CS term). The second, “three-dimensional,”contribution [19] is given by the second term in (5.4) and is nonzero only if tr R sign A dim( R ) generatesan anomalous U (1) symmetry in the four-dimensional theory. In the beginning of this Section, we found that in the three dimensional reduction of a fourdimensional chiral theory, there is freedom to have CS terms with quantized coefficients. Isthere similar freedom in the theory on R × S ? The answer can again be seen from (5.4).In Section 5.1, we assumed that the only Wilson lines turned on are those correspondingto the Cartan generators of the gauge group. We are free, however, to turn on Wilson linesof background U (1) fields gauging global chiral symmetries in four dimensions. These Wilsonlines do not break the gauge symmetry, but the symmetries they correspond to are usuallyanomalous, hence we can use eqn. (5.4) to infer the CS coefficient induced when they areturned on. It is clear from (5.4) that, generally, the value of the CS coefficient induced in thenonzero holonomy phase by these “flavor” Wilson lines would not correspond to quantizedvalues, unlike in three dimensions. However, recall that turning on Wilson lines for globalsymmetries is equivalent, by a field redefinition, to imposing non-periodic boundary conditionson the Weyl fermions in R , ψ ( x, y + L ) R = e iα R ψ ( x, y ) R , α R = A R L . (5.8) For complex representation Dirac fermions, these “chirally-twisted” boundary conditions can also be rewrit-ten as Ψ( x, y + L ) = e iα R γ Ψ( x, y ) . – 23 –onsequently, we find from (5.4), assuming that only a U (1) Wilson line, A R , is turned on,that: k ab = δ ab X R (cid:18) − α R T ( R )2 π + T ( R ) sign α R (cid:19) ≡ δ ab k ( α ) . (5.9)The induced CS term is, therefore, S CS = k ( α )4 π Z R ǫ νλκ tr( A ν ∂ λ A κ + 2 i A ν A λ A κ ) . (5.10)Note that in the case of anomalous- U (1) Wilson lines, the boundary conditions (5.8) wouldcorrespond to a symmetry of the action and measure of the theory—hence be admissible asboundary conditions—only if the Wilson lines take quantized values,2 T ( R ) α R = 2 πn, n ∈ Z , (5.11)implying that admissible boundary conditions for fermions are quantized (such that the ’tHooft vertex is invariant). Thus, the coefficients of the induced CS terms in this case alsotake quantized values.Note that the phase structure of gauge theory—massive versus perturbatively masslessphotons—is affected by turning on such discrete Wilson lines. Since the values are quantized,the one-loop potential for the Wilson line (Casimir energies) should not effect them (discreteWilson lines are known to appear in string theory, for example as disconnected componentson the moduli space of D-branes [20]). Moreover, at nonzero k , the finite action monopolesolutions (or other topological excitations, such as magnetic bions pertinent to gauge theorieson S × R ) which would render the gauge fluctuations massive nonperturbatively do notexist; see, e.g., [21]. In this sense, the two types of possible mass terms for gauge fluctuations,parity odd topological CS mass and parity even magnetic monopole or bion induced mass donot mix.To summarize, since the chiral anomalous U (1) current is parity odd, the response ofgauge theory is to produce a non-gauge invariant CS term at generic values of the backgroundWilson line. Only at admissible (discrete set of) boundary conditions for fermions, the inducedCS term is gauge invariant and sensible, and a parity odd mass term is generated for thegauge theory. At these points, the finite action topological excitation are excised from thegauge theory. If no anomalous U (1) is turned on, the photon is massless to all orders inperturbation theory, and a parity even mass term can be induced non-perturbatively viatopological excitations with zero index, either elementary or composite.The notion of the disconnected components of the gauge theory “moduli space” may findinteresting applications both in physics and mathematics. First, we formulate a QCD-likegauge theory on M × S where M is some three-manifold of arbitrary size, and S is small.Then, we impose admissible boundary conditions on the (say) adjoint Weyl fermion by That this theory is actually SYM plays no role, as one can similarly consider multi-adjoint theories.The discussion can also be generalized to Dirac fermions in complex, two-index representations. For R = – 24 –sing a “chiral twist” (5.8) obeying (5.11), by taking α = πn N , and assuming that n is apositive integer (this is to say that only a Z N is a anomaly-free remnant of the U (1) A chiralrotation, and allowed as boundary condition). Integrating out all the heavy KK-modes alongthe S circle induces, among other operators, the CS-term (5.10) with coefficient given by(5.9) and equal to k ( α ) = N − n . This means that a CS-term does not get induced for strictlyperiodic and anti-periodic (thermal) boundary conditions. Otherwise, we expect that thelong distance dynamics of these disconnected components of the “moduli space” of QCD-liketheories is described by topological CS-theory on M . This means that the theory is gapped,and is in a topologically ordered Chern-Simons phases. Up to our knowledge, this is the firstderivation of CS-theory and topological phases from QCD-like dynamics. We will pursue thisdirection in subsequent work. Acknowledgments We thank M. Shifman for useful discussions. This work was supported by the U.S. Departmentof Energy Grant DE-AC02-76SF00515 and by the National Science and Engineering ResearchCouncil of Canada (NSERC). Note added: While completing this paper, a new preprint [25] appeared, which discusses therelation between the S-duality and R-symmetry-twisted boundary conditions and CS theoriesin supersymmetric N = 4 SYM theory. In their case, the 3d CS theory arises due to mainlyS-duality twist and the non-abelian R-symmetry plays a secondary role, essentially restoringsupersymmetry. In our mechanism, a chirally-twisted boundary conditions associated witha discrete anomaly free remnant of axial U(1) symmetry induce the CS term. The twomechanism are simply different, but both yield 3d CS theory in the long distance regime. A. Another calculation of the η -invariant Here we give an alternative computation of η j [0] of eqn. (2.16). We now use the form [22, 23]: η [0] = 1 π lim m →∞ X λ Im ln λ + imλ − im = 1 π lim m →∞ Im ln det h + imh − im , (A.1)which holds because: lim m →∞ λ + imλ − im = lim m →∞ e i ( mλ ) = e iπ sign λ , { AS , S , BF , F } representations, there are respectively, 2 T ( R ) disconnected components (B.8), and for 2 T ( R ) − – 25 –nd the branch of the logarithm is defined so that ln e iφ = iφ (zero eigenvalues λ are assumedto not occur; if they do the formula (A.1) is ambiguous and needs to be modified [23]).Recall from the discussion in paragraph above eqn.(2.16) that h is the one-dimensional“massive Dirac operator” i ddy + ˆ v whose eigenvalues change sign under the combined y →− y , ˆ v → − ˆ v transformation. Together with (A.1) (or (A.2)) this implies that the spectralasymmetry (A.1) flips sign under ˆ v → − ˆ v . For our operator, λ = πnL + ˆ v , so we have: η [0] = 1 π lim m →∞ ∞ X n = −∞ Im ln πnL + ˆ v + im πnL + ˆ v − im = 1 π lim m →∞ ∞ X n = −∞ Im ln n + a + imn + a − im ,a ≡ L ˆ v π − (cid:22) L ˆ v π (cid:23) ⊂ (0 , , (A.2)where in the first line we trivially rescaled m and the second line means that a is taken tobe in the interval (0 , , 1) is the fundamental region, as η is well-defined andsmooth for all points (as opposed to the ( − / , +1 / 2) region which includes a singular point a = 0). The computation of η [0] is simplified by computing the derivative of (A.2) wrt a .Integration to recover the a -dependent part is then trivially done (note that an a -independentconstant in η [0] would be irrelevant, since a ˆ v j -independent term in η j [0] does not contributeto the sum in (2.31); furthermore it is prohibited by the “parity”-odd nature of η [0]). Thederivative of (A.2) wrt a is now given by a convergent sum, which is: dη [0] da = − π lim m →∞ m ∞ X n = −∞ n + a ) + m ≡ − π lim m →∞ m F (1 , a, m ) , (A.3)where the function F (1 , a, m ) is implicitly defined by the last equality and is computed, e.g.,in eqn. (81) of [24], F (1 , a, m ) = √ πm ( √ π + 4 ∞ P p =1 ( πpm ) cos(2 πpa ) K (2 πpm )). When m → ∞ ,only the first term in F (1 , a, m ) survives, lim m →∞ mF (1 , a, m ) = π , thus dη [0] da = − 2, whichdetermines η j [0] up to an integration constant: η j [0] = − a j + c = − (cid:18) L ˆ v j π − (cid:22) L ˆ v j π (cid:23)(cid:19) + c. (A.4)This periodic (in ˆ v j ) result can be made “parity”-odd by taking c = 1, giving back (2.24). B. Index for higher representation fermions R : The Callias index theorem on R can be obtained by restricting the sum in (2.15) to p = 0.Fund . : I F ( n , . . . , n N − ) = − N X i =1 sign(ˆ v i )( n i − n i − )= − N − X i =1 n i h sign(ˆ v i ) − sign(ˆ v i +1 ) i = n j ∗ . (B.1)– 26 –here ˆ v j ∗ < < ˆ v j ∗ +1 . Since there is no axial anomaly in d = 3, there is no other contributionto the index and this is the final result. A better way to express ( B.1), which is easilygeneralizable to arbitrary representation of the gauge group SU ( N ) is: I F ( n , . . . , n N − ) = − tr[sign( A ) · ˆ B ] . (B.2)where sign( A ) is the sign matrix and ˆ B = P N − i =1 n i ( α i H ) is the space independent part ofeq.(1.17). For an arbitrary representation R , this formula generalizes as: I R ( n , . . . , n N − ) = − tr R [sign( A ) · ˆ B ] . (B.3)Our main interest is in fermionic matter in two index representations of gauge group, namely,adjoint, antisymmetric (AS), symmetric (S) of SU ( N ) and bi-fundamental (BF) representa-tion of SU ( N ) × SU ( N ). For the adjoint, the index is:Adjoint : I Adj ( n , . . . , n N − ) = − N X i,j =1 sign(ˆ v i − ˆ v j ) h ( n i − n i − ) − ( n j − n j − ) i = − N − X j =1 N − X i =1 n i h sign(ˆ v i − ˆ v j ) − sign(ˆ v i +1 − ˆ v j ) i = N − X j =1 n j . (B.4)This means that in the background of each elementary monopole, there are two fermioniczero modes. For the other two-index representations, the expressions are:BF : I BF ( n , . . . , n N − , n , . . . , n N − ) = − N X i,j =1 sign(ˆ v i − ˆ v j ) h ( n i − n i − ) − ( n j − n j − ) i , AS : I AS ( n , . . . , n N − ) = − N X i>j sign(ˆ v i + ˆ v j ) h ( n i − n i − ) + ( n j − n j − ) i , S : I S ( n , . . . , n N − ) = − N X i ≥ j sign(ˆ v i + ˆ v j ) h ( n i − n i − ) + ( n j − n j − ) i . (B.5)It is more convenient to express the index for AS/S representations as: I AS/S = − N X i,j =1 sign(ˆ v i + ˆ v j ) h ( n i − n i − ) + ( n j − n j − ) i ± N X i sign(2ˆ v i )( n i − n i − ) . = − N − X i,j =1 n i h sign(ˆ v i + ˆ v j ) − sign(ˆ v i +1 + ˆ v j ) i ± N − X i n i h sign(2ˆ v i ) − sign(2ˆ v i +1 ) i (B.6) R × S : This formulae can be straightforwardly generalized to R × S by repeatingour derivations for the fundamental. The non-integer contributions to the index from the– 27 –opological charge cancel the corresponding non-integer part of the η -invariant, yielding ageneral expression; further, if the definition of ˆ B is extended to include the affine root (1.14),ˆ B = P Ni =1 n i ( α i H ), with n , . . . , n N − are the monopole numbers of the background and n N —the KK-monopole number, this equation can be also extended to also include the KKmonopole: I R [ n , . . . , n N ] = 2 T ( R ) n N − tr R (cid:22) A L π (cid:23) · ˆ B . 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