M. Stone, 2018-2020 Euclidean-Signature Dirac

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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, Géométrie quantique linéaire et déplacement parallèle, 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 Schrödinger-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- Weitzenböck identity for the square of the Hodge laplacian. This analogue is usually named for AndréLichnerowicz who used it in 1963 to show that manifolds with everywhere positive scalar curvature can have no Dirac-operator zero modes3. However Lichnerowicz’ identity appears as the very last equation in a paper by Erwin Schrödinger 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 connection. 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 boundary) 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. n X The Grassmann variablesχ ¯n and χn are independent, and not related by any notion of complex conjugation, but when (...) is applied to an expression containing ψ(x) we understand that it not only transposes and complex con- jugates matrices and the spinor functions un(x) but it also changes any χn’s intoχ ¯n’s. The Euclidean action functional for the Dirac ﬁeld can therefore be taken as S[ψ, ψ¯]= ddx√g 1 ψ¯(Dψ) (Dψ)ψ + mψψ¯ . 2 6 − 6 Z n o Equivalently

S[ψ, ψ¯]= ddx√g 1 ψγ¯ a(D ψ) (D ψ¯)γaψ + mψψ¯ , 2 a − a Z † where the covariant derivative Da acting on conjugate spinors ψ or ψ¯ is

¯ µ ¯ 1 bc Daψ = e ψ ←−∂ µ σ ωbcµ a − 2 ¯ ¯ ¯ with ψ←−∂ µ = ∂µψ. The second form has the advantage of treating ψ and ψ symmetrically. On inserting the eigenfunction expansions and using the eigenfunction orthonormality to evaluate the space-time integrals, the Euclidean action functional becomes diagonal

S[ψ, ψ¯] = ddx√g 1 ψ¯(Dψ) (Dψ)ψ + mψψ¯ 2 6 − 6 Z n o = (iλn + m)¯χnχn. n X The vacuum-amplitude partition function is now formally given by the Berezin integral

Z = d[ψ¯]d[ψ] exp S[ψ, ψ¯] { } Z 6 = d[¯χ ]d[χ ] exp (iλ + m)¯χ χ n n { n n n} n n Z Y X = (iλn + m) n Y = Det(D + m). 6 Here Det(D + m) is the Matthews-Salam functional determinant5. The infinite product6 over the eigenvalues usually needs some form of regularization, but for the moment we will ignore this requirement and assume that the regulated determinant has the same properties expected of a finite determinant. The Berezinian version of the Jacobean determinants involved in the ¯ change of integration measure from d[ψ(x)]d[ψ(x)] to d[¯χn]d[χn] cancel one another because we are, in effect, performing a unitary similarity transformation (D + m)= U †diag(iλ + m)U 6 n in which Det(U) = [Det(U †)]−1. This formal cancellation is not affected by some of the λn being zero. In even d =2N dimensions, so that

Γ def= γ2N+1 =( i)N γ γ 5 − 1 ··· 2N is deﬁned, the determinant n(iλn + m) should be a real number. This is because when un is an eigenvector with non-zero eigenvalue λn then Γ5un is also an eigenvector and hasQ eigenvalue λ . Thus − n Det(D + m)= mn (λ2 + m2), 6 n λYn6=0 where n is the total number of λ = 0 modes. The zero modes do not have to come in pairs so when m is negative the determinant is not always positive. In odd dimensions the determinant is typically complex. The propagator is 1 ψ¯ (y)ψ (x) = d[¯χ ]d[χ ]ψ¯ (y)ψ (x) exp S[ψ, ψ¯] h α β i Z n n α β { } Z n †Y u (y)un,β(x) = n,α , iλ + m n n X 5P. T. Matthews, A. Salam, Propagators of quantized ﬁeld, Il Nuovo Cimento 2 (1955) 120-134.

7 and obeys d (D + m) ′ ψ¯ (y)ψ ′ (x) = δ (x y)δ . 6 x ββ h α β i g − αβ In these expressions we have displayed the α, β spinor indices that we usually keep implicit.

Vector gauge ﬁelds: There is little changed in the presence of a vector gauge ﬁeld A iAa λˆ with the λˆ the Hermitian matrix generators of µ ≡ µ a a su(N) or some other Lie algebra. The Dirac operator becomes

D(A)= γaeµ ∂ + 1 σbc ω + A 6 a µ 2 bcµ µ and remains skew Hermitian in L2(M). In ﬂat space the eigenfunctions obey

µ γ (∂µ + Aµ)un = iλnun (∂ u† u† A )γ = iλ u† , µ n − n µ µ − n n and the un have an extra index corresponding to those on the λˆa matrices.

Index and axial anomaly: Now restrict to an even number d = 2N of Euclidean dimensions and recall that

Γ =( i)N γ γ 5 − 1 ··· 2N 2 is Hermitian, obeys (Γ5) = 1, and anticommutes with all the other γ’s. We know that the Euclidean, non-chiral, Dirac operator D on a closed 6 spin manifold possesses a complete set of eigenfunctions un such that

µ γ Dµun = iλnun, D u† γµ = iλ u† , µ n − n n where † † 1 bc Dµu = u ←−∂ µ σbc ω µ , n n − 2 In terms of these eigenfunctions the propagator is

u† (y)u (x) ψ¯(y)ψ(x) = n n . h i iλ + m n n X

8 Put x = y in this expression and use u† Γ γµu (D u† )Γ γµu + u† Γ (γµD u ) n 5 n = µ n 5 n n 5 µ n ∇µ iλ + m iλ + m n n ( D u† γµ)Γ u + u† Γ (γµD u ) = − µ n 5 n n 5 µ n iλn + m 2iλ u† Γ u = n n 5 n iλn + m † (iλn + m) † unΓ5un = 2 unΓ5un 2m (iλn + m) − iλn + m to read oﬀ that6

ψ¯Γ γ ψ = 2m ψ¯Γ ψ + 2(ρ ρ ) ∇µh 5 µ i − h 5 i + − − where ρ ρ = u† (x)Γ u (x). + − − n 5 n n X When m=0, the classical equations of motion for the ψ ﬁeld lead us to expect ¯ ¯ µ ψΓ5γµψ = 0 so the axial current ψΓ5γµψ conserved. But when ρ+ ρ− ∇is non-zeroh thei current is not conserved. This quantum non-conservation− is the most basic example of an anomaly. What is the origin of the anomaly? There are two answers to this, a mathematical one and a physical one. We begin with the mathematical one. Because Γ anticommutes with D, if the eigenmode u has eigenvalue λ 5 6 n n then Γ5un has eigenvalue λn. If λn is non-zero this means that Γ5un is 2 − perpendicular to un in L (M) i.e.

ddx√gu† (x)Γ u (x)=0, λ =0. n 5 n n 6 ZM 2 I On the other hand when λn = 0thenΓ5un also has zero eigenvalue. As Γ5 = we can diagonalized Γ in the space of zero modes and it has eigenvalues 1. 5 ± Consequently

n n = ddx√g u† (x)Γ u (x) + − − n 5 n n ZM X 6 † With our Hermitian γµ convention each term unΓ5γµun is purely imaginary, but the sum ψ¯Γ5γµψ is real. h i 9 counts the number of of positive chirality (Γ5un =+un) zero modes minus the number of negative chirality (Γ u = u ) zero modes of D. This number 5 n − n 6 is the index of the operator D, and the position dependent quantity 6 ρ (x) ρ (x)= u† (x)Γ u (x) + − − n 5 n n X is the index density. The integer-valued Dirac index is a topological invariant: smooth deformations of the metric or gauge field do not change the index. When the index is non-zero the numbers of left- and right-handed Dirac eigenmodes are different. Consequently the diagonalized path integral mea- ¯ sure n d[χn]d[χn] is not invariant under global chiral phase rotations ψ exp iβΓ5 ψ, ψ¯ ψ¯ exp iβΓ5 . Q 7→ { } 7→ { } Both n d[¯χn] and n d[χn] are multiplied by the same factor, so the measure changes by Q Q d[¯χ ]d[χ ] exp 2iβ(n n ) d[¯χ ]d[χ ]. n n → {− + − − } n n n n Y Y Kazuo Fujikawa’s great insight was to identify this phase as a Jacobian that prevents the chiral symmetry of the classical action being shared by the output of the path integral7. As a result of the Jacobian, the Matthews- Salam determinant is not invariant under global chiral transformations and this is the mathematical origin of the non-conservation of the chiral charge. We would like to be able to interchange the order of the sum and integral to write ? n n = ddx√g u† (x)Γ u (x) . + − − n 5 n M n Z X The sum is not necessarily uniformly convergent, however, and to get the same answer as when we integrate before summing a suitable regularization scheme has to be chosen. A gauge-invariant regulator that allows the interchange is provided by Fujikawa’s heat kernel cutoff

2 def d † −tλn Tr Γ5 = lim d x√g un(x)Γ5un(x)e { } t→0 M n Z X n o d −tD6 †D6 = lim d x√g tr Γ5e . t→0 { } Z 7K. Fujikawa, Path-Integral Measure for Gauge-Invariant Fermion Theories, Phys. Rev. Lett. 42 (1979) 1195; Path integral for gauge theories with fermions, Phys. Rev. D 21 (1980) 2848; Erratum: 22 (1980) 1499.

10 −tD6 †D6 On a closed compact manifold the integrals of tr Γ5e are necessarily independent of t as only the zero modes contribute{ — all other} eigenmodes of D†D come in opposite chirality pairs in which each member has an iden- 6 6 2 tical exp tλn regulator factor. The t dependence in the pre-integration {− } −tD6 †D6 expression for tr Γ5e is very useful however. In ﬂat space and with no gauge ﬁeld the heat{ kernel}

′ 2 2 1 x x x et∇ x′ = exp | − | h | | i (4πt)d/2 − 4t is very short-ranged and tends to δd(x x′) as t becomes small. In d Eu- clidean dimensions the Dirac heat kernel− behaves similarly, and explores the geometry and gauge ﬁelds only for points x very close to x′. As a consequence the position-diagonal matrix element

† † 2 x e−tD6 D6 x = x n n e−tD6 D6 n n x = u (x)e−tλn u† (x) h | | i h | ih | | ih | i n n n n X X has a short-time asymptotic expansion of the form

∞ † 1 x e−tD6 D6 x A (x)tm, h | | i ∼ (4πt)d/2 m m=0 X where the Am(x) are spin-space (and group representation space, if present) matrix-valued functions. Only the t-independent m = d/2 term should survive after taking the trace with Γ5 and integrating over a compact manifold. Heat kernel computations of the Am(x) can become tedious, but some basic techniques for doing them will be discussed later. Here we just state the result for a single ﬂavour Dirac fermion on a closed four-dimensional spin manifold: 1 1 n n = tr F 2 + tr R2 + − − 8π2 { } 192π2 { } Z αβγδ Z αβγδ 1 4 ε 1 4 ε a b = d x√g tr FαβFγδ + d x√g R bαβR aγδ . 32π2 √g { } 768π2 √g Z Z In the ﬁrst line above the F and R are matrix-valued two-forms 1 1 F = λˆ F a dxµ dxν , R = Ra dxµ dxν. 2 a µν ∧ 2 bµν ∧

11 on M. In the second line we have written out all the space-time indices explicitly and inserted factors of √g so that the expressions in the large parentheses are tensors. The objects being integrated are called characteristic classes, and their integrals are topological invariants. For example

1 σ(M) tr R2 = 192π2 { } − 8 Z where σ is the Hirzebruch signature of the 4-Manifold M which depends on how certain closed submanifolds of M intersect one another. † Both the notion of an index and the integral of tr Γ e−tD6 D6 still make { 5 } sense when the manifold is non-compact or has a boundary. In this case, however, the integral may depend on t. This is because to be able to talk about eigenmodes in an inﬁnite system we need to impose some self-adjoint boundary conditions at large distance and such boundary conditions typically mix left and right handed modes. As a consequence, when u(x) satisﬁes these conditions there is no guarantee that Γ5u(x) will satisfy them, and the pairing of λn eigenmodes may fail. The index must then be obtained as the t limit± (rather than the t 0 limit) of the regulated trace. The t 0 limit→∞ is → → still the one that captures the divergence of the axial current.

Twisted mass, zero modes, and the θ-vacuum: Suppose that we replace the standard d = 4 Dirac mass term by a chirally rotated version consisting of a mixed scalar and pseudo-scalar mass so that

D + m D + m + iγ m , 6 → 6 1 5 2 where m1 = m cos φ, m2 = m sin φ with m real and positive. In many ways the angle φ acts analogously to the phase of superconducting order parameter. In particular, space and time gradients of φ induce charge densities and and current ﬂows8. For now, though, we only the consider the situation where m and φ are constant. The mass term is no longer proportional to the identity matrix

meiφ 0 m m + iγ m = m(cos φ + iγ sin φ)= → 1 5 2 5 0 me−iφ 8J. Goldstone. F. Wilczek, Fractional Quantum Numbers on Solitons, Phys Rev Lett 47 (1981) 986-989

12 and the Euclidean eigenvalue problem becomes

λu =(D + m + iγ m )u. 6 1 5 2 Consider separately the case of chiral zero modes and of non zero-modes. In the presence of of a zero mode Du = 0, γ u = u , we read off directly 6 0 5 0 ± 0 that (D + m + iγ m )u = me±iφu . 6 1 5 2 0 0 When Du = iλ u , λ = 0, the eigenmode becomes a linear combination 6 n n n n 6 aun + bγ5un. In this two-dimensional space the eigenvalue problem simplifies to iλ + m im a a n 1 2 = µ im iλ + m b b 2 − n 1 The resulting eigenvalues differ from the φ = 0 ones, but for the Matthews- Salam determinant we only need the product of the two eigenvalues

iλ + m im µ µ = n 1 2 = λ2 + m2 + m2 = λ2 + m2, 1 2 im iλ + m n 1 2 n 2 − n 1

and so, for each gauge-ﬁeld conﬁguration we have

Z = Det(D + m + iγ m )= mn++n− ei(n+−n−)φ (λ2 + m2), F 6 1 5 2 n λYn6=0 where n+ and n− are the number of right and left-handed chiral zero modes. Let us temporarily assume that there are no zero-mode factors and compute ψψ¯ from h i ′ 2 2 2 ZF = (λn + m1 + m2) λYn6=0 as

¯ 1 ∂ 2 2 2 ψψ = ln (λn + m1 + m2) h i V ∂m1 ! λYn6=0 1 ∞ 2m = 1 V λ2 + m2 n=1 n X 1 ∞ 1 = cos φ V iλ + m n=−∞ n X 13 Here we are thinking of the eigenvalue λn as positive for n > 0, and that λ = λ is its negative partner. −n − n Similarly

¯ 1 ∂ 2 2 2 i ψγ5ψ = ln (λn + m1 + m2) h i V ∂m2 ! λYn6=0 1 ∞ 1 = sin φ. V iλ + m n=−∞ n X In the large space-time volume limit the sum goes over to an integral 1 ∞ 1 ∞ ρ(λ) dλ V iλ + m → iλ + m n=−∞ n −∞ X Z where 1 ∞ ρ(λ)= δ(λ λ ) V − n n=−∞ X is the local density of eigenvalues. The Sokhotski-Plemelj theorem tells us that for infinitesimal ǫ ∞ ρ(λ) ∞ ρ(λ) dλ dλ =2πρ(0), iλ + ǫ − iλ ǫ Z−∞ Z−∞ − and so provides a generalization ψψ¯ = πρ(0) cos φ, i ψ¯Γ ψ = πρ(0) sin φ. h im→0+ h 5 im→0+ of the Banks-Casher relation9 for the massless limit of the expectation values. In an interacting theory a non-zero value for ψψ¯ signals the h im→0+ spontaneous breakdown of chiral symmetry. In deriving these Banks-Casher formulæ we have tacitly assumed a fixed background gauge field, and therefore a fixed set of eigenvalues. The Banks- Casher formulæ therefore apply most transparently in the lattice gauge theory quenched approximation where we ignore the feedback from the Matthews- Salam determinant on the gauge field configurations. In the quenched approximation we simply replace ρ(λ) by its average over the gauge configurations. 9T. Banks, A. Casher, Chiral symmetry breaking in confining theories, Nucl. Phys. B169, (1980) 103-125.

14 When we do include the eﬀect of the determinant, and in the presence of a topological θ term

iθ iθ exp i(n n )θ = exp tr F 2 = exp d4xεαβγδtr F F , { +− − } 8π2 { } 32π2 { αβ γδ } Z Z the gauge conﬁgurations are weighted by

2 2 (n++n−)/2 i(θ+φ)(n+−n−) 2 2 (m1 + m2) e (λn + m ). λYn6=0 This expression depends on the chiral phase angles only in the combination θ + φ. When we have many fermion flavours ψi the total effective θ angle is ¯ θ = θ + i φi. There is of course no easy way to estimate the effects of the determinant weighting.P An exception occurs when we consider a theory with a single fermion flavour and calculate ψψ¯ in the limit m 0. This is because h i → in this limit we expect only winding number 1 gauge configurations contribute. For winding number 1 we have, for each± gauge field configuration, a single Γ u = u zero mode.± After the determinant weighting this mode 5 0 ± 0 contributes

† 1 u0(x)(1 γ5)u0(x) ±i(φ+θ) 2 2 1 ±±iφ me (λn + m ) , 2 me × Z0 λYn6=0 to ψ¯(1 γ )ψ , where u is localized near the instanton. The normalization h ± 5 i 0 factor Z0 is the partition function in the zero winding-number sector — all other sectors giving zero at m = 0 because of the zero-eigenvalue factors. After the integration over the gauge ﬁelds ψ¯(1 γ )ψ must be independent h ± 5 i of x, so we change nothing if we integrate over x and divide by the space-time volume V . Because the zero mode u0 is normalized this leads to

′ 1 ¯ 1 iθ Z1 ψ(x)(1 γ5)ψ(x) = e , 2h ± i 2 VZ0

′ where Z0 is the partition function in the winding 1 sector with the zero- eigenvalue factor omitted. Let us now deﬁne ±

Z′ Σ= 1 VZ0

15 in terms of which we have ψψ¯ (x) = 1 ψ¯(x)(1+Γ )ψ(x) + 1 ψ¯(x)(1 γ )ψ(x) h i 2 h 5 i 2 h − 5 i 1 iθ −iθ = 2 Σ(e + e ), = Σ cos θ. Similarly i ψγ¯ ψ(x) = i 1 ψ¯(x)(1 + γ )ψ(x) i 1 ψ¯(x)(1 γ )ψ(x) h 5 i 2 h 5 i − 2 h − 5 i = i 1 Σ(eiθ e−iθ). 2 − = Σ sin θ − Note that the φ angle of the chiral mass no longer appears, and the condensate depends only on the θ angle of the topological winding number term. This is quite different from what happens in the quenched approximation. Now in an infinite system the above argument is not really valid. Al- though the contributions of winding number n are suppressed by powers of m|n| the actual dimensionless factor is (mΣV )|n| and large V will overwhelm small m. However, if the winding number arises from a dilute gas of instantons that are more widely spaced than the range of the zero modes tied to them, then at any point x only the zero mode near that point will contribute and our formula remains correct. A generating function that captures these muti-zero mode contributions is 2 2 Z(m1, m2, θ) = exp m1 + m2 cos(θ + φ)V Σ . q This approximate partition function depends only on the chiral angles in the combination θ + φ — as we know it should. The expression in the exponent ′ −1 is also dimensionally correct because V Σ = Z1/Z0 has dimension (mass) . −1 Using φ = tan (m2/m1), we find 1 ∂ ψψ¯ = ln Z(m1, m2, θ) h i V ∂m1

∂ 2 2 = m1 + m2 cos(θ + φ)Σ ∂m1 q m ∂φ = 1 cos(θ + φ) m2 + m2 sin(θ + φ) Σ 2 2 1 2 m + m − ∂m1 ( 1 2 q ) = (cospφ cos(θ + φ) + sin(θ + φ) sin φ)Σ = Σ cos θ.

16 Similarly 1 ∂ i ψγ¯ 5ψ = ln Z(m1, m2, θ) h i V ∂m2

∂ 2 2 = m1 + m2 cos(θ + φ)Σ ∂m2 q m ∂φ = 2 cos(θ + φ) m2 + m2 sin(θ + φ) Σ 2 2 1 2 m + m − ∂m2 ( 1 2 q ) = (sin φ cos(θ + φ) sin(θ + φ) cos φ)Σ p − = Σ sin θ. − Both expressions coincide with those from the single-instanton calculation and so contain exactly the same physics. It is interesting to see how the φ dependence drops out in these computations10. The approximate partition function Z(m1, m2, θ), a φ = 0 version of which was first introduced in11, is a dilute instanton gas approximation and is valid when the instantons are sufficiently far apart that tunneling between them does not split the zero modes away from zero. Being dilute means that mΣV is small, and if we accept Banks-Casher’s formula Σ = πρ(0) this allows m to be much closer to zero than the mean separation [ρ(0)V ]−1 of the iλ eigenvalues on the imaginary axis. Recall that validity of the ± n Banks-Casher formula requires m to be appreciably further away from the imaginary axis than the mean eigenvalue separation so that it perceives the discrete eigenvalues as a continuum with density ρ(0). The papers cited above contain more discussion of on the apparent conflict between the approximate partition function and the Banks-Casher relation.

Vector gauge currrents: The anomaly discussion is little changed in the a ˆ ˆ presence of a non-abelian vector gauge ﬁeld Aµ iAµλa with the λa the Hermitian generators of a Lie algebra. As said earlier≡ the Dirac operator

D(A)= γaeµ ∂ + 1 σbc ω + A 6 a µ 2 bcµ µ 10J. J. M. Verbaarschot, T. Wettig, Dirac spectrum of one-ﬂavor QCD at θ = 0 and continuity of the chiral condensate, Phys. Rev. D 90 (2014)116004 (12pp); M. Kieburg, J. J .M. Verbaarschot, T. Wettig, Dirac spectrum and chiral condensate for QCD at ﬁxed θ-angle, Phys. Rev. D99, (2019) 074515 (21pp), and its sequels 11For example: H. Leutwyler, A. Smilga, Spectrum of Dirac operator and role of winding number in QCD, Phys. Rev. D46 (1992) 5607-5632.

17 remains skew Hermitian in L2(M) and the ﬂat space the eigenfunctions obey

µ γ (∂µ + Aµ)un = iλnun (∂ u† u† A )γ = iλ u† , µ n − n µ µ − n n where the un have an extra index corresponding to those on the λâ matrices. The Dirac index Tr Γ5 is now a topological invariant of both the manifold and the gauge bundle{ . } What is a little different is that we now have the option of replacing the ¯ µ gauge-singlet current ψΓ5γ ψ axial current by a matrix-valued axial current µ J5 that is defined by taking its trace with with the generators

tr λˆ J µ = ψ¯Γ γµλˆ ψ. { a 5 } 5 a Then similar algebra as before gives us a regulated expression for the anomaly as † tr λˆ J µ = tr λˆ Γ e−tD6 D6 , { ah∇µ 5 i} { a 5 } where J µ def= ∂ J µ + [A ,J ] ∇ν 5 ν 5 ν 5µ is the gauge covariant derivative. For four ﬂat dimensions the heat kernel gives 1 tr λˆ J µ = εαβγδtr λˆ F F , { a∇µ 5 } 16π2 { a αβ γδ} ˆ a where Fαβ = λaFαβ the non-abelian gauge ﬁeld curvature.

Chiral gauge fields: Unlike a vector gauge field, a chiral gauge field is one that couples differently to the left and right-handed parts of ψ. A chiral gauge field has the form µ +Γ5 µ, where µ is its vector part and Γ5 µ is its axial part. We can alsoV decomposeA it intoV its chiral components A

R = + , L = , µ Vµ Aµ µ Vµ −Aµ where Rµ couples only to the right-handed fermions and Lµ only to the left handed fermions. Both Lµ and Rµ are either pure imaginary or skew- Hermitian matrices. When the axial current fails to be conserved in a theory with vector gauge ﬁelds, it simply means that particles are being created ex nihilo. When the non-conserved axial current couples to an axial gauge ﬁeld it portends

18 disaster: the effective action is no longer gauge invariant, and if the gauge field is to be integrated over (i.e. dynamical) this makes the theory internally inconsistent. Not only are axial gauge fields an existential threat to the theory, but, because the gauge field couples differently to the left- and right-handed fermions, the Euclidean operator D is no longer skew-Hermitian and he mathematics behind the anomaly6 is rather more intricate. Consider four dimensions. We use the representation

1 0 0 1 iσ Γ = , γ = , γ = i , 5 0 1 0 1 0 i iσ −0 − i and then 0 D (L) D(L, R)= L 6 DR(R) 0 where

D (L) = (∂ + L ) iσ (∂ + L ), L 0 0 − i i i DR(R) = (∂0 + R0)+ iσi(∂i + Ri).

Under a combined vector and axial gauge transformation we have

D(R, L) e{−iθV +iΓ5θA}D(R, L)e{iθV +iΓ5θA} 6 7→ 6 or

−i(θV −θA) +i(θV +θA) 0 DL(L) e 0 0 DL(L) e 0 −i(θV +θA) +i(θV −θA) , DR(R) 0 7→ 0 e DR(R) 0 0 e e−iθL 0 0 D (L) e+iθR 0 = L , 0 e−iθR D (R) 0 0 e+iθL R 0 e−iθL D (L)eiθL = L , e−iθR D (R)eiθR 0 R 0 D (LgL ) = L , D (RgR ) 0 R where

θ = θ θ , L V − A θR = θV + θA,

19 and

gL −iθL iθL Lµ = e (Lµ + ∂µ)e , gR −iθR iθR Rµ = e (Rµ + ∂µ)e . We now see that [D(L, R)]† = D(R, L) = D(L, R). As a consequence 6 −6 6 −6 of the lack of hermiticity when L = R, the operator D(L, R) is not guaranteed to possess a complete set of6 orthonormal modes.6 What it always has, however, is a singular value decomposition and we can use this to explore the anomaly12. The singular value decomposition utilizes the eigenvalue problem for the self-adjoint operators D†D and DD† where 6 6 6 6 D (R)D (R) 0 D†D = L R 6 6 − 0 DR(L)DL(L) D (L)D (L) 0 DD† = L R . 6 6 − 0 DR(R)DL(R) These operators have two complete orthonormal sets of eigenmodes D†Dϕ = λ2 ϕ 6 6 n n n DD†φ = λ2 φ , 6 6 n n n 2 but only one set of positive-valued eigenvalues λn. This is because the the non-zero eigenvalues of D†D and DD† coincide: if 6 6 6 6 D†Dϕ = λ2 ϕ 6 6 n n n and ϕn is normalized then (DD†)Dϕ = λ2 Dϕ 6 6 6 n n6 n and (phase) φn = Dϕn 2 λn 6 is a normalized eigenfunction of DDp† with the same eigenvalue. We can choose the phase so that 6 6 Dϕ = λ φ , 6 n n n D†φ = λ ϕ 6 n n n 12 K. Fujikawa, Evaluation of the chiral anomaly in gauge theories with Γ5 couplings, Phys. Rev. D 29 (1984) 285-292.

20 where the λn are real and positive. In more detail, the ﬂat-space version of these equations are

γµ(∂ + +Γ )ϕ = λ φ , µ Vµ 5Aµ n n n γµ(∂ + Γ )φ = λ ϕ . − µ Vµ − 5Aµ n n n Taking the complex conjugate of the second of these equations, using hermiticity, and then transposing leads successively to

γ∗µ(∂ ∗ +Γ∗ ∗ )φ∗ = λ ϕ∗ , − µ −Vµ 5Aµ n n n (γµ)T (∂ T +ΓT T )(φ† )T = λ (ϕ† )T , − µ −Vµ 5 Aµ n n n † µ † φ (←−∂ µ µ +Γ5 µ)γ = λnϕ . n −V A − n The last line will be useful in the anomaly calculation. We now expand the fermion ﬁelds as

ψ(x) = ϕn(x)χn n X ¯ † ψ(x) = φn(x)¯χn n X where χn andχ ¯n are independent Grassmann variables. In terms of these expansions the action becomes 1 S = d4x √g ψ¯(Dψ)+ (D†ψ)ψ 2 6 6 Z = λmχ¯nχn. n X The Jacobean determinants in the change of measure from d[ψ¯(x)]d[ψ(x)] to d[¯χn]d[χn] will not cancel one another because we performing a singular value decomposition D = U T diag(λ )V 6 n rather than a similarity transform. We have

Z = Det(D) = det(U T )det(V ) d[¯χ ]d[χ ] exp λ χ¯ χ 6 n n { n n n} n Z X T = det(U )( λn)det(V ). n Y 21 For a fixed (i.e. non-dynamical) background gauge field, the phases do cancel in the Grassmann integral for the Green function, which gives13 1 ψ¯ (x)ψ (y) = d[¯χ ]d[χ ]ψ¯ (x)ψ (y) exp λ χ¯ χ h α β i ( λ ) n n α β { m n n} n n n Z X φ† (x)ϕ (y) = Q nα nβ . λ n n X From this we define φ† (x)Γ γµϕ (x) jµ(x) = nα 5 nα . h 5 i λ n,α n X Now we proceed in the same manner as for the non-chiral case and directly evaluate the divergence of the vacuum expectation of the axial current. We will give the details for an abelian axial current, but the algebra will be µ almost identical for a non-abelian current except that the ∂µj5 divergence µ must be replaced by the gauge-covariant divergence µJ5 . We use the explicit form of the equations for derivatives∇ of ϕ and φ† and find that the gauge fields and the gamma matrices conspire nicely to give (∂ φ† )Γ γµϕ + φ† Γ γµ(∂ ϕ ) ∂ jµ(x) = µ n 5 n n 5 µ n µh 5 i λ n n X † µ † µ φ (←−∂ µ µ +Γ5 )Γ5γ ϕn + φ Γ5γ (∂µ + µ +Γ5 µ)ϕn = n −V A n V A λ n n X † µ † µ φ (←−∂ µ µ +Γ5 )γ Γ5ϕn + φ Γ5γ (∂µ + µ +Γ5 µ)ϕn = − n −V A n V A λ n n X † † = ϕnΓ5ϕn + φnΓ5φn . n X The axial anomaly therefore requires an evaluation of

Q (x)= ϕ† (x)Γ ϕ (x)+ φ† (x)Γ φ (x) . 5 { n 5 n n 5 n } n X 13As there is no way to include a standard mass term in the action and at the same time preserve gauge invariance we will assume that there are no exact zero modes. This is not a problem as the chiral gauge anomaly is related to the failure of gauge invariance rather than the particle-creation eﬀect of zero modes.

22 This we can Fujikawa-regulate as

2 2 Q (x) = ϕ† (x)Γ e−tλn ϕ (x)+ φ† (x)Γ e−tλn φ (x) 5 { n 5 n n 5 n } n X † † = ϕ† (x)Γ e−tD6 D6 ϕ (x)+ φ† (x)Γ e−tD6 D6 φ (x) { n 5 n n 5 n } n X −tD6 †D6 −tD6 D6 † = Tr Γ5 e + e . n o Observe that under gauge transformations both D†D and DD† undergo similarity maps, so we have a gauge invariant regularization.6 6 6 As6 a consequence the resulting non-conservation equation involves the gauge fields and their derivatives only in covariant combinations and is therefore known as the covariant anomaly. Now tDL(R)DR(R) tDL(L)DR(L) † † e 0 e 0 e−tD6 D6 + e−tD6 D6 = + 0 etDR(L)DL(L) 0 etDR(R)DL(R) etDL(R)DR(R) 0 etDL(L)DR(L) 0 = + 0 etDR(R)DL(R) 0 etDR(L)DL(L) and the regulated Γ5 trace of the last line reduces to a sum of one-half each of the anomalies of two Dirac fields vector-coupled to fields Rµ and Lµ. The anomaly thus reduces to the usual case when Lµ = Rµ. Similarly, we have ∂ ψγ¯ µψ = φ† (x)φ (x) ϕ† (x)ϕ (x) , µh i n n − n n n X so the vector current also has an anomaly given by Q(x)= φ† (x)φ (x) ϕ† (x)ϕ (x) . n n − n n n X This we regulate as

† † Q(x) = Tr e−tD6 D6 e−tD6 D6 − Now n o tDL(L)DR(L) tDL(R)DR(R) † † e 0 e 0 e−tD6 D6 e−tD6 D6 = − 0 etDR(R)DL(R) − 0 etDR(L)DL(L) etDL(L)DR(L) 0 etDL(R)DR(R) 0 = 0 etDR(L)DL(L) − 0 etDR(R)DL(R) − − 23 which, on taking the trace, is one half of the regulated difference of the axial anomalies of two ordinary vector couplings of gauge fields Lµ and Rµ. For the covariant anomaly, therefore, we can treat the left and right Weyl fermions as independent of each other; each having one-half of the full Dirac anomaly but with opposite signs. In four dimensional flat space, for example, and for a left- or right-handed Weyl particle interacting only with its corresponding left or right gauge field we have 1 tr λ J µ = εαβγδtr λ F F . { a∇µ } ±32π2 { a αβ γδ} where is for left- or right-handed fermions and gauge field and F = λaF a ± αβ αβ is the corresponding Lie-algebra-valued gauge field curvature. Another route: There is an alternative approach to non-Hermitian Dirac operators14. For motivation, consider first a finite-dimensional non-Hermitian matrix M acting on a space V . Although such a matrix is not guaranteed to be diagonalizable, it will be in the generic case in which all its eigenvalues are distinct. It then possesses complete sets of left and right eigenvectors un, vn obeying

Mun = λnun, † † vnM = λnvn.

We have not distinguished between the left and right eigenvalues because the two sets of eigenvalues coincide: they are determined by the same characteristic equation. Although a right (left) eigenvector with eigenvalue λn is no longer orthogonal to all right (left) eigenvectors with a diﬀerent eigenvalue λm, it remains true that a left eigenvector with eigenvalue λn is orthogonal to every right eigenvector with a diﬀerent eigenvalue λm. This is because

† † † † λnvnum =(vnM)um = vn(Mum)= λmvnum, so 0=(λ λ )v† u . n − m n m This equation yields a proof of the equality of the left and right eigenvalues that does not require an appeal to the ﬁnite-dimensional notion of a characteristic polynomial: if um did not have a corresponding vm with the

14L. Alvarez-Gaum´e, P. Ginsparg,The Structure of Gauge and Gravitational Anomalies, Annals of Physics 161 (1985) 423-490.

24 same eigenvalue then um would be perpendicular to all vectors in V — in contradiction with the non-degeneracy of the inner product. We may choose the phases and normalization of the two eigenfunction sets so that † vnum = δmn † whence vnMum = λnδnm which we can write as

† V MU = diag(λ1,λ2,...).

Although neither of the eigenvector families compose an orthonormal set, the matrices U and V obey V†U = I so we still have

† det M = det (V MU)= λn. n Y The left eigenvalue equation can be written as

† ∗ M vn = λnvn, and if we assume that these properties of ﬁnite matrices continue to hold for our non-Hermitian Dirac operator we will have complete sets of eigenfunctions ϕn, φm obeying

Dϕ = λ ϕ , 6 n n n D†φ = λ∗ φ , 6 n n n where φ ,ϕ = δ . h n mi mn The expansions of ψ, ψ† are formally the same as before

ψ(x) = ϕn(x)χn, n X ¯ † ψ(x) = φn(x)¯χn, n X but the the new ϕ’s and φ’s will not in general coincide with the older functions. Further, the eigenvalues λn are in general complex whereas the previous singular values λn were real and positive. Indeed the eigenvalues of

25 the non-Hermitian D are not simply related to its singular values. It is true, however, that for a6 ﬁnite N-by-N matrix that

N N λold = λnew = det M n | n | | | n=1 n=1 Y Y Now, if we like, we can include a mass in the action

S[ψ, ψ¯] = ddx√g 1 ψ¯(Dψ) (Dψ)ψ + mψψ¯ 2 6 − 6 Z n o = ddx√g 1 ψ¯(Dψ)+ (D†ψ)ψ + mψψ¯ 2 6 6 Z n o = (λn + m)¯χnχn. n X The computation of the divergence of the axial current now follows on much the same lies as before, and leads to

jµ = 2m ψ¯Γ ψ + Q˜ (x) ∇µ 5 − h 5 i 5 where ˜ † Q5(x)=2 φn(x)Γ5ϕn(x). n X The problem is that it not obvious how to regularize this last sum with a Fujikawa-style heat kernel. This is because

† 2 φ† e−tD6 D6 Γ ϕ = φ† e−t|λn| Γ ϕ . n 5 n 6 n 5 n Instead we have 2 † −tD6 D6 † −tλn φne Γ5ϕn = φne Γ5ϕn, which, because the λn are complex, does not necessarily provide a controlled cutoﬀ on the sum. Denied our previous route to regulate Q˜5(x), we must approach the problem from another direction. We observe that unlike singular values which are invariant under both vector and axial gauge transformations our left/right eigenvalues λn are not invariant under axial gauge transformations

D eiΓ5β(x)DeiΓ5β(x) = D + iβ(x)Γ , D + O(β2). 6 → 6 6 { 5 6 }

26 We can compute the change in λn via the left/right eigenvector version of first order perturbation theory. We start from Dϕ = λ ϕ δDϕ + Dδϕ = δλ ϕ + λ δϕ 6 n n n ⇒ 6 n 6 n n n n n and take matrix elements φ δD ϕ + φ D δϕ = δλ φ ϕ + λ φ δϕ . h n| 6 | ni h n|6 | ni nh n| ni nh n| ni † As φn D δϕn = D φn δϕn = λn φn δϕn and φn ϕn = 1, we conclude thath |6 | i h6 | i h | i h | i δλ = φ δD ϕ . n h n| 6 | ni The change in the eigenvalue λ of D is therefore given by n 6 δλ = ddxφ† (x) iβ(x)Γ , D ϕ (x)=2iλ ddxφ† (x)β(x)Γ ϕ (x). n n { 5 6 } n n n 5 n Z Z The change in W ( , ) = ln(Det(D)) = ln λ eff V A 6 n n X is therefore formally given by δλ δ ln(Det(D)) = n { 6 } λ n n X d † = 2i d x φn(x)β(x)Γ5ϕn(x). n Z X d = 2i d x β(x)Q˜5(x). Z As we saw earlier, the quantity ˜ † Q5(x)= φn(x)Γ5ϕn(x) n X µ is the expression we would get for µj5 by manipulating the Green function. Now however, instead of attemptingh∇ i to regulate the ill-defined sum in Q˜5(x), we have the option of bypassing the eigenfunction mode expansion entirely in favour of regulating the fermion determinant which only depends on the eigenvalues. For this we may use a zeta-function regulator

def −s ζD6 (s) = λn . n X 27 When the zeta function can be analytically continued to s = 0, we can deﬁne

def ′ ln(Det(D)) eff [ , , ] = lim ζD6 (s). 6 ≡ W V A − s→0 Here the prime denotes a derivative of the zeta function with respect to s. Given the regularized determinant, we extract the axial current as15 δ [ , , ] jµ(x)= Weff V A . 5 − δ (x) Aµ Under the axial gauge transformation we have δ µ(x)= i µβ(x), and after an integration by parts this relates the anomalyA to the failure∇ of the zeta- defined effective action to be gauge invariant: δ [ , , ] δ ln(Det(D)) = Weff V A δ (x)dx = i jµ β(x) ddx. β{ 6 } δ (x) Aµ h∇µ 5 i Z Aµ Z This route leads to a non gauge-invariant effective action W ( , ) rather eff A V than a gauge-covariantly regularized current and it generates an expression for the anomaly that differs from the covariant anomaly. As we explain in the next section it satisfies the Wess-Zumino consistency condition and consequently is known as the consistent anomaly. Notice also that the eigenvalues are invariant under a vector gauge transformation, so, in contrast with the covariant anomaly, there is no consistent anomaly in the vector current. In the early days of of anomaly evaluation a number of authors16 computed the short time expansion of exp t[D( , )]2 and treated it as Fujikawa- { 6 V A } style “regulator’ either without the authors realizing that it does not provide a convergence factor, or when they did they artificially replaced by i 2 A A so as to make D Hermitian and its eigenvalues positive — but still gauge- dependent. These6 calculations made sense, and to their authors seemed preferable to other calculations because their output was the consistent anomaly. This comes about because the identity ∞ 1 1 2 = ts−1e−tλ dt λ2s Γ(s) Z0 15 µ µ µ The minus sign comes from our definition j5 = ψ¯Γ5γ ψ which has the γ and Γ5 in the opposite order than that output by the functional derivative. 16For example: M. B. Einhorn, D. R. T. Jones, Comment on Fujikawa’s path-integral derivation of the chiral anomaly, Phys. Rev. D 29 (1984) 331-9; S-K. Hu, B-L. Young, D. W. McKay, Functional integral and anomalies in theories with S, P , V , and A currents, Phys. Rev. D 30, (1984) 836-839.

28 shows that the ζ-regularized determinant (in which s rather than t is playing the role of the regulator) makes use of the same information as the small t expansion of exp t[D( , )]2 . The reason why the D2 calculations give different answers than{ 6 theV AD†D} calculations was explained6 by Fujikawa in his previously cited paper on6 anomalies6 with γ5 couplings. However the gauge non-covariant calculation is performed, the resulting anomaly for a fermion interacting with only an L or an R field is 1 1 tr λ J µ = εαβγδtr λ ∂ (A ∂ A + A A A . { a∇µ } ±24π2 a α β γ δ 2 β γ δ The is for L or R. This expression is not expressible in terms of covariant quantities,± and, even in the abelian case where it reduces to the covariant- looking expression 1 ∂ J µ = εαβγδF F , µ ±96π2 αβ γδ it differs from the covariant anomaly by being three times smaller. Anomaly inflow and the Callan-Harvey mechanism: The two methods of computing the anomaly for chiral gauge fields — regulating the gauge current or regulating the fermion determinant — give different answers. We have referred to them, respectively, as the covariant or consistent anomalies. Which is correct? The answer is that both are “correct,” but they compute the divergence of different currents. Understanding this distinction will give us a clue as to the physical origin of anomalies. While the non-zero divergence of a current associated with a global symmetry is a sign of particle creation ex nihilo, the failure of the conservation law of a gauge current indicates a loss of gauge invariance. If S[A] is the a action functional of a theory with gauge field Aµ = λaAµ, where the matrices µ λa are the generators of su(N), the vacuum expectation value J (x) of the matrix-valued gauge current can be read off from the variation

δS[A]= ddx tr J µ(x)δA (x) . { µ } Z Under a gauge transformation the ﬁeld changes as

A Ag = g−1A g + g−1∂ g, µ → µ µ µ where g SU(N). For an inﬁnitesimal transformation g = 1 ǫ the transformation∈ becomes A A + δ A , where δ A = ([A , ǫ]+−∂ ǫ) ǫ. µ → µ ǫ µ ǫ µ − µ µ ≡ −∇µ 29 The corresponding change in S[A] is therefore

δ S = ddx tr J µ([A , ǫ]+ ∂ ǫ) ǫ − { µ µ } Z = ddx tr ǫ(∂ J µ + [A ,J µ]) , { µ µ } Z where we have integrated by parts and use the cyclic property of the matrix trace to get the second line. As ǫ is arbitrary, the covariant divergence

J µ ∂ J µ + [A ,J µ] ∇µ ≡ µ µ is zero if and only if S[A] is gauge invariant. We know that even when the original action is gauge invariant, this invariance can be lost when we integrate out the fermions to produce an eﬀective action. Then

J µ = G(A), ∇µ where the anomaly G(A) is a local polynomial in the Aµ and their derivatives. Because this anomaly is found as the variation of the functional S[A], and because (δ δ ′ δ ′ δ )S = δ ′ S ǫ ǫ − ǫ ǫ [ǫ,ǫ ] the diﬀerential polynomial G[A] satisﬁes the Wess-Zumino consistency con- dition17

′ x ′ d (δ tr ǫ G(A) δ ′ tr ǫG[A] ) dd = tr [ǫ, ǫ ]G(A) d x. ǫ { } − ǫ { } { } Z Z This is why G(A) is called the “consistent” anomaly. The right hand side of the (non) conservation equation is not gauge covariant, however, and so neither is the left. The gauge current itself is therefore not covariant, and the physical meaning of the (non) conservation equation is quite unclear. On the other hand, when we regulate the current so as to preserve its gauge covariance the right hand side of the non-conservation law is expressed in terms of gauge covariant quantities but the resulting anomaly does not satisfy the Wess-Zumino consistency condition and so cannot be obtained as the derivative of the vacuum functional. 17J. Wess, B. Zumino, Consequences of anomalous Ward identities, Physics Letters 37 B (1971) 95-97.

30 Bill Bardeen and Bruno Zumino18 showed that the covariantly regulated current could be obtained by adding to the non-covariant, action-derivative current, a second non-covariant differential polynomial in A. This addition to the current is known as the Bardeen polynomial. Although apparently not noticed by Bardeen and Zumino, this contribution to the current has a both a simple mathematical origin and a physical interpretation. The basic idea is that a theory with an anomaly might actually be a low energy effective theory of chiral fermions that are trapped on defects or as surface modes of a higher dimensional anomaly-free theory. Curt Callan and Jeff Harvey19 constructed explicit examples of where this happens. Their models are anomaly-free but have position dependent Higgs-induced mass terms in which bounding surfaces, vortices or domain walls trap low energy modes whose effective theory is chiral and anomalous. Because the bulk- plus-defect theory is anomaly free, any charge non-conservation or failure of gauge invariance in the defect-trapped modes must be result of gauge currents flowing into the defects from the bulk. A four-dimensional example illustrates the idea and en passant explains the factor of one third between the consistent and covariant abelian anomalies. Let us imagine that our four-dimensional universe is the boundary of a five dimensional manifold N which hosts a Dirac fermion with a large mass M. The boundary will trap massless chiral fermions whose wavefunctions de- 5 cay as e−Mx as we move into the bulk and away from the x5 = 0 surface. The bulk∼ Dirac equation is

5 5 (D + γ ∂ 5 + M)ψ(t, x, x )=0 6 d=4 x −Mx5 and, for example, a zero energy surface energy state ψ = u0e needs

5 −Mx5 (γ ∂x5 + M)u0e =0

5 so the trapped boundary fermions must be left- or right-handed (i.e. γ u0 = u0) depending on the sign of the bulk mass M. If the bulk fermions are coupled± to an abelian gauge ﬁeld A then a computation shows the low energy (E m) response of the bulk fermions to variations in A is captured by an ≪ 18W. A. Bardeen, B. Zumino, Consistent and covariant anomalies in gauge and gravitational theories, Nucl. Phys. B244 421-453 (1984). 19C. G. Callan, J. A. Harvey, Anomalies and Fermion zero modes on strings and bound- aries Nucl. Phys. B250 427-436 (1985)

31 action sgn(M) C[A]= ελµρστ A F F d5x. 96π2 λ µν στ ZN When we vary A A + δA to compute the bulk current, and after an → integration by parts, we get equal contributions from the explicit A and also from the two A’s in the two F = dA factor. The bulk variation is therefore sgn(M) δC[A] = ελµρστ δA F F d5x, bulk 32π2 λ µν στ Zn and corresponds to a current sgn(M) J λ = ελµρστ F F . 32π2 µρ στ This bulk current leads to charge flowing into the boundary space-time at x5 = 0 at a rate per unit 3-volume of sgn(M) J 5 = ε5µρστ F F . 32π2 µρ στ This inflow from outside our universe is the source of the magically appearing charge on the RHS of the covariant anomaly. Under a gauge transformation A A + ∂ ε, however, the F are λ → λ λ µν unchanged, and the coefficient in sgn(M) δC[A] = (∂ ǫ)ελµρστ F F d5x gauge 96π2 λ µρ στ ZN sgn(M) = ∂ (ǫελµρστ F F )d5x 96π2 λ µρ στ ZN sgn(M) = ǫ(x)(ε5µρστ F F )d4x − 96π2 µρ στ Z∂N is three-times smaller than that in δC[A]bulk. This gauge variation compensates for the gauge non-invariance of the four-dimensional action, and, since the consistent anomaly is the gauge variation of the four dimensional action, the latter coincides up to a sign with the former, and hence is only one-third of the covariant anomaly. The difference in the currents whose divergence is being computed arises from the integration by parts we performed when computing the bulk current. We usually ignore integrated-out terms, but in this case the surface term

1 5µρστ δC[A]surface = sgn(M) 2 ǫ δAµ(AρFστ ) 24π 5 Zx =0 32 makes a contribution 1 Xµ = sgn(M) ǫ5µρστ A F 24π2 ρ στ to the surface current although it arises from the variation of the bulk action. Now sgn(M) ∂ Xµ = ǫ5µρστ F F . µ 48π2 µρ στ When this divergence added to that of the current arising from varying the only the gauge non-invariant surface action we have 1 1 1 + = , 48π2 96π2 32π2 so it accounts for the difference between consistent and covariant anoma- µ µ µ µ lies. Thus Jcovariant = Jconsistent + X , and X is an example of the Bardeen polynomial. Non-abelian generalization: In dealing with the non-abelian gauge fields the algebra is greatly simplified by using the calculus of differential forms. In this language the gauge potential A is a Lie-algebra-valued one-form A = a µ λaAµdx and the field strength F is the Lie-algebra-valued two-form 1 F = dA + A2 = F dxµdxν . 2 µν For even-dimensional manifolds an important object is the 2n-form Chern- character 1 iF n ch [F ]= tr n n! 2π which is the closed differential form whose integral over a spin manifold N computes the gauge-field contribution to the integer valued index. A non abelian Chern-Simons form ω2n−1 can be be constructed so that dω2n−1 = tr F n . For example, { } 2 ω (A) = tr AdA + A3 , 3 { 3 } 1 = tr AF A3 , { − 3 } and 3 3 ω (A) = tr A(dA)2 + A3dA + A5 5 { 2 5 } 1 1 = tr AF 2 F A3 + A5 . { − 2 10 } 33 2n−1 The F -free last term A in each of the second expressions for ω2n−1 has coefficient ∝ n!(n 1)! c =( 1)n−1 − . n − (2n 1)! − If we make a gauge transformation

A Ag = g−1Ag + g−1dg, → then

ω (Ag)= ω (A)+ c tr (g−1dg)2n−1 + dα (A, g), 2n−1 2n−1 n { } 2n−2 where, for example α = tr dgg−1A 2 − { } and 1 1 α (A, g)= tr (dgg−1)(AdA+dAA+A3) (dgg−1)A(dgg−1)A (dgg−1)3A . 4 −2 { −2 − } The Chern-Simons functional C[A] is deﬁned by setting

i n 1 C[A]=2π ω (A), 2π n! 2n−1 ZN where M is some (2n 1)-dimensional manifold. When M is closed, i.e. has no boundary, the coeﬃcient− in front of the integral ensures that

i n (n 1)! C[Ag] C[A]=2π − tr (g−1dg)2n−1 , − 2π (2n 1)! { } − ZN is 2π times an integer whenever g GL(n, C) or any of its compact subgroups ∈ such as SU(N) and N is a spin manifold. Consequently, when Chern-Simons functional appears in a functional integral

Z = d[A] exp ikC[A]+ { ···} Z then k being an integer ensures invariance under “large” (topologically non- trivial) gauge transformations. Such complex terms in the Euclidean effective action occur when we integrate out massive Dirac fermions in odd

34 dimensional spacetime where there is no Γ5 pairing of eigenstates of opposite chirality to make the fermion determinant real. The Chern-Simons functional is gauge invariant on closed manifolds but not on a manifold with boundary. The variation is given by the integral of the α2n over the boundary ∂M. Given an anomalous (and therefore inconsistent) gauge theory on an even dimensional manifold we can seek an N such that the even manifold is ∂N and where the theory on N has a Chern-Simons functional whose gauge variation cancels the variation of the anomalous theory on ∂N. Consider again a four-dimensional theory whose anomaly can be cancelled by attaching the four dimensional space a to a ﬁve dimensional bulk theory with a k = 1 Chern-Simons term 1 3 3 C[A]= tr A(dA)2 + A3dA + A5 . −24π2 { 2 5 } ZN Then

δ C[A] d4x tr ǫ J µ − ǫ ≡ { ∇µ } Z∂N 1 = tr dǫ(AdA + dAA + A3) −48π2 { } Z∂N 1 = tr ǫ∂ (A (∂ A )+(∂ A )A + A A A ) εµνστ d4x. 48π2 { µ ν σ τ ν σ τ ν σ τ } Z∂N 1 1 = tr ǫ∂ A ∂ A + A A A εµνστ d4x. 24π2 µ ν σ τ 2 ν σ τ Z∂N So 1 1 tr λ J aµ = tr λ ∂ A ∂ A + A A A εµνστ . { a∇µ } 24π2 a µ ν σ τ 2 ν σ τ Because this anomaly is found as (minus) the variation of the functional C[A], it does satisfy the Wess-Zumino consistency condition. It coincides with the consistent anomaly from the previous section. The full bulk-plus-boundary theory is gauge invariant and the non-zero divergence of the boundary current is being sourced by an inﬂow of gauge current from the higher dimensional bulk. This bulk current is covariant, 1 tr λ J λ = tr λ F F ελµνστ , { a } 32π2 { a µν στ } 35 and comes from the variation 3 δ ω = 3 tr δAF 2 + tr δA(AdA + dAA + A3) 5 { } { 2 } Z ZN Z∂N 1 = 3 tr δAF 2 + tr δA(AF + F A A3) . { } { − 2 } ZN Z∂N We ignore the boundary term when computing a bulk current, but in the total bulk-plus-boundary theory we must retain it as it provides a contribution to the current in the boundary of 1 tr λ Xµ def= tr λ (A F + F A A A A ) εµνστ . { a } 48π2 { a ν στ νσ τ − ν σ τ } This quantity is exactly the Bardeen polynomial that has to be added to the consistent current to obtain the covariant anomaly 1 tr λ (J µ + Xµ) = tr λ F F ε5µνστ . { a∇µ WZ } 32π2 { a µν στ }

µ µ µ The new current Jtot = JWZ + X is now gauge-covariant, and its covariant anomaly divergence is entirely accounted for by the Callan-Havey anomaly inﬂow mechanism. Similarly, in two dimensions we ﬁnd that 1 J µ = εµν∂ A ∇µ WZ 4π µ ν is the consistent anomaly, and 1 Xµ = εµνA 4π ν is the Chern-Simons term’s contribution to the boundary current. Then 1 1 (J µ + Xµ) = εµν ∂ A + εµν(∂ A + [A , A ]) ∇µ WZ 4π µ ν 4π µ ν µ ν 1 = εµν (∂ A ∂ A + [A , A ]) 4π µ ν − ν µ µ ν 1 = εµν F , 4π µν is the covariant anomaly.

36 Summary: An anomalous gauge theory with dynamical gauge ﬁelds can make physical sense if it captures the physics of surface- or defect-trapped states of a non-anomalous bulk theory. The currents seen by a surface-trapped observer are functional derivatives of the entire bulk+surface theory, and hence the covariant currents. The exchange of surface trapped charge with the bulk is therefore the covariant anomaly. On the contrary, if an anomalous gauge theory with dynamical gauge ﬁelds is not a bulk-compensated low energy surface theory, then the failure of gauge invariance makes it physically inconsistent, and its failure to be gauge invariant is captured by the consistent anomaly.

Including Higgs-induced Masses: In this section we will work in Minkowski space in which ψ(x) is an operator. We then have

ψ¯ = ψ†γ0 where 0 1 1 0 γ0 = , γ5 = , 1 0 0 1 − and ψ† is the adjoint of ψ in the many-body Hilbert space. We can introduce a mass matrix † M = URdiag(m1,...,mn)UL and set

† † † Hmass = ψRMψL + ψLM ψR 0 M ψ = ( ψ† ψ† ) R R L M † 0 ψ L 0 1 0 M ψ = ( ψ¯ ψ¯ ) R L R 1 0 M † 0 ψ L M † 0 ψ = ( ψ¯ ψ¯ ) R L R 0 M ψ L † = ψ¯LM ψR + ψ¯RMψL = ψ¯ 1 (M + M †)+ 1 γ5(M M †) ψ { 2 2 − } is invariant under

ψ V ψ , ψ V ψ , U V U , U V U . L → L L R → R R L → L L R → R R 37 As an illustration of such a mass-term UL, UR parametrization, consider the electroweak interactions where G = SU(2) U(1) and G = U(1) . L ⊗ Y R Y Here the U(1)Y in both GL and GR is the weak-hypercharge rotation. With one generation of quarks, these groups act as

∗ iθ/3 uL a b e 0 uL VL : − ∗ iθ/3 , dL 7→ b a 0 e dL i4θ/3 uR e 0 uR VR : −i2θ/3 , dR 7→ 0 e dR 2 2 where a + b = 1 so that the matrix is SU(2). The phases in the U(1)Y actions| reﬂect| | | the weak hypercharge assigments∈ Y = (1/3, 1/3, 4/3, 2/3) W − for (uL,dL,uR,dR) respectively. The quark mass term for a general gauge is † then parametrized as UR diag(m) UL by

e+i4θ/3 0 m 0 a∗ b∗ e−iθ/3 0 u ( u† d† ) u L +h.c. R R 0 e−i2θ/3 0 m b a 0 e−iθ/3 d d − L T The Higgs doublet (ϕ1,ϕ2) in this convention (upper component gains expectation) is assigned hypercharge 1, and is parametrized by − ∗ −iθ ϕ1 a b e 0 φ −iθ a ϕ = = −∗ −iθ h i = φ e . ϕ2 b a 0 e 0 h i b

When b = 0, the Higgs ﬁeld is left ﬁxed by the action of QEM = T3 + Y/2 because e2iαQEM acts to take θ θ + α, ϕ e2iα(σ3/2)ϕ. We can now write the mass term as → → u u H = y u† ϕ† L + y d† ϕ˜ † L + h.c. mass u R d d R d L L ∗ ∗ T where mu,d = yu,d ϕ and ϕ˜ =( ϕ2,ϕ1) . This makes the gauge invariance manifest and is theh wayi the Yukawa− coupling to the Higgs is usually displayed in the Standard Model literature. When there are three generations, then mu, md are replaced by 3-by- 3 matrices Mu, Md with indices acting on the generation labels. We then diagonalize

M V u†M V u = diag(m ), M V d†M V d = diag(m ), V = V uV d†. u → R u L u d → R d L d CKM L L

38 With a mass term included the action becomes

d ¯ ¯ MDL(L) ψR S = d x ( ψL ψR ) † , DR(R) M ψL Z and for the operator MDL(L) = † D DR(R) M our singular-value equations become

ϕ = [γµ(∂ + i + iΓ5 )+ 1 (M + M †)+ 1 Γ5(M M †)]ϕ = λ φ D n µ Vµ Aµ 2 2 − n n n †φ = [ γµ(∂ + i iΓ5 )+ 1 (M + M †) 1 Γ5(M M †)]φ = λ ϕ . D n − µ Vµ − Aµ 2 − 2 − n n n The last equation can be recast as

† µ 1 † 1 5 † † φ [(←−∂ µ i µ + iΓ5 )γ + (M + M )+ Γ (M M )] = λnϕ . n − V A 2 2 − − n Applying these to the vector current

φ† (x)γµϕ (x) jµ(x) ψγ¯ µψ = n n ≡h i λ n n X gives ∂ jµ(x)= φ† (x)φ (x) ϕ† (x)ϕ (x) µ { n n − n n } n X exactly as in the massless case. We can again regulate the index as

† † † † Q = ddx (φ† (x)e−tDD φ (x) ϕ† (x)e−tD Dϕ (x)) = Tr e−tDD e−tD D t n n − n n { − } n Z X or as20 κ2 κ2 Q = Tr κ † + κ2 − † + κ2 DD D D 5 We think of Qt or Qκ as the generalized Γ˜ index of the doubled operator

0 ˜ = D . D † 0 D 20N. V. Krasnikov, V. A. Rubakov, V. F. Tokarev, Zero-fermion modes in models with spontaneous symmetry- breaking. J. Phys A 12 (1979) L343-346.

39 21 22 that is introduced in and then used in to show that there is no θweak. The formulæ for this non-standard index show that will have zero modes in D the presence of gauge-ﬁeld instantons provided that they are accompanied by suitably-winding vortices in the Higgs ﬁeld.

Takagi diagonalization: If M is an n-by-n complex symmetric matrix, then there exists23 a unitary matrix Ω such that

T Ω MΩ = diag(m1,...,mn) where the numbers mi are real and non-negative. This result is useful for diagonalizing Majorana-mass matrices. Proof : the matrix N = M †M is Hermitian and non-negative, so there is a unitary matrix V such that V †NV is diagonal with non-negative real entries. Thus C = V T MV is complex symmetric with C†C V †NV real. Writing ≡ C = X+iY with X and Y real symmetric matrices, we have C†C = X2+Y 2+ i[X,Y ]. As this expression is real, the commutator must vanish. Because X and Y commute, there is a real orthogonal matrix W such that both W XW T and W Y W T are simultaneously diagonal. Set U = WV T then U is unitary and the matrix UMU T is complex diagonal. By post-multiplying U by another diagonal unitary matrix, the diagonal entries can be made to be real and non-negative. Since their squares are the eigenvalues of M †M, they coincide with the singular values of M. If A is a complex skew-symmetric matrix, one can use the same strategy to show there exists a unitary matrix Ω such that24

0 λ ΩT AΩ= i diag(0,..., 0), λi 0 ⊕ i M − 21A. A. Anselm, A. A. Johansen, Baryon nonconservation in standard model and Yukawa interaction, Nuclear Physics B 407(1993) 313-327 22A. A. Anselm, A. A. Johansen, Can the electroweak θ-term be observable?, Nuclear Physics B 412 (1994) 553-573. 23Léon Autonne, Sur les matrices hypohermitiennes et sur les matrices unitaires, Ann. Univ. Lyon, 38 (1915) 1-77; Teiji Takagi, On an algebraic problem related to an analytic theorem of Carathéodory and Fejér, and an allied theorem of Landau, Japan J. Math. 1 (1925) 83-93; Issai Schur, Ein Satz ueber quadratische Formen mit komplexen Koeffizien- ten, Amer. J. Math., 57 (1945), 472-480. 24Loo Keng Hua, On the theory of automorphic functions of a matrix variable I — geometrical basis, Amer. J. Math., 66 (1944) 470-488.

40 2 † and the λi are the positive roots of the eigenvalues λi of A A.

Gamma matrices and C, P, T in any dimension In Euclidean d =2k dimensions we can construct a representation of the generators γi of the Cliﬀord algebra

γiγj + γjγi =2δij

† by using fermion annihilation and creation operatorsa ˆn, aˆn, n =1,...,k that obey aˆ , aˆ =0= aˆ† , aˆ† , aˆ , aˆ† = δ . { n m} { n m} { n m} nm and by setting

2n−1 † γ =a ˆn +ân, γ2n = i(â† aˆ ). n − n When they act on the Fock space built on a “vacuum vector” 0 such that i k | ki aˆn 0 = 0, n =1,...,k, the γ are represented by a set of 2 -by-2 Hermitian matrices| i that are symmetric for odd i and antisymmetric for even i. We also define γ2k+1 =( i)kγ1 ...γ2k − † which is equal to ( 1)P aˆnaˆn and in the Fock basis, is real, diagonal, and so always symmetric. − Assuming this, we define25

i i C1 = γ , C2 = γ , i even iYodd Y and use them to construct matrices and such that C T γi −1 = (γi)T , C C − γi −1 = +(γi)T . T T We ﬁnd that for k=0, mod 4: = C1 symmetric, = C2 symmetric. Both commuteC with γ2k+1. T

25H. Murayama: http://hitoshi.berkeley.edu/230A/cliﬀord.pdf

41 k=1, mod 4: = C2 antisymmetric, = C1 symmetric. Both anticommuteC with γ2k+1. T k=2, mod 4: = C1 antisymmetric, = C2 antisymmetric Both commuteC with γ2k+1. T k=3, mod 4: = C2 symmetric, = C1 antisymmetric. Both anticommuteC with γ2k+1. T Under a change of basis γµ AγµA−1 the matrices and will no longer → C T given by the explicit product expressions C1 and C2, but instead transform as AT A, AT A. C → C T → T The symmetry or antisymmetry of , is unchanged, and is thus a basis- C T independent property. Another way to think of this symmetry is by making use of the transpose of the the deﬁning transformations. We then see that −1 T and −1 T commute with all γµ. As our Fock-space gamma rep- resentationC C isT clearlyT irreducible, Schur’s lemma tells us that both , are proportional to their transpose so C T

T = λ C C with λ basis independent. Then, transposing again,

= λ T = λ2 C C ⇒ C C showing that λ = 1. Similarly T = . ± T ±T If we restrict to transformations in which A is unitary the Euclidean γµ remain Hermitian and a similar argument shows that † is proportional to the identity. As † is a positive operator, the factorC ofC proportionality is C C real and positive. Consequently (and ) can be scaled by real numbers so as to be unitary. We will assumeC that weT have done this.26 If γi constitute a representation of the Cliﬀord algebra, so do (γi)T . In d =2k dimensions the Dirac representation is unique, so these three± representations must be equivalent. Consequently, even if we did not have the explicit

26One sometimes sees assertions such as “ 2 = I” — implicitly in Bjorken and Drell eq. (15.134) for example — but the manner in whichT and transform under γµ AγµA−1 shows that there is no basis-independent notionT of a productC of or matrices→ with themselves. Such formulæ are not operator identities therefore, Tand canC only hold in speciﬁc bases. There is no such problem with −1 T = I, etc. C C ±

42 construction given above, the existence of , is guaranteed. In odd dimensions, however, there are two inequivalentT representationsC of the Cliﬀord algebra that correspond to the choice of sign in γ2k+1 = ( i)N γ1 γ2k. The existence of , matrices that transpose all the γi is no± therefore− ··· longer assured. We haveT toC ask whether the even-dimensions , continue to com- T C mute or anticommute with the extra gamma matrix. The table below summarizes the outcome of this examination. It shows whether the and matrices exist, whether they are symmetric (S), or antisymmetricT (A), and,C for d even, the sign appearing in

γd+1 −1 = γd+1 −1 = (γd+1)T . C C T T ± The table repeats mod 8.

d 012345678910 SSS AAA SSS T S AAA SSS A γdC+1 + + + − − −

When is symmetric we can find a unitary matrix U with which to Takagi T diagonalize ′ = U T U to a basis in which ′ = I. In this basis all the Euclidean gammaT → T matricesT are symmetric but stillT Hermitian and therefore all real. When is symmetric we can find a basis in which = I and all the C C Euclidean gamma matrices are antisymmetric, still Hermitian and therefore purely imaginary. Numbering convention for Minkowski-space gamma matrices: We labelled our Euclidean gamma matrices as γ1,γ2,...,γ2k with γ2k+1 being a product of the 2k lower-numbered matrices. In contemporary physics usage four-dimensional Minkowski-space gamma matrices are universally numbered as γ0,γ1,γ2,γ3 with γ0 associated with x0 = t. For historical reasons their product still called γ5 — although there is no γ4. In Minkowski signature γ0 and γ5 have special roles and renaming either γ0 γ1 or γ5 γ4 to avoid the “γ4 gap” is likely to generate more fog than light.→ It seems→ simplest to keep the chirality operator as Γ5 γ2k+1, and when “γ0” appears in the familiar definition ψ¯ = ψ†γ0 it should≡ be born mind that the Minkowski signature “γ0” corresponds to the Euclidean γ4. Charge conjugation: In a Euclidean-signature path integral ψ¯ and ψ are independent Grassmann variables. Nonetheless it is useful to define the

43 Euclidean-signature charge-conjugate ﬁelds ψ¯c and ψc so as to be consistent with the Minkowski-signature operator language in which ψ¯ = ψ†γ0. We arrange for this by setting

ψc = −1ψ¯T C ψ¯c = ψT . − C To obtain the motivating Minkowski version we first note that in this signature we can use exactly the same and matrices as in Euclidean space — the insertion of factors of i in someT of theC γµ does not affect the formula for their transposition so no i’s need be inserted in and . T C Consider first the mostly-minus “West-Coast” Minkowski metric (+, , ,...) in which γ0 is Hermitian and obeys (γ0)2 = 1. Then with ψ¯T =(ψ†γ0)T−, and− writing ψ∗ for the quantum Hilbert-space adjoint of ψ without the column row operation implicit in ψ†, we have → ψc = −1ψ¯T = −1(γ0)T ψ∗ (ψc)† = ψT (γ0)T , C C ⇒ C as remains unitary in Minkowski space. We define C (ψ¯)c = (ψc) = (ψc)†γ0 = ψT (γ0)T γ0 C = ψT γ0 −1 γ0 − C C C = ψT . − C In the mostly-plus “East-Coast” Minkowski metric ( , +, +,...), in which γ0 is skew Hermitian and obeys (γ0)2 = 1, we have− (ψc)† = ψT (γ0)T and − − C

(ψ¯)c (ψc) ≡ = (ψc)†γ0 = ψT (γ0)T γ0 − C = ψT γ0 −1 γ0 C C C = ψT (γ0)2 C = ψT . − C In both signatures, therefore, (ψc) (ψ¯)c = ψT . ≡ − C 44 From these results, and with anticommuting c-number ψ’s, we ﬁnd that

ψ¯cγµψc = [ ψT ]γµ[ −1ψ¯T ]= ψT (γµ)T ψ¯T = ψγ¯ µψ, − C C − so the number current changes sign. The spin-current density transforms as

ψ¯cγ0[γi,γj]ψc = ψγ¯ 0[γj,γi]ψ = ψγ¯ 0[γi,γj]ψ, (i, j = 0), − 6 and is left unchanged. Similarly

ψ¯cψc = ψT ψ¯T = ψψ.¯ − In Euclidean signature, and using the anticommuting property of the Grass- mann ﬁelds, the action for Dirac fermions interacting with a vector gauge ﬁeld A has the property

S = dnx ψ[γµ(∂ + A )+ m]ψ = dnx ψ¯c[γµ(∂ AT )+ m]ψc. µ µ µ − µ Z Z T The Aµ are the Lie algebra representation-valued ﬁelds in the the conjugate − c representation to that of Aµ, and so ψ has the opposite “charge” to ψ. Minkowski-signature Majorana Fermions: We have deﬁned

ψc = −1ψ¯T = −1(γ0)T ψ∗ C C so, with T = λ we ﬁnd (in both mostly-plus and mostly-minus metrics) C C (ψc)c = −1(γ0)T ( −1(γ0)T ψ∗)∗ C C = −1(γ0)T T (γ0)†ψ C C = λ −1(γ0)T (γ0)†ψ C C = λγ0(γ0)†ψ − = λψ. − We can therefore consistently impose the Minkowski Majorana condition that ψc = ψ only if is antisymmetric: i.e. in 2, 3, 4 (mod 8) dimensions. C The equal-time anti-commutator of an operator-valued Majorana ﬁeld can be taken to be

′ 0 −1 d−1 ′ ψ (x), ψ (x ) ′ = [γ ] δ (x x ) { α β }t=t C αβ − where γ0 −1 = −1(γ0)T is symmetric when is antisymmetric. C −C C 45 We can regard the map C : ψ ψc = −1[γ0]T ψ∗ as an antilinear27 map C : V V where V is the gamma-matrix7→ C representation space. If C2 = id, → this is real structure on the complex V space. Vectors that are left ﬁxed by C are regarded “real” because there is a basis in which their components are real — even even though these components will be complex in other bases. The antilinear map C commutes with the gamma matrices only in the mostly plus East Coast metric. With this metric choice, and in the basis in which the Majorana spinor components are real, the gamma matrices become purely real and so preserve the reality condition. In the the West- Coast-metric Majorana representation the gamma’s are purely imaginary and we have to remove a factor of i to get matrices that commute with the antilinear C. This does not matter though, because it is the Dirac equation that must preserve the reality of of the spinor solutions, and in the West Coast Minkowski metric the Dirac equation is

( iγµ∂ + m)ψ = 0 (West Coast). − µ This version of Dirac puts the necessary factor of i with the γ’s, while on the East Coast the equation reads

µ (γ ∂µ + m)ψ =0, (East Coast), where there is no factor of i. To verify that C commutes with the γi in the East Coast metric we begin by observing that (γi)† = γ0γiγ0 in both conventions. Then

−1(γ0)T (γiψ)∗ = −1(γ0)T (γi∗ψ∗) C C = −1(γ0)T (γ0γiγ0)T ψ∗ C = −1(γ0)T (γ0)T (γi)T (γ0)T ψ∗ C = −1(γ0)T −1(γ0)T −1(γi)T −1(γ0)T ψ∗ C CC CC CC = ( γ0)( γ0)( γi) −1(γ0)T ψ∗ − − − C = (γ0)2γi −1(γ0)T ψ∗. − C Thus Cγi = γiC, or equivalently

−1(γ0)T (γiψ)∗=γi −1(γ0)T ψ∗ C C 27Charge conjugation is antilinear only when acting on the ﬁeld components. It is a linear map when acting on the states in the many-body Hilbert space.

46 holds only if (γ0)2 = 1. Minkowski-signature− pseudo-Majorana fermions: We can define an alternative “conjugation” operation ψτ = −1ψ¯T , T ψ¯τ = ψT . T This operation reverses the current, again leaves the spin unchanged, but flips the sign of ψψ¯ . Almost identical algebra to the conventional charge conjugation case shows that the condition ψτ = ψ is consistent only when is symmetric, hence in d= 8, 9, 10 (mod 8). Fermions such that ψτ = ψ T are said by some to be pseudo-Majorana.28 In his online lecture notes29 José Figueroa-O’Farril also uses the term pseudo-Majorana spinors, but by this I believe he means the purely imaginary gamma matrices of the West-Coast Majorana representation. Repeating the algebra for the C conjugation, but with replaced by , C T gives an extra minus sign. Consequently in the mostly-minus West-Coast metric the gamma matrices of a pseudo-Majorana representation can be chosen to be real, while in a Majorana representation they are pure imaginary. It is the other way around in the mostly-plus East-Coast metric. Because this “conjugation” flips ψψ¯ , these pseudo-Majorana fermions are necessarily massless. Indeed the absence of the mass term is necessary for the real gamma matrices in the West Coast pseudo-Majorana representation and the pure imaginary gamma matrices in the East Coast pseudo-Majorana representation to avoid conflict with the Dirac equation. Euclidean-signature Majorana fermions: We now explore to what ex- tent the constraints on the Minkowski signature space-time dimensions in which Majorana and pseudo-Majorana fermions exist are consistent with Euclidean-signature Grassmann-variable path integration. Assume that the gauge fields in the Dirac operator

D = γµ(∂ + A ) 6 µ µ are in real representations. Then if , we have an eigenfunction such that

Du = iλ u , 6 n n n 28For example: Jeong-Hyuck Park, Lecture note on Cliﬀord algebra, sec 2.2. http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/ModaveI/gamma.pdf. 29J. Figueroa-O’Farril, Majorana Spinors, https://www.maths.ed.ac.uk/ jmf/Teaching/Lectures/Majorana.pdf

47 complex conjugation gives D∗u∗ = iλ u∗ . 6 − n n This can be written as D −1u∗ = iλ u∗ , C6 C n n n or D −1u∗ = iλ −1u∗ . 6 C n nC n −1 ∗ Thus un and un are both eigenfunctions of D with the same eigenvalue. They will beC orthogonal, and therefore linearly6 independent, when is an- C tisymmetric — something that happens in d = 2, 3, 4 (mod 8) Euclidean dimensions. These are precisely the dimensions in which Minkowski space Majorana spinors can occur. This suggests that we can take the Euclidean Majorana-Dirac action to be30 1 S[ψ]= ddx ψT (D + m)ψ. 2 C 6 Z Expanding the ﬁelds out as

ψ(x) = [ξ u (x)+ η ( −1u∗ (x))], n n n C n n X ψT (x) = [ξ uT (x) η (u† (x) −1)], n n − n n C n X where ξn and ηn are Grassmann variables we find that 1 S = ddx (ξ uT η u† −1) (iλ + m)(ξ u + η −1u∗) 2 i i − i i C C j j j jC j i,j Z X 1 = ddx ξ η (uT u∗)+ ξ η (u†u ) (iλ + m) 2 i j i j j i i j j i,j Z X n o = ξiηi(iλi + m). i X As the Grassmann integration uses only one copy of the doubly degenerate eigenvalue, we obtain a square-root of the full Dirac determinant Det(D+m). 6 We anticipate that the resulting partition function is the Pfaffian Z = Pf[ (D + m)] C 6 30ψT is called by Peter van Nieuwenhuizen the “Majorana adjoint.” C 48 of the skew-symmetric kernel (D + m). To confirm that the kernelC is6 skew symmetric we take a transpose and find

[ (γµ∂ + m)]T = (∂T (γµ)T + m) T C µ µ C = [( ∂ )( γµ −1)+ m]( ) − µ −C C −C = (γµ∂ + m). −C µ Note that ∂T = ∂ because its x-basis matrix element is δ′(x x′) is skew. This result continues− to hold for D in curved space or with gauge− ﬁelds in real representations. 6 A further step is needed to conﬁrm that

Pf[ (D + m)] = (iλ + m). C 6 n n Y We know that under a basis change Pf[BT QB] = Pf[Q] det[B] and the mode expansion ψ (x)= [ξ u (x)+ η ( −1u∗ (x))], α n αn n Cαβ βn n X is a linear change of variables in the Grassmann integral corresponding to a matrix B with indices Bαn;βx, where the range of n is extended include both the labels on u and on −1u∗ . To show that diagonalizing D has not changed n C n 6 the value of the Pfaﬃan we need to show that this matrix is unimodular — or at least does not depend on the background ﬁelds. However 1 1 ddx ψT (x) ψ(x)= ξ η = (ξ η η ξ ) 2 C i i 2 i i − i i i i Z X X holds for any set of eigenmodes un. In other words

BT ( I)B = ˜I C ⊗ C ⊗ d ′ where [ I] = δ and [ ˜I] ′ = δ (x x ), The matrices B C ⊗ αn;βm Cαβ nm C ⊗ αx;βx Cαβ − from different background fields can differ only by left factors that preserve the symplectic form ( I). Such symplectic matrices are automatically unimodular. C ⊗

49 We could redeﬁne the mode expansion as31

ψ (x)= [ξ u (x)+ η eiθn ( −1u∗ (x))], α n αn n Cαβ βn n X iθn where the e are arbitrary phases, and so apparently replace (iλn + m) iθn → e (iλn + m) in the eigenvalue product, but in that case the B matrix is no longer unimodular and it cancels the phase factors leaving Pf[ (D + m)] C 6 unaffected. We therefore disagree with the claim by Golterman and Shamir that the phase of the Pfaffian is ambiguous. Majorana-Weyl Fermions: In 2 (mod 8) dimensions Γ γ8k+3 obeys 5 ≡ Γ5 −1 = (Γ )T and therefore DΓ5 = (Γ5)T D. We can thus decompose C C − 5 C6 C6 ψT (D + m)ψ = ψT (D)ψ + ψT (D)ψ + m(ψT ψ + ψT ψ ). C 6 RC 6 R L C 6 L RC L L C R If m = 0, we may retain only one of the the right or left fields, in which case we have a Euclidean Majorana-Weyl fermion. 4-d Dirac vs. Majorana: In four dimensions we can relabel our Dirac fermions as32 ψ χ ψ = R = 2R ψ χ L 1L and define χ1R and χ2L by

ψ¯ = [ψ¯ , ψ¯ ] = [ χT , χT ] . L R − 1R − 2L C If all gauge ﬁelds are in real representations we can rewrite the kinetic part of the Dirac action density

¯ ¯ 0 DL ψR = [ψL, ψR] 6 L D 0 ψL 6 R 31M. Golterman, Y. Shamir, Phase ambiguity of the measure for continuum Majorana fermions, Phys. Rev. D 100, 034507 (2019); arXiv:1904.08600. 32 T If we write ψ = [ψR, ψL] then in the Minkowski operator language we have ψ¯ = † † ¯ † ¯ [ψL, ψR]. It is a common convention to label the entries in ψ so that ψR ψR and † → ψ ψ¯L making ψ¯ = [ψ¯L, ψ¯R]. L →

50 as

1 T T 0 DL χ2R T T 0 DL χ1R = [χ1R, χ2L] 6 + [χ2R, χ1L] 6 L −2 C D 0 χ1L C D 0 χ2L 6 R 6 R 1 T T 0 DL χ1R T T 0 DL χ2R = [χ1R, χ1L] 6 + [χ2R, χ2L] 6 −2 C D 0 χ1L C D 0 χ2L 6 R 6 R 1 = χ¯ Dχ +χ ¯ Dχ . (1) 2 { 16 1 2 6 2} Hereχ ¯ =[¯χ , χ¯ ] = [ χT , χT ] and both χ and χ are Majorana because L R − R − L C 1 2 χ¯ = χT . The averaging in the first line comes from antisymmetry of D. A− DiracC mass term C6 ψ mψψ¯ = m[ψ¯ , ψ¯ ] R L R ψ L becomes m m χ¯ χ +χ ¯ χ = χσ¯ χ 2 { 1 2 2 1} 2 1 where the σ1 acts on the “flavour” indices 1,2. We can do a flavour diagonalization by

5 iπσ1γ /4 1 5 χ e χ = (1 + iσ1γ )χ → √2 5 iπσ1γ /4 1 5 χ¯ χe¯ =χ ¯ (1 + iσ1γ ) → √2 that takes

5 χσ¯ χ χσ¯ eiπσ1γ /2χ =χi ¯ (σ )2γ5χ 1 → 1 1 to get m χ¯ (iγ5)χ +χ ¯ (iγ5)χ . 2 1 1 2 2 The transformation is unimodular for both eigenvalues of γ5, and so it seems as if we have two Pfaffians (and therefore a determinant) of fields with an imγ5 chiral mass. As every eigenvalue occurs twice this means that the product of the two identical Pfaffians, which should reproduce the Dirac determinant, has become

m n++n− ei(π/2)(n+−n−) (m2 + λ2 ). | | n n Y 51 This expression has apparently acquired a factor of ( 1) for each pair of zero modes when compared to the original Dirac determinant,− which does not contain the factor ei(π/2)(n+−n−). However, for the Grassman integral to give the product of the two Pfaﬃans, we need reorder the measure factors to get all the dχ2’s behind the dχ1’s. For each mode number n we have

dψ¯ dψ dψ¯ dψ = det[ ]−1dχ dχ dχ dχ , n,L n,R n,R n,L C n,1R n,2R n,2L n,1L where the order of the factors in the measure on the LHS is mandated so that we get Det(D + m). The factors of det[ ]−1, one for each mode n, 6 C cancel the “metric” factor Det[C] = Det[ I] that always occurs when we use eigenvalues of an operator to computeC the⊗ Pfaﬃan of a skew symmetric matrix. If all modes are present the remaining factors on the RHS can be rearranged to get the Pfaﬃan without changing the sign. If there is a zero mode then (taking into account that each zero mode occurs twice) the factors of ( 1) that arise from each exchange pairs of dχ’s cancel the extra sign from the−iγ5 mass. This resolves the apparent paradox that motivated the paper by Golterman et al. cited above. Euclidean-signature pseudo-Majorana fermions: If

Du = iλ u 6 n n n then D∗u∗ = iλ u∗ 6 n − n n or, assuming that any gauge ﬁelds are in real representations,

D −1u∗ = iλ −1u∗ . 6 T n − nT n If λ = 0 we will have u and −1u∗ orthogonal because they have diﬀerent n 6 n T n eigenvalues. Let let us choose un to be the positive-λn eigenfunctions and consider the cases d=8, 9, 10 (mod 8) in which is symmetric and we can consistently impose the Minkowski signature pseudo-MajoranaT condition and the action to be 1 S = ddx ψT Dψ. 2 T 6 Z Note that if is skew-symmetric, then T 1 ddx ψT Dψ =0. 2 T 6 Z 52 Consequently euclidean-signature pseudo-Majoranas are only available in the same dimensions as Minkowski-signature pseudo-Majoranas. If there are no zero modes we can expand

ψ(x) = [ξ u (x)+ η ( −1u∗ (x))] n n n T n n X ψT (x) = [ξ uT (x)+ η (u† (x) −1)] n n n n T n X and find that 1 ddx ψT Dψ = (iλ )η ξ . 2 T 6 n n n Z λXn>0 As in the ordinary Majorana case, the partition function is the product of only half the eigenvalues, so again we get a square-root of the full Dirac determinant which we expect to identify with the Pfaffian Pf[ D] of the skew-symmetric kernel D. T 6 T 6 What is different from the ordinary Majorana case is that we cannot add a mass term by taking 1 S = ddx ψT (D + m)ψ 2 T 6 Z because ψT ψ = 0 by the symmetry of . Consequently Euclidean pseudo- T T Majorana fermions are necessarily massless — just as are their Minkowski bretheren. As second consequence of ψT ψ = 0 is that establishing that the essential T unimodularity of the diagonalizing matrices requires a slightly different tac- tic. We need to replace the anticommuting ξn and ηn by commuting variables Xn and Yn and define

−1 ∗ φα(x) = [Xnuαn(x)+ Yn( αβ uβn(x))] n T X φT (x) = [X uT (x)+ η (u† (x) −1)]. α n αn n βn Tβα n X Then ddxφT (x) φ(x)= (X Y + Y X ), T n n n n n Z X independently of particular form of the un eigenfunctions. All diagonalizing transformations B therefore preserve the same (non-positive deﬁnite)

53 symmetric form and can differ only by factors drawn from some orthogonal group. Such orthogonal matrices obey det[B]2 = 1, so, unlike the ordinary Majorana case where the B matrices differer by symplectic (and therefore unimodular) matrix factors, here we have the possibility det[B] = 1 and the Pfaffian changing sign. Indeed we have already seen an inherent− sign ambiguity because we arbitrarily assigned the positive eigenvalue λn to un and ξ , rather than to −1u∗ and η . This ambiguity is a potential source of n T n n global anomalies: if we smoothly interpolate between two gauge-equivalent background fields (which necessarily have the same set of λn), and an odd number of λn change sign during the interpolation then the partition function changes sign and the theory inconsistent33. Symplectic Majorana fermions: In d =5, 6, 7 (mod 8) dimensions neither Majorana nor pseudo-Majorana fermions can be constructed. There is however the option of Symplectic Majorana fermions. To construct these we start from a pair of fermions ψ1, ψ2 and in for d = 7 (mod 8) where is symmetric set C 1 S = ψT ǫab(D + m)ψ ddx. 2 a C 6 b Z In d = 5 (mod 8) where is antisymmetric we can set T 1 S = ψT ǫabDψ ddx. 2 a T 6 b Z iIn d = 6 (mod 8) we can use either of or — and if in addition the mass vanishes we can have symplectic Majorana-WeylC T fermions. Rokhlin’s theorem: Recall that in d = 4(mod8) the matrix is antisymmetric and with Γ5 γ8k+5 we have T ≡ Γ5 −1 = +(Γ5)T = (Γ5)∗. T T Assume that no gauge fields are present (i.e gravity only), then similar algebra to the previous section shows that if un obeys Du = iλ u , 6 n n n then D −1u∗ = iλ −1u∗ . 6 T n − nT n 33E. Witten, An SU(2) anomaly, Phys. Lett. B 117, 324-328 (1982); E. Witten, Fermion Path Integrals And Topological Phases, Rev. Mod. Phys. 88 (2016) 35001-40; arXiv:1508.04715.

54 −1 ∗ In particular, if λn = 0 then un is also a zero mode and is orthogonal to u . Further, as Γ5( −1u∗ )T = −1(Γ5u )∗, it has the has the same Γ5 n T n T n eigenvalue as un. It follows that chiral zero modes come in pairs, and so the Dirac index

n n = Aˆ(R) + − − ZM is an even integer. In dimension 4 this result implies Rokhlin’s theorem that the signature of a 4-dimensional spin manifold is divisible by 16. This is because when d =4 1 Aˆ = p , Dirac Index, 1 −24 1 1 L = + p , Signature, 1 3 1 where 1 1 p = tr R R , 1 −(2π)2 2 ∧ is the four-form Pontryagin class. We see that the Aˆ-genus evaluates to minus one-eighth of the signature. Observe that iterating the antilinear map u −1u∗ twice gives n → T n u −1[ −1u∗ ]∗ = u , n → T T n − n where we have used ∗ = [ −1]T = −1. Thus our map gives rise to a quaternionic structureT — Ti.e. an antilinear−T map that squares to minus the identity — on the zero mode space. This is how Rokhlin’s theorem is explained in the mathematics literature. We could, of course, have deduced the doubling of the zero modes from the Majorana doubling given by the antisymmetric . This also gives rise to a quaternionic structure. Indeed in d = 4 (mod 8) weC have

= Γ5, C T so we can take 1 Q def= (1 Γ5) ± 2T ± to be pair of independent quaternionic structures, one for each of the Γ5 1 subspaces of chiral zero-modes. → ± 55 Intrinsic parity of Dirac and Majorana Fermions: In even space-time dimensions parity is deﬁned by P : (t, x) (t, x). In the mostly-minus 7→ − metric P is implemented on spinors as

P : ψ(t, x) ψp(t, x)= ηγ0ψ(t, x), 7→ − where the phase η is the particle’s intrinsic parity. We usually take η = 1 so that that P2 = id. However if P is to be compatible with the Majorana± condition ψc = ψ and if we require (ψc)p = (ψp)c then we must have same parity transformation rule for ψ and ψc. Let us see what this requires. Using (ψp(t, x))∗ = η∗(γ0)∗ψ∗(t, x) we have − ηγ0[ −1(γ0)T ψ∗(t, x)] = −1(γ0)T η∗(γ0)∗ψ∗(t, x) C − C − which reduces to η γ0 −1(γ0)T = η∗(γ0)T (γ0)∗, C C or η(γ0)T = η∗(γ0)∗. − Since γ0 is Hermitian in the mostly minus metric we see that η∗ = η, and so for a Majorana particle we must have η = i and so P2 = 1. − If we have the freedom to allow ± −

(ψc)p(t, x)= ηcγ0ψc(t, x) − then the same algebra shows that ηc = η∗. For particles that are distinct from their antiparticles we are therefore allowed− to have η to be 1, but then the parity of an antiparticle is minus that of the particle. ± In the mostly-plus metric parity is usually implemented by34

P : ψ(t, x) ψp(t, x)= iηγ0ψ(t, x) 7→ − The reason for the extra factor of i is that when η = 1 we again want P2 = id, and the extra i compensates for (γ0)2 = 1. For Majorana fields we still find that η∗ = η. − − 34Steven Weinberg’s The Quantum They of Fields is the only text that I know that uses the mostly plus convention, and he has this “i” factor. Mark Srednicki’s Quantum Field theory claims to use mostly-plus, but he defines his Clifford algebra by γµγν + γν γµ = 2gµν, so his are the mostly-minus gamma matrices. −

56 R symmetry: In odd space-time dimension d = 2k + 1, the standard parity operation (t, x) (t, x) is an SO(2k) rotation, so “parity” is instead 7→ − defined as the inversion of an odd number of the spatial coordinates. In the case that we flip only one direction Witten calls it R symmetry35. Let us define R to invert x1 so

R :(t, x , x ,...x ) (t, x , x ,...x ) (t, x˜). 1 2 2k 7→ − 1 2 2k ≡ In Euclidean signature the natural way flip the sign of γ1 only is by the using the Clifford algebra “twisted map” reflection in the plane perpendicular to the x1 axis: R : γµ ( γ1)γµγ1 =γ ˜µ 7→ − To get R : ψ¯(x)γµψ(x) ψ¯(x˜)˜γµψ(x˜) 7→ we must therefore set

R : ψ(x) γ1ψ(x˜) 7→ R : ψ¯(x) ψ¯(x˜)( γ1). 7→ − In Euclidean signature the R : u (x) γ1u (x˜) anticommutes with D and n 7→ n 6 so changes the sign of the corresponding eigenvalue λn In Minkowski space consider ﬁrst the mostly plus metric (γ1)2 = 1 and γ1 is Hermitian. When ψ(t, x) ηγ1ψ(t, x˜) 7→ we have

ψ¯(t, x) ηγ1ψ(t, x˜)= η∗ψ†(t, x˜)(γ1)†γ0 = η∗ψ¯(t, x˜)( γ1) 7→ − and so we have R : ψγ¯ µψ ψ¯γ˜µψ. 7→ In the mostly-minus metric when R acts on the fermi ﬁelds as

ψ(t, x) ηγ1ψ(t, x˜) 7→ then ψ¯(t, x) ηγ1ψ(t, x˜)= η∗ψ∗(t, x˜)(γ1)†γ0 = η∗ψ¯(t, x˜)γ1. 7→ 35E. Witten, Fermion Path Integrals And Topological Phases, Rev. Mod. Phys. 88 (2016) 35001-40; arXiv:1508.04715.

57 This appears to diﬀer from the Euclidian R, but (γ1)2 = 1, so we still have − R : ψγ¯ µψ ψ¯γ˜µψ. 7→

As ∂x ∂x˜ both signature versions of R leave the kinetic part of the Dirac action7→ invariant. However

R : ψψ¯ ψψ.¯ 7→ − so the mass term is not invariant. In even space-time dimensions we can undo the ﬂip with a Γ5, but this option is not available in odd space-time dimensions, where a mass is unavoidably parity-violating. Requiring that reﬂection commutes with charge conjugation leads to ηc = η∗. To see this compare − [ψc(x)]r = η γ1[ −1γ0T ψ∗(˜x)] c C with

[ψr(x)]c = −1γ0T [η∗γ1∗ψ∗(˜x)] C = γ0 −1γ1∗ψ∗(˜x) −CC C = η∗γ0 −1γ1∗ψ∗(˜x) − C = +η∗γ0 −1γ1T ψ∗(˜x) C = η∗γ0 −1 γ1 −1ψ∗(˜x) − C C C = η∗γ0γ1 −1ψ∗(˜x) − C = +η∗γ1γ0 −1ψ∗(˜x) C = η∗γ1[ −1γ0T ψ∗(˜x)]. − C Time reversal: At ﬁrst sight time reversal should simply be an R map applied to x0 t rather than x1. However such an R reverses the direction ≡ of particle trajectories in time and so converts particles into antiparticles. The conventional particle-physics time reversal operation does not charge- conjugate and so T is deﬁned by composing an R with a compensating charge conjugation operation. There is still a problem: time reversal does not play nicely with the passage from Euclidean to Minkowski signature because, as in non-relativistic quantum mechanics, time reversal must be implemented on the many-particle Hilbert space by an antiunitary operator I.

58 An operator Ω is said to be antiunitary with respect to a conjugate- symmetric sesquilinear inner product , if h i Ωa, Ωb = a, b ∗ = b, a . h i h i h i Consider the vector

X = Ω(αa + βb) α∗(Ωa) β∗(Ωb). − − Using the definition of antiunitarity and the antilinearity of , in its first slot and linearity in the second, we can expand out X 2 =h X, Xi and find that it is zero. For a positive definite inner product ak vanishingk h normi implies that X = 0, and so for such a product we have

Ω(αa + βb)= α∗(Ωa)+ β∗(Ωb).

Thus an antiunitary operator acting on a positive-definite Hilbert space is necessarily antilinear. One consequence of the antilinearity is that there is no way to define an adjoint Ω†. The standard definition Ω†a, b = a, Ωb leads to h i h i b, a = Ωa, Ωb =? Ω†Ωa, b h i h i h i and a contradiction: the leftmost expression is antilinear in b while the right- most is linear in b. A similar issue leads to

b, Ωa = a, Ω−1b . h i h i Another useful result is that if A is a linear operator then so is Ω−1AΩ, and we can compute its adjoint as follows

∗ y, (Ω−1AΩ)x = Ωy, AΩx ∗ = A†Ωy, Ωx = (Ω−1A†Ω)y, x h i h i h i h i 36Some sources—for example the Wikipedia article on antiunitary operators—deﬁne an “Ω†” by equating it to Ω−1, but I think that this notation is dangerously confusing.

59 so (Ω−1AΩ)† = Ω−1A†Ω. In the mostly minus Minkowski metric the time reversal operator I is usually taken to acts on Dirac ﬁeld operators as

I−1ψ(x, t)I = η ψ(x, t) T T − I−1ψ¯(x, t)I = η∗ ψ¯(x, t) −1, T − T where ηT is a phase. Despite the antilinearity of I the ﬁeld operator is not Hermitian-conjugated: time reversal changes the sign of momentum and the spin, but does not change particle to antiparticle. This, however, is the action on the ﬁeld operator. The action of I on wavefunctions does involve complex conjugation, and will be described later. We can decompose the action of I into a composition of T = CR followed by complex conjugation: R ψ(x, t) γ0ψ(x, t) 7→C − γ0ψ(x, t) η −1(γ0)T (γ0)∗ψ∗(x, t)= η −1ψ∗(x, t) − 7→ T − T − η −1ψ∗(x, t) ∗ η∗( −1)∗ψ(x, t) T − 7→ T − = λη∗ ψ(x, t), T − where T = λ . We have elected to use the version of charge conjugation ratherT than theT version because a conjugationT inverts ψψ¯ and so undoes C T the ψψ¯ inversion due to the R. As a result

ψ¯(x)ψ(x, t) I−1ψ¯(x, t)ψ(x, t)I = Iψ¯(x, t)I−1I−1ψ(x, t)I = ψ¯(x, t)ψ(x, t). 7→ − − A Γ5 chiral mass term does change sign. The transformation of ψ¯ follows that of ψ via

(I−1AI)∗ = I−1A∗I.

We use instead of to indicate that the Hermitian adjoint in the quantum- ∗ † state Hilbert space does not transpose column-matrix spinors to row-matrix spinors. Then

I−1ψ(x, t)I = η ψ(x, t) I−1ψ∗(x, t)I = η∗ ∗ψ∗(x, t). T T − ⇒ T T −

60 Transposing and using antilinearity

I−1ψ†(x, t)γ0I = η∗ ψ†(x, t) †(γ0)∗ T − T = η∗ ψ¯(x, t)γ0 −1(γ0)∗ T − T = η∗ ψ¯(x, t) −1 γ0 −1(γ0)∗ T − T T T = η∗ ψ¯(x, t) −1(γ0)T (γ0)∗ T − T = η∗ ψ¯(x, t) −1 T − T The time reversal of the current is

I−1ψ¯(x, t)γµψ(x, t)I = I−1ψ¯(x, t)II−1γµψ(x, t)I = I−1ψ¯I(γµ)∗I−1ψ(x, t)I = ψ¯(x, t) −1(γµ)∗ ψ(x, t) − T T − = ψ¯(x, t) −1(γµ†)T ψ(x, t) − T T − = ψ¯(x, t) −1(γµ)T ψ(x, t) ± − T T − = ψ¯(x, t)γµψ(x, t). ± − − Here γµ† = γµ so we have + for the Hermitian γ0, so the charge is not altered, and ±1 for the antiHermitian γa which changes the sign of the the spatial current.− If we act twice we ﬁnd

I−2ψ(x, t)I2 = η 2 ∗ψ(x, t). | T | T T Now −1 = † =( T )∗ = λ ∗ so T T T T I−2ψ(x, t)I2 = λψ(x, t).

Thus conjugating by I twice gives a 1 in 4, 5, 6 (mod 8) dimensions in which is antisymmetric. As the vacuum− is left ﬁxed T I 0 = 0 | i | i and the ﬁeld operators change the fermion number by 1, the 1 means ± − that when I acts on the many-particle Hilbert space we have I2 =( 1)F id where F is the fermion number. We get a +1, and hence I2 = id, in− 0, 1, 2 (mod 8) dimensions in which is symmetric. T

61 In 3 and 7 (mod 8) dimensions the matrix does not exist. We can however use the charge-conjugation operationT to deﬁne an alternative time C reversal

J−1ψ(x, t)J = η ψ(x, t) T C − J−1ψ¯(x, t)J = η∗ ψ¯(x, t) −1, − T − C at the expense of flipping the the sign of ψψ¯ . Thus a mass term is necessarily time-reversal-symmetry violating in 3 and 7 (mod 8) dimensions. Acting twice, this time reversal gives a ( 1)F in 3 (mod 8) dimensions and a plus sign in 7 (mod 8). − T and anomaly inflow: Consider a chiral (Weyl) fermion in space-time dimension d =2k and interacting with an abelian gauge field Aµ. This is an anomalous theory in which the anomaly can be be accounted for by current flowing into the d = 2k dimensional surface from the D = 2k + 1 bulk at a rate 1 J 2k+1 = sgn(M)ǫ(2k+1)i1...i2k F F . (2π)kk! i1i2 ··· i2k−1i2k Here M is the large Dirac mass in the 2k+1 dimensional theory. If we reverse time, the direction of this flow will reverse. How does this reversal relate to our discussion so far? In Minkowski signature time reversal acts on the components of the gauge field as A A , A A. 0 7→ 0 7→ − From this we see that the “electric field”

F = ∂ A ∂ A 0i 0 i − i 0 is unaﬀected by time reversal, but all other (magnetic) F ij change sign. Consequently the gauge-ﬁeld 2k-form F k is time-reversal invariant in 3 and 7 (mod 8) bulk dimensions and changes sign in 1 and 5 dimensions (mod 8). The reversal of J 2k+1 is therefore accounted for by the mass changing sign in 3 and 7 (mod 8) and by the F k factor changing sign in the other odd dimensions. A change in sign of the 2k + 1 bulk theory Dirac mass should also cause a change in the Γ5 chirality of the surface-trapped 2k dimensional fermions. An inspection of the table shows that it is precisely in 2 and 6 dimensions

62 that Γ5 anticommutes with both and , and so the change in chirality is consistent with the C T ψ(t, x) ( or )ψ( t, x) 7→ C T − time reversal transformation. The action of T on wavefunctions: The c-number spinor wavefunction corresponding to a single particle state φ is | i φ(x, t)= 0 ψ(x, t) φ . h | | i The wavefunction of the time reversed state is then

0 I−1 v 0 (I−1 v )= v (I 0 )= v 0 = 0 v ∗. h | | i≡h | | i h | | i h | i h | i The result is that, in contrast with the transformation of the operator, the single-particle wavefunction is complex-conjugated.

C and T in Condensed Matter37 C symmetry: Suppose that we have a many-fermion hamiltonian

ˆ † H =ΨαHαβΨβ where the N-by-N one-particle Hamiltonian matrix Hαβ is traceless and obeys CH∗C−1 = H − 37S. Ryu, A. Schnyder, A. Furusaki, A. Ludwig, Topological insulators and super- conductors: ten-fold way and dimensional hierarchy; New J. Phys. 12, 065010 (2010); arXiv:0912.2157.

63 for some unitary matrix C. Then Hu = λ u HCu∗ = λ Cu∗ , n n n ⇒ n − n n so, when λ is non zero, the single-particle eigenfunctions come in opposite- eigenvalue pairs. In the absence of zero energy states the ground state gnd has all negative-energy states occupied and is non-degenerate. | i We deﬁne the action of a unitary particle-hole operator C on the many- body Fock space by C empty = empty and | i | i C C−1 † C † C−1 † Ψβ =ΨαCαβ, Ψβ = CβαΨα. When C acts on the Hamiltonian we have C ˆ C−1 C † C−1 H = ΨαHαβΨβ C † C−1C C−1C C−1 = Ψα Hαβ Ψβ C † C−1 C C−1 = Ψα Hαβ Ψβ † † = CαρΨρHαβΨσCσβ = Ψ† C H C† Ψ − σ σβ αβ αρ ρ = Ψ† C HT C† Ψ − σ σβ βα αρ ρ = Ψ† C H∗ C† Ψ − σ σβ βα αρ ρ † = +ΨσHσρΨρ. = H.ˆ Thus the one-particle transformation on H leaves the many-particle hamiltonian invariant. We used C∗† = CT and the tracelessness (line 5 6) and hermiticity of H in the above manipulations. More importantly, and→ despite the appearance of the complex conjugation “ ” in H∗ = C−1HC, the many-body operator C must act on the Fock space∗ linearly: − C(λ ψ + µ ψ )= λC ψ + µC ψ . | 1i | 2i | 1i | 2i The linearity is required in the step

−1 CHαβC = Hαβ. If we write Ψα = unαaˆn n X 64 with n > 0 corresponding to positive energy and n < 0 to negative, then in the absence of zero energy states the ground state gnd is speciﬁed up to | i phase by aˆ gnd =ˆa† gnd =0, n> 0. n| i −n| i Let CT = λC, λ = 1. From this we deduce that Ca C−1 = λa† and hence ± n −n that C gnd = gnd . We also have that | i | i C(Ψ† Ψ N/2)C−1 =Ψ Ψ† N/2= (Ψ† Ψ N/2), β β − α α − − α α − so the sign of the normal-ordered charge operator 1 Qˆ = (Ψ† Ψ Ψ Ψ† )=Ψ† Ψ N/2 2 β β − β β β β − is reversed. T symmetry: Again consider a many-fermion hamiltonian

ˆ † H =ΨαHαβΨβ, but now assume that the one-particle Hamiltonian matrix Hαβ obeys

TH∗T −1 = H for some unitary matrix T . This condition tells us that if un(x, t) obeys

∂ i H u (x, t)=0 ∂t − n then Tu∗ (x, t) obeys the same equation: n − ∂ i H Tu∗ (x, t)=0. ∂t − n − We deﬁne the action of an anti-unitary time reversal operator T on the many-body Fock space by

T T−1 † T † T−1 † Ψβ = TβαΨα, Ψβ =ΨαTαβ.

Observe that when T is symmetric acting twice by T takes Ψα Ψα. If the T matrix is skew symmetric then acting twice with T takes Ψ → Ψ . α → − α

65 T † T−1 † Also ΨβΨβ =ΨβΨβ, so the sign of the charge is unchanged, and

T ˆ T−1 T † T−1 H = ΨαHαβΨβ T † T−1T T−1T T−1 = Ψα Hαβ Ψβ T † T−1 ∗ T T−1 = Ψα Hαβ Ψβ ∗ † = ΨρTραHαβTβσΨσ † = ΨρHρσΨσ. = H.ˆ

Again the transformation of H leaves the many-particle hamiltonian invariant. We required T to be a anti-linear map because we need

T T−1 ∗ Hαβ = Hαβ.

Consider the antisymmetric case, then given an eigenvector Hun = Enun, ∗ ∗ we have that HTun = EnT nn. The skew symmetry then tells us that un and ∗ Tun are orthogonal so all levels are doubly degenerate. This is Kramers theorem. Assuming that H is a 2N-by-2N matrix let us deﬁne

un, for 0 < n N ψ˜n = ≤ Tu∗ , for N

Then the orthonormality of the ψn gives us

0 I ψ˜T T −1ψ˜ = N ni ij jm I 0 − N nm As H∗ = HT and

U T T (UHU −1)T −1(U T )−1 =(U T T U)H(U T T U)−1 =(UHU −1)T a change of basis H UHU −1 takes T U T T U. We can use this freedom to make → → 0 I T = N , I 0 − N in which case the matrix ψ˜in is an element of the unitary symplectic group

Sp(N) Sp(2N, C) U(2N) ≡ ∩ 66 Fujikawa heat-kernel technique: Suppose we wish to compute the quantum mechanics matrix element

x e−tH(ˆp,xˆ) y , [ˆx, pˆ]= i. h | | i We use x pˆ ψ = i∂ x ψ , x xˆ ψ = x x ψ , h | | i − xh | i h | | i h | i to proceed as follows

x e−tH(ˆp,xˆ) ψ = e−tH(−i∂x,x) x ψ , h | | i h | i dk = e−tH(−i∂x,x) x k k ψ , 2π h | ih | i Z dk = e−tH(−i∂x,x)eikx k ψ , 2π h | i Z dk = eikxe−tH(−i∂x+k,x) k ψ , 2π h | i Z dk = eikx k ψ e−tH(−i∂x+k,x)1. 2π h | i Z Now set ψ = y and k y = e−iky to get | i | i h | i dk x e−tH(ˆp,xˆ) y = eik(x−y)e−tH(−i∂x+k,x)1 h | | i 2π Z where the ∂x derivatives act on everything to their right until they reach ∂x1 = 0. Generalizing this for our Dirac matrix-valued operators H = D2 we have 1 I and −6 7→ ddk Tr Γ e−tH(−i6∂,x) = tr Γ x e−tH(−i6∂,x) x dx = ddx tr Γ e−tH(−i6∂+6k,x)I . 5 { 5h | | i} (2π)d 5 Z Z Z Now 2 e−tH(−i6∂+6k,x) = e−tk e−tO(6∂,6k,x). We can (laboriously) expand exp tO(∂, k, x) in a power series in t, use ′ {− 6 6 } ∂xf(x) = f(x)∂x + f (x) to commute all the derivatives to the right where

67 they die by acting act on I, and lastly perform the k integral. The result is a short time asymptotic expansion

1 ∞ x e−tH(ˆp,xˆ) x A (x)tn, h | | i ∼ (4πt)d/2 n n=0 X where the An are spin-space matrix-valued functions of x. For a d = 2N Dirac operator, when we compute Tr Γ exp(tD2) only the t independent { 5 6 } term containing AN (x) should survive after the integration over a compact manifold.

Quantum mechanics illustration of Fujikawa’s method: A Laplace transform ∞ e−λt x e−tH x dt = x (H + λI)−1 x h | | i h | | i Z0 relates the heat kernel to the resolvent. Consider the relatively simple case of a Schr¨odinger operator H =p ˆ2 + V (x). If we write the short time expansion as

−tH 1 1 2 1 3 x e x 1 b1(x)t + b2(x)t b3(x)t + , h | | i ∼ √4πt − 3 − 3.5 ··· we ﬁnd that 1 b (x) b (x) b (x) x (H + λI)−1 x 1 1 + 2 3 + , h | | i ∼ √ − 2λ (2λ2) − (2λ)3 ··· 2 λ

The bn(x) in the resolvent are the well-known Gelfand-Dikii coeﬃcients

b1(x) = V (x), 3 V (x)2 V ′′(x) b (x) = , 2 2 − 2 5 V (x)3 5 V ′(x)2 5 V (x) V ′′(x) V (4)(x) b (x) = + , 3 2 − 4 − 2 4 35 V (x)4 35 V (x) V ′(x)2 35 V (x)2 V ′′(x) 21 V ′′(x)2 b (x) = + 4 8 − 4 − 4 8 7 V ′(x) V (3)(x) 7 V (x) V (4)(x) V (6)(x) + + , 2 4 − 8 . . ,

68 which Mathematica rapidly computes from their recursion formula 1 b (x)= V (x), ∂ b = V (x)∂ + ∂ V (x) ∂3 b . 1 x n+1 x x − 2 xxx n

Let’s see how eﬃcient Fujikawa’s method is at computing the bn(x). Fujikawa gives

dk 2 2 x e−tH x = e−tk et[∂x+2ik∂x−V (x)]1 h | | i 2π Z dk 2 1 = e−tk 1+ t[∂2 +2ik∂ V (x)] + t2[∂2 +2ik∂ V (x)]2 + 1 2π x x − 2 x x − ··· Z 1 = 1 tV (x)+ 1 √4πt { − ···}

2 The terms that need to be integrated against e−tk have a simple recursion relation t G (k, x)= tV (x), G (k, x)= [∂2 +2ik∂ V (x)]G (k, x). 1 − n n x x − n−1 With some eﬀort and the aid of Mathematica we can expand the exponen- 8 tial to order t . We ﬁnd that this only gives the bn(x) correct for n 4, as, after integration over k, the tn[...]n/n! terms in the exponential≤ for n =2, 3, 4, 5, 6, 7, 8 give contributions that start with powers t2, t2, t3, t3, t4, t4 and t5 respectively. Naturally, the correctly-evaluated terms agree with the Gelfand-Dikii computation.

Lai-Him Chan’s variant: We can work directly with the resolvent38. Again we illustrate in one dimension:

x (ˆp2 + V (x)+ λ)−1 y = ( ∂2 + V (x)+ λ)−1 x y h | | i − x h | i dk = ( ∂2 + V (x)+ λ)−1 x k k y 2π − x h | ih | i Z dk = ( ∂2 + V (x)+ λ)−1eikxe−iky 2π − x Z dk = eik(x−y)( [∂ + ik]2 + V (x)+ λ)−1 2π − x Z 38Lai-Him Chan, Eﬀective action Expansion in Perturbation theory, Phys. Rev Lett. 54 (1985) 1222-1225.

69 so the diagonal term becomes dk G(x, x)= (k2 + V (x)+ λ [∂2 +2ik∂ ])−1. 2π − x x Z The denominator contains terms that do not mutually commute, but we can use the identity

(A B)−1 = A−1 +(A B)−1BA−1 − − to recursively generate the Born series

(A B)−1 = A−1 + A−1BA−1 + A−1BA−1BA−1 + . − ··· The integrand thus becomes 1 1 1 + [∂2 +2ik∂ ] + k2 + V (x)+ λ k2 + V (x)+ λ x x k2 + V (x)+ λ 1 1 1 [∂2 +2ik∂ ] [∂2 +2ik∂ ] + . k2 + V (x)+ λ x x k2 + V (x)+ λ x x k2 + V (x)+ λ ···

Again the ∂x derivatives act on everything to their right. We have the recursion relation 1 F (k, x)= , F (k, x)= F (k, x)[∂2 +2ik∂ ]F (k, x). 1 k2 + V (x)+ λ n 1 x x n−1 The evaluation of the derivatives is still rapid although there are many terms. Mathematica, however, ﬁnds the integrations challenging and this variant method appears less eﬃcient.

SUSY-QM and the Atiyah-Patodi-Singer theorem: As an application of the Gelfand-Dikii formulæ we can consider one-dimensional SUSY QM in which 0 ∂ + h Q = i(σ ∂ + ih(x)σ )= i x . 1 x 2 ∂ h 0 x − 2 2 ′ 2 2 2 ′ ∂x + h + h 0 Q = ∂ + h + σ3h = − . − x 0 ∂2 + h2 h′ − x − Here h(x) is a potential that tends to constants h( ) at x . ±∞ → ±∞ Inspection of the first few Gelfand-Dikii expansion coefficients bn(x) for V (x)=(h(x))2 h′(x) shows that the h′ expressions differer only by total ± ± 70 derivatives. If this observation remains true for all n, and because h tends to a constant at large distance, we can evaluate

−tQ2 Tr σ3e n o in closed form. This is because only terms with exactly one derivative can contribute, and they give

∞ ∞ n 2 1 ( t) Tr σ e−tQ = tr σ − (h2 + σ h′)n + ... dx 3 √ 3 n! 3 2 πt ( −∞ " n=0 # ) n o Z X 1 ∞ ∞ ( 1)ntn = tr σ − σ n(h2)n−1h′ dx √ 3 n! 3 2 πt ( −∞ "n=0 # ) Z X ∞ n n 1 ( 1) t ∞ = − h2n−1 √ (n 1)!(2n 1) −∞ πt n=0 X − − 1 ∞ = sgn(h)erf(√t h ) , −2 | | −∞ where x 2 −ξ2 1 at x , erf(x)= e dξ = →∞ √π 0 0 at x = 0. Z n It is reasonable to doubt the validity of the process of extracting a convergent series out of an asymptotic series, but we can derive this result more rigorously. Firstly we know that 1 ∞ n n = sgn(h) + − − −2 −∞ is the diﬀerence between the number of σ3 1 zero-eigenvalue bound states. Secondly we can use the fact that for non-zero→ ± eigenvalues λ2 we can map the solutions of (∂ + h)( ∂ + h)u =( ∂2 + h′ + h2)u = λ2u x − x − x to those of ( ∂ + h)(∂ + h)v =( ∂2 h′ + h2)v = λ2v − x x − x − by setting v ( ∂x + h)u and u ( ∂x + h)v. If we impose Dirichlet boundary conditions∝ − at some large∝ distance− L this allows us to compute ± 71 the diﬀerence in the density of eigenvalue λ2 = k2 +h2( ) scattering states for the hamiltonians H = ∂2 + h2 h′ as39 ±∞ ± − x ± 1 h ∞ ρ ρ = . + − − 2π k2 + h2 −∞ With this information we have ∞ ∞ ∞ −tQ2 1 dk h −t(k2+h2) Tr σ3e = sgn(h) + 2 2 e −2 −∞ −∞ 2π k + h n o Z −∞ 1 ∞ 1 ∞ = sgn(h) + sgn(h)(1 erf(√t h )) , −2 2 − | | −∞ −∞ which is the same result as before. We can write this as ∞ 1 −tQ2 1 ∞ sgn(h) = Tr σ3e η(t) −∞ −2 −∞ − 2 | n o with η(t) = sgn(h)erfc(√t h ), | | where erfc(x) denotes the complementary error function 0 at x , erfc(x)=1 erf(x)= →∞ − 1 at x = 0. n In terms of the zero modes we have

2 1 n n = Tr σ e−tQ η(t) ∞ . + − − 3 − 2 |−∞ n o This is the simplest example of the key formula in the Atiyah-Patodi-Singer (APS) theorem40 The APS theorem is of use when we consider a manifold of the form R M 2N−1 with metric × 2 0 2 µ ν ds =(dx ) + gµνdx dx .

39M. Stone, Zero Modes, Boundary Conditions and Anomalies on the Lattice and in the Continuum, Annals Of Physics 155 (1984) 56-84. 40M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral Asymmetry and Riemannian Ge- ometry I, Mathematical proceedings of the Cambridge Philosphical Society, 77 (1976) 43-69.

72 2N−1 Here gµν is the metric of M , an odd-dimensional compact manifold without boundary. We think of the the R coordinate x0 as Euclidean time. Now consider the skew-Hermitian massless 2N dimensional Dirac operator

0 ∂ 0 + H D = x , 6 ∂x0 H 0 − where H is a Hermitian operator on M 2N−1 in which we think of x0 as a parameter. 0 The Weyl-like chiral hamiltonian H will have eigenvalues En(x ) and replacing h(x) in the SUSY QM example by the eigenvalues of H(x0) we have 2 1 n n = Tr Γ etD6 η(t, H) ∞ + − − 5 − 2 |−∞ where n o η(t, H)= sgn(E )erfc(√t E ). n | n| n X and n are the numbers of normalizable zero modes of D with ± 6 1 0 Γ = 5 0 1 − having eigenvalues 1. These zero-mode numbers are determined by the 0 ± 0 number of En(x ) that change sign as x goes from to + . For D we have the usual short-time heat kernel expansion−∞ ∞ 6 1 ∞ x exp tD2 x A (x)tn h | { 6 }| i ∼ (4πt)N n n=−N X in which AN (x) is the index density. On taking the matrix trace with Γ5 only this term will survive. Taking this fact into account, our zero-mode index formula becomes 1 T 1 n n = A (x)d2N−1x dx0 η(t, H) T . + − − (4π)N N − 2 |−T Z−T ZM The LHS is an integer and on a compact manifold the integral involving AN (x) will also be an integer. Here, however, the manifold is not compact in the x0 direction so there is no reason for the integral to have an integer

73 “instanton number.” The extra term η(t, H) therefore appears as a complementary correction that rounds-up the integral to a whole number. There is an obvious ﬂaw in this interpretation however: both the LHS and the integral are independent of the heat-kernel parameter t while the η(t) term is not. How do we get around this problem? The escape is that the short time expansion is only an asymptotic series in t and is therefore ambiguous up to non- perturbative terms such as exp 1/t . {− } If the set E is the Dirac-like spectrum of an unbounded operator, and { n} f(t) is a cutoﬀ that obeys limt→0 f(t) = 1, limt→∞ f(t) = 0, a sum

η(t) = lim sgn(λ)f(t En ) t→0 | | ( En ) X deﬁnes the spectral asymmetry. The t 0 limit does not usually depend on → the detailed form of f(t) but the rate of convergence to the limit may do so. Take, for example, En = n + α and consider for the the sum

∞ η (t, α)= sgn(α + n) exp t n + α . exp {− | |} n=−∞ X

The cutoﬀ ensures that ηexp(t, α) = ηexp(t, α + m) for any m Z, and for 0 <α< 1 we can evaluate the sum in closed form as ∈ sinh[(1 2α)t/2] η (t, α)= − . exp sinh t/2

This expression converges to (1 2α) as t 0. Taking into account the − → periodicity, we see that the graph of ηexp(0+,α) is sawtooth. With the exponential cutoﬀ the convergence to the limit is not very rapid: there are O(t2) corrections. The “erfc” regulated sum behaves diﬀerently. It is not easy to evaluate the sum

∞ η (t, α)= sgn(α + n) erfc(√t n + α ) erfc | | n=−∞ X directly, but as long as we avoid the jumps that occur when α passes through an integer its derivative with respect to α can be re-expressed via Poisson

74 summation to give

∂η 4t ∞ erfc = exp t(n + α)2 ∂α − π {− } r n=−∞ X ∞ (2πn)2 = 2 exp +2πinα . − − 4t n=−∞ X The n = 0 term gives is the expected d (1 2α)= 2. The next correction, dα − − from the n = 1 terms, is exponentially vanishing at small t and so is invisible to the± small-t asymptotic expansion. That this rapid convergence holds for all Dirac-like spectra is asserted just after eq (2.24) in the APS 2 paper. Presumably the exp (2π) /4t in the En = n + α example will be replaced by exp (L2)/4t {−where L is} some length in the compact manifold {− } that determines the characteristic level spacing of the discrete eigenvalues. Accepting APS’s claim, the apparent t-dependence paradox is avoided. The net result is that we can think of 1 η(0 ,H) as being the fractional − 2 + part of the chiral charge created by the topological winding captured by AN (x).

Connection forms in geodetic coordinates: In locally geodetic coordinates, and in any number of Euclidean dimensions, we have 1 g (x) = δ R (0)xσxτ + O( x 3), µν µν − 3 µσντ | | 1 Γλ (x) = (R (0) + R (0))xτ + O( x 2). µν −3 λνµτ λµντ | | Similarly we we can construct local vielbein frames in which we have 1 e∗a(x) = δ R (0)xσxτ + O(x2), µ aµ − 6 aσµτ 1 ωa (x) = Ra (0)xτ + O( x 2). bµ −2 bµτ | |

Gravitational index from Heat kernel: We wish to compute the index

† Ind(D) = Tr Γ = ddx u† (x)Γ u (x)= ddx tr Γ e−tD6 D6 . 6 { 5} n 5 n { 5 } n Z X Z

75 By scaling to the small t limit41 Getzler shows that we only need compute

tr Γ eQ { 5 } where ∂ i 2 Q = tΩ xj − ∂xi − 4 ij i X 1 Ω = γaγbR , ij 2 abij and in this limit we can regard the γ”s as being classical anticommuting objects γaγb = γbγa for all a and b. As the pair-products− γaγb now commute we can formally imagine that we have skew-diagonalized Ωij to the form

0 ω − 1 ω1 0  .  Ωij = ..  0 ω   N   ω −0   N    where after division by 2π the notional quantities ωi/2π are the Chern roots xi = ω/2π. Then Q becomes a sum of N terms of the form

∂ i 2 ∂ i 2 ∂2 ∂2 1 i ∂ ∂ tωy + tωx = + + t2ω2(x2+y2)+ tω y x − ∂x − 4 − ∂y 4 − ∂x2 ∂y2 16 2 ∂x − ∂y for which Mehler gives

1 tω/2 [eQ0 ] = . x,x 4πt sinh tω/2

Taking into account that iω are the eigenvalues of Ω we see that full heat ± kernel of the reduced form is

1 tΩ/2 det , (4πt)N sinh tΩ/2 s 41E. Getzler, A Short Proof of the Local Atiyah-Singer Index Theorem, Topology, 25 111-117 (1986).

76 where now we can restore the γa’s as genuine matrices. We can expand this 0 out to find the coefficient of t , and then take the trace with Γ5. The net effect of the 2π’s and the i’s in Γ5 is to make

Ind(D)= Aˆ 6 Z where only the form of the right degree is understood.

Chern roots, Aˆ-genus, and Pontryagin classes: There are various matrix identities such as tr(ln A) = ln(det[A]) and the expansion 1 det(1 + A)=1+tr(A)+ (tr A)2 tr(A2) + ... 2 − that require complicated combinatorics to prove for general A, but become almost obvious when A is diagonal. Now these relations are invariant under A V −1AV and as almost all (in the measure theory sense) matrices are → diagonalizable by such a transformation, and because algebraic relations that are true almost everywhere in fact hold everywhere, we only ever need to consider diagonal A. Indeed we don’t even have to diagonalize A, we only to imagine that we have diagonalized it. We can similarly imagine that the skew-symmetric matrix-valued curvature two-form 1 R = R dxµdxν ij 2 ijµν can be written in the canonical form 0 x − 1 x1 0  .  Rij =2π ..  0 x   n/2   x − 0   n/2    The notional skew eigenvalues xi are called the Chern roots. They are are only notional because the Rij only become numbers after we plug vectors µ into the dx . Nonetheless the xi enable easy derivations of otherwise hard- to-prove relations between characteristic classes as they are also invariant under R V −1RV transformations. → 77 For example the Pontryagin class becomes

n/2 p(R) def= det (1 R/2π)= (1 + x2). − i i=1 Y while the Dirac index involves

n/2 R/4πi x /2 Aˆ(R) def= det = i . sinh R/4πi sinh x /2 s i=1 i Y

Both quantities are symmetric functions of the xi and expanding out and comparing coeﬃcients of the same degree in the xi leads to the relations

R/4πi Aˆ[R] def= det sinh R/4πi s 1 1 1 1 = 1+ tr R2 + (tr R )2 + tr R4 + (4π)2 { } (4π)4 288 { } 360 { } ··· 1 1 = 1 p + (7p2 4p )+ ..., − 24 1 5760 1 − 2 that remain true, but require hard combinatorics to show, when R is not in its skew-diagonal canonical form. Second Covariant Derivative: The expressions

2 = D D Γλ D ∇µν µ ν − νµ λ and 2 = D D ωc D ∇ab a b − ba c are examples of the “second covariant derivative” which is deﬁned for general vector ﬁelds U, V , by

2 = . ∇U,V ∇U ∇V −∇∇U V The second term gets rid of the derivative of V so that

2 = U µV ν ∇U,V ∇µ∇ν

78 with an extra connection term understood in µ to deal with the index“ν”. For a torsion-free connection where V ∇U = [U, V ], we have ∇U −∇V ( 2 2 )W = [ , ]W ( )W ∇U,V −∇V,U ∇U ∇V − ∇∇U V −∇∇V U = [ , ]W W = R(U, V )W. ∇U ∇V −∇[U,V ] We have used that is linear in X. ∇X

Absence of anomalies in the standard model While anomalies are allowed in global symmetries such as baryon number B and lepton number L, it is necessary for consistency that the gauge currents in the SU(3) SU(2) U(1) standard model be free from anomalies. C × W × Y That this is so requires some bizarre cancellations among the group-generator traces in the triangle diagram. Here is a table from a talk by Hitoshi Mu- rayama42:

U(1)3 : 3 2 1 3 +3 2 3 +3 1 3 +2 1 3 + (1)3 =0. · 6 − 3 3 − 2 U(1)(gravity)2 : 3 2 1 +3 2 +3 1 +2 1 +(1)= 0. · 6 − 3 3 −2 U(1)(SU(2))2 : 3 2 1 +2 1 =0. · 6 − 2 U(1)(SU(3))2 : 3 2 1 +3 2 +3 1 =0. · 6 − 3 3 (SU(3))3 : #3 #3=2¯ 1 1=0 − − − The (SU(2))3, (SU(3))2SU(2), and SU(3)(SU(2))2 anomalous triangle dia- grams are also zero. The first because tr (σ σ , σ ) = 0, and the other two a{ b c} from the vanishing trace in the single factor. There is also a potential global anomaly in SU(2) which is avoided because the number of SU(2) doublets in each generation is 3 + 1 = 4, which is even. These cancellations work separately in each fermion generation. The peculiar cancellations in the table can be understood by extending the standard model to a Grand Unified Theory (GUT) based on embedding the model first in SU(5), and then in the anomaly-free group SO(10). The SO(10) group is anomaly-free in four dimensions and for all representations because it is not possible to build an invariant tensor for the triangle-diagram group factor

tr(T T , T ), T = T , ij{ kl mn} ij − ji 42http://hitoshi.berkeley.edu/129A/GUT+SUSY.pdf

79 having the correct symmetries and using only the basic SO(10) invariants, which are δij and the ten-index Levi-Civita symbol. This impossibility is in contrast to SO(6), for example, for which we can have tr(T T , T ) ǫ . ij{ kl mn} ∝ ijklmn SU(5) GUT fermion allocations: Consider the subgroup decomposition SU(5) [SU(3) SU(2) U(1) ]/Z → C × W × Y 6 2πi/6 2πi/2 −2πi/3 2πi/2 where the Z6 arises because e = e e and e SU(2) and e−2πi/3 SU(3). ∈ The∈ deﬁning irrep 5 of SU(5) breaks up in the obvious way as 5 3 2 → ⊕ where the 3 comprises the ﬁrst three components and the 2 the second two. Label the SU(3)C indices as R,G,B (Red, Green, Blue) and the SU(2)W indices as P, Q (Peach and Quince). Georgi and Glashow show that all fermions for a single generation of quarks and leptons can be accommodated in the 10 5¯ of SU(5). To achieve this they take all fermions to be left handed ⊕ by using the action of CP to replace the right-handed SU(2)W -singlet quarks c c uR, dR by their left handed antiparticles uL and dL. The antisymmetric 10 dimensional representation 10 = of SU(5) then has particle allocations

uL RP GP BP dL RQ GQ BQ c uL RG GB BR (B¯) (R¯) (G¯) c eL P Q (0)

The colour labels in parenthesis are the eﬀective gauge charges after contract- abc ab ing with the invariant tensors ε or ε for SU(3)C and SU(2)W respectively. The 5¯ of SU(5) has Young tableau and we assign

c ¯ ¯ ¯ dL R G B νL Q¯ eL P¯

Charges under U(1)EM are given by the Gell-Mann-Nishijima formula 1 Q =(T ) + Y, EM 3 W 2 80 where Y is the weak hypercharge. The SU(5) generator

2 3 − 2 3  −  5 2 Y = 2 = λ , tr λ =2. − 3 3 24 { 24}  1  r    1      supplies what Wilczek calls43 the “peculiar” Y values:

uL dL uR dR eR eL νL Y 1/3 1/3 4/3 -2/3, -2 -1 -1 T3 1/2 -1/2 0 0 0 -1/2 1/2 QEM 2/3 -1/3 2/3 -1/3 -1 -1 0

c For example uL : RG 2/3 2/3 = 4/3, so uR : Y = +4/3. Similarly c → − − − eL : P Q 1+1 = 2, so eR : Y = 2. The 5/3 scaling factor between Y and the→λ that has the same normalization− as T can be absorbed into 24 p 3 gW /gY coupling constant ratio, and hence determines the value of the Wein- 2 15 berg angle sin θW = 3/8 at the GUT scale 10 GeV compared to the 2 ∼ measured value sin θW = .2312 at MZ . SU(N) Spin(2N): The curious 10 5¯ combination arises naturally when we embed⊂ SU(5) as a subgroup of SO(10),⊕ or more properly Spin(10). The general case of SU(N) Spin(2N) can be understood from the gamma matrix construction ⊂

† γi = ai + ai γ = i(a† a ). i+1 i − i The generators of the Cartan algebra of Spin(2N) are 1 h def= [γ ,γ ]= a†a 1 , i =1,...,N, i 4i i+1 i i i − 2 and the chiral subspaces determined by

† Γ =( 1)Pi ai ai 5 − 43F. Wilczek, hep-ph/9702371. My fermion tables are adopted from this talk.

81 are those of odd and even “fermion number.” Let us deﬁne ω as the so(2N) element with N copies of 0 1 iσ2 = − − 1 0 arranged on the diagonal. Then

ω = (a†a 1 ). i i − 2 i X The subgoup of SO(2N) that commutes with ω is generated by ω and the other fermion number conserving operators ˆ ij † λa = λa ai aj, where the λa are the traceless matrix generators of SU(N). This is one way to understand SO(2N) Sp(2N, R) U(N) [SU(N) U(1)]/Z ∩ ≃ ≃ × N with ω as the skew-symmetric matrix deﬁning Sp(2N, R). If we write a typical element of the reducible non-chiral 32-dimensional spin representation of SO(10) as ψ = 0 ψ + a† 0 ψ + 1 a†a† 0 ψ + 1 εijklma† a†a† 0 ψ¯ | i | i 0 i | i i 2 i j| i ij 2·3! k l m| i ij + 1 εjklmna† a†a† a† 0 ψ¯ + a† a†a† a† a† 0 ψ¯ , 4! k l m n| i j 1 2 3 4 5| i 0 we have the decomposition into even and odd fermion-number chiral irreps ψ ψ = + , ψ − where ¯ T ¯ ¯ T ψ+ =(ψ0, ψij, ψj) = 16 ψ− =(ψ0, ψij, ψj) = 16, exhibits the SU(5) U(1) decompositions × X 16 1 10 5¯ , 16 1¯ 10 5 . → −5 ⊕ −1 ⊕ 3 → 5 ⊕ 1 ⊕ −3 The subsripts are the U(1)X charges where the X generator is

5 X = (2a†a 1). i i − i=1 X 82 In the even-fermion-number chiral irrrep 16 of SO(10) we have the follow- ing weight-particle assignments which are compatible with the SU(5) U(1) × restriction and the SU(5) SU(3) SU(2)W U(1)Y restriction. Here → 1× × ± denotes the SO(10) weights hi 2 , and we just display typical weights for the restricted irrep: → ± ( + +) 6 (u ,d ) − − | − L L 10 : ( ++ ) 3¯ uc  − | − − L ( ++) 1 ec  −−−| L  ( ++ + +) 3¯ dc 5¯ : L (+++− | +) 2¯ (e , ν ) | − L L 1 : ( ) 1 N − − −| − − The last particle, N, is a (Standard Model) sterile neutrino that can acquire a GUT-scale Majorana mass and precipitate the see-saw mechanism for the masses of the usual neutrinos.

Domain-wall femions and Ginsparg-Wilson Let the d + 1 dimensional Dirac operator be

D =Γ ∂s + D m(s) 6 d+1 5 6 d − and let H =Γ (D m(s)) 5 6 d − be the Hamiltonian for propagation in the s direction. The anti-hermiticity of D makes H† = H. 6 Define H D =(1+Γ5ǫ(H)), ǫ(H)= √H†H and observe that (ǫ(H))2 =1 Now (D 1)†(D 1) = ǫ(H)Γ Γ ǫ(H)† =1, − − 5 5 so D =1+ V where V is unitary. The eigenvalues of D therefore lie on a unit circle surrounding the point 1. We have the Ginsparg-Wilson relation 1 Γ D + DΓ = DΓ D 5 5 M 5 83 Berezin Integrals: Determinants vs. Pfaffians: The path integral for fermions requires a formal integration over Grassmann-valued fields. Felix Berezin’s recipe for this process is purely algebraic but is called “integration” because its output mirrors, up to signs, the result of the corresponding analytic operation on real and complex variables. The general Grass- mann/Berezin integral requires sophisticated mathematics44, but the only cases we require are (the function-space extensions of) finite “Gaussian” integrals, and these are relatively straightforward. β If ψ¯α and ψ , α, β = 1,...,N are a set of anticommuting Grassmann variables, we define their Berezin integral by setting N [dψdψ¯ ] ψ¯ ψ1 ψ¯ ψN dψ¯ dψα ψ¯ ψ1 ψ¯ ψN =1. 1 ··· N ≡ α 1 ··· N "α=1 # Z Z Y To obtain a non-zero answer all 2N anticommuting variable must be present in the integrand, and in numerical order to get a +1. Each interchange of adjacent variables gives a factor of 1. Under linear changes of variables− ψα ψ′α = Aα ψβ → β ψ¯ ψ¯′ = ψ¯ Bβ α → α β α we have d[ψ] [dψ′] = det[A]−1d[ψ], → d[ψ¯] d[ψ¯′] = d[ψ¯] det[B]−1 → in which the Jacobean factors are the inverse of the commuting variable version. α If L, with entries L β, is an N-by-N matrix representing a linear map L : V V , we expand the exponential function in the first line below and → use the definition to get

Z(L) = [dψdψ¯ ] exp ψ¯ Lα ψβ , { α β } Z1 = ǫ ǫβ1...βN Lα1 LαN N! α1...αN β1 ··· βN = det [L].

44See for example: Martin R. Zirnbauer, Riemannian symmetric superspaces and their origin in random-matrix theory, J. Math. Phys. 37 (1996) 4986; arXiv:math-ph/9808012.

84 The integral for the two-variable correlator 1 ψ¯ ψj def= [dψ¯][dψ]ψ¯ ψj exp ψ¯ Lα ψβ , h i i Z(L) i { α β } −1 j Z = [L ] i

j follows because the explicit ψ¯iψ factor forces the omission of the term con- i taining L j in the expansion of the exponential, and from the formula for the inverse of a matrix 1 L−1 = Adj[L], det [L] where Adj[L] is the adjugate matrix i.e. the transposed matrix of the co- factors. We can check the sign and index placement by observing that the claimed expression gives

ψ¯ Li ψj = Li [L−1]j = tr I = N, h i j i j i { N } i j which is correct because inserting an explicit factor of ψ¯iL jψ into the integral means that we need to expand the exponential only to order N 1 to get all the ψ’s, and hence we get N!/(N 1)! = N times the integral without− − the explicit factor. Linear maps are naturally associated with eigenvectors and and eigenvalues. When L is diagonalizable — i.e. possesses suﬃcient eigenvectors un to form a basis — the determinant is the product of the eigenvalues

det[L]= λn. n=1 Y We can extract formula from the integral by diagonalizing L A−1LA = −1 → diag(λ1,...,λn) and then by setting B = A in the change of variables formulæ given above. For Majorana fermions we require an integral containing a skew symmetric matrix Qij representing a skew bilinear (symplectic) form Q : V V C. As the matrix Q is equipped with two lower indices, we no longer× need→ dis- α ¯ α tinguish between ψ and ψα. For a 2N-by-2N matrix we have ψ , α = 1,..., 2N, and the deﬁning integral becomes

[dψ]ψ1 ψ2N =1. ··· Z 85 Again all ψα must be present and in numerical order to get +1. Using this deﬁnition we evaluate 1 Z(Q) = [dψ] exp ψαQ ψβ 2 αβ Z 1 = ǫα1...α2N Q Q 2N N! α1α2 ··· α2N−1α2N = Pf [Q].

The last two lines serve to deﬁne the Pfaﬃan of the skew symmetric matrix Q. The two-variable correlator is now

1 1 ψiψj = [dψ] ψiψj exp ψαQ ψβ h i Z(Q) 2 αβ Z = [Q−1]ji.

Again we check the sign and index placement by computing 1 1 1 ψiQ ψj = Q [Q−1]ji = tr I = N 2h ij i 2 ij 2 { 2N } Regarding Q simply as a numerical matrix there is a well-known identity

(Pf Q)2 = det [Q], which implies that the Pfaffian of a matrix is a square-root of its determinant. We need to interpret this statement with care. A linear map L and a symplectic form Q are different mathematical objects. A linear map L : V V → possesses eigenvalues and eigenvectors while a bilinear form Q : V V C does not. The placement of the indices on their entries indicates tha× t their→ matrix representatives respond differently to a change of basis in the vector space V :

L B−1LB, (Similarity transformation), → Q BT QB, (Congruence transformation). → From det [BT QB] = det [Q] det [B]2

86 we see that a bilinear form does not possess a basis-independent determinant. We will show later that Pf[BT QB] = Pf[Q] det[B], so a skew bilinear form does not possess a basis-independent Pffafian. To convert a linear map into a bilinear form, or vice-versa, we need to α have some sort of “metric” to lower or raise the first index on L β or Qαβ. For relativistic Majorana fermions in spacetime dimensions 2, 3, 4 (mod 8) this role is played by the antisymmetric charge-conjugation matrix and Cαβ its inverse [C−1]αβ. For (pseudo) Majorana fermions in spacetime dimensions 0, 1, 2 (mod 8) the role is taken by the symmetric time-reversal matrix Tαβ and its inverse [T −1]αβ. The two cases are different. In the finite dimensional veriosn of the first we are given a 2N-by-2N non-degenerate skew-symmetric matrix C with entries Cij together with a self-adjoint linear operator represented by a 2N-by-2N i k hermitian matrix L with entries L j, such that their product Qij = CikL j is skew symmetric. The simplest case is 0 1 λ 0 0 λ = . 1 0 0 λ λ 0 − − We wish to evaluate Pf[Q] = Pf[CL] in terms of the eigenvalues of L. Each eigenvalue occurs twice and, after some algebra that we will display later, we find that we need only one of the pair in the result

Pf[Q] = Pf[C] λn. n=1 Y In the second case C is replaced by a 2N-by-2N non-degenerate symmet- k ric matrix T with entries Tij such that Qij = TikL j is skew symmetric. The simplest example is 1 0 0 λ 0 λ = . 0 1 λ 0 λ 0 − − We can again evaluate Pf[Q] = Pf[T L] in terms of the eigenvalues of L, but the result is more complicated. The eigenvalues occur in λ pairs, and if ± n we arbitrarily select λn to the positive eigenvalue, we ﬁnd N Pf[T L]= ( 1)N det[T ] ( λ ). ± − − n n=1 p Y 87 It is not possible to decide what sign to take for the without more information. In the simplest example above, we need the minus± sign if λ is positive and the plus sign if λ is negative. The source of the diﬀerence between the C case and the T case is that after reducing C to a standard symplectic form, the matrices B in the subsequent normal-form reduction

N 0 λ Q BT QB = n → λn 0 n=1 M − belong to Sp(2N) and symplectic matrices are automatically unimodular. After reducing T to a standard metric the matrices B lie in SO(N, N) and orthogonal matrices can have either 1 as their determinant. The proofs of these Pfaﬃan formulæ are given below.± Pfaﬃan to Determinant: We can rewrite the linear operator “action” as

1 0 L ψ¯ ψLψ¯ = [ψ,¯ ψ] . 2 LT 0 ψ − When L is N-by-N we can now compute the Pfaffian of the skew symmetric matrix and so find that 0 L Pf =( 1)N(N−1)/2det[L], LT 0 − − The sign comes from the need to rearrange the dψ and dψ¯ so as to put all the dψ¯’s before the dψ’s instead of in adjacent pairs. Some proofs: To show that Pf[BT QB] = det[B]Pf[Q], start from the definition of the Pfaffian 1 Pf [Q]= ǫj1...j2N Q Q 2N N! j1j2 ··· j2N−1j2N and recall that

ǫj1,...j2N det[B]= ǫi1,...,i2N B B . j1i1 ··· j2N i2N Thus 1 Pf [BT QB] = ǫi1...i2N [BT QB] BT QB] 2N N! i1i2 ··· i2N−1i2N 88 1 = ǫi1...i2N B Q B B Q B 2N N! j1i1 j1j2 j2i2 ··· j2N−1i2N−1 j2N−1j2N j2N i2N 1 = ǫj1...j2N det[B]Q Q 2N N! j1j2 ··· j2N−1j2N = det[B]Pf [Q].

To show that (Pf Q)2 = det[Q] remember that given a 2N-by-2N non-degenerate skew-symmetric matrix Q with entries in a ﬁeld, we can repeatedly complete squares to ﬁnd a linear map B that reduces Q to the canonical form45

N 0 1 Q = BT JB, J = . 1 0 1 M − Taking the Pfaﬃan of this equation we get

Pf[Q] = det[B]Pf[J] = det[B].

Taking the determinant gives

det[Q] = det[(BT JB] = det[B]2 = (Pf Q)2.

To evaluate Pf[CL] where C is skew symmetric and L is a 2N-by-2N k hermitian matrix such that Qij = CikL i is skew symmetric, observe that the hermiticity of L implies that L∗ = LT , and hence

Lu = λu LC−1u∗ = λC−1u∗ . n n ⇒ n n −1 −1 ∗ The skew symmetry of C guarantees that un and C un are mutually orthogonal † −1 ∗ ∗T −1 ∗ un(C un)= un C un =0, so each eigenvalue of L is therefore doubly degenerate and we can assume that −1 ∗ un and C un, n =1,...,N, together constitute a complete orthonormal set. 45See section A.7.2 in M. Stone, P. Goldbart, Mathematics for Physics: a guided tour for graduate students..

89 Let us introduce vectors X˜ = (˜x1, y˜1, x˜n, y˜n) and X = (x1,y1, xn,yn) Using the orthonormality we have ··· ···

˜ T T −1 ∗ T −1 ∗ X B (CL)BX = (˜xnun +˜ynC un) CD(xnun + ynC un) n X = (˜x uT y˜ u† C−1)CD(x u + y C−1u∗ ) n n − n n n n n n n X = λ (˜x y y˜ x ) n n n − n n n X = X˜ T ΛX.

Here N 0, λ Λ= n , λn 0 n=1 M − and B is the 2N-by-2N matrix

B = [u ,C−1u∗, ,u ,C−1u∗ ]. 1 1 ··· n n We have reduced CL to a canonical form, and taking the Pfaﬃan we have

T Pf[B (CL)B] = Pf[CL]det[B]= λn. n Y We need to ﬁnd an expression for det[B]. To do this replace L by I2N while keeping the un unchanged. This results in

N 0 1 BT CB = J = . 1 0 n=1 M − But Pf[J] = 1 so det[B] = Pf[C]−1. The end result is that

Pf[CL] = Pf[C] λn. n Y In the second case we are given 2N-by-2N non-degenerate symmetric k matrix T with entries Tij such that Qij = TikL i is skew symmetric. An example to bear in mind is 1 0 0 λ 0 λ = . 0 1 λ 0 λ 0 − − 90 To evaluate Pf[Q] = Pf[T L] in terms of the eigenvalues of L we use a similar strategy as before. In this case the hermiticity of L gives us Lu = λ u LT −1u∗ = λT −1u∗ , n n n ⇒ n − n −1 ∗ and if λn is non zero T un is orthogonal to un because they have different eigenvalues. The non-zero-mode eigenvectors of L therefore come in −1 ∗ opposite eigenvalue pairs. If there are no zero modes the un and T un, n = 1,...,N, together constitute a complete orthonormal set. We will take λn to be the positive eigenvalue. Again set X˜ = (˜x1, y˜1, xñ, yñ) and ··· −1 X = (x1,y1, xn,yn) and use the orthonormality and symmetry of T to conclude that··· ˜ T T −1 ∗ T −1 ∗ X B (T L)BX = (˜xnun +˜ynT un) T L(xnun + ynT un) n X T † −1 −1 ∗ = (˜xnun +˜ynunT )T L(xnun + ynT un) n X = ( λ )(˜x y y˜ x ) − n n n − n n n X = X˜ T ( Λ)X − where N 0, λ Λ= n , λn 0 n=1 M − and B is the 2N-by-2N matrix B = [u , T −1u∗, ,u , T −1u∗ ] 1 1 ··· n n Thus Pf[BT (T L)B] = Pf[T L]det[B]= ( λ ). − n n Y In this case, however we cannot take the Pfaffian after replacing L by I because T is not skew symmetric. We can still replace L I and find that → T XB˜ TBX = (˜xnyn +˜ynxn)= XGX˜ n X where N 0, 1 G = . 1 0 n=1 M 91 We can now take the determinant to conclude that

det[BT T B] = det[G]=( 1)N − and so det[B]2detT = ( 1)N . Hence − Pf[T L]= ( 1N )det[T ] ( λ ), ± − − n n p Y where the sign comes from the need to take det[B]2. The uncertainty as to which root± to take is inevitable. We arbitrarily assigned u to the positive p n eigenvalue rather than to the negative. If we make the opposite choice for some eigenvalue pair, the product formula must change sign whilst Pf[Q] itself is indiﬀerent to our choice. The source of the diﬀerence between the C and T cases is that after reducing C to the standard symplectic form J the diagonalizing matrices preserve J and hence belong to Sp(2N) — and symplectic matrices are automatically unimodular. After reducing T to the standard form G the diagonalizing matrices preserve G and so lie in SO(N, N) — and orthogonal matrices can have determinant 1. ±