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S.G. Scott and K.P. Wojciechowski GAFA Geom funct anal c Birkhauser Verlag Basel Vol X GAFA Geometric And Functional Analysis THE DETERMINANT AND QUILLEN DETERMINANT FOR A DIRAC OPERATOR ON A MANIFOLD WITH BOUNDARY SG Scott and KP Wojciechowski Abstract Let D denote a Dirac op erator on a compact o dddimensional mani fold M with b oundary Y The elliptic b oundary value problem D is P the op erator D with domain determined by a b oundary condition P from the smo oth selfadjoint Grassmannian Gr D It has a well dened determinant see Wo The determinant line bundle over Gr D has a natural trivialization in which the canonical Quillen determinant section b ecomes a function denoted by det D equal C P to the Fredholm determinant of a naturally asso ciated op erator on the space of b oundary sections In this pap er we show that the regularized determinantdet D is equal to det D mo dulo a natural P C P multiplicative constant Intro duction Since the early stages of Quantum Mechanics there has b een a fundamental need for a rigorous and workable denition of the determinan tofaninvert ible linear op erator acting on an innitedimensional space The Fredholm determinant provides a natural extension from the niterank case to a lin ear op erator T HHacting on a separable Hilb ert space such that T Id is an op erator of trace class It is dened in a small neighb ourho o d of the identity Id by the formula Tr det e e Fr where denotes an op erator of trace class and for general T Id with traceclass by the absolutely convergent series X k det Id Tr Fr k In particular the Fredholm determinant retains the characteristic multi plicative prop erty of the niterank algebraic determinant det T T det T det T Fr Fr Fr SG SCOTT AND KPWOJCIECHOWSKI GAFA The traceclass condition is however very restrictive and the class of op erators with Fredholm determinants certainly do es not include any class of elliptic dierential op erators Nevertheless imp ortant applications in P hy sics particularly in Quantum Field Theory see the fundamental pa p ers MS led to the idea that the Fredholm determinantallows one to study the ratio of the determinant of an elliptic op erator to the determi nant of a comparable basep oint op erator for instance the ratio of the determinant of a Hamiltonian with p otential and the determinant of the free Hamiltonian In this waya regularized determinant relativetoachoice of basep oint op erator may b e dened On the other hand in many imp ortant problems such as in quantizing gauge theories it is necessary to discuss directly a regularized determinant of an elliptic op erator The Heuristic Approach to the determinant in this context was rst prop osed by mathematicians for the case of a p ositive denite secondorder elliptic dierential op erator L C M S C M S acting on the sections of a smo oth vector bundle S over a closed mani M The op erator L has a discrete sp ectral resolution and so formally fold has determinant equal to the innite pro duct of its eigenvalues The start ing p oint in dening a regularized pro duct is the following formula for an invertible niterank linear op erator T s d ln det T fTr T gj s ds For large Res the function of the op erator L is just the trace o ccurring on the right side of Z s tL s t Tr e dt sTr L L s It is a holomorphic function of s for Res dim M and has a mero morphic extension to the whole complex plane with only simple p oles see d Se In particular s is not a p ole Hence f sgj is L s L ds welldened and wemay dene the determinant by L det L e This denition was intro duced in in a famous pap er of Rayand Singer RS in order to dene Analytic Torsion the analytical counterpart of the top ological invariant FranzReidemeister Torsion The equalityof the two torsions was subsequently proved indep endently by Je Cheeger and Werner Muller see C Mu Since then there have b een numerous applications of the determinantinphysics and mathematics b eginning with the Hawking pap er H on quantum gravity Vol PROJECTIVE EQUALITY OF DETERMINANTS For p ositivedenite op erators of Laplace typ e over a closed manifold the determinantprovides a generally satisfactory regularization metho d Though the fundamental multiplicative prop erty of the Fredholm deter minant no longer holds if L and L denote two p ositive elliptic op erators of p ositive order on a Hilb ert space H then in general det L L det L det L We refer to KV for a detailed study of the socalled Multiplicative Anomaly In manyphysical applications however suchasthequantization of Fermions one encounters the more problematic task of dening the de terminant of a rstorder Dirac operator These are not p ositive op erators and it is at this p oint that anomalies may arise due to the phase of the deter minantAtS For a Dirac op erator D C M S C M S acting on the sections of a bundle of Cliord mo dules over a closed o dddimensional manifold M one pro ceeds in the following way see BoW for an intro duction and notation The op erator D is an elliptic selfadjoint rstorder op erator and hence has innitely many p ositive and negative eigenvalues Let f g denote the set of p ositive eigenvalues and f g denote k k N k k N s the set of negativeeigenvalues Once again sTrD iswelldened D and holomorphic for Res dim M and wehave X X s s s s D k k k k s s s s s s s s X X s k k k k k k k k k k which can b e written as s s s s D D D D s s D P P s s is the function of the op erator D intro where s D k k k k duced byAtiyah Pato di and Singer see AtPS Once again it is holomor phic for Res large and has a meromorphic extension to the whole complex plane with only simple p oles There is no p ole at s and therefore we can study the derivativeof sats Wehave D d D D s D f g D ds s s The ambiguity in dening ie a choice of sp ectral cut now leads to an ambiguity in the phase of the determinant Wehave s i s e and we pickthe sign This leads to the following formula for the SG SCOTT AND KPWOJCIECHOWSKI GAFA determinant of the Dirac op erator D i D D D det D e e s We come back to the discussion of the choice of sign in in the nal section of the pap er The purp ose of this pap er is to explain a direct and precise identitybe tween the determinant of a selfadjoint elliptic b oundary value problem for the Dirac op erator over an o dddimensional manifold with b oundary and a regularization of the determinant as the Fredholm determinantofa canonically asso ciated op erator over the b oundaryWe consider an innite dimensional Grassmannian of elliptic b oundary conditions commensurable with the AtiyahPato diSinger condition The latter regularization is natu rally understo o d in the sense explained b elow as the ratio of the determi nant of the elliptic b oundary value problem to the determinant of a base p oint elliptic b oundary value problem It is a regularization canonically constructed from the top ology of the asso ciated determinant line bundle and hence called the canonical determinant The canonical determinantis a robust algebraic op eratortheoretic ob ject while the determinantisa highly delicate analytic ob ject and so it is surprising that they coincide Though the equality of the torsions mentioned ab ove at least suggests that the determinantmay b e somehow related to Fredholm determinants To formalise the construction of taking the ratios of determinants used to dene the canonical determinantwe use the machinery of the determi nant line bundle This was intro duced in a fundamental pap er of Quillen Q for a family of CauchyRiemann op erators acting on a Hermitian bundle over a Riemann surface as the pullbac k of the corresp onding universal determinant bundle over the space of Fredholm op erators on a separable Hilb ert space Without making further choices the determinant arises not as a function on the parameter space of op erators but as a canonical section A det A of the asso ciated determinant line bundle DE T More precisely max max det A lives in the complex line Det A Ker A Coker A and is nonzero if and only if A is invertible Using function regularization Quillen constructed a natural Hermitian metric on the determinant bun dle for a family of Cauchy Riemann op erators and computed its curvature This was extended by Bismut and Freed to the context of general families of Dirac op erators on closed manifolds see BF and the curvature identied with the form comp onent of the families index density It did not though provide a straightforward corresp ondence b etween the determinantand the Quillen determinant section More precisely given that det A is de Vol PROJECTIVE EQUALITY OF DETERMINANTS ned the problem is to identify the nonzero section of DE T such that det A det A A Clearly the global existence of such a section is equivalent to the triviality of the determinant line bundle From this view p oint the principal result of this pap er is the exact identication of the basep oint section for the class of elliptic b oundary value problems considered here In order to link this up with Fredholm determinants we use a construction of the determinant line bundle due to Segal Seg We do not discuss in this pap er the corresp onding problem for a closed manifold Perhaps surprisingly it is easier to discuss the relation b etween the determinant and the Quillen determinant section on a manifold with b oundary b ecause of reduction to the boundary integral see BoW In early the second author as the
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