# A lattice formulation of the Atiyah-Patodi-Singer index

Hidenori Fukaya, Naoki Kawai, Yoshiyuki Matsuki, Makito Mori, Katsumasa Nakayama, Tetsuya Onogi, Satoshi Yamaguchi

aa r X i v : . [ h e p - l a t ] J a n A lattice formulation of the Atiyah-Patodi-Singerindex ∗ Hidenori Fukaya a † , Naoki Kawai ‡ a § , Yoshiyuki Matsuki a ¶ Makito Mori a k , KatsumasaNakayama ab ∗∗ , Tetsuya Onogi a †† , and Satoshi Yamaguchi a ‡‡ a Department of Physics, Osaka University, Toyonaka, Japan b NIC, DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany

Atiyah-Singer index theorem on a lattice without boundary is well understood owing to the semi-nal work by Hasenfratz et al . But its extension to the system with boundary (the so-called Atiyah-Patodi-Singer index theorem), which plays a crucial role in T-anomaly cancellation between bulk-and edge-modes in 3+1 dimensional topological matters, is known only in the continuum theoryand no lattice realization has been made so far. In this work, we try to non-perturbatively deﬁnean alternative index from the lattice domain-wall fermion in 3+1 dimensions. We will show thatthis new index in the continuum limit, converges to the Atiyah-Patodi-Singer index deﬁned on amanifold with boundary, which coincides with the surface of the domain-wall. ∗ The original title of the talk was “Atiyah-Patodi-Singer index theorem on a lattice.” † E-mail: [email protected] ‡ Speaker. § E-mail: [email protected] ¶ E-mail: [email protected] k E-mail: [email protected] ∗∗ E-mail: [email protected] †† E-mail: [email protected] ‡‡ E-mail: [email protected] c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ tiyah-Patodi-Singer index theorem on a lattice

Naoki Kawai

1. Introduction

The Atiyah-Patodi-Singer(APS) index theorem [1] is an extesion of the Atiyah-Singer(AS) in-dex theorem [2] to a manifold with boundary. Let us consider a four-dimensional closed Euclideanmanifold X with a three-dimensional boundary Y . We assume that X is extending in the region x > x =

0. Then the index is given byInd ( D ) = π Z x > d x ε µνρσ F µν F ρσ − η (cid:0) iD (cid:1) , (1.1)where iD is the three-dimensional Dirac operator on Y , and η ( H ) is the so-called η -invariantwhich is deﬁned by a regularized summation of signs of eigenvalues of H : η ( H ) ≡ ∑ sgn ( λ ) = Tr H / √ H , (1.2)where λ is eigenvalues of H . The ﬁrst term of Eq. (1.1) is equivalent to an integral of the instantonnumber density, which is not an integer generally on a manifold with boundary. The second termof (1.1) contains the Chern-Simons term, which is not an integer, either. The APS index theoremclaims that the sum of two terms is always an integer.Recently the APS index theorem is used to the study of condensed matter physics. It is re-lated to the physics of 3 + + η -invariant of the three-dimensional Dirac operator on the edge: Z edge = Det (cid:0) iD (cid:1) ∝ exp (cid:2) − i πη (cid:0) iD (cid:1) / (cid:3) . (1.3)Similarly the partition function of the bulk is Z bulk ∝ exp (cid:20) i π π Z x > d x ε µνρσ F µν F ρσ (cid:21) . (1.4)As we mentioned, each of these two factors is complex in general, which means that these partitionfunctions break the T-symmetry. But the total partition function Z total ∝ exp (cid:20) i π π Z x > d x ε µνρσ F µν F ρσ − i πη (cid:0) iD (cid:1) / (cid:21) = exp [ i π Ind ( D )] (1.5)is real, therefore time-reversal symmetry is protected. Namely, the APS index theorem describesthe bulk-edge correspondence in the symmetry protected topological insulator by T-anomaly can-cellation between bulk and edge [3].However, the original set-up by APS is quite different from topological insulators. APS con-sidered a Dirac operator for massless fermions with a non-local boundary condition(APS boundarycondition), under which the edge-localized modes are not allowed to exist. On the other hand, theelectron in a topological insulator is massive in the bulk, and the edge-localized modes appear. It is,therefore, a mathematical puzzle of why the APS index is related to the massive fermion systems.To ﬁll the gap, three of the authors proposed a new formulation of the APS index theorem [4]using domain-wall fermion [5]. Since there exist massive fermions in the bulk and massless edge1 tiyah-Patodi-Singer index theorem on a lattice Naoki Kawai localized modes on the kink, the domain-wall fermion shares similar properties with the topologicalinsulator. They showed that in continuum theory the η -invariant of the domain-wall Dirac operator H c DW = γ ( D − M sgn ( x )) with appropriate regularization(Pauli-Villars(PV) regularization) coin-cides with the APS formula: − η ( H c DW ) PV reg . = π Z x > d x ε µνρσ F µν F ρσ − η (cid:0) iD (cid:1) . (1.6)This new formulation of the APS index does not require any non-local boundary conditions. Re-cently, its mathematical justiﬁcation was given by [6].In [4], they also showed that the AS index is given by a massive Dirac operator:Ind AS ( D ) = − η ( γ ( D − M )) PV reg . , (1.7)where the Pauli-Villars mass has an opposite sign to M . This fact that the index theorems can bereformulated by massive Dirac operators is quite suggestive since in the reformulation the chiralsymmetry is not very important.Keeping the unimportance of the chiral symmetry in mind, let us revisit the lattice formulationof the Atiyah-Singer index theorem established by Hasenfratz et al. [7]. Using the Dirac operatorwhich satisﬁes Ginsparg-Wilson(GW) relation γ D + D γ = aD γ D [8], the AS index can be givenby Ind AS ( D ) = Tr γ ( − aD / ) , (1.8)where a is lattice spacing. As an example, let us take the overlap Dirac operator [9] aD ov = + γ H W / q H , H W = γ ( D W − M ) , M = / a , (1.9)where D W is the Wilson-Dirac operator. The convergence of Eq. (1.8) with the overlap Diracoperator to the AS index in the continuum limit was conﬁrmed by [10, 11, 12, 13, 14].Simply substituting Eq. (1.9) into Eq. (1.8), we can easily show that the index is equivalent tothe η -invariant of massive Wilson-Dirac operator,Ind AS ( D ) = − η ( H W ) = −

12 Tr H W / q H , (1.10)which is a naive lattice discretization of Eq. (1.6) by the Wilson-Dirac operator.The original AS and APS indices require exact chiral symmetry to deﬁne the chiral zero modes.The overlap fermion meets this requirement by the GW relation. But for our new formulation withthe η -invariant of massive Dirac operator, the chiral symmetry is less important. The fact thatthe index of the overlap Dirac operator is the same as the η -invariant of the massive Wilson-Dirac operator strongly supports a hypothesis that the naive discretization of the η -invariant ofthe domain-wall Dirac operator agrees with the APS index in the continuum limit. As shown in[15], the original APS boundary condition is difﬁcult to realize on a lattice with the overlap Diracoperator. Also, any boundary condition would break the GW relation, which makes it impossibleto deﬁne the APS index on a lattice by the chiral zero modes.2 tiyah-Patodi-Singer index theorem on a lattice Naoki Kawai

The η -invariant of the massive Dirac operator, gives a uniﬁed view of the index theorems. Inthe continuum theory, the APS index theorem is given by just adding a kink structure to the mass inthe AS formula. For the lattice version of the AS index, we only need the Wilson-Dirac operator.The application to the APS index is therefore straightforward. Note that H DW is a four-dimensionalhermitian operator, η ( H DW ) / H DW do not cross zero.In the following, we will see that the above observation is correct, showing − η ( H DW ) = π Z x > d x ε µνρσ F µν F ρσ − η (cid:0) iD (cid:1) , (1.11)in the continuum limit.

2. Lattice set-up

We consider the Wilson-Dirac operator with a mass term having a kink structure: H DW = γ h D W − M ε (cid:16) x + a (cid:17) + M i , (2.1)where ε ( x ) = sgn ( x ) . The domain-wall is located at x = a /

2. Since the index is deﬁned on acompact manifold, we should consider in a compact space but here we proceed as if we were on aninﬁnite lattice to make the presentation simpler. See [15] for more precise treatment.The key of this work is to ﬁnd a good complete set to evaluate the η -invariant as − η ( H DW ) = −

12 Tr (cid:20) H DW q H (cid:21) = ∑ x , n Φ † n ( x ) (cid:20) H DW q H (cid:21) Φ n ( x ) . (2.2)We cannot use simple plane waves due to the loss of translational symmetry in the x -direction. Forthe three directions, which does not have domain-walls, the plane wave set ψ p = e ippp · xxx / ( π ) / isstill useful. We denote the momentum by ppp = ( p , p , p ) and we assign two-spinor componentsto this wave functions.We consider a complete set given by a direct product ψ p ⊗ φ ( x ) which is the eigenfunctionsof the squared free domain-wall Dirac operator. Denoting s i = sin ( p i a ) and c i = cos ( p i a ) , thesquared free domain-wall Dirac operator is expressed by a ( H DW ) = s i + θ ( x + a / ) { M + − ( + M + )( a ∇ ∗ ∇ ) } , + θ ( − x − a / ) { M − − ( + M − )( a ∇ ∗ ∇ ) } , + M a ( P + δ x , − a S + − P − δ x , S − ) , (2.3) M ± = ∑ i = , , ( − c i ) ∓ M a + M a , (2.4)where θ ( x ) = ( ε ( x ) + ) / ∇ µ and ∇ ∗ µ denote the forward and backwarddifference operators respectively, P ± = ( + γ ) / S ± µ is a shift operator operating as S ± µ f ( x ) = f ( x ± ˆ µ a ) . We have three types of the eigenfunctions in the x -direction of a ( H DW ) : (1) edge-localized modes at x =

0, (2) plane wave modes in the region x ≥

0, and (3) plane wave modes atany x . 3 tiyah-Patodi-Singer index theorem on a lattice Naoki Kawai

To simplify the computation, we take the Wilson parameter unity, and take the limit where M + M → ∞ while the difference is ﬁxed to a ﬁnite value M − M = M >

0. After taking thislimit, the system is equivalent to the Shamir-type domain-wall fermion [16, 17]. After taking thislimit three types of eigenfunctions are reduced to the type (1) and (2) only which still make acomplete set. More explicitly, we have φ edge − ( x ) = p − M + ( + M + ) / ae − Kx , (2.5) φ ω + ( x ) = √ π ( e i ω ( x + a ) − e − i ω ( x + a ) ) , (2.6) φ ω − ( x ) = √ π ( C ω e i ω x − C ∗ ω e − i ω x ) , (2.7)in the region x ≥

0, where the subscript ± denotes the eigenvalue of γ = ± K = − ln ( + M + ) / a , C ω = − ( + M + ) e i ω a − | ( + M + ) e i ω a − | . (2.8)Due to the normalizability of the edge-localized modes, we can get the fermion mass condition as | + M + | < < Ma <

3. The evaluation of the η -invariant Using the complete set which is derived in the previous section, we can completely decomposethe η -invariant into bulk contribution and edge contribution. − η ( H DW ) = −

12 Tr bulk (cid:20) H DW / q H DW (cid:21) −

12 Tr edge (cid:20) H DW q H DW (cid:21) . (3.1) For the bulk contribution, we consider the density of the η -invariant: −

12 tr (cid:20) H DW / q H DW (cid:21) ( x ) bulk = − ∑ g = ± Z π / a d ω Z π / a − π / a d p ( [ ψ p ( ~ x ) ⊗ φ ω g ( x )] † tr (cid:20) P g (cid:18) H DW / q H DW (cid:19) P g (cid:21) [ ψ p ( ~ x ) ⊗ φ ω g ( x )] (cid:27) . (3.2)Here the trace is taken over color and spinor indices only. We decompose the squared domain-wall Dirac operator into the free part ( H DW ) and the other part ∆ H DW which has the gauge ﬁelddependence: H DW = ( H DW ) + ∆ H DW , (3.3) ∆ H DW = − ∑ µ , ν [ γ µ , γ ν ][ ˜ D µ , ˜ D ν ] − γ µ [ ˜ D µ , ˜ R ] + · · · , (3.4)4 tiyah-Patodi-Singer index theorem on a lattice Naoki Kawai where · · · are the terms having no γ structures, and˜ D µ = a (cid:2) e ip µ a ( U µ ( x ) S + µ − ) − e − ip µ a ( S − µ U µ ( x ) † − ) (cid:3) , (3.5)˜ R = − a ∑ µ (cid:2) e ip µ a ( U µ ( x ) S + µ − ) + e − ip µ a ( S − µ U µ ( x ) † − ) (cid:3) , (3.6)assuming them to operate on [ ψ p ( ~ x ) ⊗ φ edge − ( x )] and denoting U µ ( x ) as the link variables.Expanding 1 / q H in ∆ H which is higher-order term in a , we can show that many termsvanish due to the spinor structure. The only surviving term is −

12 tr (cid:20) H DW / q H DW ( x ) bulk (cid:21) = (cid:0) I ( Ma ) + I DW ( Ma , x ) (cid:1) π ε µνρσ tr F µν F ρσ (3.7)upto O ( a ) corrections. The ﬁrst term is I ( Ma ) = a π Z π / a − π / a d pd ω ∏ µ c µ − M ′ + + ∑ ν s ν / c ν (cid:16) s µ + (cid:2) ∑ µ ( − c µ ) − Ma (cid:3) (cid:17) / , (3.8)which was already evaluated in [13], and I ( Ma ) = / M . Since I ( Ma ) + I DW ( Ma , x ) = + O ( / M ) , weobtain the standard curvature term from the bulk contribution. For the edge contribution, ﬁrst we consider the domain-wall Dirac operator in U = aH DW = γ [ − aP − ∇ + aP + ∇ ∗ + a γ i D i ( x ) + M + ( x )] . (3.9)Note that D i ( x ) and M + ( x ) have x dependence through the link variables. We assume that x dependence of link variables is mild. Then we evaluate the edge contribution in adiabatic approxi-mation ( || U † ∂ x U || / M ≪ H DW is written in φ ( x ) = φ λ ( ) ( xxx ) ⊗ φ edge − ( x ) , where φ λ ( ) ( xxx ) is an eigenfunction of i σ i D i ( x = ) with the eigenvalue λ ( ) . φ edge − ( x ) satisﬁes − aP − ∇ φ edge − ( x ) = − M + ( ) φ edge − ( x ) , then the operation of H DW to φ edge − ( x ) is aH DW φ edge − ( x ) = a i σ i D i ( ) ! φ edge − ( x ) . (3.10)Similarly, we can evaluate the higher order of the adiabatic approximation, and we can concludethat it is suppressed by 1 / M .Therefore the edge contribution becomes −

12 Tr edge (cid:20) H DW / q H DW (cid:21) = − ∑ λ ( ) sgn λ ( ) = − η ( i σ i D i ) | x = , (3.11)upto O ( / M ) corrections. 5 tiyah-Patodi-Singer index theorem on a lattice Naoki Kawai

4. Summary

In this work, we have formulated the Atiyah-Pstodi-Singer index theorem on a lattice. Wehave shown that the eta invariant of massive Dirac operator gives a uniﬁed view of both the Atiyah-Singer and the APS index theorems in the continuum and lattice theory. To compute the η -invariantof the domain-wall Dirac operator, we have obtained a good complete set which consists of bulkplane wave modes and edge-localized modes, in the Shamir-type limit. We have computed thecontribution from bulk and edge separately, then we have conﬁrmed that the η -invariant of theWilson-Dirac operator with domain-wall mass converges to the APS index in the continuum limit.Acknoledgemants: We thank H. Suzuki for his instruction on the computation of I ( M ) . Thiswork was supported in part by JSPS KAKENHI Grant Number JP15K05054, JP18H01216, JP18H04484,JP18J11457, JP18K03620, and JP19J20559. The authors thank the Yukawa Institute for Theoreti-cal Physics at Kyoto University. Discussions during the YITP workshop YITP-T-19-01 on “Fron-tiers in Lattice QCD and related topics" were useful to complete this work. T.O. would also like tothank YITP for their kind hospitality during his stay. References [1] M. F. Atiyah, V. K. Patodi and I. M. Singer, Math. Proc. Cambridge Phil. Soc. , 43 (1975).doi:10.1017/S0305004100049410[2] M. F. Atiyah and I. M. Singer, Annals Math. , 484 (1968). doi:10.2307/1970715[3] E. Witten, Rev. Mod. Phys. , no. 3, 035001 (2016) doi:10.1103/RevModPhys.88.035001,10.1103/RevModPhys.88.35001.[4] H. Fukaya, T. Onogi and S. Yamaguchi, Phys. Rev. D , no. 12, 125004 (2017)doi:10.1103/PhysRevD.96.125004.[5] D. B. Kaplan, Phys. Lett. B , 342 (1992) doi:10.1016/0370-2693(92)91112-M [hep-lat/9206013].[6] H. Fukaya, M. Furuta, S. Matsuo, T. Onogi, S. Yamaguchi and M. Yamashita, arXiv:1910.01987[math.DG].[7] P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. Lett. B , 125 (1998)doi:10.1016/S0370-2693(98)00315-3[8] P. H. Ginsparg and K. G. Wilson, Phys. Rev. D , 2649 (1982). doi:10.1103/PhysRevD.25.2649[9] H. Neuberger, Phys. Lett. B , 141 (1998). doi:10.1016/S0370-2693(97)01368-3[10] Y. Kikukawa and A. Yamada, Phys. Lett. B , 265 (1999) doi:10.1016/S0370-2693(99)00021-0.[11] M. Luscher, Nucl. Phys. B , 515 (1999) doi:10.1016/S0550-3213(98)00680-4.[12] K. Fujikawa, Nucl. Phys. B , 480 (1999) doi:10.1016/S0550-3213(99)00042-5.[13] H. Suzuki, Prog. Theor. Phys. , 141 (1999) doi:10.1143/PTP.102.141.[14] D. H. Adams, Annals Phys. , 131 (2002) doi:10.1006/aphy.2001.6209.[15] H. Fukaya, N. Kawai, Y. Matsuki, M. Mori, K. Nakayama, T. Onogi and S. Yamaguchi,arXiv:1910.09675 [hep-lat].[16] Y. Shamir, Nucl. Phys. B , 90 (1993) doi:10.1016/0550-3213(93)90162-I.[17] V. Furman and Y. Shamir, Nucl. Phys. B , 54 (1995) doi:10.1016/0550-3213(95)00031-M., 54 (1995) doi:10.1016/0550-3213(95)00031-M.