Dirac Operators, APS Boundary Conditions, and Spectral Flow on a Finite Warped Cylinder

Dirac Op erators, APS Boundary Conditions, and Sp ectral Flo w on a Finite W arp ed Cylinder T aro Kim ura ∗ , Sanc hita Sharma ∗ Abstract W e study the Dirac operator on a finite w arp ed cylinder coupled to a bac kground U (1) gauge field. W e iden tify the in trinsic endpoint op erators defining the Atiy ah–Patodi–Singer (APS) b oundary condition and deriv e a determinant characterization of the modewise APS sp ectrum. In the constan t-gauge, inv ertible setting, the endp oin t reduced η con tributions cancel, so the APS index v anishes. F or smooth gauge families, the APS pro jector becomes discon tinuous when a b oundary mo de crosses zero. W e therefore introduce a regularized APS-type family of self-adjoin t endp oin t conditions that remains contin uous across such crossings. This regularized family admits a real-symplectic boundary formulation within the standard spectral-flow/Maslo v framew ork: for nondegenerate regularization, the zero-mode set coincides with the boundary- zero set, and transverse b oundary zeros give isolated regular crossings. Con ten ts 1 In tro duction and Overview 2 2 Dirac op erator on a finite t wo-dimensional w arp ed manifold 4 2.1 Spinor bundle and Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Coupling to a background U (1) gauge field . . . . . . . . . . . . . . . . . . . . . . . 6 3 Bulk sp ectrum 7 3.1 Reduction to Heun type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 APS b oundary conditions on the w arp ed cylinder . . . . . . . . . . . . . . . . . . . . 9 3.3 Bulk APS sp ectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 The APS b oundary correction: η -in v arian t and cancellation 15 4.1 The b oundary η -inv arian t and the reduced in v arian t ξ . . . . . . . . . . . . . . . . . 15 4.2 F redholm c hiral APS problem and the APS index . . . . . . . . . . . . . . . . . . . . 17 5 Regularized b oundary families, Maslov index, and zero-mo de crossings 20 5.1 Boundary zeros and regularized endpoint conditions . . . . . . . . . . . . . . . . . . 21 5.2 Real formulation, b oundary form, and Lagrangians . . . . . . . . . . . . . . . . . . . 23 5.3 The regularized b oundary and the Maslo v formalism . . . . . . . . . . . . . . . . . . 25 5.4 Explicit zero-mo de criterion and proof of the main theorem . . . . . . . . . . . . . . 26 5.5 Three explicit A ( s )-paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.6 Numerical plots: branch tracking and crossings . . . . . . . . . . . . . . . . . . . . . 29 ∗ Univ ersit´ e Bourgogne Europ e, CNRS, IMB UMR5584, Dijon, F rance 1 A Heun reduction of the radial Dirac equation 30 A.1 Decoupled radial equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A.2 Liouville transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A.3 Exponential change of v ariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.4 The z -equation and its singular p oin ts . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.5 F rob enius analysis at z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.6 F rob enius analysis at z = ± i √ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A.7 F rob enius analysis at z = ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A.8 Heun classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 A.9 Role of the Heun reduction in the sp ectral analysis . . . . . . . . . . . . . . . . . . . 34 B Complement: Gorokho vsky–Lesch gauge-conjugation picture 34 C Numerical implemen tation and plots 36 1 In tro duction and Ov erview In the context of Dirac-type operators on manifolds with b oundary , the A tiyah-P ato di-Singer (APS) b oundary condition [ 1 , 2 , 3 ] is a fundamen tal elliptic b oundary condition that has far-reac hing implications for v arious fields, including ph ysics and mathematics. It is defined from the sp ectral decomp osition of the induced self-adjoint b oundary Dirac op erator and enters b oth the APS index theorem and later sp ectral-flo w and Maslov-index formulations; see, for example, [ 4 , 5 , 6 , 7 , 8 ]. Recen tly , APS-t yp e phenomena hav e also been revisited from the ph ysical, bulk-b oundary , and domain-wall viewpoints. In particular, F uk a ya, Onogi, and Y amaguchi sho w ed that the APS index can be reco vered from a domain-w all Dirac operator in a ph ysically natural setup [ 9 ], and this p erspective was subsequen tly dev elop ed further in a more systematic mathem atical form in [ 10 ]; see also F uk ay a’s review [ 11 ] for a broader ov erview of massive-fermion and domain-w all reformulations of index theory . Related bulk-boundary and anomaly-inflow interpretations app ear in the w ork of Witten and Y onekura [ 12 ] and of Kobay ashi and Y onekura [ 13 ]. In a tw o-dimensional setting, close in spirit to the presen t pap er, Onogi and Y o da related the reform ulated APS index to Berry-phase and bulk-b oundary con tributions in a domain-wall picture [ 14 ]. Complemen tary lattice studies of curv ed domain-wall fermions by Aoki, F uk ay a, and collab orators are also relev an t to the geometric setting considered here [ 15 , 16 , 17 , 18 ]. More recently , Zh u established a generalized APS/domain- w all formula without assuming the in vertibilit y of the boundary Dirac operator [ 19 ], while Aoki et al. ga ve a lattice/domain-w all realization showing how APS-type index information can b e captured discretely [ 20 ]. This pap er studies the aforementioned structures on a finite w arp ed cylinder coupled to a bac kground U (1) field. In abstract form, these structures are well understo o d [ 4 , 5 , 6 , 7 , 8 ]. In explicit curv ed mo dels, how ev er, the bulk Dirac op erator, the intrinsic b oundary operator, the b oundary η -terms, and the zero-mo de crossing are less often written out simultaneously in a single concrete setting. In this model, we can compute the coupled Dirac op erator explicitly and iden tify the b oundary op erators entering the APS pro jector [ 7 , 21 ]; reduce the bulk spectra to ordinary differen tial equations and related c haracteristic/determinant form ulations [ 22 ]; prov e cancellation of the endp oin t reduced η -inv ariant con tributions in the constan t gauge case under the inv ertibilit y assumption [ 23 , 21 , 24 ]; and form ulate the parameter-dependent zero-mo de problem in a symplectic b oundary space, in the spirit of sp ectral flow and Maslov index theory [ 25 , 26 , 27 , 24 , 28 , 29 , 30 ]. A t the level of the reduced mo de equations, the presen t model also sits in a broader analytic con- text. After F ourier decomp osition, the warped-cylinder Dirac problem b ecomes a one-dimensional 2 first-order coupled system, and hence, a scalar second-order equation; for the warped function con- sidered here, this equation is of general Heun type. Reductions of Dirac equations in curved or w arp ed geometries to Heun-t yp e equations o ccur in several settings, including quantum-corrected and black-hole-t yp e backgrounds [ 31 , 32 ], one-dimensional Dirac systems with v arying interaction profiles [ 33 ], and related recen t analyses of Dirac systems via Heun equations [ 34 , 35 , 36 ]. Our fo cus, how ever, is the explicit w arp ed-cylinder b oundary-v alue problem and its APS and sp ectral consequences. W e w ork on a finite warped cylinder M = [0 , T ] × S 1 , g = dt 2 + f ( t ) 2 dθ 2 , (1.1) coupled to a bac kground U (1) gauge field. This mo del is simple enough to admit explicit form ulas, y et rich, exhibiting non trivial APS b oundary op erators, mo dewise reduction of the bulk problem, reduced η -inv ariant contributions, and a parameter-dependent zero-mo de crossing problem. F or the warping function f ( t ) = e t + αe − t , the scalar mo de equation is of general Heun type. W e explicitly w ork with the finite warped cylinder coupled to a background U (1) field, and com- pute the bulk Dirac op erator, iden tify the in trinsic endp oint op erators defining the APS b oundary condition, characterize the mo dewise APS sp ectrum b y a b oundary determinan t, and prov e can- cellation of the endp oint reduced η -inv ariant con tributions in the constant gauge case under the in vertibilit y assumption k + A  = 0 for all allow ed mo des. F or parameter-dep enden t gauge families, w e do not claim contin uit y of the APS pro jector across the b oundary-zero set k + A ( s ) = 0. Instead, w e in tro duce a regularized APS-type b oundary family , contin uous across those p oin ts, and use it to place the corresp onding zero-mo de problem into the standard sp ectral-flow/Maslo v formalism. Main results. (i) F or each F ourier mo de k , the coupled Dirac op erator reduces to a one-dimensional first-order system; for warping function f ( t ) = e t + αe − t , the asso ciated scalar second-order equation is of general Heun type. (ii) The in trinsic self-adjoint b oundary op erators for APS condition are identified explicitly . F or m = k + A  = 0, w e characterize the mo dewise APS sp ectrum by a b oundary determinant condition F k ( λ ) = 0. (iii) In the constan t gauge case, assuming k + A  = 0 for all allow ed mo des, the endp oin t reduced η -in v ariant contributions cancel. Th us, ind( D + APS ) = 0 . (1.2) (iv) F or smooth gauge families A = A ( s ), the zero-mo de problem admits a real symplectic b ound- ary form ulation. F or the con tin uous regularized APS-t yp e b oundary family introduced in Section 5 , the corresp onding zero-mo de problem fits the standard sp ectral-flo w/Maslov for- malism. Under the assumption δ  = 2  ( T ) ,  ( T ) = Z T 0 dτ f ( τ ) . The zero-mo de set is exactly the boundary-zero set k + A ( s ) = 0, and isolated regular crossings are precisely the transverse b oundary zeros. Sections 2 - 4 discuss the APS index. Section 5 concerns the regularized APS-type family intro- duced only to study parameter v ariation across the b oundary zeros k + A ( s ) = 0. All spectral-flow and Maslov statements in Section 5 refer to the regularized family . 3 Organization of the pap er. W e organize the pap er as follows. In Section 2 , w e deriv e the explicit Dirac op erator on the warped cylinder, fix conv entions for the spinor bundle and Clifford m ultiplication, and include the coupling to the background U (1) field. In Section 3 , we p erform the F ourier-mo de reduction, derive the decoupled scalar equation, explain the reduction to Heun t yp e for the specific warped function f ( t ) = e t + α e − t , and form ulate the APS b oundary con- dition together with the determinant sp ectral condition. In Section 4 , w e study the c hiral APS index, compute the relev ant reduced η -in v ariants, and prov e the cancellation of the t wo endp oin t con tributions in the constant gauge inv ertible assumption. In Section 5 , we treat one-parameter gauge families, introduce the symplectic boundary formalism and the regularized APS-t yp e La- grangian family , and describ e the resulting Maslo v/sp ectral-flo w framework together with explicit mo del paths A ( s ). The app endices discuss the longer ordinary differential equation reductions and auxiliary computations. Ac kno wledgments. This work was supp orted by EIPHI Graduate School (No. ANR-17-EURE- 0002) and the Bourgogne-F ranc he-Comt ´ e region. 2 Dirac op erator on a finite tw o-dimensional warped manifold In this section, w e deriv e the Dirac op erator on a tw o-dimensional warped cylinder, fix the geometric and Clifford conv entions, and include the coupling to a background U (1) gauge field. Throughout, w e work in the Euclidean signature. Let M = [0 , T ] × S 1 , T > 0 , (2.1) b e equipp ed with co ordinates ( t, θ ), where t ∈ [0 , T ] and θ ∈ [0 , 2 π ). W e endo w M with the warped pro duct metric g = dt 2 + f ( t ) 2 dθ 2 , (2.2) where f ∈ C ∞ ([0 , T ]) satisfies f ( t ) > 0 for all t ∈ [0 , T ]. Th us, g is a smo oth Riemannian metric on M . The b oundary of M is ∂ M = { 0 } × S 1 ⊔ { T } × S 1 . (2.3) W e write Y 0 = { 0 } × S 1 , Y T = { T } × S 1 . (2.4) Hence, M is a compact Riemannian manifold with b oundary . Next, w e fix an orthonormal frame on M . F ollowing standard conv en tions, Greek indices ( µ, ν ) refer to curved co ordinates, whereas Latin indices ( a, b ) refer to the orthonormal frame. Using e a = e a µ dx µ , we choose the orthonormal coframe e 1 = dt, e 2 = f ( t ) dθ , (2.5) with the dual frame e 1 = ∂ t , e 2 = 1 f ( t ) ∂ θ . (2.6) The metric comp onents in the co ordinate basis are recov ered from the vielb ein by g µν = e a µ e b ν δ ab . This frame is adapted to the warped manifold: e 1 is normal to the b oundary , while e 2 is the tangen t to the circle, scaled by the warping factor. W e orien t M by dt ∧ dθ , that is, b y the ordered orthonormal frame ( e 1 , e 2 ). The inw ard unit normal is ∂ t at t = 0 and − ∂ t at t = T . In Section 3.2 , this sign change will induce the corresp onding sign change in the b oundary op erators on Y 0 and Y T used in the APS pro jector. F or brevity , we often write f in place of f ( t ). 4 2.1 Spinor bundle and Dirac op erator Let S → M denote the complex spinor bundle asso ciated with the chosen spin structure on M . The tw o-dimensional complex spinor bundle has rank t wo. Since M = [0 , T ] × S 1 is a cylinder, once a spin structure is fixed, the spinor bundle can b e trivialized in the chosen orthonormal frame. Spinors can therefore b e represented by C 2 -v alued functions, with the choice of spin structure reflected in the p erio dic or anti-perio dic b oundary condition in the θ -direction. W e c ho ose a represen tation of the complex Clifford algebra Cl 2 in C 2 b y Hermitian 2 × 2 matrices γ 1 , γ 2 satisfying { γ a , γ b } = 2 δ ab . (2.7) the asso ciated Hermitian Dirac op erator is, D = i γ a ∇ e a , (2.8) where i = √ − 1. This conv ention is chosen so that the Clifford action of the induced b oundary is defined b elow. c ( X ) = − i γ ( N ) γ ( X ) , X ∈ T ( ∂ M ) , (2.9) is Hermitian. Here, T ( ∂ M ) denotes the tangent v ector bundle along the b oundary of M . W e represen t the Clifford algebra using Pauli matrices, where γ 1 = σ 1 and γ 2 = σ 2 . With the orien tation fixed ab o ve, w e define the chiralit y matrix by γ 3 = iγ 1 γ 2 = − σ 3 . (2.10) Therefore, γ 1 =  0 1 1 0  , γ 2 =  0 − i i 0  , γ 3 =  − 1 0 0 1  . (2.11) The spin connection on S is determined by the Levi-Civita connection of g , with resp ect to the orthonormal frame ( e 1 , e 2 ) introduced ab o ve. The only non-zero co efficien ts of the Levi-Civita connection are ∇ e 2 e 1 = f ′ f e 2 , ∇ e 1 e 2 = − f ′ f e 1 . (2.12) Th us, the only indep enden t non-v anishing spin connection one-form is ω 1 2 = − f ′ f e 2 = − ω 2 1 . (2.13) These spin connection one-forms satisfy Cartan’s first structure equation de a + ω a b ∧ e b = 0. The spinor cov arian t deriv ative on S is denoted by ∇ S . In the chosen trivialization, it is given by ∇ S e a = ∂ e a + 1 2 ω a bc Σ bc , Σ bc = 1 4 [ γ b , γ c ] , (2.14) where ∂ e a denotes directional differentiation along e a , acting comp onen twise on spinors. F or the w arp ed metric under consideration, this gives; ∇ S e 1 = ∂ t , ∇ S e 2 = 1 f ∂ θ − 1 2 f ′ f γ 1 γ 2 . (2.15) 5 F unction spaces and sections. F or a v ector bundle V → M , w e write Γ( V ), L 2 ( M ; V ), and H 1 ( M ; V ) for the spaces of smo oth, square-integrable, and first Sob olev sections of V , defined using the Riemannian volume measure and the Hermitian bundle metric. The same notation applies to eac h b oundary comp onen t Y t 0 . On [0 , T ], w e use the standard scalar Sob olev spaces L 2 ([0 , T ]) and H 1 ([0 , T ]). F or each b oundary comp onen t Y t 0 , let S Y t 0 denote the induced b oundary spinor bundle. W e then write Γ( S Y t 0 ) , L 2 ( Y t 0 ; S Y t 0 ) , H 1 ( Y t 0 ; S Y t 0 ) (2.16) for the corresponding spaces of smo oth, square-in tegrable, and first Sob olev b oundary spinors. W e also write H 1 / 2 ( Y t 0 ; S Y t 0 ) (2.17) for the standard fractional Sob olev trace space on the b oundary comp onen t Y t 0 . Therefore, H 1 / 2 ( Y t 0 ; S Y t 0 ) is the natural target of the trace map H 1 ( M ; S ) − → H 1 / 2 ( Y t 0 ; S Y t 0 ) (2.18) giv en by restriction of spinors to Y t 0 . Chiralit y splitting. The tw o-dimensional spinor bundle splits as follows. S = S + ⊕ S − , (2.19) where, under the chosen trivialization, S + ∼ = M × C  0 1  , S − ∼ = M × C  1 0  . (2.20) S ± are the ± 1-eigenbundles of the chiralit y op erator γ 3 . In our representation, γ 3  1 0  = −  1 0  , γ 3  0 1  = +  0 1  . (2.21) 2.2 Coupling to a background U (1) gauge field Let E → M b e a Hermitian complex line bundle, and fix a global unitary trivialization in whic h the unitary connection takes the form ∇ E = d + iA dθ , (2.22) where A ∈ R is constan t. W e then consider the spinor bundle S ⊗ E coupled to the bac kground gauge field and the corresp onding Dirac op erator D : Γ( S ⊗ E ) → Γ( S ⊗ E ) . (2.23) F rom this p oin t onw ard, all spinor bundles, b oundary spinor bundles, and the asso ciated spaces of sections, Sob olev spaces, and L 2 -spaces are understo od to b e coupled to E . F or brevit y , we suppress E from the notation and contin ue to write S , S ± , and S Y t 0 in place of S ⊗ E , S ± ⊗ E , and S Y t 0 ⊗ E | Y t 0 , resp ectively . In gauge theory notation, the connection one-form is A µ dx µ = A dθ , so that A t = 0 and A θ = A is constant. 6 The Dirac op erator is defined using the tensor pro duct connection. ∇ S ⊗ E = ∇ S ⊗ 1 + 1 ⊗ ∇ E (2.24) on S ⊗ E . Th us, D = iγ a ∇ S ⊗ E e a . (2.25) In this trivialization, one obtains ∇ S ⊗ E e 1 = ∂ t , ∇ S ⊗ E e 2 = 1 f ( ∂ θ + iA ) − f ′ 2 f γ 1 γ 2 . (2.26) Com bining these expressions, we obtain the full Dirac op erator: D = iγ 1  ∂ t + f ′ 2 f  + iγ 2 1 f ( ∂ θ + iA ) . (2.27) With Hermitian Euclidean gamma matrices, D is self-adjoint. F or the APS b oundary condition, ho wev er, the relev ant op erator is not the tangential term appearing in the bulk decomposition, but the intrinsic self-adjoin t b oundary Dirac op erator defined on each b oundary component b elo w. After plugging in the Pauli matrices in the previous equation, we obtain D = i    0 ∂ t + f ′ 2 f − i f ( ∂ θ + iA ) ∂ t + f ′ 2 f + i f ( ∂ θ + iA ) 0    . (2.28) The term f ′ 2 f enco des the effects of warping geometry . Th us, we obtain the Dirac op erator on the w arp ed cylinder coupled to the background gauge field. 3 Bulk sp ectrum W e observ e that the metric and background connection are in v ariant in the θ -direction. Therefore, the op erator decomp oses in to a one-dimensional radial equation parametrized b y the angular mo de k . Remark 3.1. W e allow either spin structure on S 1 : for the p erio dic spin structure, k ∈ Z , whereas for the anti-perio dic spin structure, k ∈ Z + 1 2 . In what follows, k denotes an arbitrary allo wed mo de; the deriv ation is identical in b oth cases. Th us, the bulk sp ectral equation reduces to a mo dewise family of radial equations parametrized b y m = k + A. (3.1) W e no w consider the eigenv alue equation and the asso ciated Hilb ert space of eigenfunctions. D ψ = λψ , λ ∈ R . (3.2) 7 Mo de Hilb ert spaces. F or a fixed F ourier mode k , the natural radial Hilb ert space induced from L 2 ( M ; S ⊗ E ) is H k = L 2 ([0 , T ] , f ( t ) dt ; C 2 ) , (3.3) with the corresp onding Sob olev space W k = H 1 ([0 , T ] , f ( t ) dt ; C 2 ) . (3.4) Accordingly , all mo dewise adjoin tness and self-adjointness statements in this sec tion are understoo d with respect to the weigh ted radial inner pro duct unless an explicit unitary conjugation to the un weigh ted space is made later. W riting ψ ( t, θ ) = e ikθ  u ( t ) v ( t )  , (3.5) w e hav e i    0 ∂ t + f ′ 2 f + k + A f ∂ t + f ′ 2 f − k + A f 0     u ( t ) v ( t )  = λ  u ( t ) v ( t )  , (3.6) leading to coupled differential equations v ′ + f ′ 2 f v + k + A f v = − iλu, (3.7a) u ′ + f ′ 2 f u − k + A f u = − iλv . (3.7b) The coupled system decouples into second-order scalar equations for the tw o spinor comp onents. W riting m = k + A , we obtain for u ( t ) u ′′ + f ′ f u ′ +  f ′′ 2 f − f ′ 2 4 f 2 + mf ′ f 2 − m 2 f 2 + λ 2  u = 0 . (3.8) The corresp onding equation for v ( t ) is obtained by replacing m with − m . 3.1 Reduction to Heun type T o place the resulting second-order equation in to a standard F uchsian form, we perform a sequence of transformations. First, we apply a Liouville transformation, whic h remo ves the first-deriv ative term and rewrites the equation in Schr¨ odinger form. In particular, if the decoupled equation is written as u ′′ ( t ) + P ( t ) u ′ ( t ) + R ( t ) u ( t ) = 0 , (3.9) then setting u ( t ) = e − 1 2 R t P ( s ) ds w ( t ) pro duces w ′′ ( t ) +  R ( t ) − 1 2 P ′ ( t ) − 1 4 P ( t ) 2  w ( t ) = 0 . (3.10) In our case P ( t ) = f ′ ( t ) /f ( t ), hence u ( t ) = f ( t ) − 1 / 2 w ( t ). The explicit effective p otential obtained from ( 3.10 ) is given in App endix A . No w we substitute f ( t ) = e t + αe − t with α > 0, and introduce z = e t . Since f = z 2 + α z , d dt = z d dz , (3.11) 8 on the original real domain t ∈ [0 , T ], we hav e z = e t ∈ [1 , e T ] ⊂ R > 0 . F or singularity analysis, we consider the normalized equation as a complex ODE in the v ariable z . Its co efficien ts are rational functions of z , so the equation is regular on C \ { 0 , ± i √ α } , (3.12) with singular p oints at z ∈ { 0 , ± i √ α, ∞} . (3.13) T o place these four singular p oin ts in a standard configuration, let β = √ α , and then w e apply the M¨ obius transformation x = z − iβ z + iβ , (3.14) whic h maps z ∈ { 0 , ± iβ , ∞} to x ∈ {− 1 , 0 , 1 , ∞} . In the x -v ariable, the equation can b e written in the form W ′′ ( x ) + 2 x x 2 − 1 W ′ ( x ) + Q ( x ) W ( x ) = 0 . (3.15) The explicit expressions for Q ( x ), the Liouville-transformed potential, and the rational form of the z -equation are given in App endix A . Finally , we remov e the double p oles b y factoring off the F rob enius gauge. W ( x ) = x ρ 0 ( x − 1) ρ 1 ( x + 1) ρ − 1 y ( x ) , (3.16) where ρ 0 , ρ ± 1 are c hosen from the indicial equations so that the ( x − a ) − 2 terms cancel for a ∈ { 0 , ± 1 } . The resulting equation for y has four regular singular p oints at x ∈ {− 1 , 0 , 1 , ∞} , so the mo de equation is of general Heun type. W e used this observ ation only as structural motiv ation; the actual sp ectral condition will b e formulated later using the APS b oundary conditions and the asso ciated b oundary determinant. 3.2 APS b oundary conditions on the warped cylinder In the previous subsection, w e reduced the eigenv alue equation D ψ = λψ to a second-order mo de equation. T o obtain a discrete sp ectrum of a finite cylinder, w e need to sp ecify an elliptic b oundary condition. F or this purp ose, we imp ose APS b oundary conditions, which require iden tifying the induced b oundary op erator on each b oundary comp onen t. The b oundary is ∂ M = Y 0 ⊔ Y T with Y 0 = { 0 } × S 1 and Y T = { T } × S 1 . 3.2.1 Normal v ector and b oundary Clifford m ultiplication Let { e 1 , e 2 } b e the orthonormal frame e 1 = ∂ t , e 2 = 1 f ( t ) ∂ θ . F ollowing the conv en tions of [ 6 ], we use the inw ard-p oin ting unit normal N along eac h b oundary comp onen t. A t Y 0 the inw ard unit normal is N = + e 1 , whereas, at Y T the inw ard unit normal is N = − e 1 . W e contin ue to write γ ( · ) for the Hermitian gamma-matrix representation in the chosen orthonormal frame, so that γ ( e 1 ) = γ 1 , γ ( e 2 ) = γ 2 . (3.17) Consequen tly , γ ( N ) = γ 1 at Y 0 , γ ( N ) = − γ 1 at Y T . (3.18) F or the w arp ed metric, the bulk Dirac op erator is not literally of pure pro duct form near the b oundary , b ecause its normal part contains the additional zeroth-order term f ′ ( t ) 2 f ( t ) . (3.19) 9 Consequen tly , we define the APS b oundary op erator in trinsically on each boundary component, fol- lo wing the hypersurface formalism of [ 6 ]. T o define a Hermitian Clifford action along the b oundary , w e set c ( X ) = − i γ ( N ) γ ( X ) , X ∈ T ( ∂ M ) . (3.20) Since X ⊥ N , the Hermitian matrices γ ( N ) and γ ( X ) anticomm ute, so γ ( N ) γ ( X ) is sk ew- Hermitian; hence c ( X ) is Hermitian. Moreo ver, c ( X ) satisfies the Clifford relations for the induced b oundary metric. W e equip L 2 ( Y t 0 ; S Y t 0 ) with the natural L 2 -inner pro duct ⟨ φ, ψ ⟩ L 2 ( Y t 0 ) = Z 2 π 0 ⟨ φ ( θ ) , ψ ( θ ) ⟩ C 2 f ( t 0 ) dθ . (3.21) With resp ect to this inner pro duct, the tangential co v ariant deriv ative is sk ew-adjoint and the induced b oundary Dirac op erator is self-adjoint. In dimension t wo, the in trinsic b oundary Dirac op erator ˜ D Y t 0 is related to the bulk Dirac op erator by the hypersurface identit y , ˜ D Y t 0 ψ = 1 2 H t 0 ψ − γ ( N ) Dψ − ∇ N ψ , (3.22) for spinors ψ obtained by restriction from the bulk. Here H t 0 denotes the mean curv ature of the b oundary comp onen t Y t 0 , computed with resp ect to the chosen inw ard unit normal. In the present mo del, this in trinsic b oundary op erator is iden tified with the self-adjoin t b oundary op erator B t 0 in tro duced b elow, and it is this op erator that enters the APS pro jector. Therefore, for the unit tangent vector U = e 2 | t = t 0 on Y t 0 one obtains c ( U ) = ( + σ 3 , t 0 = 0 , − σ 3 , t 0 = T . (3.23) Crucially , the sign c hanges b et ween the t wo ends b ecause the inw ard normal changes sign. 3.2.2 Self-adjoin t b oundary Dirac op erator With the line bundle E fixed, w e suppress it from the notation here as well. The co v arian t deriv ative along the unit tangent at Y t 0 is ∇ U = 1 f ( t 0 ) ( ∂ θ + iA ) . (3.24) The op erator ∇ U is skew-adjoin t on L 2 ( Y t 0 ; S Y t 0 ), so we in tro duce the Hermitian tangential op erator D t 0 = − i ∇ U = 1 f ( t 0 )  − i∂ θ + A  , (3.25) whic h has a real sp ectrum. W e define the intrinsic b oundary Dirac op erator by B t 0 = c ( U ) D t 0 . (3.26) With the explicit expression for c ( U ) from ( 3.23 ), this b ecomes B 0 = 1 f (0) σ 3  − i∂ θ + A  , B T = − 1 f ( T ) σ 3  − i∂ θ + A  . (3.27) Eac h B t 0 is self-adjoint on L 2 ( Y t 0 ; S Y t 0 ) with the natural domain H 1 ( Y t 0 ; S Y t 0 ). 10 W e obtain the explicit mo de matrices B 0 = m f (0)  1 0 0 − 1  , B T = − m f ( T )  1 0 0 − 1  , (3.28) with m = k + A . In the in vertible case relev ant to the APS discussion below, henceforth w e as sume m  = 0. Therefore, the boundary k ernel is trivial, and b oth B 0 and B T are in vertible. The boundary eigen v alues are real and given by µ k, ± ( t 0 ) = ± m f ( t 0 ) . (3.29) The assignmen t of the p ositiv e eigenv alue to the upp er or low er spinor component is reversed b et w een t = 0 and t = T . 3.2.3 APS b oundary conditions Let P > 0 ( B t 0 ) denote the L 2 -orthogonal pro jection on to the direct sum of eigenspaces of B t 0 with p ositiv e eigenv alues. Since w e are w orking in the case m  = 0, the b oundary op erators are inv ertible and P > 0 ( B t 0 ) = P ≥ 0 ( B t 0 ). The APS b oundary condition is P > 0 ( B 0 )  ψ | t =0  = 0 , P > 0 ( B T )  ψ | t = T  = 0 . (3.30) Since B 0 and B T are diagonal in ( 3.28 ), the APS boundary condition reduces to a single comp onen t constrain t for each mo de m  = 0. Explicitly: • If m > 0, then µ > 0 corresp onds to the upp er comp onen t when t = 0 and to the lo wer comp onen t when t = T , hence u (0) = 0 , v ( T ) = 0 . (3.31) • If m < 0, then µ > 0 corresp onds to the lo wer comp onen t when t = 0 and to the upper comp onen t when t = T , hence v (0) = 0 , u ( T ) = 0 . (3.32) With these b oundary conditions imp osed at b oth ends, each F ourier mo de defines a F redholm b oundary equation for the bulk. The compatibility of the bulk solutions with the APS constraints determines the sp ectral condition and hence the sp ectrum. 3.3 Bulk APS sp ectrum W e in tro duce first-order differen tial op erators A + = ∂ t + f ′ 2 f + m f , A − = ∂ t + f ′ 2 f − m f , (3.33) so that the equations ( 3.7 ) b ecome A + v = − iλu and A − u = − iλv . 11 3.3.1 Decoupling and reconstruction Eliminating v yields the decoupled second-order equation for u , A + A − u + λ 2 u = 0 . (3.34) Similarly , eliminating u yields A − A + v + λ 2 v = 0. F or λ  = 0, the second comp onen t can b e reconstructed from u using ( 3.7 ): v ( t ) = i λ  u ′ ( t ) +  f ′ ( t ) 2 f ( t ) − m f ( t )  u ( t )  , λ  = 0 . (3.35) An y non trivial solution of ( 3.34 ) determines a unique spinor solution of ( 3.7 ) after fixing an ov erall normalization. 3.3.2 APS b oundary conditions and b oundary constrain ts The APS b oundary condition imp oses one scalar constraint at each b oundary comp onen t, and in the present basis, it takes the simple mo de-b y-mo de form: m > 0 : u (0) = 0 , v ( T ) = 0 , m < 0 : v (0) = 0 , u ( T ) = 0 . (3.36) Alternativ ely , there exist linear functionals b 0 , b T ∈ ( C 2 ) ∗ (3.37) suc h that b 0  ψ (0 , λ )  = 0 , b T  ψ ( T , λ )  = 0 . (3.38) With resp ect to the basis of the dual space ( C 2 ) ∗ , these functionals are m > 0 : b 0 = (1 0) , b T = (0 1) , (3.39a) m < 0 : b 0 = (0 1) , b T = (1 0) . (3.39b) Prop osition 3.2 (Mo dewise APS op erator) . Fix an allo wed F ourier mo de k and assume m = k + A  = 0. Let D k = i  0 A + A − 0  (3.40) act on H k = L 2 ([0 , T ] , f ( t ) dt ; C 2 ) with domain Dom( D k, APS ) = (  u v  ∈ W k : u (0) = 0 , v ( T ) = 0  , m > 0 ,  u v  ∈ W k : v (0) = 0 , u ( T ) = 0  , m < 0 . (3.41) Then D k, APS is a self-adjoint op erator on H k with compact resolven t. In particular, its sp ectrum is real, discrete, and of finite multiplicit y . Pr o of. Since f ∈ C ∞ ([0 , T ]) and f ( t ) > 0 on [0 , T ], the w eighted Sob olev space W k = H 1 ([0 , T ] , f ( t ) dt ; C 2 ) (3.42) coincides with H 1 ([0 , T ]; C 2 ) with the same norms. Let χ =  u v  , ν =  p q  ∈ W k . (3.43) 12 A direct integration by parts in the weigh ted inner pro duct on H k giv es ⟨ D k χ, ν ⟩ H k − ⟨ χ, D k ν ⟩ H k = if (0)  v (0) p (0) − u (0) q (0)  − if ( T )  v ( T ) p ( T ) − u ( T ) q ( T )  . (3.44) If m > 0, the APS b oundary conditions are u (0) = 0 , v ( T ) = 0 , (3.45) and the b oundary form v anishes on Dom( D k, APS ). If m < 0, the APS b oundary conditions are v (0) = 0 , u ( T ) = 0 , and again the b oundary form v anishes on Dom( D k, APS ). Thus D k, APS is symmetric. The asso ciated b oundary trace space is a t wo-dimensional subspace of C 2 ⊕ C 2 , and in eac h case, it is maximal isotropic for the full b oundary form ab ov e. Therefore, the symmetric op erator is self-adjoint; see, for example, [ 7 , 8 ]. Finally , on the compact in terv al [0 , T ], the graph norm of D k, APS is equiv alent to the H 1 -norm on the domain, and the embedding H 1 ([0 , T ]; C 2 )  → L 2 ([0 , T ]; C 2 ) (3.46) is compact. Hence D k, APS has compact resolven t. The sp ectral conclusions follo w. 3.3.3 Determinan t formulation of the sp ectral condition Let ψ 1 ( t, λ ) and ψ 2 ( t, λ ) b e tw o linearly indep enden t solutions of ( 3.7 ) on [0 , T ]. Then an y solution can b e written in the form ψ ( t, λ ) = C 1 ψ 1 ( t, λ ) + C 2 ψ 2 ( t, λ ) . (3.47) Imp osing the tw o APS constrain ts gives a homogeneous linear system for ( C 1 , C 2 ), and a nontrivial solution exists if and only if the asso ciated 2 × 2 boundary matrix is singular. Accordingly , w e define F k ( λ ) = det  b 0 ψ 1 (0 , λ ) b 0 ψ 2 (0 , λ ) b T ψ 1 ( T , λ ) b T ψ 2 ( T , λ )  . (3.48) Prop osition 3.3 (Determinant c haracterization of the mo dewise APS sp ectrum) . Fix an allow ed F ourier mo de k and assume m = k + A  = 0. Then λ ∈ Spec APS ( k ) ⇐ ⇒ F k ( λ ) = 0 . (3.49) Moreo ver, the zero set of F k is indep endent of the chosen fundamental pair ( ψ 1 , ψ 2 ). Pr o of. An y solution of ( 3.7 ) can b e written uniquely as ψ ( t, λ ) = C 1 ψ 1 ( t, λ ) + C 2 ψ 2 ( t, λ ) . (3.50) Imp osing the APS boundary conditions at t = 0 and t = T gives a homogeneous linear system in ( C 1 , C 2 ). A nontrivial solution exists if and only if the asso ciated b oundary matrix is singular, whic h is exactly the condition F k ( λ ) = 0. If another fundamental pair is obtained by ( e ψ 1 , e ψ 2 ) = ( ψ 1 , ψ 2 ) M ( λ ) , M ( λ ) ∈ GL 2 ( C ) , (3.51) then e F k ( λ ) = det( M ( λ )) F k ( λ ) , (3.52) so the zero set is unchanged. 13 λ F k ( λ ) Figure 1: Mo dewise c haracteristic function F k ( λ ) for the APS boundary problem at α = 1, A = 0 . 3, k = 1, and T = 1 . 5. The zeros of F k ( λ ) are the APS eigenv alues in this mo de. In comp onents, this reads: m > 0 : F k ( λ ) = u 1 (0 , λ ) v 2 ( T , λ ) − u 2 (0 , λ ) v 1 ( T , λ ) , (3.53a) m < 0 : F k ( λ ) = v 1 (0 , λ ) u 2 ( T , λ ) − v 2 (0 , λ ) u 1 ( T , λ ) . (3.53b) 3.3.4 Bulk Sp ectrum F or a fixed F ourier mo de k , let m = k + A and consider the reduced first-order Dirac system ( 3.7 ). T o determine the APS spectrum n umerically , w e solv e this system with the left APS boundary condition imp osed at t = 0 and define a scalar terminal residual at t = T . When m > 0 w e imp ose the left APS condition u (0) = 0 and define F k ( λ ) = v ( T ; λ ) , (3.54) whereas when m < 0 we imp ose the left APS condition v (0) = 0 and define F k ( λ ) = u ( T ; λ ) . (3.55) Th us, F k ( λ ) = 0 is exactly the remaining APS b oundary condition, so the zeros of F k coincide with the APS eigenv alues in mo de k . The plotted curv e in Figure 1 is therefore not itself an eigenv alue branc h, but a mo dewise characteristic function whose zero set is the sp ectrum. Prop osition 3.4 (Absence of zero mo des in the in vertible case) . Fix an allo wed F ourier mo de k and assume m = k + A  = 0. Then λ = 0 is not an APS eigenv alue for the k -th mo de. Pr o of. F or λ = 0, the system ( 3.7 ) reduces to A + v = 0 , A − u = 0 . (3.56) 14 Hence v ( t ) = C 1 exp  − Z t 0  f ′ 2 f + m f  ds  , u ( t ) = C 2 exp  − Z t 0  f ′ 2 f − m f  ds  . (3.57) If m > 0, the APS conditions are u (0) = 0 and v ( T ) = 0, whic h force C 2 = 0 and C 1 = 0. If m < 0, the APS conditions are v (0) = 0 and u ( T ) = 0, which again force C 1 = C 2 = 0. Therefore, the only solution is the trivial solution, so λ = 0 is not an APS eigenv alue. 4 The APS b oundary correction: η -in v arian t and cancellation So far, we hav e considered the self-adjoint APS eigenv alue equation for the full Dirac op erator D , whose mo dewise APS conditions constrain b oth comp onen ts of the spinor. Now we switch to the F redholm APS problem for the chiral op erator D + . This shift is natural b ecause the APS index theorem is form ulated for the chiral Dirac op erator, whereas the previous section concerned the self-adjoin t sp ectral problem for the full Dirac op erator. D + : Γ( S + ) → Γ( S − ) . (4.1) F or the chiral Dirac op erator, the APS pro jector is imp osed only on the S + b oundary , while the complemen tary b oundary condition app ears in the adjoint problem for D − . By Section 3.2.1 , the unit normal N is taken to b e inw ard-p oin ting along each b oundary com- p onen t, and all b oundary operators and APS pro jections b elo w are understo od in this conv en tion. As in the h yp ersurface formalism abov e, the APS condition is defined from the spectral resolu- tion of the intrinsic self-adjoin t boundary Dirac op erator on eac h boundary component. In the w arp ed-cylinder mo del, these are the op erators B 0 and B T in ( 3.27 ). By Remark 3.1 , the allow ed F ourier mo des are K = Z in the p erio dic case, K = Z + 1 2 in the anti-perio dic case. (4.2) 4.1 The b oundary η -inv arian t and the reduced in v arian t ξ Let B b e a self-adjoin t first-order elliptic op erator on a closed manifold. F or ℜ ( s ) ≫ 1, its η -function is defined by η ( B , s ) = X µ ∈ Spec( B ) \{ 0 } sign( µ ) | µ | − s . (4.3) It admits a meromorphic contin uation to s ∈ C which is regular at s = 0. W e write η ( B , 0) for its v alue at s = 0. In APS index theory , the b oundary correction is expressed in terms of the reduced inv ariant ξ ( B ) = η ( B , 0) + h ( B ) 2 , h ( B ) = dim ker B . (4.4) When B is inv ertible, one has h ( B ) = 0, hence, ξ ( B ) = η ( B , 0) 2 . (4.5) W e begin with the η -in v ariant computation for the circle op erator and then return to the w arp ed cylinder to show cancellation of the tw o APS b oundary contributions when the gauge is constan t. 15 Circle mo del. The intrinsic b oundary Dirac op erator is, B A e ikθ = ( k + A ) e ikθ , k ∈ K . (4.6) Assuming k + A  = 0 for ev ery k ∈ K , (4.7) equiv alently 0 / ∈ Sp ec( B A ). W e now justify the reduction to a unique parameter ρ ∈ (0 , 1). The op en in terv al is forced by the inv ertibilit y assumption 0 / ∈ Sp ec( B A ), since ρ = 0 or ρ = 1 would place 0 in the sp ectrum in the representation ( 4.13 ). If K = Z , write A =  + ρ,  ∈ Z , ρ ∈ (0 , 1) . (4.8) Then Sp ec( B A ) = { n + A : n ∈ Z } = { n +  + ρ : n ∈ Z } = { m + ρ : m ∈ Z } , (4.9) after relab elling m = n +  . If K = Z + 1 2 , then Sp ec( B A ) = n n +  A + 1 2  : n ∈ Z o . (4.10) Since 0 / ∈ Sp ec( B A ), the num b er A + 1 2 is not an integer. Hence, there is a unique decomp osition A + 1 2 =  + ρ,  ∈ Z , ρ ∈ (0 , 1) , (4.11) and therefore Sp ec( B A ) = { m + ρ : m ∈ Z } , (4.12) after relab elling the F ourier index. Thus, in either spin structure, there is a unique ρ ∈ (0 , 1) such that Sp ec( B A ) = { n + ρ : n ∈ Z } . (4.13) No w we compute the η -function explicitly . Using ( 4.13 ), for ℜ ( s ) > 1 one has absolute conv er- gence and can split the p ositiv e and negative parts of the sp ectrum: η ( B A , s ) = X n + ρ> 0 ( n + ρ ) − s − X n + ρ< 0 | n + ρ | − s = ∞ X n =0 ( n + ρ ) − s − ∞ X n =0 ( n + 1 − ρ ) − s . (4.14) Indeed, the negative eigenv alues are precisely − ( n + 1 − ρ ) , n = 0 , 1 , 2 , . . . . (4.15) Therefore, for ℜ ( s ) > 1, η ( B A , s ) = ζ ( s, ρ ) − ζ ( s, 1 − ρ ) , (4.16) where ζ ( s, a ) denotes the Hurwitz zeta function. Each Hurwitz zeta function admits a meromorphic con tinuation to all s ∈ C , with a simple p ole at s = 1 of residue 1 [ 37 ]. Hence b oth ζ ( s, ρ ) and ζ ( s, 1 − ρ ) ha ve the same p ole at s = 1. It follo ws that in the difference ( 4.16 ) the p ole parts cancel. Hence η ( B A , s ) admits a meromorphic contin uation to all s ∈ C and is regular at s = 0. Ev aluating at s = 0 and using the standard formula ζ (0 , a ) = 1 2 − a, a ∈ (0 , 1) , (4.17) 16 one obtains η ( B A , 0) =  1 2 − ρ  −  1 2 − (1 − ρ )  = 1 − 2 ρ, (4.18) and therefore ξ ( B A ) = 1 − 2 ρ 2 . (4.19) Relation to the actual endp oin t op erators. The op erator B A ab o v e is a normalized op erator on the scalar b oundary mo de space o ver S 1 . After identifying the b oundary comp onents Y 0 and Y T with S 1 via the co ordinate θ , the actual p ositiv e-chiralit y endp oin t op erators are B + 0 = − 1 f (0) B A , B + T = + 1 f ( T ) B A . (4.20) Since f (0) , f ( T ) > 0, p ositiv e rescaling do es not change the reduced η -in v ariant: η ( cB , 0) = η ( B , 0) , ξ ( cB ) = ξ ( B ) ( c > 0) , (4.21) pro vided B is inv ertible. Thus, only the sign matters, and therefore ξ ( B + 0 ) = − ξ ( B A ) , ξ ( B + T ) = ξ ( B A ) . (4.22) This is the mechanism behind the cancellation of the t wo APS boundary contributions on the finite cylinder. 4.2 F redholm chiral APS problem and the APS index W e no w return to the chiral APS problem on the finite warped cylinder. M = [0 , T ] × S 1 . (4.23) Let D + : Γ( S + ) → Γ( S − ) denote the p ositiv e-chiralit y part of the Dirac op erator. Because the metric and connection are S 1 -in v ariant, b oth D + and the induced tangential b oundary op erators are diagonal in F ourier mo des. A spinor may b e decomp osed into F ourier mo des as ψ ( t, θ ) = e ikθ  u ( t ) v ( t )  , k ∈ K . (4.24) In our matrix con ven tion, S + is represented by the second comp onen t v , while S − is represented b y the first comp onen t u . The Dirac op erator ( 2.28 ) takes the form, D = i  0 A + A − 0  , A ± = ∂ t + f ′ ( t ) 2 f ( t ) ± m f ( t ) , m = k + A. (4.25) Th us, D + = i A + : Γ( S + ) → Γ( S − ) , D − = i A − : Γ( S − ) → Γ( S + ) . (4.26) Let B + 0 and B + T denote the tangential op erators induced by D + at the b oundary comp onents t = 0 and t = T , resp ectiv ely . Similarly , let B − 0 and B − T denote the tangential op erators induced b y D − . Under natural identification of both b oundary circles with S 1 , the change in boundary orien tation together with the rescaling of the tangential metric implies that the endp oin t op erators differ by a sign and a p ositiv e constan t factor. The op erators B t 0 in tro duced earlier act on the full b oundary spinor bundle S Y t 0 = S + Y t 0 ⊕ S − Y t 0 . (4.27) In the chiral APS problem, w e write B ± t 0 for the restrictions of B t 0 to the corresp onding chiral summands. 17 Chiral decomp osition of the b oundary op erator. Since S Y t 0 = S − Y t 0 ⊕ S + Y t 0 , (4.28) and in our con ven tion S − is the upper comp onent while S + is the lo wer comp onen t, the full b oundary op erator decomp oses diagonally as B t 0 =  B − t 0 0 0 B + t 0  . (4.29) Th us B ± t 0 are simply the restrictions of B t 0 to the corresp onding chiral summands. In particular, the mo dewise eigenv alues of B ± t 0 are read off directly from ( 3.28 ), and therefore B + 0 ( k ) = − m f (0) , B + T ( k ) = + m f ( T ) , B − 0 ( k ) = + m f (0) , B − T ( k ) = − m f ( T ) . (4.30) Here U : L 2 ( Y 0 ; S Y 0 ) − → L 2 ( Y T ; S Y T ) (4.31) denotes the unitary induced b y the identification of b oth b oundary comp onen ts with S 1 via the co ordinate θ ; in the chosen trivialization, it acts by ( U φ )( θ ) = φ ( θ ) . (4.32) Hence, after iden tifying the tw o b oundary Hilb ert spaces by the unitary map U , the endp oin t op erators satisfy B ± T = − c U B ± 0 U − 1 , c = f (0) f ( T ) > 0 . (4.33) Global APS domain. The APS chiral op erator is, D + APS : Dom( D + APS ) ⊂ H 1 ( M ; S + ) − → L 2 ( M ; S − ) , (4.34) with domain Dom( D + APS ) = n ψ ∈ H 1 ( M ; S + ) : P > 0 ( B + 0 )  ψ | t =0  = 0 , P > 0 ( B + T )  ψ | t = T  = 0 o . (4.35) APS theory implies that the op erator D + APS is F redholm. F or the p ositiv e-c hirality k th F ourier mo de, we write H + k = L 2 ([0 , T ] , f ( t ) dt ; C ) , W + k = H 1 ([0 , T ] , f ( t ) dt ; C ) . (4.36) Mo dewise description. Due to S 1 -in v ariance, D + APS decomp oses into F ourier modes. On the k -th mo de, one obtains the scalar op erator D + APS ,k = i A + : Dom( D + APS ,k ) ⊂ W + k → H + k , (4.37) acting on the v -comp onen t. F or m  = 0, the APS b oundary condition b ecomes, m > 0 : v ( T ) = 0 (no condition at 0) , m < 0 : v (0) = 0 (no condition at T ) . (4.38) The adjoint APS conditions for D − imp ose the complementary endp oin t conditions on u : m > 0 : u (0) = 0 , m < 0 : u ( T ) = 0 . (4.39) Th us, the c hiral APS conditions are compatible with the tw o-comp onen t self-adjoint op erator used in Section 3.3.2 ; mo dewise, they imp ose complementary endp oin t conditions on u and v . 18 4.2.1 APS index form ula and cancellation for constan t gauge APS index formula. By the Atiy ah-P ato di-Singer index theorem for Dirac op erators on mani- folds with boundary (see [ 1 , Theorem 4.2]; for bac kground on con ven tions, see also [ 4 , 5 ]), the index is given by ind( D + APS ) = i 2 π Z M F ∇ E − ξ ( B + 0 ) − ξ ( B + T ) . (4.40) In constant gauge, the connection is flat, so F ∇ E = 0. Geometric relation betw een the t wo endp oin t op erators. Equation ( 4.33 ) shows that, after identifying the tw o b oundary circles, the tangen tial op erators at t = 0 and t = T are related b y a sign and a p ositiv e scale factor: B ± T = − c U B ± 0 U − 1 , c = f (0) f ( T ) > 0 . (4.41) In particular, B + T is unitarily equiv alen t to − c B + 0 . Lemma 4.1 (Sign/scale flip implies ξ -cancellation) . Let B b e a self-adjoint elliptic op erator with discrete sp ectrum, and assume that B is inv ertible. If c > 0, then η ( cB , 0) = η ( B , 0) , η ( − B , 0) = − η ( B , 0) . (4.42) Consequen tly , if B T = − c U B 0 U − 1 , c > 0 , (4.43) for some unitary U , then ξ ( B T ) = − ξ ( B 0 ) , and hence ξ ( B 0 ) + ξ ( B T ) = 0 . (4.44) Pr o of. When, ℜ ( s ) ≫ 1 η ( cB , s ) = c − s η ( B , s ) , (4.45) so by meromorphic contin uation η ( cB , 0) = η ( B , 0). Similarly , η ( − B , s ) = − η ( B , s ) , (4.46) hence η ( − B , 0) = − η ( B , 0). Unitary conjugation do es not change the sp ectrum, so η ( U B U − 1 , 0) = η ( B , 0) . (4.47) If B is inv ertible, then h ( B ) = 0 and therefore ξ ( B ) = η ( B , 0) / 2. Applying these facts to B T = − c U B 0 U − 1 yields ( 4.44 ). Prop osition 4.2 (V anishing of the APS b oundary correction for constan t gauge) . Assume that k + A  = 0 for every k ∈ K . Then the endp oin t op erators B + 0 and B + T are inv ertible, and ξ ( B + 0 ) + ξ ( B + T ) = 0 . (4.48) Consequen tly , ind( D + APS ) = 0 . (4.49) 19 Pr o of. By assumption, one has m = k + A  = 0 for ev ery k ∈ K , so the mo dewise eigenv alues in ( 4.30 ) never v anish. Th us, b oth B + 0 and B + T are inv ertible. By ( 4.33 ) and Lemma 4.1 , ξ ( B + T ) = − ξ ( B + 0 ) . (4.50) Hence, the APS b oundary correction cancels: ξ ( B + 0 ) + ξ ( B + T ) = 0 . (4.51) Since F ∇ E = 0, the index formula ( 4.40 ) reduces to ind( D + APS ) = 0 . (4.52) Remark 4.3 (W all mo de and parameter v ariation) . When A is constan t, no wall mo de o ccurs precisely when k + A  = 0 for every k ∈ K . The wall case m = 0 b ecomes relev ant when one v aries A (or, more generally , a b oundary family) so that A crosses the lattice of allow ed shifts; dep ending on the spin structure, this lattice is Z or Z + 1 2 . In that situation, 0 ∈ Sp ec( B ) and the APS pro jection is no longer canonical on ker B . W e return to this issue in the discussion of sp ectral flow. Remark 4.4 (No APS b oundary condition at infinit y) . The cancellation ab o ve is a finite-cylinder statemen t: it uses the fact that M has tw o genuine b oundary comp onen ts, and hence tw o APS b oundary op erators B 0 and B T . If one passes formally to the half-infinite cylinder [0 , ∞ ) × S 1 without imp osing additional asymptotic conditions, there is no second APS boundary contribution and therefore no literal “ η -inv ariant at infinity” canceling η ( B 0 , 0). Consequen tly , throughout this pap er, the phrase “ T = ∞ ” is in terpreted only as the large- T limit of the finite APS problem on [0 , T ] × S 1 , not as an APS boundary condition imp osed at infinit y . In particular, for a constan t A under the inv ertible assumption, the APS index v anishes for every finite T , and therefore also in the large- T limit. 5 Regularized b oundary families, Maslo v index, and zero-mo de crossings Our motiv ation for the analysis in this section is to isolate the crossing mec hanism asso ciated with the b oundary-zero set k + A ( s ) = 0 . (5.1) In the t wo-boundary APS setup studied in Sections 2 - 4 , the endp oin t reduced η -in v ariant con tri- butions cancel in the constant gauge inv ertible case, so the APS index do es not directly provide a con tinuous framework for analyzing these b oundary-zero cross ings. F or this reason, we introduce a regularized family of self-adjoin t b oundary conditions whose parameter dep endence is contin uous. This makes the standard sp ectral-flo w/Maslov formalism accessible and enables direct analysis of the zero-mo de crossing set. Assumption. Throughout this section, we fix a F ourier mo de k ∈ K and a smo othing parameter δ > 0. Unless explicitly stated otherwise, all op erators and b oundary conditions b elo w refer to the regularized family D ( δ ) s,k . W e assume endp oin t inv ertibilit y at s = s 1 , s 2 . Whenev er a signed lo cal crossing count is used, we assume the relev ant crossings are isolated and simple. 20 Theorem 5.1 (Zero-mo de criterion for the regularized family) . Fix a mo de k ∈ K and assume δ  = 2  ( T ) ,  ( T ) = Z T 0 dτ f ( τ ) . (5.2) Let { D ( δ ) s,k } s ∈ [ s 1 ,s 2 ] denote the regularized k -mo de op erator family defined in Section 5.3 . Then 0 ∈ Sp ec  D ( δ ) s,k  ⇐ ⇒ k + A ( s ) = 0 . (5.3) Th us, zero mo des of the regularized family o ccur exactly at the b oundary zeros. If moreo ver A ′ ( s ∗ )  = 0 at suc h a p oin t s ∗ , then s ∗ is an isolated regular crossing for the regularized family , and the lo cal sp ectral-flo w/Maslov crossing formalism applies. Corollary 5.2 (Global zero-mo de criterion) . Under the same assumptions, the global regularized family has a zero mo de at parameter s if and only if there exists k ∈ K such that k + A ( s ) = 0 . (5.4) Steps to pro ve Theorem 5.1 First, w e iden tify the b oundary-zero set and define the regu- larized endp oint co efficien ts. Then, we rewrite the mo de equation in a real first-order form and describ e the asso ciated b oundary symplectic space and Lagrangians. Next, we show that the re- sulting regularized b oundary problem defines a contin uous self-adjoint F redholm family to whic h the standard sp ectral-flo w/Maslov corresp ondence applies. Finally , w e compute the zero-mo de matc hing condition explicitly and deduce the zero-mo de criterion and the regularity of transverse crossings. 5.1 Boundary zeros and regularized endp oint conditions The regularization replaces the discon tinuous sign c hoice in the APS b oundary condition by a con- tin uous parameter m ( s, k ) = k + A ( s ). F or m ( s, k )  = 0, the regularized endp oin t family conv erges as δ → 0 to the corresp onding APS choice, while remaining contin uous through m ( s, k ) = 0. Let A = A ( s ) b e a smo oth one-parameter family . The endp oin t b oundary op erators are B s (0) = 1 f (0) σ 3  − i∂ θ + A ( s )  , B s ( T ) = − 1 f ( T ) σ 3  − i∂ θ + A ( s )  , s ∈ [ s 1 , s 2 ] . (5.5) Since these op erators are S 1 -in v ariant, they are diagonal in F ourier mo des. W riting K = ( Z , p eriodic spin structure , Z + 1 2 , anti-perio dic spin structure , m ( s, k ) = k + A ( s ) , k ∈ K , (5.6) their restriction to the k -th F ourier mo de is B s (0)   k = m ( s, k ) f (0)  1 0 0 − 1  , B s ( T )   k = − m ( s, k ) f ( T )  1 0 0 − 1  . (5.7) Hence, the b oundary eigen v alues in mo de k are µ k, ± ( s ; t 0 ) = ± m ( s, k ) f ( t 0 ) , t 0 ∈ { 0 , T } . (5.8) 21 A b oundary-zer o for mo de k is a parameter s ∗ suc h that m ( s ∗ , k ) = k + A ( s ∗ ) = 0 . (5.9) W e assume endp oin t in vertibilit y , 0 / ∈ Sp ec( B s 1 ( t 0 )) , 0 / ∈ Sp ec( B s 2 ( t 0 )) , t 0 ∈ { 0 , T } , (5.10) so no b oundary-zero o ccurs at s = s 1 or s = s 2 . A b oundary-zero s ∗ is called tr ansverse if ∂ s m ( s ∗ , k ) = A ′ ( s ∗ )  = 0 . (5.11) In the present scalar mo dewise setting, every transverse b oundary-zero is simple and hence regular for the lo cal Maslo v crossing form. If A ′ ( s ∗ ) = 0, then the b oundary-zero is nontransv erse; it may b e a genuine sign-changing crossing or merely a touc hing point. A sufficient lo cal criterion for touc hing is m ( s ∗ , k ) = 0 , ∂ s m ( s ∗ , k ) = 0 , ∂ 2 s m ( s ∗ , k )  = 0 , (5.12) equiv alently , k + A ( s ∗ ) = 0 , A ′ ( s ∗ ) = 0 , A ′′ ( s ∗ )  = 0 . (5.13) Because A ( s ) is b ounded on the compact interv al [ s 1 , s 2 ], only finitely many mo des satisfy k + A ( s ) = 0 for some s ∈ [ s 1 , s 2 ]. Fix δ > 0. On the scalar b oundary mo de space ov er S 1 , set M s = − i∂ θ + A ( s ) , (5.14) and defined by functional calculus α 0 ( s ) = tanh  M s δ  , α T ( s ) = − α 0 ( s ) . (5.15) On the k -th mo de, α 0 ( s, k ) = tanh  m ( s, k ) δ  , α T ( s, k ) = − α 0 ( s, k ) . (5.16) Remark 5.3 (Why this regularization is chosen) . The choice α 0 ( s, k ) = tanh  m ( s, k ) δ  , α T ( s, k ) = − α 0 ( s, k ) , (5.17) is a c hoice rather than a canonical one. It gives a smo oth in terp olation b et w een the t wo APS endp oin t choices as the sign of m ( s, k ) changes, and the identit y 1 + tanh z 1 − tanh z = e 2 z (5.18) mak es the zero-mo de matching equation explicitly solv able. 22 5.2 Real form ulation, b oundary form, and Lagrangians W e now pass to a real first-order formulation adapted to the b oundary symplectic form. W rite the original mo de spinor as ψ =  u v  . (5.19) Let U = diag(1 , i ) , (5.20) and define U : L 2 ([0 , T ] , f ( t ) dt ; C 2 ) − → L 2 ([0 , T ] , dt ; C 2 ) , ( U ψ )( t ) = f ( t ) 1 / 2 U ∗ ψ ( t ) . (5.21) If φ = U ψ =  ˜ u w  , ˜ u = f ( t ) 1 / 2 u, w = − i f ( t ) 1 / 2 v , (5.22) then U is unitary . W e relab el ˜ u as u , and write φ =  u w  . (5.23) Conjugating by U transforms the mo de differential expression into D s,k =     0 − ∂ t − m ( s, k ) f ( t ) ∂ t − m ( s, k ) f ( t ) 0     . (5.24) Hence, D s,k φ = λφ ⇐ ⇒ u ′ ( t ) = m ( s, k ) f ( t ) u ( t ) + λ w ( t ) , w ′ ( t ) = − m ( s, k ) f ( t ) w ( t ) − λ u ( t ) , t ∈ [0 , T ] . (5.25) The co efficients are real. In these v ariables, the regularized endp oint conditions are (1 + α 0 ( s, k )) u (0) + (1 − α 0 ( s, k )) w (0) = 0 , (1 + α T ( s, k )) u ( T ) + (1 − α T ( s, k )) w ( T ) = 0 . (5.26) Lemma 5.4 (Real and complex zero mo des) . Fix s , k , and δ > 0, and let D ( δ ) s,k denote the operator asso ciated with ( 5.24 ) and the endp oin t conditions ( 5.26 ). Then, complex conjugation preserves Dom( D ( δ ) s,k ) and commutes with D ( δ ) s,k . Consequen tly , k er C  D ( δ ) s,k  = k er R  D ( δ ) s,k  ⊗ R C . (5.27) In particular, if the real k ernel is one-dimensional, then the complex k ernel is one-dimensional as a complex vector space. Pr o of. The co efficien ts of ( 5.24 ) and the endp oin t relations ( 5.26 ) are real. Therefore, complex conjugation preserves both the domain and the action of the op erator. If x is a complex zero mo de, then its real and imaginary parts are real zero mo des satisfying the same endpoint conditions, which giv es the stated complexification identit y . 23 Let x ( t ) =  u ( t ) w ( t )  , y ( t ) =  p ( t ) q ( t )  , (5.28) with x, y ∈ H 1 ([0 , T ]; R 2 ). In tegration b y parts gives ⟨D s,k x, y ⟩ L 2 − ⟨ x, D s,k y ⟩ L 2 =  u ( t ) q ( t ) − w ( t ) p ( t )  t = T t =0 . (5.29) If J ′ =  0 − 1 1 0  , (5.30) define ω ( ξ , η ) = ⟨ J ′ ξ , η ⟩ R 2 , ξ , η ∈ R 2 . (5.31) On the endp oint trace space H k = R 2 ⊕ R 2 , ( x 0 , x T ) = ( x (0) , x ( T )) , (5.32) set Ω  ( x 0 , x T ) , ( y 0 , y T )  = ω ( x T , y T ) − ω ( x 0 , y 0 ) . (5.33) Then ( 5.29 ) b ecomes ⟨D s,k x, y ⟩ L 2 − ⟨ x, D s,k y ⟩ L 2 = Ω  ( x (0) , x ( T )) , ( y (0) , y ( T ))  . (5.34) Th us, separated self-adjoint b oundary conditions are encoded by Lagrangian subspaces of ( H k , Ω). W e no w define the tw o Lagrangians relev an t for the zero-mo de crossing problem. First, Λ 0 k ( s ) = n x = ( u, w ) T ∈ R 2 : (1 + α 0 ( s, k )) u + (1 − α 0 ( s, k )) w = 0 o , (5.35a) Λ T k ( s ) = n x = ( u, w ) T ∈ R 2 : (1 + α T ( s, k )) u + (1 − α T ( s, k )) w = 0 o , (5.35b) and Λ bc ,k ( s ) = Λ 0 k ( s ) ⊕ Λ T k ( s ) ⊂ H k . (5.36) Lemma 5.5. F or each fixed s and k , the subspace Λ bc ,k ( s ) ⊂ H k is Lagrangian. Pr o of. Eac h of Λ 0 k ( s ) and Λ T k ( s ) is a one-dimensional isotropic subspace of ( R 2 , ω ). Their direct sum is therefore a tw o-dimensional maximal isotropic subspace of ( H k , Ω). Second, at λ = 0 the system ( 5.25 ) decouples: u ′ = m ( s, k ) f ( t ) u, w ′ = − m ( s, k ) f ( t ) w . (5.37) Let  ( t ) = Z t 0 dτ f ( τ ) . (5.38) Then the transfer matrix is x ( T ) = Φ s,k, 0 ( T ) x (0) , Φ s,k, 0 ( T ) =  e m ( s,k ) ℓ ( T ) 0 0 e − m ( s,k ) ℓ ( T )  . (5.39) Since Φ s,k, 0 ( T ) is symplectic with resp ect to ω , its graph Λ s,k (0) = graph  Φ s,k, 0 ( T )  = n ( x 0 , x T ) ∈ H k : x T = Φ s,k, 0 ( T ) x 0 o (5.40) is Lagrangian. 24 Lemma 5.6. F or each fixed s and k , the subspace Λ s,k (0) ⊂ H k is Lagrangian. Pr o of. The graph of a symplectic map is Lagrangian in the pro duct symplectic space ( H k , Ω). Lemma 5.7 (Limit of the regularized endp oin t family) . Fix a mo de k ∈ K and a parameter v alue s such that m ( s, k ) = k + A ( s )  = 0 . (5.41) Then, as δ ↓ 0, the endp oin t lines Λ 0 k ( s ) and Λ T k ( s ) conv erge to the corresp onding APS endp oin t lines: m ( s, k ) > 0 = ⇒ Λ 0 k ( s ) → { u = 0 } , Λ T k ( s ) → { w = 0 } , (5.42) and m ( s, k ) < 0 = ⇒ Λ 0 k ( s ) → { w = 0 } , Λ T k ( s ) → { u = 0 } . (5.43) Pr o of. If m ( s, k )  = 0, then α 0 ( s, k ) = tanh  m ( s, k ) δ  − → sign( m ( s, k )) ( δ ↓ 0) , (5.44) and α T ( s, k ) = − α 0 ( s, k ). Substituting these limits into ( 5.26 ) yields the claim. 5.3 The regularized b oundary and the Maslov formalism F or fixed s , k , and δ > 0, let D ( δ ) s,k denote the op erator asso ciated with ( 5.24 ) on L 2 ([0 , T ] , dt ; C 2 ) and the regularized endp oin t conditions Dom( D ( δ ) s,k ) =  x =  u w  ∈ H 1 ([0 , T ]; C 2 ) : (1 + α 0 ( s, k )) u (0) + (1 − α 0 ( s, k )) w (0) = 0 , (1 + α T ( s, k )) u ( T ) + (1 − α T ( s, k )) w ( T ) = 0  . (5.45) Define the corresp onding op erator in the original mo de v ariables by D ( δ ) s,k = U − 1 D ( δ ) s,k U . (5.46) A t the global level, D ( δ ) s = d M k ∈K D ( δ ) s,k , (5.47) but all arguments b elo w are mo dewise. Prop osition 5.8 (Fixed- s regularized op erator) . Fix k ∈ K , δ > 0, and s ∈ [ s 1 , s 2 ]. Then D ( δ ) s,k is a self-adjoint op erator with separated b oundary conditions asso ciated with the mo de differential expression ( 5.24 ). Pr o of. By ( 5.34 ), the op erator with separated b oundary conditions is symmetric exactly when its endp oin t trace space is isotropic in ( H k , Ω). F or fixed s , the regularized endp oin t conditions define the trace space Λ bc ,k ( s ) = Λ 0 k ( s ) ⊕ Λ T k ( s ) , (5.48) whic h is Lagrangian by Lemma 5.5 . Hence, the op erator is symmetric and maximally isotropic, therefore self-adjoint. Corollary 5.9 (Compact resolv ent) . F or each fixed s ∈ [ s 1 , s 2 ], the op erators D ( δ ) s,k and D ( δ ) s,k ha ve compact resolven t. In particular, their sp ectra are real, discrete, and consist of eigenv alues of finite m ultiplicity . 25 Pr o of. The op erator is first order on the compact interv al [0 , T ] with H 1 -domain and separated homogeneous b oundary conditions. Hence the graph norm is equiv alen t to the H 1 -norm on the domain, and the embedding H 1 ([0 , T ]; C 2 )  → L 2 ([0 , T ]; C 2 ) (5.49) is compact. Unitary equiv alence preserves compactness of the resolven t and the sp ectrum. Lemma 5.10 (Contin uity of the regularized mo de family) . Fix k ∈ K and δ > 0. Then the maps s 7− → Λ bc ,k ( s ) , s 7− → Λ s,k (0) (5.50) are con tinuous in the Lagrangian Grassmannian of ( H k , Ω), and the corresp onding self-adjoin t family s 7− → D ( δ ) s,k (5.51) it is contin uous in the gap top ology . Equiv alently , the family s 7− → D ( δ ) s,k (5.52) is gap-contin uous. Pr o of. The co efficients of ( 5.24 ) dep end contin uously on s through m ( s, k ) = k + A ( s ). The b oundary co efficien ts α 0 ( s, k ) = tanh  m ( s, k ) δ  , α T ( s, k ) = − α 0 ( s, k ) (5.53) therefore v ary con tin uously in s , so Λ bc ,k ( s ) v aries con tin uously . The same is true for the zero-mode propagator Φ s,k, 0 ( T ) and hence for its graph Λ s,k (0). Standard results for separated self-adjoint first-order systems on a compact interv al then yield gap contin uity; see, for example, [ 27 , 8 ]. Corollary 5.11 (Mo dewise sp ectral flo w equals Maslov index) . Fix k ∈ K and δ > 0. Then SF  { D ( δ ) s,k } s ∈ [ s 1 ,s 2 ] ; 0  = µ Maslov  Λ s,k (0) , Λ bc ,k ( s )  . (5.54) Pr o of. By Prop osition 5.8 , Corollary 5.9 , and Lemma 5.10 , the family { D ( δ ) s,k } s ∈ [ s 1 ,s 2 ] is a con tinuous path of self-adjoin t F redholm operators in the standard separated-b oundary setting. The usual sp ectral-flo w/Maslo v corresp ondence therefore applies; see, for example, [ 27 , Theorem 3.14]. Remark 5.12 (Scope) . The spectral-flow/Maslo v corresp ondence is used here only for the con tinu- ous regularized family D ( δ ) s,k . It is not applied to the APS family across b oundary zeros k + A ( s ) = 0, since the APS pro jector is discon tinuous there. 5.4 Explicit zero-mo de criterion and pro of of the main theorem W e no w compute the zero-mo de crossing condition. By the previous subsections, 0 ∈ Sp ec  D ( δ ) s,k  ⇐ ⇒ Λ s,k (0) ∩ Λ bc ,k ( s )  = { 0 } . (5.55) Equiv alently , there exists a nonzero solution of the zero-mo de system satisfying the regularized endp oin t conditions ( 5.26 ). 26 W e rep eatedly use the non-degenerate condition δ  = 2  ( T ) ,  ( T ) = Z T 0 dτ f ( τ ) . (5.56) Cho ose a nonzero spanning v ector for the left endp oin t line Λ 0 k ( s ): v 0 ( s, k ) =  1 − α 0 ( s, k ) − (1 + α 0 ( s, k ))  ∈ Λ 0 k ( s ) . (5.57) Let x ( t ; s, λ, k ) =  u ( t ; s, λ, k ) w ( t ; s, λ, k )  (5.58) denote the unique solution of ( 5.25 ) with initial condition x (0; s, λ, k ) = v 0 ( s, k ) . (5.59) Define the scalar matching function F ( s, λ, k ) = (1 + α T ( s, k )) u ( T ; s, λ, k ) + (1 − α T ( s, k )) w ( T ; s, λ, k ) . (5.60) Then F ( s, λ, k ) = 0 if and only if the corresp onding solution satisfies b oth regularized endp oint conditions. Prop osition 5.13 (Explicit zero-mo de criterion for the regularized family) . Fix a mo de k and write m ( s, k ) = k + A ( s ). F or λ = 0, the matching function ( 5.60 ) is F ( s, 0 , k ) =  1 − tanh m ( s, k ) δ  2 e m ( s,k ) ℓ ( T ) −  1 + tanh m ( s, k ) δ  2 e − m ( s,k ) ℓ ( T ) . (5.61) Equiv alently , F ( s, 0 , k ) = 0 ⇐ ⇒ m ( s, k )   ( T ) − 2 δ  = 0 . (5.62) In particular, under ( 5.56 ), F ( s, 0 , k ) = 0 ⇐ ⇒ m ( s, k ) = 0 . (5.63) Hence, zero mo des of the regularized family o ccur exactly at the b oundary zeros. Pr o of. A t λ = 0, the system decouples: u ( t ) = u (0) e m ( s,k ) ℓ ( t ) , w ( t ) = w (0) e − m ( s,k ) ℓ ( t ) , (5.64) where  ( t ) = Z t 0 dτ f ( τ ) . (5.65) With x (0; s, 0 , k ) = v 0 ( s, k ) =  1 − α 0 ( s, k ) − (1 + α 0 ( s, k ))  , α 0 ( s, k ) = tanh  m ( s, k ) δ  , (5.66) w e get u ( T ) = (1 − α 0 ( s, k )) e m ( s,k ) ℓ ( T ) , w ( T ) = − (1 + α 0 ( s, k )) e − m ( s,k ) ℓ ( T ) . (5.67) 27 Since α T ( s, k ) = − α 0 ( s, k ), substitution into ( 5.60 ) gives ( 5.61 ). Using 1 + tanh z 1 − tanh z = e 2 z , (5.68) the equation F ( s, 0 , k ) = 0 is equiv alent to e 2 m ( s,k ) ℓ ( T ) =  1 + tanh( m ( s, k ) /δ ) 1 − tanh( m ( s, k ) /δ )  2 = e 4 m ( s,k ) /δ , (5.69) whic h is exactly ( 5.62 ). The last statement follo ws from ( 5.56 ). Corollary 5.14 (T ransverse b oundary zeros are isolated regularized crossings) . Assume ( 5.56 ). Let s ∗ b e a b oundary-zero for mo de k , and assume A ′ ( s ∗ )  = 0. Then s ∗ is an isolated simple zero of the scalar function s 7→ F ( s, 0 , k ). More precisely , ∂ s F ( s, 0 , k )   s = s ∗ = 2 A ′ ( s ∗ )   ( T ) − 2 δ   = 0 . (5.70) Pr o of. Differen tiate ( 5.61 ) with resp ect to s , and use m ( s ∗ , k ) = 0 together with ∂ s m ( s ∗ , k ) = A ′ ( s ∗ ). Pr o of of The or em 5.1 . The zero-mo de statement is exactly Prop osition 5.13 . If moreov er A ′ ( s ∗ )  = 0, then Corollary 5.14 sho ws that s ∗ is an isolated regular crossing. Corollary 5.11 then places this crossing in the standard mo dewise sp ectral-flow/Maslo v formalism. Pr o of of Cor ol lary 5.2 . By the orthogonal decomp osition ( 5.47 ), 0 ∈ Sp ec  D ( δ ) s  ⇐ ⇒ ∃ k ∈ K such that 0 ∈ Sp ec  D ( δ ) s,k  . (5.71) The result, therefore, follows from Theorem 5.1 . Remark 5.15. Theorem 5.1 identifies exactly when regularized zero mo des o ccur and when the corresp onding crossings are isolated and regular. W e do not pursue here a general closed sign form ula for arbitrary nontransv erse crossings. 5.5 Three explicit A ( s ) -paths W e illustrate the regularized crossing mechanism for three explicit one-parameter gauge families: • Example 1: A ( s ) = s − 1 2 , s ∈ [0 , 1]. • Example 2: A ( s ) = 3 s − 1 4 , s ∈ [0 , 1]. • Example 3: A ( s ) = sin(4 π s ), s ∈ [ ε, 1 − ε ], with ε = 1 24 . F or these plotted examples, w e restrict to k ∈ Z . Since the global problem decomp oses in to F ourier mo des, the figures displa y the mo dewise branc hes corresp onding to the finite n umber of mo des that can contribute crossings on the chosen parameter in terv al. In each case, the theoretical zero-mo de set is determined by k + A ( s ) = 0 (5.72) 28 0.0 0.2 0.4 0.6 0.8 1.0 - 0.2 - 0.1 0.0 0.1 0.2 s λ ( s ) k = 0 Figure 2: Example 1 ( A ( s ) = s − 1 2 ): track ed branc hes λ ( s, k ) for representativ e modes k . Boundary- zero lo cations are mark ed. under the non-degenerate assumption. The numerical contin uation is illustrative: it visualizes nearb y branches F ( s, λ, k ) = 0 and distinguishes transverse crossings from nontransv erse touc hing p oin ts. In Examples 1 and 2, every interior b oundary-zero is transverse, since A ′ ( s ) = 1 and A ′ ( s ) = 3, resp ectiv ely . In Example 3, non transverse boundary zeros o ccur only in the p eriodic spin structure, for the mo des k = ± 1, at s ∗ ∈  1 8 , 3 8 , 5 8 , 7 8  . (5.73) A t these p oints A ′ ( s ∗ ) = 0, so they are touc hing p oin ts rather than transv erse crossings. The cutoff s ∈ [ ε, 1 − ε ] ensures endp oin t in vertibilit y in Example 3. 5.6 Numerical plots: branc h tracking and crossings Figures 2 - 4 display mo dewise branches s 7→ λ ( s, k ) for selected F ourier mo des k . A zero-mo de crossing o ccurs exactly when a branc h meets λ = 0, equiv alently , when F ( s, λ, k ) = 0 with λ = 0 . (5.74) Th us, the figures provide a numerical visualization of the criterion prov ed ab o ve. F or branc h trac king, we use the Riccati reform ulation of the ratio r = w /u , whic h is n umerically more stable than solving F ( s, λ, k ) = 0 from scratch at eac h step, esp ecially near degenerate configurations such as Example 3. This is numerically equiv alent to the same b oundary-v alue problem with the same endp oin t conditions ( 5.26 ). The figures are pro duced n umerically from the Riccati reformulation, with the same endp oin t conditions as in ( 5.26 ). Boundary zeros m ( s, k ) = 0 are used as canonical starting points for branc h con tinuation, in agreemen t with Prop osition 5.13 . 29 0.0 0.2 0.4 0.6 0.8 1.0 - 0.2 - 0.1 0.0 0.1 0.2 s λ ( s ) k =- 2 k =- 1 k = 0 Figure 3: Example 2 ( A ( s ) = 3 s − 1 4 ): trac ked branc hes λ ( s, k ) for selected mo des. Conclusion of the regularized analysis. Under the non-degenerate assumption, the regular- ized zero mo des o ccur exactly at k + A ( s ) = 0, and transv erse b oundary zeros yield isolated regular crossings to which the standard mo dewise sp ectral-flo w/Maslov formalism applies. A Heun reduction of the radial Dirac equation The main text uses only the following structural facts ab out the radial mo de equation: (i) after decoupling, it b ecomes a second-order scalar ODE, (ii) under the explicit warped function f ( t ) = e t + αe − t its co efficients b ecome rational after z = e t , and (iii) the resulting equation has four r e gular singular p oints , hence b elongs to the general Heun class. W e therefore do not need closed- form Heun solutions; what matters for the sp ectral problem is the singularit y structure and the asso ciated F rob enius exp onents. Throughout, we fix a F ourier mo de k and write. m = k + A, f ( t ) > 0 , (A.1) where f is the warping function, m  = 0 and λ ∈ R is the sp ectral parameter in D ψ = λψ . Our goal in this app endix is to show the Heun reduction in a self-con tained w ay: § A.1 decoupling of the first-order Dirac system to a scalar ODE, § A.2 Liouville substitution to remov e the first- deriv ative term, § A.3 - § A.4 rational z -equation and identification of its singular p oints, § A.5 - § A.7 F rob enius exp onents at z = 0, z = ± i √ α and z = ∞ , § A.8 - § A.9 Summary of the Heun classification and its role in the sho oting formulation used in Section 3.3. The singularit y analysis and extraction of c haracteristic exponents uses the standard F rob enius metho d for second-order linear ODEs with regular singular p oin ts, together with the usual reduction of the p oint z = ∞ via z = 1 /ζ ; see, e.g., Ince [ 38 ]. Finally , the fact that a second-order equation with four regular singularities b elongs to the general Heun class is standard; see Ronv eaux [ 39 ]. 30 0.2 0.4 0.6 0.8 - 0.2 - 0.1 0.0 0.1 0.2 s λ ( s ) k =- 1 k = 0 k = 1 Figure 4: Example 3 ( A ( s ) = sin(4 πs ) on [ ε, 1 − ε ], ε = 1 24 ): trac ked branches for the selected mo des, with endp oin t cutoff ensuring inv ertibility at the ends. A.1 Decoupled radial equation W e b egin b y decoupling the first-order radial Dirac system in to a single scalar equation for u , whic h will serve as the master ODE go verning eac h F ourier mo de. In Section 3.3 w e obtained the coupled system v ′ + f ′ 2 f v + m f v = − iλ u, (A.2a) u ′ + f ′ 2 f u − m f u = − iλ v . (A.2b) Eliminating v yields a second-order equation for u . Differen tiating ( A.2b ) and using ( A.2a ) to substitute for v and v ′ giv es u ′′ + f ′ f u ′ + " λ 2 − m 2 f 2 + mf ′ f 2 + f ′′ 2 f − ( f ′ ) 2 4 f 2 # u = 0 . (A.3) This is the master scalar equation gov erning each F ourier mo de of the bulk sp ectrum. A.2 Liouville transformation In this section, we remo ve the first-deriv ativ e term b y the standard Liouville substitution u = f − 1 / 2 w ; to this end, w e apply the Liouville substitution u ( t ) = f ( t ) − 1 / 2 w ( t ) . (A.4) A direct computation shows that ( A.3 ) is equiv alent to the Sc hr¨ odinger-t yp e equation w ′′ + V ( t ) w = 0 , (A.5) 31 where the effective p oten tial is V ( t ) = λ 2 − m 2 f ( t ) 2 + mf ′ ( t ) f ( t ) 2 . (A.6) In particular, ( A.5 ) isolates the singular structure coming from the w arp factor f . A.3 Exp onen tial c hange of v ariables No w we conv ert the equation in to a rational-co efficien t ODE via z = e t . The w arp factor for our metric is f ( t ) = e t + αe − t ( α > 0) , (A.7) and introduce the exp onen tial co ordinate z = e t , z ∈ (1 , e T ) . (A.8) Then f ( t ) = z + α z = z 2 + α z , d dt = z d dz . (A.9) Under this change of v ariables, ( A.5 ) b ecomes d 2 w dz 2 + 1 z dw dz + 1 z 2 Q ( z ) w = 0 , (A.10) where Q ( z ) = λ 2 − m 2 z 2 ( z 2 + α ) 2 + m ( z 2 − α ) ( z 2 + α ) 2 . (A.11) A.4 The z -equation and its singular p oin ts After the Liouville substitution u = f − 1 / 2 w and the change of v ariable z = e t , the mo de equation tak es the form w z z + 1 z w z + 1 z 2 Q ( z ) w = 0 , (A.12) where Q ( z ) is the rational function display ed in ( A.11 ). The singular p oin ts of ( A.12 ) are exactly the p oin ts where the co efficients fail to b e analytic: z = 0, the zeros of z 2 + α (namely z = ± i √ α ), and z = ∞ . Eac h of these p oin ts is a r e gular singular p oin t (see Section A.5 - A.7 b elow). Hence ( A.12 ) is a F uc hsian second-order equation with four regular singularities, which places it in the (general) Heun class. R emark. W e will not need the explicit canonical Heun parameters in the main text. What matters for our purp oses is the singularity structure and the fact that the sp ectral problem can b e handled via the sho oting form ulation in Section 3.3 without inv oking closed-form Heun solutions. A.5 F rob enius analysis at z = 0 Here, w e discuss the F rob enius exp onents at z = 0, whic h determine the lo cal b eha vior of solutions near the left endp oin t t → −∞ in z = e t co ordinate. As z → 0 w e hav e z 2 + α = α + O ( z 2 ), hence expanding Q ( z ) gives Q ( z ) = λ 2 − m α + O ( z 2 ) . (A.13) 32 Therefore, the leading mo del near z = 0 is w z z + 1 z w z + 1 z 2  λ 2 − m α  w = 0 . (A.14) Substituting a F rob enius ansatz w ∼ z ρ yields the indicial equation ρ ( ρ − 1) + ρ +  λ 2 − m α  = 0 ⇐ ⇒ ρ 2 +  λ 2 − m α  = 0 . (A.15) Hence, the characteristic exp onen ts at z = 0 are ρ ± 0 = ± i r λ 2 − m α when λ 2 − m α > 0 , (A.16) and ρ ± 0 = ± r m α − λ 2 when λ 2 − m α < 0 (A.17) A.6 F rob enius analysis at z = ± i √ α Let z ∗ ∈ { + i √ α, − i √ α } and write z = z ∗ + ζ . Since z 2 + α has a simple zero at z = z ∗ , we hav e z 2 + α = 2 z ∗ ζ + O ( ζ 2 ) , (A.18) W e observ e 1 z 2 Q ( z ) has a double pole at ζ = 0. In particular, each of z = ± i √ α is a r e gular singular p oint . W riting w ∼ ζ ρ and collecting the most singular terms yield an indicial equation of the form ρ ( ρ − 1) + p ∗ ρ + q ∗ = 0 , (A.19) where p ∗ , q ∗ dep end explicitly on ( m, α ) through the principal part of the Lauren t expansion of Q ( z ) at z = z ∗ . W e record the corresp onding c haracteristic exp onen ts as ρ ± ∗ = 1 − p ∗ 2 ± r  1 − p ∗ 2  2 − q ∗ . (A.20) (Explicit expressions for p ∗ , q ∗ can b e obtained b y a direct expansion of Q at z = z ∗ , but are not required for the sp ectral analysis in the main text.) A.7 F rob enius analysis at z = ∞ T o analyze z = ∞ , set z = 1 /ζ and rewrite ( A.12 ) in the v ariable ζ → 0. Since Q ( z ) = λ 2 + O (1 /z 2 ) as z → ∞ , the leading mo del is w z z + 1 z w z + λ 2 z 2 w = 0 , (A.21) whic h sho ws that z = ∞ is again a r e gular singular p oint . The corresp onding asymptotic b ehaviors are w ( z ) ∼ z ± iλ z → ∞ (A.22) This confirms that z = ∞ is again a regular singular p oin t and that solutions ha ve p o w er- la w/oscillatory b eha vior controlled b y λ . 33 A.8 Heun classification Equation ( A.12 ) has four regular singular p oin ts z = 0 , z = + i √ α, z = − i √ α, z = ∞ . (A.23) A second-order F uchsian ODE with four regular singular p oin ts is equiv alent, after a M¨ obius c hange of v ariable and a gauge transformation removing c haracteristic exp onen ts, to a (general) Heun equation. This justifies describing the mo de equation as Heun-typ e and explains wh y closed-form expressions for the bulk sp ectrum are not exp ected in general. A.9 Role of the Heun reduction in the sp ectral analysis The Heun classification is used here only to clarify the analytic complexity of the reduced radial equation. Our sp ectral analysis do es not require explicit Heun solutions: instead, w e use the sho oting formulation in Section 3.3 , where eigen v alues are characterized as zeros of a scalar endp oin t condition. The F rob enius data ab o v e remain useful for (i) c hecking lo cal b ehavior, (ii) designing stable numerical integration, and (iii) dev eloping asymptotic regimes (e.g. large | m | or large T ). B Complemen t: Gorokho vsky–Lesch gauge-conjugation picture A complemen tary gauge-conjugation viewpoint. This discussion is logically indep enden t of the regularized APS/Maslov analysis ab o ve. Its purp ose is to record a complementary gauge- conjugation interpretation for lo c al el liptic self-adjoint b oundary c onditions preserv ed b y the rele- v ant unitary gauge transformation. It should therefore b e read as a parallel viewp oin t, not as a second pro of of the regularized crossing results. Consider a lo cal elliptic self-adjoin t b oundary condition F = ( F 0 , F T ) , F 0 ⊂ H 1 / 2 ( Y 0 ; S Y 0 ) , F T ⊂ H 1 / 2 ( Y T ; S Y T ) , (B.1) suc h that the asso ciated bulk Dirac op erator is self-adjoint and F redholm. W riting γ : H 1 ( M ; S ⊗ E ) → H 1 / 2 ( Y 0 ; S Y 0 ) ⊕ H 1 / 2 ( Y T ; S Y T ) (B.2) for the trace map, we set Dom( D A, F ) = { ψ ∈ H 1 ( M ; S ⊗ E ) : γ ( ψ ) ∈ F 0 ⊕ F T } , (B.3) and denote by D A, F : Dom( D A, F ) ⊂ L 2 ( M ; S ⊗ E ) → L 2 ( M ; S ⊗ E ) (B.4) the resulting self-adjoint F redholm op erator. Let g : S 1 → U (1) (B.5) b e a smo oth unitary , extended trivially in the t -direction to M = [0 , T ] × S 1 , and assume that F is g -inv arian t: g ( F 0 ⊕ F T ) = F 0 ⊕ F T . (B.6) 34 Then multiplication by g preserves Dom( D A, F ). W e fix A 0 ∈ R . The Gorokhovsky-Lesc h gauge-conjugation path is D A 0 , F ( τ ) = D A 0 , F + τ g − 1 [ D A 0 , g ] , τ ∈ [0 , 1] . (B.7) Since g dep ends only on θ , [ D A 0 , g ] = iγ 2 1 f ( t ) ∂ θ g , g − 1 [ D A 0 , g ] = iγ 2 1 f ( t ) g − 1 ∂ θ g , (B.8) so this is a b ounded zeroth-order multiplication op erator. Hence ( B.7 ) is a norm-contin uous path of self-adjoint F redholm op erators with common domain Dom( D A 0 , F ), and D A 0 , F (1) = g − 1 D A 0 , F g . (B.9) Let P A 0 , F = 1 [0 , ∞ ) ( D A 0 , F ) . (B.10) No w we define the T o eplitz op erator T g = P A 0 , F g P A 0 , F : Ran( P A 0 , F ) → Ran( P A 0 , F ) . (B.11) Under the hypotheses ab o ve, T g is F redholm, and the Gorokhovsky-Lesc h theorem giv es SF  { D A 0 , F ( τ ) } τ ∈ [0 , 1]  = ind( T g ); (B.12) see [ 30 ]. W e no w sp ecialize to g N ( θ ) = e iN θ , N ∈ Z , (B.13) whose winding num b er is wind( g N ) = 1 2 π i Z S 1 g − 1 N d g N = N . (B.14) Equiv alently , g N represen ts the class N ∈ K 1 ( S 1 ) ∼ = Z . In our cylinder mo del, g − 1 N ( ∂ θ + iA ) g N = ∂ θ + i ( A + N ) , (B.15) so D A + N = g − 1 N D A g N . (B.16) If F is also g N -in v ariant, then D A + N , F = g − 1 N D A, F g N . (B.17) Therefore, whenever A ( s 2 ) − A ( s 1 ) = N ∈ Z , (B.18) the endp oint op erators D A ( s 1 ) , F and D A ( s 2 ) , F are gauge-equiv alent, and the GL theorem yields SF  { D A ( s 1 ) , F ( τ ) } τ ∈ [0 , 1]  = ind  P A ( s 1 ) , F g N P A ( s 1 ) , F  , (B.19) where P A ( s 1 ) , F = 1 [0 , ∞ ) ( D A ( s 1 ) , F ) . (B.20) In the presen t circle mo del, this index dep ends only on the homotopy class of g N , hence, only on the winding num b er N . The GL construction is a gauge-c onjugation computation for gauge-equiv alent endpoint op- erators under lo cal b oundary conditions. It is therefore complementary to, but not literally the same path as, the parameter family s 7→ D A ( s ) , F . In particular, it applies when the endp oint shift is an in teger and the chosen lo cal elliptic self-adjoint b oundary condition is preserved b y the corresp onding unitary g N . 35 Three explicit examples Example 1: A ( s ) = s − 1 2 , s ∈ [0 , 1] . Here A (1) − A (0) = 1 2 − ( − 1 2 ) = 1 . (B.21) Hence one chooses g 1 ( θ ) = e iθ , wind( g 1 ) = 1 . (B.22) If F is g 1 -in v ariant, then D A (1) , F = g − 1 1 D A (0) , F g 1 , (B.23) and SF  { D A (0) , F ( τ ) } τ ∈ [0 , 1]  = ind  P A (0) , F g 1 P A (0) , F  . (B.24) Example 2: A ( s ) = 3 s − 1 4 , s ∈ [0 , 1] . Here A (1) − A (0) = 11 4 −  − 1 4  = 3 . (B.25) Hence one chooses g 3 ( θ ) = e i 3 θ , wind( g 3 ) = 3 . (B.26) If F is g 3 -in v ariant, then D A (1) , F = g − 1 3 D A (0) , F g 3 , (B.27) and SF  { D A (0) , F ( τ ) } τ ∈ [0 , 1]  = ind  P A (0) , F g 3 P A (0) , F  . (B.28) Example 3: A ( s ) = sin(4 πs ) , s ∈ [ ε, 1 − ε ] , ε = 1 24 . The cutoff excludes endp oint b oundary zeros and therefore guarantees endp oin t inv ertibility . In this case, A (1 − ε ) − A ( ε ) = sin(4 π − 4 π ε ) − sin(4 π ε ) = − 2 sin(4 π ε ) = − 1 . (B.29) Hence one chooses g − 1 ( θ ) = e − iθ , wind( g − 1 ) = − 1 . (B.30) If F is g − 1 -in v ariant, then D A (1 − ε ) , F = g − 1 − 1 D A ( ε ) , F g − 1 , (B.31) and SF  { D A ( ε ) , F ( τ ) } τ ∈ [0 , 1]  = ind  P A ( ε ) , F g − 1 P A ( ε ) , F  . (B.32) C Numerical implemen tation and plots Prop osition C.1 (Liouville/W ronskian identit y) . F or any fundamen tal matrix ψ ( t ) we ha ve f ( t ) det ψ ( t ) = constan t in t. (C.1) Equiv alently , for any tw o solutions ψ (1) = ( u 1 , v 1 ) and ψ (2) = ( u 2 , v 2 ), f ( t )  u 1 ( t ) v 2 ( t ) − u 2 ( t ) v 1 ( t )  (C.2) is t -indep endent. 36 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.000 0.002 0.004 0.006 0.008 t 10^8 | f ( t ) det Φ ( t ) - 2 | 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 0.000 0.002 0.004 0.006 0.008 t 10^8 | f ( t ) det Φ ( t ) - 2 | Figure 5: Numerical illustration of Proposition C.1 for the parameter v alues α = 1, A = 0 . 3, k = 1, and T = 1 . 5, matc hing the setup of Section 3.3.4 , with represen tative sp ectral parameter λ = 2. The plotted quantit y is 10 8 | f ( t ) det ψ ( t ) − 2 | , where ψ ( t ) is the fundamental matrix normalized by ψ (0) = I . Since the exact conserv ed v alue is f (0) = 2, the graph measures only n umerical deviation from the Liouville/W ronskian identit y . Righ t: zo omed view on a shorter subinterv al sho wing the oscillatory structure more clearly . Pr o of. Liouville’s form ula gives d dt det ψ ( t ) = tr  A ( t ; λ )  det ψ ( t ) . (C.3) In our system, tr  A ( t ; λ )  = ( − q + p ) + ( − q − p ) = − 2 q ( t ) = − f ′ ( t ) f ( t ) . (C.4) Th us, d dt log det ψ ( t ) = − f ′ ( t ) f ( t ) , hence det ψ ( t ) = C /f ( t ) and f ( t ) det ψ ( t ) = C is constant. Numerical illustration of Prop osition C.1 . T o complement C.1 , w e plot in Figure 5 the n umerical deviation of the conserved quan tity f ( t ) det ψ ( t ) (C.5) from its exact constant v alue. Here ψ ( t ) is the fundamental matrix of the first-order system, normalized by ψ (0) = I , so the exact constant is f (0) det ψ (0) = f (0) = 2 (C.6) for the parameter choice α = 1. T o match the bulk-sp ectrum example of Section 3.3.4 , we use α = 1 , A = 0 . 3 , k = 1 , T = 1 . 5 , (C.7) and fix the representativ e sp ectral v alue λ = 2. Since the exact quantit y is constant, the plotted oscillations represent only numerical error. F or visibilit y , we therefore plot the scaled error 10 8   f ( t ) det ψ ( t ) − 2   . (C.8) W e also include a zo omed view ov er a shorter subin terv al to displa y the fine oscillatory structure more clearly . 37 References [1] M. F. Atiy ah, V. K. P ato di, and I. M. Singer, “Sp ectral Asymmetry and Riemannian Geometry . I,” Math. Pr o c. Camb. Philos. So c. 77 no. 1, (1975) 43–69 . [2] M. F. Atiy ah, V. K. P ato di, and I. M. Singer, “Sp ectral Asymmetry and Riemannian Geometry . I I,” Math. Pr o c. Camb. Philos. 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