A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification
Hidenori Fukaya, Mikio Furuta, Shinichiroh Matsuo, Tetsuya Onogi, Satoshi Yamaguchi, Mayuko Yamashita
AA physicist-friendly reformulation of theAtiyah-Patodi-Singer index and its mathematicaljustification ∗ Hidenori Fukaya † a ‡ , Mikio Furuta b § , Shinichiroh Matsuo c ¶ , Tetsuya Onogi a (cid:107) , SatoshiYamaguchi a ∗∗ , and Mayuko Yamashita d †† a Department of Physics, Osaka University, Toyonaka, Japan b Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan c Graduate School of Mathematics, Nagoya University, Nagoya, Japan d Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
The Atiyah-Patodi-Singer index theorem describes the bulk-edge correspondence of symmetryprotected topological insulators. The mathematical setup for this theorem is, however, not di-rectly related to the physical fermion system, as it imposes on the fermion fields a non-localand unnatural boundary condition known as the "APS boundary condition" by hand. In 2017,we showed that the same integer as the APS index can be obtained from the η invariant of thedomain-wall Dirac operator. Recently we gave a mathematical proof that the equivalence is not acoincidence but generally true. In this contribution to the proceedings of LATTICE 2019, we tryto explain the whole story in a physicist-friendly way. ∗ The original title of the talk was “Domain-wall fermion and Atiyah-Patodi-Singer index.” † Speaker. ‡ E-mail: [email protected] § E-mail: [email protected] ¶ E-mail: [email protected] (cid:107)
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/ a r X i v : . [ h e p - l a t ] J a n physicist-friendly APS index Hidenori Fukaya
1. Introduction
The Atiyah-Singer(AS) index theorem [1] on a manifold without boundary is well-known inphysics, and it has played an important role especially in high energy particle physics. But itsextension to the manifold with boundary, known as Atiyah-Patodi-Singer(APS) index theorem [2],was not discussed very much, as we were not that interested in a space-time having boundaries.Recently, the theorem is drawing attention from the condensed matter physics. This is becausethe APS index is a key to understand the bulk-edge correspondence [3, 4] of topological insulatorsfrom anomaly matching of the time-reversal(T) symmetry.It is, however, difficult to understand why the APS index “must” appear in the physics ofthe topological insulators, since the original set up of the APS index employs a very unnaturalboundary condition and is unlikely to be realized in the real electron systems.In [5], three physicist half of the authors found a different fermionic quantity, which coincideswith the APS index. We used a domain-wall Dirac operator [6, 7, 8] on a closed manifold with-out boundary, sharing a “half” of it with the original set up of the APS. We have perturbativelyshown that its η invariant, defined as a regularized difference of the number of positive and nega-tive eigenmodes, is equal to the original APS index. Since the domain-wall Dirac fermion sharesproperties of the electron systems of the topological insulators, we proposed this η invariant as a“physicist-friendly” reformulation of the APS index.In [9], three mathematician half of the authors joined the collaboration, and we succeededin a proof that the above observation of [5] is not a coincidence but generally true. Namely, weproved that for any APS index on a manifold with boundary (we denote X + and its boundary Y ),there exists a domain-wall Dirac operator on a closed manifold without boundary, where its halfcoincides with X + , and its η invariant is equal to the original APS index.The key of this work is to add a mass term to the Dirac operator. The mass term breaksthe chiral symmetry, which is apparently an essential property to describe the index theorems.Nevertheless, as we will show below, there is no problem in giving a fermionic integer, whichcoincides with the original index. This is true even on a lattice, and we proposed a non-perturbativeformulation of the APS index in lattice gauge theory in [10]. In fact, the η invariant of the massiveDirac operator gives a unified view of the index theorems including their lattice version. See alsoKawai’s contribution [11] to these proceedings.
2. Why APS index unphysical?
First we review the original work of APS and discuss how it is unnatural. We consider aDirac operator D = ∑ µ γ µ (cid:16) ∂∂ x µ + iA µ ( x ) (cid:17) , which operates on fermion fields on a four-dimensionalEuclidean flat space in the x > . Here, A µ is the SU ( N ) or U ( ) gauge field and wetake A = x = B = γ ∑ i = γ i (cid:18) ∂∂ x i + iA i ( x ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = , (2.1) In order to make the index well-defined, we have to compactify the space time. But here we perform our compu-tation as if we were in a semi-infinite space for simplicity of the presentation. physicist-friendly APS index Hidenori Fukaya such that any positive eigencomponent of B becomes zero at x =
0, which guarantees the Her-miticity of the Dirac operator. As [ B , γ ] =
0, the chiral symmetry is conserved. In fact, B can beblock-diagonal: B = diag ( iD , − iD ) , where iD is the three-dimensional Dirac operator on thesurface. Since this condition requires information of the whole eigenfunctions of B , it is non-local.Let us compute the axial U ( ) anomaly of a massless fermion with the APS boundary con-dition. The path integral measure transforms as d ψ d ¯ ψ → d ψ d ¯ ψ exp [ i α Tr γ ] , under the chiraltransformation. The trace can be evaluated by the heat-kernel regulator asTr γ e D / M ≡ lim M → ∞ (cid:90) d x tr γ e D / M . (2.2)If it were without boundary, the plane wave complete set would apply to obtain the AS index.However, we need here a different complete set, which satisfies the APS boundary condition.At the leading order of the adiabatic expansion in x , let us consider the eigenproblem of − D ψ ( x ) = Λ ψ ( x ) , where the solution is given by ψ = φ ± ( x ) ⊗ φ λ ( xxx ) , choosing the three-dimensional part φ λ ( xxx ) to be the eigenfunction of iD with the eigenvalue λ . The results are φ p ± ( x ) = u ± √ π (cid:0) e i ω x − e − i ω x (cid:1) , φ n ± ( x ) = u ± (cid:0) ( i ω ∓ λ ) e i ω x − c . c . (cid:1)(cid:112) π ( ω + λ ) , (2.3)where φ p ± ( x ) denotes the positive eigenmodes of B , while φ n ± ( x ) is the negative modes. We take γ u ± = ± u ± and ω = Λ − λ , which must be positive so that there is no edge mode allowed.Using the above complete set, Eq. (2.2) is evaluated as ∑ λ (cid:90) x > dx sgn λ e − λ / M (cid:90) d ω π (cid:18) − + i | λ | ω + i | λ | (cid:19) e − ω / M + i ω x = − ∑ λ sgn λ ( | λ | / M ) , (2.4)which gives the η invariant of iD in the M → ∞ limit. From the next-to-leading order contributionof the adiabatic expansion, we obtain the curvature term in the x > APS ( D ) = Tr γ e D / M = π (cid:90) x > d x ε µνρσ tr F µν F ρσ − η ( iD ) . (2.5)The above derivation has no problem in mathematics but physically unnatural. First of all, thecausality becomes questionable with the non-locality , as any change in the eigenfunction of B isimmediately reflected to the whole boundary, which means that the information propagates fasterthan speed of light. Second, as B involves the momentum and gauge fields in spatial directions, theAPS boundary condition does not respect the rotational symmetry on the surface. Third problem isthe fact that there is no edge-localized mode allowed to exist under the APS condition.This unnaturalness of the APS boundary condition motivated us to explore a physicist-friendlyreformulation of the APS index. What we have to give up is clear when we consider the reflectionof the fermionic particle at the boundary. As the translational invariance is lost in the x direction,the momentum in that direction is not conserved. But as the energy has to be conserved the fourthmomentum must flip at the boundary. On the other hand, the angular momentum in the x directionshould be conserved, which means that the helicity must flip by the reflection. It is, therefore,essential to give up the chirality and we have to consider the index theorem with massive fermions. In a recent work [12], the use of the APS boundary condition is justified by rotating the boundary to the temporaldirection and regarding it as an intermediate state in the partition function of massive fermion systems. Here we give analternative way, which allows us to take the boundary remaining in space-like direction. physicist-friendly APS index Hidenori Fukaya
3. Atiyah-Singer index in terms of the massive Dirac operator
Here let us consider a simpler case, an Euclidean four-dimensional space without boundary,and try to reformulate the AS index in terms of the massive Dirac operator. The Dirac fermionpartition function is given by det D + MD + M , (3.1)where we have introduced the Pauli-Villars regulator with a mass M (cid:29) M . When M and M havethe same sign, the large M → M limit leads to a trivial consequence that the above determinant isunity, which is the situation of normal insulators.When the sign of the mass is flipped, the situation is different. For the determinantdet D − MD + M , (3.2)one can flip the sign of the mass by the chiral rotation with α = π , but the measure changes asdet D − MD + M ∝ exp (cid:18) i π π (cid:90) d x ε µνρσ tr F µν F ρσ (cid:19) = exp ( i π Q ) = ( − ) Q , (3.3)which means a nontrivial θ = π vacuum. In fact, Q is the AS index.The same determinant can be evaluated in a different way asdet D − MD + M = det i γ ( D − M ) i γ ( D + M ) = ∏ i λ ( − M ) ∏ i λ ( M ) ∝ exp (cid:104) i π (cid:16) ∑ sgn λ ( − M ) − ∑ sgn λ ( M ) (cid:17)(cid:105) , (3.4)where λ ( m ) is the eigenvalue of γ ( D + m ) . Note that the exponent is nothing but an expression ofthe η invariant (with the Pauli-Villars subtraction), and therefore, we can conclude − η PV reg . ( γ ( D − M )) = Q , (3.5)which coincides with the AS index. For more mathematical proof, see our paper [9].
4. APS index and domain-wall fermion
Now let us consider the domain-wall fermion determinant,det D − M ε ( x ) D + M , (4.1)where ε ( x ) = x / | x | is the sign function. Unlike the original set up by APS, we include the x < γ Hermiticity: D † = γ D γ , the determinant is still real, andtherefore, the sign of the determinant is controlled by an integer Q .Note that the physical properties of the domain-wall fermion are similar to those of topologicalinsulators. The fermion is massive(gapped) in the bulk, while the massless(gapless) mode appearsat the wall between two physically different regions. We also note that any topological insulator inour world is surrounded by normal insulators. Therefore, it is more natural to model the physics oftopological insulators with this domain-wall fermion than the one on a manifold with boundary.3 physicist-friendly APS index Hidenori Fukaya
The domain-wall fermion determinant in Eq. (4.1) can be expressed as = det i γ ( D − M ε ( x )) i γ ( D + M ) ∝ exp (cid:20) − i π η PV reg . ( H DW ) (cid:21) , (4.2)where H DW = γ ( D − M ε ( x ))) . Therefore, we can conclude − η PV reg . ( H DW ) = Q . (4.3)In fact, this integer coincides with the APS index.In our paper [5], we evaluated the η invariant, using the integral expression for 1 / (cid:113) H DW as1 (cid:113) H DW = √ π (cid:90) ∞ dtt − / e − tH DW , (4.4)and expanding the exponential part e − tH DW in the gauge coupling. The essential point here is thateven in the zero coupling limit, H DW has a non-trivial structure as H DW = − D + M − M γ δ ( x ) , (4.5)where the last delta-function comes from the domain-wall. In fact, it has an edge-localized mode,whose eigenfunction in the x direction is φ ( x ) = C exp ( − M | x | ) , γ φ ( x ) = − φ ( x ) , (4.6)and from this edge mode, we have reproduced the second term of Eq. (2.5).On the other hand, the extended modes have higher energy than M , and therefore, their con-tribution to the η invariant reduces to the integral of a local quantity that is exactly the first termof Eq. (2.5). It is important to note that we did not assume any boundary condition at x =
5. Mathematical proof
Recently, the mathematician-half of the authors joined and we succeeded in proving the equiv-alence of the APS index and the domain-wall η invariant on a general even-dimensional manifold[9]. The main theorem we have proved is Theorem 1.
Let X be a n-dimensional closed and oriented manifold and S be a Hermitian vectorbundle (whose section corresponds to the fermion field) on X . We assume that S is Z gradedand Γ S as its Z grading operator (or γ ). Let D be a first-order and elliptic partial differentialoperator (or Dirac operator multiplied by γ ), which anti-commutes with Γ S . Let Y be a separatingsub-manifold that decomposes X into two compact manifolds X + and X − (Y is the domain-wall).We consider a step function κ , which takes ± on X ± (or ε ( x ) in the previous section).Then, there exists m > and for any m > m , Ind
APS ( D | X + ) = − η ( D − m κ Γ S ) − η ( D + m Γ S ) physicist-friendly APS index Hidenori Fukaya holds. Here, the left-hand side is the APS index on X + with the APS boundary condition on Y . Theright-hand side denotes the η invariant of the domain-wall fermion Dirac operator, regularized bythe Pauli-Villars fields, where we have chosen M = M = m. Below we give a rough sketch of the proof, for which we need three known mathematicaltheorems: 1) The APS index is equal to the AS index on a manifold with an infinite cylinderattached to the original boundary, where the gauge fields and metric are constant on the cylinder.2) Localization and product formula [13, 14]: adding a potential term, we can “localize” the zeromode eigenfunction in a vicinity of a lower-dimensional sub-manifold, and we can evaluate theindex as a product of the index in lower dimension and that in the normal direction. 3) The APSindex on an odd-dimensional manifold is expressed by the boundary η invariant only.Here we introduce a manifold R × X and t as the coordinate in the extra dimension. Then weconsider the following operator¯ D = (cid:32) ( D − m ρ Γ S ) + ∂ t ( D − m ρ Γ S ) − ∂ t (cid:33) , ρ = (cid:40) [ , + ∞ ] × X + − . (5.2)Note that ρ = κ at t = +
1, and ρ = − t = − x
Schematic picture ofthe manifold R × X . The hori-zontal axis represents the origi-nal four-dimensional X , and X + is shown in the x > Let us evaluate the index of ¯ D in two different ways. First,we use the theorem 2) taking the large m limit, then the zeroeigenmodes of ¯ D are localized on a slice, which consists of X + at t = [ , ∞ ) × Y (solid lines in Fig. 1). Us-ing the theorem 1) we can cut off the cylinder part and obtainInd ( ¯ D ) = Ind
APS ( D | X + ) . The second evaluation starts from thetheorem 1) to cut down R × X to [ − , ] × X , putting the APSboundary condition on t = ± ( ¯ D ) = Ind
APS ( ¯ D | [ − , ] × X ) = − η ( D − m κ Γ S ) − η ( D + m Γ S ) , which is the right-hand side of Eq. (5.1). Namely, Eq. (5.1) al-ways holds since its both sides are just two different expressionsof the same index Ind ( ¯ D ) .
6. Summary and discussion
We have shown that the η invariant of the domain-wallfermion Dirac operator gives a physicist-friendly reformulationof the APS index. Using a Dirac operator in a higher dimen-sions, we have given a mathematical proof that this reformula-tion is valid on any even-dimensional curved manifold. As thedomain-wall fermion shares similar properties to those of topo-logical insulators, we believe that our work gives a mathematical foundation to describe the bulk-edge correspondence of topological matters. 5 physicist-friendly APS index Hidenori Fukaya
The proof of the equivalence used Ind ( ¯ D ) on R × X or equivalently, Ind APS ( ¯ D ) on [ − , ] × X .An interesting extension is to express this index by an η invariant again,Ind APS ( ¯ D | [ − , ] × X ) = − η ( ¯ D − ¯ m ¯ κ ¯ Γ ) reg . , (6.1)where we have introduced a second mass term ¯ m ¯ Γ = diag ( ¯ m I S , − ¯ m I S ) , with I an identity operatoron S , and ¯ κ taking ¯ κ = t ∈ [ − , ] , and −
1, otherwise. Then the original edge mode localizedin 2 n − Y becomes the edge-of-edge states of ¯ D − ¯ m ¯ κ ¯ Γ , which is localizedat the junction of the first and second domain-walls. This recursive structure might be useful forthe physics of higher order topological insulators.The authors thank the organizers of the workshop Progress in the Mathematics of TopologicalStates of Matter, which triggered our collaboration. This work was supported in part by JSPS KAK-ENHI (Grant numbers: JP15K05054, JP17H06461, JP17K14186, JP18H01216, JP18H04484,JP18K03620, and 19J22404). References [1] M. F. Atiyah and I. M. Singer, Annals Math. , 484 (1968) doi:10.2307/1970715.[2] M. F. Atiyah, V. K. Patodi and I. M. Singer, Math. Proc. Cambridge Phil. Soc. , 43 (1975)doi:10.1017/S0305004100049410; Math. Proc. Cambridge Phil. Soc. , 405 (1976)doi:10.1017/S0305004100051872; Math. Proc. Cambridge Phil. Soc. , 71 (1976)doi:10.1017/S0305004100052105.[3] Y. Hatsugai, Phys. Rev. Lett. , no. 22, 3697 (1993) doi:10.1103/PhysRevLett.71.3697; Phys. Rev. B , no. 16, 11851 (1993) doi:10.1103/PhysRevB.48.11851.[4] E. Witten, Rev. Mod. Phys. , no. 3, 035001 (2016) doi:10.1103/RevModPhys.88.035001.[5] H. Fukaya, T. Onogi and S. Yamaguchi, Phys. Rev. D , no. 12, 125004 (2017)doi:10.1103/PhysRevD.96.125004.[6] R. Jackiw and C. Rebbi, Phys. Rev. D , 3398 (1976).[7] C. G. Callan, Jr. and J. A. Harvey, Nucl. Phys. B , 427 (1985).[8] D. B. Kaplan, Phys. Lett. B , 342 (1992) [hep-lat/9206013].[9] H. Fukaya, M. Furuta, S. Matsuo, T. Onogi, S. Yamaguchi and M. Yamashita, arXiv:1910.01987[math.DG].[10] H. Fukaya, N. Kawai, Y. Matsuki, M. Mori, K. Nakayama, T. Onogi and S. Yamaguchi,arXiv:1910.09675 [hep-lat].[11] H. Fukaya, N. Kawai, Y. Matsuki, M. Mori, K. Nakayama, T. Onogi and S. Yamaguchi, in theseproceedings.[12] E. Witten and K. Yonekura, arXiv:1909.08775 [hep-th].[13] E. Witten, J. Diff. Geom. , no. 4, 661 (1982).[14] M. Furuta, “Index Theorem 1,” Amer Mathematical Society, ISBN-10: 0821820974.[15] Hidenori Fukaya, SUURI-KAGAKU, Vol.58-1, pp.44-50, SAIENSU-SHA CO.,LTD., 2020., no. 4, 661 (1982).[14] M. Furuta, “Index Theorem 1,” Amer Mathematical Society, ISBN-10: 0821820974.[15] Hidenori Fukaya, SUURI-KAGAKU, Vol.58-1, pp.44-50, SAIENSU-SHA CO.,LTD., 2020.