Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Dai–Freed curvature and eta holonomy for the inverse determinant line
Statement
Let be a smooth proper submersion with closed compact even-dimensional fibers and a Spin structure on the vertical tangent. Fix a vertical metric, horizontal distribution and the associated family of chiral complex Spin Dirac operators. Let be the inverse Quillen determinant line with the inverse Bismut–Freed connection. Its curvature is
For a smooth loop , let and write for the ordinary Fredholm index of the chiral operator on the fiber over . This integer is locally constant in the parameter and hence constant along . Let be the mapping torus, put the nonbounding Spin structure on the circle, and use the adiabatic metric . With , the holonomy is
The inverse-line sign and parity factor use Dai–Freed’s Corollary 5.11 convention. This proposition concerns the analytic determinant line only; it does not identify that line with a physical anomaly theory.
Statement references
Facts & Assumptions
Given: The family and conventions of the statement, with the full AC of the analytic suppliers.
Given Spin and Spin-c structures and their spinor bundles gives the specified vertical Spin lift and its complex spinor module.
The maximal distributional domain of a geometric Dirac operator is H1 and A chiral geometric Dirac operator is Fredholm give the self-adjoint elliptic domains, compact-fiber kernels, and Fredholm complements.
The geometric Dirac heat parametrix has a graph-norm remainder gives arbitrarily differentiated local heat transport with arbitrary-order graph-norm remainder.
Radial gauge yields the Getzler oscillator as the leading operator gives radial gauge, the Clifford-degree recursion, and the normalized leading oscillator for a geometric Clifford Dirac operator.
The Mehler coefficient has A-hat times the relative Chern character and The A-hat form and relative Chern character give the exact finite-form Mehler computation and the convention .
The Dirac heat trace is smooth and its supertrace equals the index gives the smoothing heat trace and the kernel superdimension.
The Axiom of Choice supplies the AC used by these Hilbert-space and bundle constructions. The families, determinant, and holonomy statements are proved below; none is an additional named-theorem assumption.
Proof
Given: The stated geometric family, inverse determinant convention, and [F1]–[F7].
If the fibers have dimension zero, properness makes each fiber a finite set. On a base neighborhood each sheet and its Spin data are locally constant, so the vertical spinor spaces form a finite graded flat Hermitian bundle and . Thus is flat. The vertical tangent curvature vanishes, , and its fiber integral has degree zero only, proving the curvature formula in this case. The orientation-graded reference line gives at each point, so oriented integration agrees with the actual kernel superdimension, including negative points. Over a loop the mapping-torus operator is exactly on this finite bundle. For a holonomy eigenvalue , parallel coordinates give spectrum for the positive circle Dirac operator. Subtracting the leading integral in the Euler summation formula gives the Hurwitz value , hence for and exponentiated reduced eta . At the symmetric spectrum and one kernel vector give the same phase; negative chirality conjugates it. Multiplication over these finite eigenvalues therefore gives its exponentiated reduced eta as . Its parity equals , proving the stated holonomy formula. Empty fibers give the trivial determinant line and phase one. The remaining proof treats positive even fiber dimension, where [F2]'s closed Dirac hypotheses apply.
Work locally on the base with coordinate horizontal lifts , and choose a smooth orthonormal vertical frame . At the point where the following coefficients are evaluated the base frame is orthonormal and its Christoffel symbols vanish. Write Use the total metric . Its orthonormal horizontal frame is . The relevant Koszul coefficients are Each identity follows by inserting the three frame vectors into All derivatives of the displayed orthonormal-frame inner products vanish; the brackets above give exactly the listed terms. The all-vertical coefficients give the vertical Levi-Civita connection. The all-horizontal coefficients give the ordinary base connection, and tensoriality reinstates it away from the chosen normal base frame. Represent the horizontal Clifford algebra on base forms by This satisfies . The vertical Clifford operators are graded tensored with this representation, so they anticommute with the horizontal odd operators. This representation is equivalent for every to the usual exterior Clifford representation after rescaling forms; hence the ordinary total Dirac and Lichnerowicz identities apply before taking u to zero. The spin connection contribution to the total Dirac operator is The symmetric B terms sum to Indeed the symmetric contraction is -2 tr B. Adding the direct horizontal derivative term and taking u to zero gives the Hilbert-space connection This connection is unitary: Lie differentiation of the vertical volume form along is times that volume form. Differentiating the fiberwise L2 pairing therefore gives exactly the two scalar corrections , one from each argument. For T the vertical-derivative contribution is whereas the two mixed horizontal-derivative contributions together are Their sum is times the ordered alpha,beta sum, equivalently times . Put and define The limit of the total Dirac operator is consequently Twisting adds the actual given connection to both the vertical Dirac and the horizontal connection; it does not alter the universal coefficients just calculated. The Spin-c determinant-line connection is included in the vertical spinor connection and has relative curvature , so with the existing minus-curvature Chern character its factor is . Finally, rescale the vertical metric by . Vertical orthonormal frames scale by sqrt(t), hence D scales by sqrt(t), Clifford multiplication by the fixed vertical vector tau scales by , and the Hilbert connection is unchanged after the canonical volume-unitary identification. Thus the family is exactly
Use the total metric and the horizontal Clifford representation . The total Lichnerowicz formula is an identity of differential operators for every . It is an identity of Laurent-polynomial coefficients in u in these frames. The Koszul calculation shows its Dirac operator has a coefficientwise u->0 limit A. Therefore its square has limit , without any interchange of a heat limit or spectral limit. The vertical connection in the total Bochner operator has a finite limit on vertical spinors tensored with base forms. In addition to the vertical spin connection it has the terms in direction . The first is the two mixed spin-connection terms, using the first Koszul coefficient. The second follows from , namely . Thus the horizontal-curvature contribution is included in the connection; it must not be thrown away. Choose the auxiliary base metric Euclidean throughout this base chart; the geometric superconnection is independent of that auxiliary metric. There are then no all-horizontal base spin-connection coefficients. In the representation just chosen, the connection in direction is u times a connection with a finite coefficientwise limit: the mixed coefficient is multiplied by one horizontal Clifford factor , and the vertical spin coefficient is already O(u). Thus its Bochner contribution has a scalar factor and tends to zero. The horizontal part of is also in the coordinate horizontal frame and gives no limiting drift. The scalar curvature tends to the vertical scalar curvature. Taking the coefficientwise limit of the ordinary Lichnerowicz formula therefore gives the explicit identity Here is precisely the vertical connection calculated above, and replaces each horizontal covector by exterior multiplication. This identity has scalar vertical principal symbol and finite exterior coefficients. In particular its tau/4 connection term generates the actual vertical drift when the square is expanded; no independent horizontal heat operator has been retained. The limiting Clifford multiplication by a vertical covector need not be constant under radial parallel transport, but has the form in vertical normal coordinates. To see the asserted order, it equals at the center; the Clifford-connection identity transports it smoothly, so the difference vanishes there. Horizontal Clifford multiplication is exterior multiplication. These properties are sufficient for the filtration in the next paragraph. Radial gauge and the leading curvature Parallel transport the limiting connection radially in the vertical coordinates. The exact radial identity for any connection with finite exterior coefficients is Its proof differentiates , using . It works coefficientwise in the finite form algebra, with no convergence issue. The constant term is thereby removed by the radial transport itself. For constant tau this transport is the finite nilpotent exponential ; in particular it removes both the drift term and the apparent divergence, rather than neglecting the latter. Assign degree one to vertical Clifford generators and horizontal exterior generators, degree minus one to x, and degree one to vertical derivatives. The O(x)epsilon correction to has degree zero, below its constant Clifford term. Radial gauge makes the leading connection term one fourth of the curvature matrix times x. Its Clifford-degree-two symbol is the full curvature two-form of the vertical tangent connection, including its horizontal and mixed components. This last identification can be checked before taking u to zero: the Clifford curvature is , and by the usual Riemann curvature pair symmetry. With i,j vertical, the latter components are precisely the curvature of the vertical connection in the limit. The mixed Koszul coefficients above account for the horizontal components; no separate symmetry of a degenerate metric is being assumed. The scalar-curvature term has degree zero. The relative curvature term has degree two, with the horizontal Clifford factors replaced by exterior factors. Consequently the degree-two model of is where Rcal is the curvature matrix in the same sign convention as the existing radial-gauge/Getzler supplier, now with coefficients in the full exterior algebra of vertical plus horizontal covectors. The relative curvature includes the twisting connection and the Spin-c determinant curvature . Actual kernels and the finite-algebra extension The scalar vertical principal part has the ordinary normal-neighborhood Gaussian parametrix. Its transport equations and exact Duhamel remainder are those of the existing graph-norm parametrix supplier, now evaluated in the finite exterior coefficient algebra. Equivalently expand by positive base-form degree around the ordinary vertical heat kernel; the Duhamel expansion terminates. For each coefficient, insertions have order at most one after putting the square into connection-Laplace form. Their relative heat bounds are , integrable at every heat-simplex face. A fixed finite number of parameter derivatives is controlled by the differentiated transport equations and graph-norm remainder, increasing the parametrix order as necessary. These facts prove the differentiated remainder locally uniformly in base parameters; they do not postulate an elliptic operator in base directions. Apply the radial transport recursion in the resulting finite algebra. The same induction gives total Clifford/exterior degree at most 2j for its jth coefficient. Its degree-2j symbol is determined by . The Mehler calculation in the existing provider is an identity of formal power series in even curvature forms, hence remains valid after tensoring with the finite horizontal exterior algebra. It gives No new analytic Mehler theorem is used: substitution differentiates this polynomial-in-each-degree expression and verifies the heat equation and Gaussian initial coefficient. Finally use , where multiplies base degree k by . The supertrace kills every vertical Clifford monomial below degree 2m. In horizontal degree k the prefactor leaves a finite limit precisely at the coefficient with total exterior degree 2m+k. Lower possible coefficients vanish by the degree bound, and higher ones tend to zero. The differentiated arbitrary-order remainder survives this rescaling: there are only finitely many horizontal degrees, and the worst power loss is finite, so first choose its order greater than that loss and the desired derivative order. Thus the formal leading coefficient is the actual smooth heat-form limit. With the periodic normalization already fixed in the existing Mehler/relative-Chern-character provider, the result is In the Spin-c case , under the source's convention. This establishes the local form and its full parameter derivatives.
Let B have finite dimension b, and work on a compact base coordinate subset. Let D be a smooth family of odd self-adjoint compact-fiber Dirac operators, with a smooth finite-rank graded kernel bundle P H and a common positive gap on . Assume the given Hilbert connection nabla preserves the grading and is the usual smooth connection on fiberwise smooth sections. Let C be a smooth two-form with odd fiberwise endomorphism coefficients. Put All estimates below hold after any fixed number of base derivatives and in each fixed coefficient Sobolev norm. The kernel projection and its derivatives are smoothing: around a base point take a small resolvent circle inside the common gap, differentiate the resolvent identity, and integrate. Compactness supplies one circle on each of finitely many patches and one common gap. We claim The second estimate proves convergence of the eta integral at infinity. Exact diagonalization in the nilpotent base-form ideal Use the P,Q block decomposition. The zero-form part of is . All its off-diagonal terms have positive base-form degree. The degree-one off-diagonal term of nabla is a bundle homomorphism, not a base derivative: the base derivative cancels in P nabla Q and Q nabla P. Its coefficients are smoothing, since derivatives of P are smoothing. At degree k, suppose degrees below k have been made block diagonal. An even change of frame , with off-diagonal and of form degree k, changes the degree-k off-diagonal part by . On the P,Q and Q,P blocks this equation is solved by right or left multiplication by , with the sign determined by the graded product. Explicitly one solves the two equations independently; multiplication by is invertible on Q and the other diagonal block is zero. Thus is times the degree-k off-diagonal coefficient times , in its corresponding order. Its coefficients are smoothing and bounded in every fixed Sobolev norm. Changing frame introduces no lower-degree term, because all other factors in have positive form degree. Iterate . The positive-form ideal has nilpotence degree b+1, so every inverse has a finite geometric series, and after b steps the result is EXACTLY block diagonal. Write The coefficient estimates follow directly from the recursion. At degree one the new change is ; every subsequent nonzero correction contains either that factor or the original . Hence These assertions are coefficientwise in the finite form algebra; they are not operator-norm assertions about the unbounded zero-form . Differentiating the explicit powers of t in the finite recursion gives the stated derivative estimates. No infinite asymptotic conjugation is being invoked. Heat estimates on the Q block The zero-form square is with spectrum at least c t for one . The positive-form part of has finite differential order and coefficients polynomially bounded in sqrt(t); its form degree is positive. Its heat exponential is therefore the finite Duhamel series of length at most b. For each fixed term distribute a fraction of the common gap to obtain exp(-ct/2). The remaining heat factors absorb each vertical differential factor by the spectral estimate At simplex faces with zero u, commute an insertion to an adjacent positive heat interval, expanding its commutators; equivalently use the smooth heat parametrix and its differentiated graph-norm remainder. Since at least one interval has length at least 1/(b+1), a smoothing trace-class factor remains, and the other differential factors are bounded between sufficiently separated Sobolev spaces. Thus each fixed base derivative and trace norm is bounded by a polynomial in t times exp(-c' t). The same estimate holds with a fixed differential insertion, in particular . The P block is finite dimensional, so its heat exponential converges in all coefficient norms to , with error . Conjugation invariance of the smoothing supertrace now proves the first claim. The eta insertion Put . Differentiating gives The trace of the first term times is on P, by the derivative estimate, and exponentially small times a power of t on Q, by the Q-block heat estimate just established. For the second term use the Bianchi identity. Supertracing a commutator with a superconnection is its exterior derivative; hence this contribution is, up to the fixed graded commutator sign, the exterior derivative of This form and its exterior derivative are : and every fixed base derivative have that size on P, and the Q terms satisfy the preceding exponential estimate. This proves the second claim. Integrating proves large-time convergence, with all fixed base derivatives, on compact parameter subsets. Producing a kernel bundle after stabilization For a compact base, choose finitely many smooth sections of whose values, together with im at every base point. Such sections exist: choose a finite-dimensional complement at each point, extend its finitely many smooth eigenvectors locally, use continuity of surjectivity in to , and then a finite partition of unity and compact subcover. The smooth finite-rank operator given by these sections makes surjective. Its kernel is a smooth finite-rank bundle, as is seen from a local right inverse and the corresponding projection. The self-adjoint odd stabilization with positive space and negative space consequently has constant kernel dimension and a uniform spectral gap on compact base subsets. The difference from the original operator plus the auxiliary zero block is smoothing. This constructs the stabilization needed for the three block calculations just proved. To define the actual differential index one must still include the finite-dimensional auxiliary bundle and the finite-parameter eta variation; omitting that correction changes the differential class. On noncompact bases no global fixed N is asserted by this argument.
For s in a compact interval put on B times the interval. Its curvature is dot . Duhamel differentiation and graded cyclicity give To check cyclicity, cyclically move the two heat factors in each integrand; they are even, so the supertrace acquires no extra sign, and their heat times sum to one. The simplex integral then has length one. The Bianchi identity , followed by the supertrace, gives Str . This step uses trace-class cyclicity only for a smoothing product; approximate differential factors by spectral cutoffs before passing in trace norm. Writing the closed form as wedge gives . Consequently Every statement here is for finite a,b in the regular parameter region. Multiplication of each homogeneous form degree by the fixed periodic 2pi-i normalization commutes with this calculation when the exterior differential is normalized in the same way. Finite rectangle identity. For , use , and the same finite Duhamel definition. Bianchi and trace cyclicity again give a closed form on B times the rectangle. Stokes, with the actual induced boundary orientations, gives The minus sign is the two-dimensional fiber-integration sign: decompose a total form into ds dt components and check on a scalar coefficient. Thus eta forms along the four sides differ by an explicitly defined exact form. No invertibility of A's degree-zero part is needed on a finite rectangle; heat smoothing suffices. The inserted estimate can now be obtained from the established local heat expansion, without differentiating a limit or suppressing a possible term. Introduce a real base coordinate lambda, put , and keep the twisting connection, determinant connection, and original horizontal lifts fixed. The new lambda-horizontal lift is ; it has zero horizontal-curvature component with every old base lift. In the orthonormal vertical frame its bracket coefficients with are . Hence the preceding Koszul calculation gives the Hilbert connection before the volume-unitary identification. That identification multiplies sections by exp(-m lambda/2), and changes this connection exactly into . It follows directly from the scaling of D and c(tau), rather than by a formal deformation, that the geometric heat superconnection for this extended family is The vertical metric connection has no lambda curvature: in the fixed vertical frame its lambda component is -id/2 and its old components are lambda independent. Its curvature commutator with -id/2 vanishes. The relative twisting curvature is also lambda independent with no d lambda part. The local index density of the extended family consequently has zero d lambda coefficient. Apply the already proved full parameter-dependent heat expansion on a compact lambda interval. The finite-order transport expansion and rescaling give a polyhomogeneous expansion in nonnegative half-integer powers after the vanishing of all potentially negative powers. Its constant d lambda coefficient is zero by the preceding density calculation, so that coefficient is , with every fixed base derivative. On the other hand the exact one-parameter Duhamel identity gives At it follows that the eta insertion is . This is integrable at zero. Combining with the proved estimate at infinity, after finite-rank stabilization when necessary, gives a smooth eta transgression form and its exterior-derivative identity.
Write for the vertical spinor Hilbert bundle, Gamma for its grading, and D for its odd self-adjoint Dirac operator. Over a neighborhood where avoids the spectrum of , let . The inverse Quillen line is On an overlap with , the invertible odd operator D identifies the extra positive and negative band. Its determinant supplies the transition isomorphism. The product of transition isomorphisms for three cuts is the same band determinant, so these are actual line-bundle transitions. The Quillen metric is obtained from the L2 metric by the zeta-regularized determinant of the positive Laplacian in the complementary band. The needed zeta continuation follows from the existing differentiated heat parametrix: subtract finitely many small-time powers in its Mellin integral and use exponential decay on the gapped complement at infinity. Differentiating the same subtracted integral gives the connection on a cut. In unitary local frames, its imaginary complementary-band one-form is The real logarithmic derivative is the derivative of the zeta metric and disappears in unitary frames. To verify the factor and sign without a convention guess, write the finite invertible band as . Then and the inverse determinant transition has unitary logarithmic derivative minus the imaginary part of . This is exactly minus one half of the displayed supertrace. The same regularization in the infinite tail gives (1). A finite change of cut adds that finite band trace, exactly the derivative of its transition function. Thus (1) is the actual inverse Bismut--Freed spectral-cut connection; it is not a newly chosen connection with the same curvature. If the kernel K is a bundle and has a uniform gap on the compact parameter set, the local heat calculation and the small/large-time estimates imply in the unnormalized one-form convention. For a one-dimensional base the torsion two-form is zero. To check (2), square , expand its heat exponential to first base degree, and move D across that base one-form. The two minus signs cancel; trace cyclicity combines its two heat intervals to . The resulting coefficient is exactly one half of the integrand in (2). Its convergence at both endpoints was proved by the geometric scale-parameter and large-time block arguments. Mellin continuation therefore identifies (1) with minus ; there is no unproved exchange of a divergent integral. Including the fixed periodic 2pi-i normalization, the inverse determinant connection is The same identity holds locally after smoothing stabilization, subtracting its auxiliary finite determinant line.
The eta transgression identity proved in the local heat construction is with the auxiliary term added when a finite-dimensional stabilization is used. Under the declared ch(nabla)=tr exp(-F/(2pi i)) convention, the curvature of is -2pi . Taking the curvature of (3) therefore gives For completeness, the stabilization used here does not assume constant original kernel rank. Choose finitely many smooth sections which, together with im on a compact parameter neighborhood. Adding their map J from an auxiliary finite bundle makes surjective. Its kernel is a smooth finite bundle and has a common complement gap. Add the auxiliary bundle with its own connection and subtract its determinant line. Changes of this stabilization are compared through the surjective path after adding both auxiliary bundles. Finite-parameter heat transgression gives exactly the spectral-cut band transition of step 4.1. A smoothing change of D does not change (4). This assertion can also be checked directly at base degree two, rather than appealed to as locality: terms involving the perturbation in the heat expansion have a positive power of sqrt(t) and a smoothing insertion, so their traces tend to zero. The only possible degree-two term with power is the anticommutator of the odd smoothing perturbation with the torsion Clifford operator. Its supertrace is zero by graded cyclicity. Hence the limiting two-form is unchanged. This justifies the stabilized computation on every small base neighborhood. Such neighborhoods cover Z, proving (4) without a global kernel-bundle hypothesis or compactness of Z.
Pull the family back to the specified loop and make the canonical volume-unitary identification of vertical Hilbert spaces. The Koszul calculation gives the mapping-torus Dirac operator exactly as when the circle coordinate has the standard period; changing the circle length rescales epsilon and does not change the limit. The odd spinor convention is fixed here, rather than guessed from the exponential: Clifford multiplication by the positively oriented circle vector is -i Gamma. Thus its -i eigenspace is the positive vertical chirality, exactly Dai--Freed's boundary convention on PDF page5. Equivalently the oriented odd complex Clifford volume acts by +1. The mean-curvature term is already in the unitary Hilbert connection. There is no horizontal torsion two-form over a circle. Two modifications will be used: the odd vertical smoothing stabilization D->D+J+J*, and removal of the off-diagonal components of . They change (5) by a smooth vertical smoothing operator whose fiberwise ordinary trace is zero. For the first modification this follows from oddness; for the second it follows from its off-diagonal block form. Here is why the exponentiated reduced eta is unchanged, for each fixed . The variation formula follows by differentiating the absolutely convergent Mellin heat integral for Re s large, using Duhamel cyclicity, then continuing: Vertical smoothing makes the short-time trace in this formula a one-dimensional horizontal Gaussian trace with smooth vertical trace coefficients. Its expansion is To establish the remainder, insert the smooth kernel of C in the local horizontal Gaussian parametrix, expand the remaining smooth coefficients in the horizontal difference, and integrate odd differences to zero. Vertical derivatives are harmless because C is smoothing; wrapping terms are exponentially small. The same argument gives arbitrary further integer powers and differentiated remainders. The leading integral vanishes by tr . Thus (6) has value zero at . Kernel crossings can change eta/2 by an integer; including h/2 is precisely what makes exp(2pi i xi) continuous through such a crossing. This proves exact invariance of that phase. Regularity of eta at for the initial geometric odd-dimensional Dirac operator follows from the existing radial heat construction: the possible residue is its top odd-Clifford coefficient. Its leading radial model is a Clifford derivative of the even Mehler Gaussian, which vanishes at its diagonal center. All lower Clifford degrees have zero ordinary trace in the odd spin representation. The smoothing variation formula then preserves this regularity in the reductions just described. No value of eta at a pole is being used. The inverse determinant holonomy is unchanged by the same reductions. Stabilization adds its auxiliary finite determinant line, which will be canceled explicitly below. Off-diagonal connection components enter neither the projected kernel connection nor the diagonal complementary-band trace (1). A vertical smoothing homotopy has zero ds dtheta component of the local curvature by step 5.1, so parallel transport around the circle has constant holonomy along that homotopy. Thus both sides of the desired formula are reduced compatibly. After stabilization and off-diagonal removal, (5) splits exactly into The second vertical operator has a common nonzero gap. The first summand is finite dimensional.
We prove the following assertion for the gapped summand occurring in (7), including its finite-rank smoothing compression: The finite part on the right is the convergent one-form integral of (2). We give the actual joint parameter estimates needed to evaluate the left side at zero; an estimate for the heat operator on one fixed fiber would not imply them. The compressed spaces and uniform parameter norms. Write for the vertical dimension and . If is the smooth finite-rank kernel projection in step 6.1, the space used here is , , with domain and projected connection . In the case without a compression set . All estimates are on these spaces; no invertible odd extension across the unequal positive and negative kernel ranks is asserted. The kernel projections and their derivatives have smooth integral kernels. Locally, the equation , , gives a unitary identification of with for every integer ; its coefficients and derivatives are smoothing, hence bounded on every such space. Combining it with the volume-unitary horizontal transport gives a connection trivialization in which , the grading is constant, and is a smooth first-order elliptic family, up to the finite-rank smoothing terms already introduced in step 6.1. Transition maps at an interval endpoint are unitary and preserve all these norms. Fix any positive reference vertical Laplacian on this finite family of local trivializations. For let have norm and define negative norms by duality. Restrict these norms to when appropriate. Repeated elliptic estimates on the compact fiber, with a finite cover of the circle, give uniformly For even this follows by expanding and applying the ordinary elliptic estimates successively; the reverse inequality follows because each differential coefficient is bounded. The same argument using supplies odd , and duality supplies negative . Thus the constants are independent of and . Restriction to causes no loss: commutes with , and and its derivatives map every negative Sobolev space to every positive one with uniform bounds. A smoothing term is consequently retained as a bounded term in each displayed estimate, rather than falsely treated as a scalar symbol of strictly decreasing order in every vertical covariable. In these coordinates The bound and Cauchy--Schwarz show that its quadratic form is at least Since on the selected summand, this is at least for sufficiently small . The fixed- operator has the selfadjoint elliptic domain, also after a smooth finite-rank compression, by the local elliptic estimate and its adjoint estimate. Choose a Dunford contour consisting of two fixed rays outside the positive real axis and a fixed small connecting arc below . Its orientation is the usual one enclosing the positive spectrum. On this contour the spectral resolvent is defined and has the ordinary sector bound; the contour and lower arc are independent of . Actual symbol composition and graph estimates. Use the left horizontal quantization For the following operator-valued symbols, joint order means that every fixed derivative preserves order, each derivative lowers it by one, and the resulting operator maps to with a uniform bound, where is the number of derivatives and . This is a weighted operator norm condition, not an assertion that the finite-rank vertical smoothing terms have a classical isotropic symbol in all covariables. From (11) and the spectral bound, the exact fiber inverse has order on . Indeed, for , Commuting powers of gives all integer graph estimates. Differentiating the inverse gives and ; induction proves the asserted bounds for every fixed number of derivatives. The derivative has vertical order two, including its smoothing commutators. These estimates therefore hold for the actual projected family. Taylor expansion of the second amplitude in and integration by parts in give The integral Taylor remainder proves, in the weighted norms just defined, that has order when have orders . Here is the norm justification, including uniformity: write the remainder as its oscillatory integral in the horizontal difference and its dual variable; insert the integral Taylor remainder, then integrate by parts in those two variables more times than the required Sobolev order plus two. The weight ratio for a shifted covariable is bounded by a fixed power of , since . These integrations leave an integrable scalar majorant times the stated operator norm. Schur's inequality proves the operator bound. For its full graph norms, put and use unit-length intervals on the resulting circle of length . The coefficients on these intervals have uniformly bounded derivatives, and a partition with bounded overlap gives the same bound after the squared local norms are summed. The number of intervals therefore introduces no factor into operator or graph norms. Cutoff commutators off the horizontal diagonal are of arbitrarily negative parameter order: repeated integration by parts in gives this for each prescribed . This also proves (13) with actual global remainders on the twisted periodic bundle. The global quantization here is properly supported, rather than a formal replacement of a discrete torus sum by an integral. Choose and a smooth cutoff equal to one for and zero for . On this unique short circle arc insert the actual parallel transport from to into the displayed continuous- kernel. Constant changes between connection-parallel frames conjugate this kernel, so it defines an operator on the actual twisted bundle. On the diagonal its trace is EXACTLY whenever the integrated coefficient is trace class. In products, derivatives of this cutoff, or differences between two such local descriptions, are supported away from the diagonal. There integrations by parts give a factor and replace the amplitude by . There are only finitely many short-arc charts, and the weighted order decreases by . Choosing greater than therefore puts every such error into the remainder in (15), with trace estimate (16) and its holomorphic contour bounds. Thus all wrapping and partition effects are in an arbitrarily high-order actual remainder. The finite coefficients in (17) really are -independent continuous-frequency integrals; there is no missing Poisson or winding term in the residue identity. For clarity, the local graph estimate underlying this calculation does not assume an adiabatic theorem. On a fixed vertical coordinate box times a unit interval, freeze the positive scalar principal coefficients at a point. The Fourier multiplier estimate follows from Choose the box so the variation of its principal coefficients times this inverse has norm below . Move that error to the left. Lower-order terms and partition commutators are bounded by a small multiple of the two-derivative norm plus a constant times the zero-derivative norm. For large the latter is absorbed by the parameter weight; for bounded the sector spectral estimate bounds it. Differentiating this estimate and commuting the fixed derivatives proves higher integer orders, and duality gives negative orders. The coefficients' bounds on the rescaled intervals and their overlap are uniform in . The compression adds only smoothing commutators; their lower norms are controlled by the same sector estimate and cannot change the scalar principal inequality. Thus two vertical derivatives, a vertical derivative times , and are all bounded by the same graph estimate. This is the required mixed, rather than fixed-fiber, estimate. Set and . Recursively define Only occurs for , and only for . Formula (13) and induction show that has joint order . The globally patched satisfies where has order , uniformly on , including all fixed derivatives. Patch errors are included by taking their integration-by-parts order greater than . In each fixed finite list of weighted graph norms, is bounded uniformly; hence for sufficiently small the inverse of is its convergent Neumann series. Since the spectral inverse exists, this right inverse is that inverse. Consequently where has order in every fixed required graph norm. Holomorphy in is preserved by the inverse and the series. This establishes the actual uniform resolvent remainder; it is not an estimate on formal coefficients alone. Holomorphic trace remainder and the finite coefficients. Let , with . Integrate (15) against on , and multiply by . The spectral formula for is absolutely convergent in operator norm for . The high-order remainder can be traced on this same contour before continuation. In local vertical Fourier coordinates and the rescaled circle coordinate, its trace norm satisfies To verify the factor and exponent, compose with a sufficiently negative graph power on each of the fixed boxes. Its singular values are bounded by the summable Fourier sequence ; the remaining factor has bounded norm by (15). Summing or integrating that sequence gives . The horizontal frequency spacing is , so its sum is bounded by times the integral. The same bound follows for a bundle by a finite partition; on the projections have bounded norms and their smoothing corrections only improve vertical summability. More explicitly, the trace-class assertion can be obtained by factoring twice through a graph power with square-summable Fourier sequence and applying the Hilbert--Schmidt inequality; the graph norm bound controls the middle factor. Thus it does not presume a trace formula for a non-trace-class low-order resolvent. For , (16) times is integrable on the rays, uniformly for . Any fixed derivative introduces only a power of , still integrable after making larger. The lower arc is fixed and the uniform gap controls it. The traced remainder is therefore holomorphic for and is there, with all fixed derivatives. For initially large the finite coefficient operators may instead be traced after the contour integration, giving The are independent of and have meromorphic continuation with only simple poles near zero. The precise continuation argument is as follows. For the differential part, the usual vertical Fourier inversion is constructed directly by solving its principal inverse and recursively canceling lower orders with the integral Taylor formula (13), also in the vertical variables. The homogeneous terms are smooth away from and their remainders have arbitrarily negative joint order by the same graph construction as (15). After contour integration a term of degree has radial integral, for , The angular coefficient is holomorphic; the compact frequency part and the arbitrarily negative remainder are holomorphic. This proves the claimed continuation for the differential terms, without extrapolating a fixed-fiber estimate to horizontal frequencies. The finite-rank vertical terms require a separate justification, since their vertical symbol need not belong to that joint homogeneous class. Each coefficient in (14) is a finite sum of resolvent words with polynomial factors. Its heat-contour transform is the corresponding finite heat-simplex word. If one factor is a vertical smoothing operator, cyclically place that factor at the beginning of the trace. For a finite-rank factor write it as with smooth vectors; every simplex heat word then becomes a finite sum of inner products of smooth vectors with heat factors and differential operators between them. All derivatives in the individual nonnegative heat times exist at zero, uniformly on the compact parameter set: differentiate a heat factor to insert a power of , and bound the resulting expression in a sufficiently high fixed graph norm. Taylor's integral remainder gives arbitrary-order expansions uniformly up to every simplex face. A smoothing factor not of finite rank is handled by its trace-class factorization through arbitrarily high positive and negative graph powers, with the same derivative estimates. A smoothing perturbation of is reduced to such a word by the exact Duhamel identity and its iterates; the remainder contains arbitrarily many powers of the total time and obeys the same bound. For the kernel compression one uses , so its extra terms are precisely of this kind. The scalar integrals are Gaussian moments. Their explicit powers of the total time, multiplied by the simplex powers and these Taylor expansions, again give a full power expansion with an arbitrarily high remainder. The subtraction and the uncompressed differential comparison are used only for . Mellin integration on that interval gives simple poles. For retain the original, unexpanded -compressed heat-simplex words: at least one simplex coordinate is at least the reciprocal of the number of factors, and its heat factor has the actual common gap; graph weights absorb the finitely many differential insertions. Thus this long-time integral is locally uniformly convergent in , without incorrectly assigning decay to the separate uncompressed and terms. Thus the compressed and smoothing coefficients in (17) have the stated continuation as well. No isotropic-symbol assertion about a finite-rank projection has been substituted for this argument. Step 6.1 supplies regularity of the actual eta function at zero for exactly the geometric operator and smoothing reductions in use. Since the remainder in (17) is holomorphic, its residue identity is for every sufficiently small positive . A finite polynomial which vanishes on an interval has all coefficients zero. Hence each is regular at zero. This proves that contributes at zero and that the holomorphic remainder tends to zero. The argument uses actual regularity and the actual uniform remainder, rather than exchanging two divergent integrals. The two surviving coefficients. The complex-power symbol is Its term is odd in the vertical grading and has trace zero; its term is odd in . Therefore initially in the convergent half-plane and then identically by continuation. No low-order resolvent trace has been taken. Write . The first recursion gives The order-one coefficient of the product with is After trace and horizontal integration, the terms odd in vanish and the remaining derivative terms are They integrate to zero around the circle; the traced quantities are periodic even in a connection trivialization with nontrivial holonomy. The remaining term is . Carrying out its heat-contour integral gives . The Gaussian integral with measure is , so This identity is first obtained for large, where all heat and trace integrals converge, and then continues by the preceding finite-coefficient argument. At zero its scalar factor is , and (2) fixes the convergent finite part of the one-form. Equations (16)--(17) now prove (8), including the existence of the limit for the actual compressed family. There is no kernel term on this summand. Exponentiating half the eta invariant consequently gives which by (1) is exactly the complementary inverse-determinant holonomy.
Let be the unitary holonomies of nabla on the finite kernel bundles . Diagonalize each holonomy. In parallel coordinates a holonomy eigenvalue exp(2pi i alpha) gives boundary condition v(1)=exp(-2pi i alpha)v(0), so the positive circle Dirac spectrum is 2pi(k-alpha). For its eta value at zero is 2alpha-1: split the positive and negative series into their Hurwitz sums, whose value at zero is 1/2-a. That value follows directly by subtracting the leading integral in the Euler summation formula for . At the spectrum is symmetric with one kernel vector and gives the same continuous exponent. Hence Changing the sign of the circle Dirac conjugates this phase. Multiplying the positive and negative kernel eigenvalues therefore gives The parity in (10) equals . Equations (7), (9), and (10), with the exact smoothing phase reductions, prove If an auxiliary bundle F was added, its positive circle contributes hol(det F), its inverse determinant line contributes det F, and its vertical index is rank F. All three factors cancel on dividing back out. Thus this identity holds for the original family, with no constant-kernel assumption. The nonbounding circle structure is essential: it gives the periodic spectrum used in (10). It is exactly why the parity factor is present. Together with (4), this proves both formulas in the original analytic-line statement. It makes no physical anomaly identification, assumes no differential-topological index theorem, and does not purport to prove the unresolved embedding comparison for that separate target.
Depends on
Given Spin and Spin-c structures and their spinor bundles
The maximal distributional domain of a geometric Dirac operator is H1
A chiral geometric Dirac operator is Fredholm
The Dirac heat trace is smooth and its supertrace equals the index
The geometric Dirac heat parametrix has a graph-norm remainder
Radial gauge yields the Getzler oscillator as the leading operator
The Mehler coefficient has A-hat times the relative Chern character
The A-hat form and relative Chern character
The Axiom of Choice