Fall Term Exam, N^{0}. 1 (solution by L. Jeffrey and S. Wu)

Problem:

Prove via path integrals the character valued index theorem for the Dirac operator.

That is, let M be a spin manifold and let g be an isometry of M, with a lift to the spin bundles. Calculate

T r(−1)^{F} g

by path integrals.

That is, write down the path integral for sections of a suitable M bundle over a circle. Taking the circumference of the circle to be small, compute the integral explicitly and obtain the Atiyah-Bott formula for (1).

Solution:

We use a supersymmetric quantum field theory where the fields are maps X : R^{1,1} → M satisfying a periodicity condition. We write the maps as X(t, θ) where t is the even variable and θ the odd variable: decomposing into bosonic and fermionic fields we write

X(t, θ) = x(t) + θψ(t).

The Lagrangian is

L(X) = ∫ dtdθg_{IJ} \frac{dX^{I}}{dt} DX^{J}

where we have introduced a metric g_{IJ} on M and the operator

D = \frac{∂}{∂θ} − iθ\frac{∂}{∂t}.

In components the action becomes

L(X) = ∫ dt\{g_{IJ} \frac{dx^{I}}{dt} \frac{dx^{J}}{dt} + \frac{i}{2}g_{IJ}ψ^{I} \frac{Dψ^{J}}{Dt} \},

where Dψ^{J}/Dt denotes the covariant derivative (with respect to the Levi-Civita connection) in the direction of the tangent vector x_{∗}(d/dt) in M. The supersymmetry operator is

Q = \frac{∂}{∂θ} + iθ\frac{∂}{∂t}.

Upon quantization the space of states is identified with the sections H_{±} of the spin bundle S_{±} on M and the operator Q is identified with the Dirac operator D on M. We introduce the operator (−1)^{F} on H_{±} which is defined to have the values +1 on H_{+} (states with an even number of fermions) and (−1) on H_{−} (states with an odd number of fermions). The index of the Dirac operator D is then

Index(D) = Trace(−1)^{F} = Trace(−1)^{F} e^{−βH},  (1)

where we have introduced the Hamiltonian H and a positive parameter β. Notice that since the Dirac operator D gives an isomorphism between the eigenspaces H_{+}^{λ} of H on H_{+} with a positive eigenvalue λ and the corresponding eigenspaces H_{−}^{λ} of the action of H on H_{−}, the trace of (−1)^{F} e^{−βH} is in fact independent of β. This can be computed by the path integral

Trace(−1)^{F} e^{−βH} = ∫ DXe^{− ∫_{0}^{β} L(X)dt},  (2)

where we integrate over fields X(t) satisfying periodic boundary conditions X(t+1) = X(t). The corresponding character valued index is

Index_{g}(D) = Trace(−1)^{F} g = Trace(−1)^{F} ge^{−βH}.  (3)

Since the isometry g commutes with the Dirac operator, the trace Trace(−1)^{F} ge^{−βH} is again independent of β. It is given by the path integral

Trace(−1)^{F} ge^{−βH} = ∫ DXe^{− ∫_{0}^{β} L(X)dt},  (4)

which is the same as (2) but with the boundary conditions on the fields changed to

x(t + 1) = gx(t), ψ(t + 1) = gψ(t).  (5)

We now rescale the time variable t by replacing it by t′ = βt where t ∈ [0, 1]: this is equivalent to replacing the action ∫_{0}^{β} L(X)dt by

\frac{1}{β} ∫_{0}^{1}\{g_{IJ} \frac{dx^{I}}{dt} \frac{dx^{J}}{dt} + \frac{i}{2}g_{IJ}ψ^{I} \frac{Dψ^{J}}{Dt} \},  (6)

where we have also rescaled the fermionic variables ψ^{I} by multiplying them by a factor 1/\sqrt{β}. We must first identify the solutions of the Euler-Lagrange equation. This equation is

dX^{I}/dt = 0;

given the twisted periodicity condition (5) the only solutions are constant paths x_{0}(t) = x_{0} which send R to a point x_{0} in the fixed point set Z = M^{g} of g, and constant paths ψ_{0}(t) = ψ_{0} which send R to a point ψ_{0} in the tangent space ΠT_{x_{0}}M.

If X_{0} = (x_{0}, ψ_{0}) is a solution of the Euler-Lagrange equation, we write an infinitesimal variation δX at X_{0} as

δX = (α, η)

(where α is the bosonic piece, a path in T_{x_{0}}M, and η is the fermionic piece, a path in ΠT_{x_{0}}M). To avoid overcounting infinitesimal variations δX which are simply deformations within the class of solutions X_{0} of the Euler-Lagrange equation, we impose the conditions

∫_{0}^{1} α(t)dt = 0, ∫_{0}^{1} η(t)dt = 0.  (7)

We decompose the infinitesimal variations (α, η) further into

α = α^{∥} + α^{⊥}, η = η^{∥} + η^{⊥}

where α^{∥} is a path in the tangent space T_{x_{0}}Z at x_{0} to the fixed point set Z, while α^{⊥} is a path in the fibre N_{x_{0}} of the normal bundle N to Z in M. The fermionic piece η is similarly decomposed into a path η^{∥} in the tangent space T_{x_{0}}Z and a path η^{⊥} in the fibre N_{x_{0}} of the normal bundle. These components separately satisfy the condition (7). The endpoint conditions become

α^{∥}(1) = α^{∥}(0), η^{∥}(1) = η^{∥}(0)  (8)
α^{⊥}(1) = gα^{⊥}(0), η^{⊥}(1) = gη^{⊥}(0)  (9)

where in (9) g acts as an automorphism of the fibres of the normal bundle to Z in M. Expanding the Lagrangian to second order in (α, η) we find L = L^{∥} + L^{⊥}, where

L^{∥} = \frac{1}{β} ∫_{0}^{1} dt(\frac{1}{2}|\frac{dα^{∥}}{dt} |^{2} − \frac{i}{4}ψ_{0}^{I}ψ_{0}^{J}R_{IJKL}(α^{∥})^{K} \frac{d(α^{∥})^{L}}{dt} + \frac{i}{2}(η^{∥}, \frac{dη^{∥}}{dt}))  (10)

while

L^{⊥} = \frac{1}{β} ∫_{0}^{1} dt(\frac{1}{2}|\frac{dα^{⊥}}{dt} |^{2} − \frac{i}{4}ψ_{0}^{I}ψ_{0}^{J}R_{IJKL}(α^{⊥})^{K} \frac{d(α^{⊥})^{L}}{dt} + \frac{i}{2}(η^{⊥}, \frac{dη^{⊥}}{dt})).  (11)

(Because the isometry g preserves the Levi-Civita connection, the fixed point set Z is totally geodesic. The solutions ψ_{0} to the Euler-Lagrange equation are tangent to the fixed point set Z. Thus, in (10) the indices I, J, K, L run over a basis for the tangent bundle to the fixed point set Z, while in (11) the indices I, J run over a basis for the tangent space to Z while the indices K, L run over a basis for the normal bundle N to the fixed point set.)

The bosonic parts of these Lagrangians are given by a differential operator (1/β)D_{B} of order 2 acting on α^{∥} (resp. α^{⊥}), where D_{B} is defined by

(D_{B})_{KL} = −\frac{d^{2}}{dt^{2}} δ_{KL} − F_{KL} \frac{d}{dt},  (12)

where F = (F_{KL}) is the curvature (i/2)ψ_{0}^{I}ψ_{0}^{J}R_{IJKL} of the Levi-Civita connection on M. Similarly the fermionic Lagrangians are given by a differential operator (1/β) D_{F} (where D_{F} = i\frac{D}{Dt} has order 1) acting on η^{∥} (resp. η^{⊥}). (At this point we may use normal coordinates to identify the covariant derivative D/Dt with the ordinary derivative d/dt.)

We compute the path integral (4) perturbatively. Since the result is independent of β, we may evaluate the integral in the limit β → 0: however, since the propagator is multiplied by a factor β, this means that in the limit β → 0 the exact answer is given by the leading order term in perturbation theory (i.e. the ratio of determinants arising from the Gaussian integral). In other words, the path integral (4) becomes

\frac{Pf(D_{F})}{\sqrt{det D_{B}}} .  (13)

Since the operators D_{B} and D_{F} preserve the tangent bundle to Z and the normal bundle to Z (in other words D_{B} decomposes into operators

D_{B} = D_{B}^{∥} ⊕ D_{B}^{⊥},

and similarly for D_{F}: this will follow from the decomposition (10)-(11) of the Lagrangian given below) our path integral decomposes as

∫_{Z} \frac{Pf(D_{F}^{∥})}{\sqrt{det D_{B}^{∥}}} \frac{Pf(D_{F}^{⊥})}{\sqrt{det D_{B}^{⊥}}} .

The computation of

\frac{Pf(D_{F}^{∥})}{\sqrt{det D_{B}^{∥}}}

is identical to the computation presented in the lecture and yields

Â(F^{∥}) = \frac{1}{(2π)^{dim Z/2}} ∏_{a=1}^{dim Z/2} \frac{F_{a}^{∥}/2}{sinh F_{a}^{∥}/2}.  (14)

Â(Z) = Â(F^{∥}/(2π)) represents the A-hat genus of the fixed point set Z. Here, we have decomposed the curvature F^{∥} of the Levi-Civita connection of Z (which is a matrix valued two-form on Z with values in skew-symmetric square matrices of dimension dim Z) into its Chern roots: in other words we have skew diagonalized it and written it in the form

F^{∥} = ( \begin{matrix}0 & F^{1} & 0 & 0 & 0 … & 0 \\ −F^{1} & 0 & 0 & 0 & 0 … & 0 \\ 0 & 0 & 0 & F^{2} & 0 … & 0 \\ 0 & 0 & −F^{2} & 0 & 0 … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & … & … \end{matrix} )  (15)

The computation of the determinants for D_{B}^{⊥} and D_{F}^{⊥} differs from the computation presented in the lecture only because new periodicity conditions (5) for the fields have been introduced. We first observe that since g is orientation preserving, the normal bundle N to each component of the fixed point set Z of g is even dimensional. Moreover, the multiplicity of the eigenvalue −1 is even. Therefore the fiber N_{x_{0}} over x_{0} ∈ Z decomposes into subspaces V_{b} invariant under g, b = 1, …, (dim_{R} N)/2. Each V_{b} has real dimension 2 and g acts on V_{b} as

g = ( \begin{matrix}cos ϑ_{b} & sin ϑ_{b} \\ − sin ϑ_{b} & cos ϑ_{b} \end{matrix} )  (16)

Here a pair of eigenspaces of eigenvalue −1 corresponds to one V_{b} with ϑ_{b} = π. (Alternatively, identifying each eigenspace V_{b} with C the isometry g acts as multiplication by e^{iϑ_{b}}. Of course, here the ϑ_{b} need not be distinct.) Since the isometry g preserves the Levi-Civita connection, the basis of the normal bundle which skew-diagonalizes the curvature (as in (15)) may be chosen to coincide with the basis of eigenspaces of g. Thus we may compute the action of D_{B}^{⊥} and D_{F}^{⊥} acting on paths α_{b}^{⊥}, η_{b}^{⊥} taking values in one eigenspace V_{b}.

In fact, it is convenient to replace α_{b}^{⊥} and η_{b}^{⊥} by paths α_{b,±}^{⊥}, η_{b,±}^{⊥} in the complexification V_{b} ⊗ C: this decomposes naturally as V_{b} ⊗ C = V_{b}^{+} ⊕ V_{b}^{−}, where g acts on V_{b}^{±} with the eigenvalue e^{±iϑ_{b}}. Thus we may convert the endpoint conditions (5) to periodic endpoint conditions by writing

α_{b,±}^{⊥}(t) = exp(±itϑ_{b})ζ_{b}^{±}(t),  (17)
η_{b,±}^{⊥}(t) = exp(±itϑ_{b})ξ_{b}^{±}(t),  (18)

where ζ_{b}^{±} : R → V_{b}^{±} and ξ_{b}^{±} : R → ΠV_{b}^{±} satisfy the usual periodic boundary conditions ζ_{b}^{±}(t + 1) = ζ_{b}^{±}(t), ξ_{b}^{±}(t + 1) = ξ_{b}^{±}(t).

If we substitute (17) into the definition of D_{B} we find that the operator D_{B} acting on the variables ζ_{b}^{±} becomes instead (using the decomposition of F^{⊥} corresponding to (15), where the eigenspaces are the eigenspaces V_{b}^{±}: the subspaces which diagonalize the curvature coincide with the eigenspaces of g because the isometry g is assumed to preserve the Riemannian metric on M)

(D_{B})_{b}^{±} = −(\frac{d}{dt} ± iϑ_{b})^{2} ± F_{b}(\frac{d}{dt} ± iϑ_{b}).  (19)

Since the eigenvalues of \frac{d}{dt} on the ζ_{b} are 2πin for n ∈ Z, we find that the eigenvalues are given by

(2πin ± iϑ_{b})^{2} ± F_{b}(2πin ± iϑ_{b}).  (20)

To compute the determinant of D_{B} we take the product over all the eigenvalues.

The computation of the character-valued index differs from the ordinary index in that the fermionic Pfaffian now depends on the values of the eigenvalues ϑ_{b} for the isometry. Substituting (18) into D_{F} = id/dt we find that the eigenvalues of D_{F} acting on the periodic fermionic paths ξ_{b}^{±} are

2πn ± ϑ_{b}.

We shall make use of the identity

∏_{n≥1}(1 − \frac{z^{2}}{4π^{2}n^{2}} ) = \frac{sin z/2}{z/2}  (21)

Combining all factors we find

det D_{B} = ∏_{±} ∏_{n∈Z} ∏_{b}(2πin + iϑ_{b})^{2} ± F_{b}(2πin ± iϑ_{b})  (22)

while the fermionic Pfaffian is given by

(PfD_{F})^{2} = ∏_{±} ∏_{b} ∏_{n}(2πn ± ϑ_{b}).  (23)

Combining (22) and (23) we see that (up to a multiplicative constant from regularization)

\frac{(PfD_{F})^{2}}{det D_{B}} = ∏_{n∈Z} ∏_{b,±} \frac{1}{2πn ± (ϑ_{b} + iF_{b})},  (24)

= ∏_{b} \frac{1}{ϑ_{b} + iF_{b}} ∏_{±} ∏_{n≥1} \frac{1}{4π^{2}n^{2} − (ϑ_{b} + iF_{b})^{2}}

= ∏_{b} \frac{1}{2 sin((θ_{b} + iF_{b})/2)}.

The final answer for the character-valued index of the Dirac operator is then

Index_{g}(D) = \frac{1}{(2π)^{dim Z/2}} ∫_{Z} Â(F^{∥}) ∏_{b=1}^{dim_{R} N /2} \frac{1}{2 sin((ϑ_{b} + iF_{b})/2)}

= ∫_{Z} Â(Z) ∏_{b=1}^{dim_{R} N /2} \frac{1}{2 sin((ϑ_{b} + iF_{b}/(2π))/2)}.  (25)

References:

1. L. Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer index theorem, Commun. Math. Phys. 90 (1983) 161-173.

2. D. Friedan, P. Windey, Supersymmetric derivation of the Atiyah-Singer index theorem and the chiral anomaly, Nucl. Phys. B 235 (1984) 385-416.

3. M. Goodman, Proof of character-valued index theorems, Commun. Math. Phys. 107 (1986) 391-409.

4. M. Goodman, E. Witten, Global symmetries in four and higher dimensions, Nucl. Phys. B 271 (1986) 21-52.