M. Stone, 2018-2020 Euclidean-signature Dirac machinery and anomalies The Dirac operator on an N-dimensional Euclidean-signature Riemann manifold (M,g) takes the form1 D = γaeµ ∂ + 1 σbc ω = γaD . 6 a µ 2 bcµ a µ a Here the ea ea ∂µ compose an orthonormal vielbein on M, the γ are Hermitian matrices≡ obeying γa,γb =2δab, { } the ab 1 a b σ = 4 [γ ,γ ] are the skew-Hermitian spinor generators of so(N) which obey [σab, σcd] = δbcσad δacσbd + δadσbc δbdσac, − − [σab,γc] = γaδcb γbδac, − and µ 1 bc D Dea = e ∂ + σ ω a ≡ a µ 2 bc µ is the covariant derivative acting on the components of a Dirac spinor. Global requirements: The above formulæ apply only in a local coordinate patch possessing a smooth vielbein frame. For global considerations we need M to be orientable, and to be able to equip (M,g) with a spin structure. The need for a spin structure arises because in overlaps between patches the (1) (2) vielbein frame ea in patch (1) will be related to the frame eb in patch (2) by a smoothly varying SO(N) map (1) (1,2) (2) ea = Oab (x)eb . The spinor fields will then be related by ψ(1)(x)= S[O(1,2)(x)]ψ(2)(x), 1V. Fock, D. Ivanenko, G´eom´etrie quantique lin´eaire et d´eplacement parall`ele, Compt. Rend. Acad. Sci. Paris 188, 1470-1472 (1929); V. Fock, Geometrisierung der Diracschen Theorie des Elektrons, Zeits. Phys. 57, 261-277 (1929). 1 where S[O(x)] is a corresponding smoothly varying element of Spin(N). There are two Spin(N) maps ( S[O(x)]) for each element O(x) SO(N), ± ∈ and so a choice of sign for S[O] has to be made in each overlap. These choices need to be mutually consistent: in triple overlaps we will have O(1,2)O(2,3)O(3,1) = I for the vielbein maps, and consistency demands that our sign choices for O S also satisfy 7→ (1,2) (2,3) (3,1) S[O ]S[O ]S[O ]= I. If a globally consistent set of choices is made, the resulting spin structure provides a lift of the SO(N)-valued Levi-Civita connection to the two-valued Spin(N) connection. There may be more than one consistent set of choices— or perhaps none. CP 2 is an example of an orientable 4-manifold for which no spin structure is possible and hence no globally defined Dirac operator. A manifold which possesses a spin structure is said to be a spin manifold, or to be spin. Hermiticity: The natural L2(M) inner product on spinors is ψ ψ = ddx√g ψ†ψ , h 1| 2i 1 2 ZM where ψ† denotes the conjugate transpose of the complex-valued column spinor ψ(x). Given ψ Dψ = ddx√g ψ†(Dψ ) h 1|6 2i 1 6 2 ZM we wish to integrate by parts so as to compute the Hilbert space adjoint operator D† with respect to the inner product. This requires us to evaluate µ 6 ∂µ(√gea ). We find ν ν ν α ∂µ(√gea) = √g(∂µea + eaΓ αµ) ν ν α = √g(∂µea + eaΓ µα). We now set µ = ν to obtain2 µ µ µ α ∂µ(√gea ) = √g(∂µea + ea Γ µα) µ α µ = √g(∂µea + ea Γ αµ) = √g eµ = √g eµωb . ∇µ a b aµ 2Note the interchange of labels on the Christoffel symbol in the second line of the equation above! There will be extra terms when torsion is present. 2 The contribution from this derivative combines with a contribution from the necessity of re-ordering γa and σbc via 1 [γa, σbc]= 1 (δabγc δacγb) 2 2 − to show that † ddx√g ψ† γaeµ ∂ + 1 σbc ω ψ = ddx√g γaeµ ∂ + 1 σbc ω ψ ψ . 1 a µ 2 bcµ 2 − a µ 2 bcµ 1 2 Z Z † Thus ψ Dψ = Dψ ψ , so the formal adjoint is D = D. h 1|6 2i −h6 1| 2i 6 −6 Schr¨odinger-Lichnerowicz identity: In flat space the square the Dirac- operator is the Laplace operator times the identity matrix. In curved space computing D2 is not a trivial operation as various quantities have to be passed through one-another6 and the result is a useful analogue of the Bochner- Weitzenb¨ock identity for the square of the Hodge laplacian. This analogue is usually named for Andr´eLichnerowicz who used it in 1963 to show that man- ifolds with everywhere positive scalar curvature can have no Dirac-operator zero modes3. However Lichnerowicz’ identity appears as the very last equa- tion in a paper by Erwin Schr¨odinger writen some 30 years earlier4. µ a µ One way to proceed is to define γ = γ ea and note that (∂ + 1 σabω )γν = γν(∂ + 1 σabω ) γλΓν µ 2 abµ µ 2 abµ − λµ so that (D)2 = γµγν(D D Γλ D ). 6 µ ν − νµ λ λ If the connection is torsion-free, so Γ νµ is symmetric in µ, ν, we can proceed as follows: (D)2 = 1 γµ,γν (D D Γλ D )+ 1 [γµ,γν]D D 6 2 { } µ ν − νµ λ 2 µ ν = gµν(D D Γλ D )+ 1 [γµ,γν][D ,D ]. µ ν − νµ λ 4 µ ν = gµν(D D Γλ D )+ 1 [γµ,γν] 1 σabR µ ν − νµ λ 4 2 abµν = gµν(D D Γλ D )+ 1 σabσcdR µ ν − νµ λ 2 abcd 1 µν 1 ab cd = Dµ√gg Dν + σ σ Rabcd. √g 2 3A. Lichnerowicz, Spineurs harmonique, Compt. Rend. Acad. Sci. Paris, S´er. A 257, (1963) 7-9. 4E. Schr¨odinger, Diracsches Elektron im Schwerefeld I , Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl. 11, (1932) 105-128. (Available online at: https://edition-open- sources.org/media/sources/10/15/sources10chap13.pdf) 3 µν 2 The first term in the last-but-one line is often just written as g µ ν , with the use of , the covariant derivative acting on the components∇ ∇ ≡∇ of ∇ spinors and/or tensors as appropriate, tacitly implying the extra Christoffel symbol. We can also use the same technique to show that (D)2 = γaγb(D D ωc D ). 6 a b − ba c We then use 1 cd [ ea , e ] e e = σ R . ∇ ∇ b −∇[ a, b] 2 cdab with the torsion-free condition giving the first equality in c c [e , e ]= ea e e e = e (ω ω ). a b ∇ b −∇ b a c ba − ab We find (D)2 = δab(D D ωc D )+ 1 σabσcdR . 6 a b − ba c 2 abcd We can now use the symmetries of the Riemann tensor to simplify the a b c d curvature term: Consider γ γ γ γ Rabcd. By antisymmetry properties of R we know that a = b and c = d. Suppose that b is not equal to either c abcd 6 6 or d, then we can write a b c d 1 a b c d a c d b a d b c γ γ γ γ Rabcd = 3 (γ γ γ γ + γ γ γ γ + γ γ γ γ )Rabcd 1 a b c d = 3 γ γ γ γ (Rabcd + Radbc + Racdb) = 0, by the first Bianchi identity for the Riemann curvature of a torsion-free con- nection. To get a non-zero answer, therefore, there are two possibilities: b must equal c (and not d), or it must equal d (and not c). Thus γaγbγcγdR = γaγcR γaγcR abcd abbc − abcb = 2γaγcR − ac = 2R − where Rab = Racbc is the (symmetric) Ricci tensor, and R = Raa is the scalar curvature. We thus have the identity 1 (D)2 = 2 R, 6 ∇ − 4 4 where 2 1 µν Dµ√gg Dν ∇ ≡ √g is the “rough”, or “connection” laplacian acting on spinors. As a consequence we have the identity 1 Dψ Dψ = ddx√ggµν(D ψ)†(D ψ)+ ddx√gR ψ 2. h6 |6 i µ ν 4 | | Z Z As the first term on the RHS is 0, if R is everywhere positive this prohibits ≥ the existence of a ψ0 such that Dψ0 = 0. This is Lichnerowicz’ theorem. One can actually make a slightly6 stronger statement: suppose that R is non-negative and non-zero at a point, then there can be no zero mode. To see this suppose Dirac zero mode exists, i.e. Dψ0 = 0. Lichnerowicz’ identity then shows that D ψ = 0 for all a. This in turn6 tells us that ψ is constant. a 0 | 0| Now R being positive at a point (and hence in a neighbourhood of the point) makes the R-integral positive and gives us a contradiction. Lichnerowicz and other mathematicians call a Dirac-operator zero mode a harmonic spinor, hence the title of his paper. Euclidean action and propagators: On a closed (compact without bound- ary) spin d-dimensional spin manifold the skew-adjoint Dirac operator D = γaD = γaeµ ∂ + 1 σbc ω 6 a a µ 2 bcµ will possess a complete orthonormal set of c-number spinor eigenfunctions un(x) with the properties Du = iλ u , ddx√gu† (x)u (x)= δ , u (x)u† (x′)= I δd(x x′). 6 n n n n m mn n n g − n Z X Here the the λn are real, I is the identity matrix in spinor space, and the distribution δd(x x′) obeys g − ddx√g δd(x y)=1. − Z In Euclidean signature there is no preferred “γ0” and therefore no inherent need to distinguish between ψ†(x) and ψ¯(x), but when we use the eigenmodes 5 to expand out the Grassmann-valued Fermi fields it is convenient to write ψ(x) = un(x)χn, n X ¯ † ψ(x) = un(x)¯χn.