DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 309–317

CLIFFORD BUNDLES OVER MAPPING SPACES

AKIRA ASADA

Abstract. Let X be compact spin manifold, E an Hermitian vector bundle over X, D a non-degenerate 1-st order self-adjoint elliptic pseudo-differential operator. Fixing the Sobolev metric on W^{k}(X), the k-Sobolev space of section of E, by D, we have developped calculus of differential forms, including (∞ − p)–forms, on an open set of W^{k}(X) ([3], [4]) and constructed Clifford algebra over W^{−k}(X) with ∞–spinor (γ_{5}) together with its representation in algebra of bounded linear operators on ∧W^{−k}(X) ⊕ ∧W^{k}(X) ([5]). By using these results, (∞ − p)–forms and Hodge operators on a mapping space Map(X, M) (and some of its extensions) are investigated and Clifford bundle is constructed. If D is positive definite, such construction is always possible, while D is Dirac type, vanishing of string class Map(X, M) is necessary to the construction of Clifford bundle. Difficulties with the construction of spinor bundle and definition of Dirac-Ramond operator are investigated.

1. Regularized dimension of W^{k}(X)

Let X be a d–dimensional compact spin manifold, E an Hermitian vector bundle over X. L^{2}(X) the Hilbert space of section of E. We fix the norm and inner product of L^{2}(X) and denote them by ||f|| and (f, g), respectively. Let D be a non-degenerate 1-st order self-adjoint elliptic pseudo-differential operator acting on smooth sections of E. Then D allows spectral decomposition

(1) Df = ∑ λ(f, e_{λ})e_{λ}, De_{λ} = λe_{λ} {e_{λ}} is an O.N. − basis of L^{2}(X) .

By (1), we can define operators |D|, D_{+} and D_{−} by

|D|f = ∑ |λ|(f, e_{λ})e_{λ} ,

D_{+} = (|D| + D)/2, D_{−} = (|D| − D)/2 . (2)

1991 Mathematics Subject Classification. 58D15, 58G25, 15A63, 81T30. Key words and phrases. Mapping space, Clifford bundle, Regularized dimension, Connections

of differential operators.

This paper is in final form and no version of it will be submitted for publication elsewhere.

D_{+} and D_{−} are positive, hence their k–th powers (k > 0) are defined. Since D is non-degenerate, D has the Green operator G, |G|, G_{+}, and G_{−} are defined similarly. We also set

P_{+} = G_{+}D_{+}, P_{−} = G_{−}D_{−} , J = P_{+} − P_{−} = G|D| . (3)

By using D, we fix the k–th Sobolev metric ||f||_{k} of a section of E (and (−k)–th Sobolev metric ||f||_{−k}) by

||f||_{k} = ||D^{(k)}f|| , ||f||_{−k} = ||G^{(k)}f|| . (4)

The k–th Sobolev space and (−k)–th Sobolev space of sections of E are denoted by W^{k}(X) and W^{−k}(X), respectively. There is the natural pairing ⟨u, f⟩ of u ∈ W^{−k}(X) and f ∈ W^{k}(X) given by

⟨u, f⟩ = (G^{k}u, D^{k}f) . (5)

Let η(D, s), ζ(|D|, s), etc., the spectral eta or zeta function of D, |D|, etc. They are meromorphic on whole complex plane with possible poles at s = d, d−1, …, 1, are at most order 1, and holomorphic at s = 0 ([10]). We say ν = ζ(|D|, 0) the regularized dimension of W^{k}(X). We also set

ν_{+} = ζ(D_{+}, 0), ν_{−} = ζ(D_{−}, 0) . (6)

Since we have

ζ(D_{±} + P_{±}, 0) ≡ ν_{±} − mRes_{s=1}ζ(D_{±}, s) + … + (−1)^{d} \frac{m^{d}}{d} Res_{s=d}ζ(D_{±}, s) mod.2 , (7)

selecting constants m, n, and replacing D by D + mI + nJ, we may assume ν and ν_{−} both integers with arbitrary parity.

The Ray-Singer determinant det|D| = exp(−ζ′(|D|, 0)) can be thought to be the volume of cube {∑ c_{λ}e_{λ}|0 ≤ c_{λ} ≤ |λ|} ⊂ W^{k}(X), k < −d/2. Since we have

(8) det|D^{k}| = (det|D|)^{k} , det(t|D|) = t^{ν}det|D| ,

we may define the regularized volume of a cube {∑ c_{λ}e_{λ}|0 ≤ c_{λ} ≤ |λ|^{k}} by using det|D|. detD is defined to be (−1)^{ν_{−}}det|D|.

2. Differential forms on an open set of W^{k}(X)

Since {e_{λ}} is an O.N.-basis of L^{2}(X), setting e_{λ,k} = sgnλ |λ|^{−k}e_{λ}, {e_{λ,k}} becomes an O.N.-basis of W^{k}(X). Then taking {e_{λ}} and {e_{λ,k}} the coordinates of L^{2}(X) and W^{k}(X), we have

\frac{∂f}{∂e_{λ,k}} = sgnλ · |λ|^{k} \frac{∂f}{∂e_{λ}} (9)

By (9) and df = ∑ ∂f/∂e_{λ}de_{λ} = ∑ ∂f/∂e_{λ,k}de_{λ,k} , we may identify de_{λ,k} and e_{λ,−k} = sgnλ|λ|^{k}e_{λ}. This shows that a 1-form on U, an open set of W^{k}(X), is a smooth map from U to W^{−k}(X). Hence a p–form on U is a smooth map from U to ∧^{p}W^{−k}(X), the Sobolev (−k)–space of alternating sections of \overbrace{E × ⋯ × E}^{k} (on

\overbrace{X × ⋯ × X}^{k}).

An (∞−p)–form on U is a smooth map from U to ∧^{p}W^{k}(X). Since the volume form on U is (detD)^{−k−d/2}, an ∞–form on U is a scalar function U multiplied by (detD)^{−k−d/2} ([4], [7]).

By using Sobolev duality, we define the wedge product of a p–form u and an (∞ − q)–form f, p ≤ q. The sign convension is

u ∧ f = (−1)^{p(ν−q)}f ∧ u , (10)

that is, ∞ is replaced by ν. Then to be consistent in this definition, we need to assume ν to be an integer.

If f(x) = f(x, x_{1}, …, x_{p}) is an (∞ − p)–form, its Fréchet differential is denoted by F-df. Regarding x_{1}, …, x_{p−1} to be parameters, F-df(x, x_{1}, …, x_{p}) is a linear operator on W^{k}(X). Under the assumption this operator to be trase class, the exterior differential df of f is defined to be

df(x, x_{1}, …, x_{p−1}) = (−1)^{p−1}trF−df(x, x_{1}, …, x_{p−1}, x), p ≥ 1 . (11)

One of the striking property of (∞ − p)–form is the following Theorem ([3], [4])

Theorem 2.1. If an (∞ − p)–form f is exterior differentiable, then f is globally exact.

As a conseqence, d is not nilponent on the space of (∞ − p)–forms.

In the rest, we denote by ∧W^{−k}(X) and ∧W^{k}(X) the Grassmann algebras over W^{−k}(X) and W^{k}(X). We consider them to be Hilbert spaces with the norms

|| ∑ u^{p}||^{2} = ∑ ||u^{p}||_{−k}^{2} , u^{p} ∈ ∧^{p}W^{−k}(X) ⊂ W^{−k}(X × ⋯ × X) ,
(12) || ∑ f^{p}||^{2} = ∑ ||f^{p}||_{k}^{2}, f^{p} ∈ ∧^{p}W^{k}(X) ⊂ W^{k}(X × ⋯ × X) .

The Grassmann algebra over W^{−k}(X) with (∞ − p)–form is the Hilbert space ∧W^{−k}(X) ⊕ ∧W^{k}(X).

3. Clifford algebras over Sobolev spaces

The Clifford algebra C(W^{−k}) over W^{−k}(X) is the algebra over C generated by {e_{λ}} with the relation

e_{λ}e_{μ} + e_{μ}e_{λ} = −2sgnλ |λ|^{−2k}δ_{λ,k} . (13)

It has the representation r : C(W^{−k}) → B(∧W^{−k}), the algebra of bounded linear operators on ∧W^{−k}. Let 1 be the unit of C = ∧^{0}W^{−k} ⊂ ∧W^{−k}. Then set s(a) = r(a) · 1, we have the module isomorphism

s : C(W^{−k}) ≅ ∧W^{−k} . (14)

We define ||a|| by ||s(a)||. Precisely saying, we consider C(W^{−k}) to be the completion of the algebra generated by {e_{λ}}.

If the regularized dimension ν of W^{k}(X) is an integer, the infinite spinor e^{∞}(γ_{5}) thought to be the infinite product ⋯ e_{λ_{1}} ⋯ e_{λ_{2}} ⋯, ⋯ ≤ λ_{1} ≤ ⋯ ≤ λ_{2} ≤ ⋯, must satisfy

e_{λ} · e^{∞} = (−1)^{ν−1} · e_{λ}, e^{∞} · e^{∞} = (−1)^{ν(ν+1)/2+ν_{−}}(det|D|)^{−2k} . (15)

The algebra C(W^{−k}[e^{∞}]) is said to be the Clifford algebra over W^{−k}(X) with an infinite spinor. A representation R : C(W^{−k})[e^{∞}] → B(∧W^{−k} ⊕ ∧W^{k}), the algebra of bounded linear operators on ∧W^{−k} ⊕ ∧W^{k}, is given by

R(e_{λ}) = ( \begin{matrix}r(e_{λ}) & 0 \\ 0 & (−1)^{ν−1}G^{2k}D^{2k} \end{matrix} ) (16)

(17) R(e^{∞})

= ( \begin{matrix}0 & (−1)^{\frac{1}{2} (ν_{−}+ \frac{ν(ν+1)}{2} )}(det|D|)^{−2k}D^{2k} \\ (−1)^{\frac{1}{2} (ν_{−}+ \frac{ν(ν+1)}{2} )}G^{2k} & 0 \end{matrix} ) .

The isometry s : C(W^{−k})[e^{∞}] → ∧W^{−k} ⊕ ∧W^{k} is defined by

s(y) = R(y) ( \begin{matrix}1 \\ 0 \end{matrix} ) . (18)

Then we have

s(C(W^{−k})) = ∧W^{−k}, s(C(W^{−k})e^{∞}) = ∧W^{k} . (19)

4. Sobolev structures of mapping spaces

Let M be an N–dimensional smooth manifold. We denote Map(X, M) the space of Sobolev k–class maps from X to M, k > d/2. In the rest, we only consider the subspace of Map(X, M) consisting of 0-homotopic maps. For sake of simplicity, this space is also denoted by Map(X, M).

If f ∈ Map(X, M) is homotopic to a constant map, then its graph

Gr(f) = {(x, f(x))|x ∈ X} ⊂ X × M (20)

is homotopic to X×∗, ∗ ∈ M. Since the normal bundle of X×∗ in X×M is trivial, the normal bundle of Gr(f) in X × M is trivial. Hence there is a diffeomorphism h_{u} such that

h_{u} : U(Gr(f)) ≅ Gr(f) × R^{N}, U(Gr(f)) is a neighborhood of Gr(f)in X × M . (21)

By (21), Map(X, M) is a Sobolev manifold modeled by W^{k}(X) ⊗ R^{N}, where W^{k}(X) is the k–Sobolev space of scalar fields on X.

If M is real algebraic manifold defined by the equations

F_{1}(X_{1}, …, X_{m}) = ⋯ = F_{i}(X_{1}, …, X_{m}) = 0 , (22)

in R^{m}, Map(X, M) is a connected component of the algebraic set in W^{k}(X) ⊗ R^{m} determined by

F_{1}(f_{1}, …, f_{m}) = ⋯ = F_{i}(f_{1}, …, f_{m}) = 0, f_{i} ∈ W^{k}(X), 1 ≤ i ≤ m . (23)

Hence Map(X, M) is an algebraic set in W^{k}(X) ⊗ R^{m}.

Since F_{1}, …, F_{i} are polynomials, replacing W^{k}(X) by the k–Sobolev space of spinor fields W^{k}(X, E) on X, the equations (23) have meanings. Then Map(X, M) is contained in a connected compact C of the algebraic set in W^{k}(X, E) ⊗ R^{m} defined by the equations (23), and, there is a neighborhood U of Map(X, M) in C such that U is a non-singular manifold modeled by W^{k}(X, E) ⊗ R^{N}. Taking U sufficiently small, we may assume Map(X, M) is the deformation retract of U. For simply, this U is also denoted by Map(X, M) in the rest.

5. Connection with respect to D

Let F : U → S , U an open set of M, be a function, then

F^{#}(f)(x) = F (f(x)), x ∈ X, f ∈ Map(X, U) , (24)

is a map F^{#} : Map(X, U) → Map(X, S).

Let E = {g_{uv}} be a G–bundle over M, G a Lie group, then {g_{uv}^{#}} defines a Map(X, G)–bundle over Map(X, M). This bundle is denoted by E^{#}. When G is a unitary group (or an orthogonal group), we may consider B^{#} to be a loop group bundle ([2], cf. [8]). Especially, if T is the tangent bundle of M, T^{#} becomes the tangent bundle of Map(X, M), and we may consider T^{#} to be a loop group bundle.

Let B = {b_{uv}} be a Map(X, G)–bundle. We assume G acts on E, the Hermitian vector bundle over X and D acts on its sections. Let D(0) be the space of 0-th order pseudo-differential operators acting on the sections of E. Then a collection {A_{u}}; A_{u} : U → D(0), such that

(D + A_{u})b_{uv} = b_{uv}(D + A_{v}) (25)

is said to be a connection of B with respect to D. {A_{u}} always exist and if G is a unitary group, we may assume A_{u} is Hermitian for any x ∈ U.

Taking B to be the (co)tangent bundle of Map(X, M) and D the operator by which the Sobolev metric of W^{k}(X) is determined, we have the following facts (cf. [2], [6]).

(i) If D positive definite, then D has a connection {A_{u}} such that D + A_{u}(x) is non-degenerate for any U and x ∈ U.

(ii) If D is of the Dirac type, then D + A_{u}(x) degenerates for some U and x ∈ U, whenever the tangent bundle of Map(X, M) is non-trivial as a loop group bundle.

By (25), the spectral type of D + A_{u}(x) does not depend on U. Hence the regularized dimension ν(x) = ζ(|D + A_{u}(x)|, 0) is a globally defined function on Map(X, M). Therefore selecting mass-term m = m(x), we may assume ν is a constant integer on Map(X, M). While if D is a Dirac type, we can take both ν and ν_{−} to be constant integers if and only if structure group of the tangent bundle of Map(X, M) is reduced to the connected component of the identity (of the loop group, cf. [6]).

6. (∞ − p)–forms and Hodge operators on mapping space

If |m| is small, det(D_{±} + mP_{±}) is holomorphic in m. Its analytic continuation is an entire function on whole m–plane. In this n^{∘}, we denote this function by det(D_{±} + mP_{±}) and set

(26) det(D + mP) = det(D_{+} + mP_{+})det(D_{−} + mP_{−}) .

If (D + A_{u}(x) has 0-modes for some x ∈ U, we choose a scalar m = m_{u} such that D + A_{u}(x) + mI does not have 0-modes for any x ∈ U, and set

det_{m}(D + A_{u}) = det((D + A_{u}(x) + mI) − mI) . (27)

Then {γ_{uv}} = {det_{m}(D+A_{u})/det_{m}(D+A_{v})} defines a line bundle over Map(X, M). It is shown this bundle is determined by the tangent bundle of Map(X, M) and does not depend on the choice of {A_{u}} and {m_{u}} ([3]).

By the local definition of ∞–forms f^{∞} on Map(X, M) takes the form

f^{∞} = {f_{u}det_{m}(D + A_{u})^{−k−d/2}} . (28)

Hence {f_{u}} should be a cross-section of {γ_{uv}}^{k+d/2}. Therefore, to define ∞–form on Map(X, M) we need to assume the bundle {γ_{uv}}^{k+d/2} can be defined. That is, denoting C the Chern class of {γ_{uv}}, (k +d/2)C should be an integral class. In the rest, we denote {γ_{uv}}^{k+d/2} by τ and call it the determinant bundle of Map(X, M).

Let T be the tangent bundle of Map(X, M), then we say a cross-section of the associated ∧^{p}W^{k}(X)–bundle of T ⊗ τ to be an (∞ − p)–form of Map(X, M).

We assume ν and ν_{−} are both integers. Let u^{p} be a p–form of Map(X, M) and f^{∞−p} = {f_{u}}, f_{u} = fdet_{m}(D+A_{u})^{k+d/2} be an (∞−p)–form of Map(X, M). Then we define the Hodge ∗–operator by

∗u^{p} = (−1)^{p}det_{m}(D + A_{u})^{k+d/2}G_{A_{u}}^{2k} ,

G_{A_{u}}^{2k} is the Green operator of D + A_{u},

∗f^{∞−p} = (−1)^{νp+ν_{−}}det_{m}(D + A_{u})^{2k}det_{m} (D + A_{u})^{−(k+d/2)}f_{u} . (29)

If D + A_{u}(x) has 0-modes, then G_{A_{u}(x)} can not be defined. Hence ∗ = ∗(x) can not be defined. But if u^{p} takes the values in ∧^{p}L^{2}(X) (⊂ ∧^{p}W^{−k}(X)), near x then the pairing ⟨u^{p}, f^{∞−p}⟩ is defined at x.

We note that if dim ker(D + A_{u}(x)) = 1, then u^{1} ∧ ∗u^{1} becomes integrable near x if k < d/2 + 1. Since k is larger than d/2, the appropriate selection of k is

k = (d + 1)/2 . (30)

By (30), we have k + d/2 = d + 1/2. Therefore τ is defined on Map(X, M) if and only if the Chern class of {γ_{uv}} is divisible by 2.

By (29), if ∗ is defined, then we have

∗^{2}u^{p} = (−1)^{p(ν−p)+ν_{−}}u^{p} , ∗^{2}f^{∞−p} = (−1)^{(ν−p)p+ν_{−}}f^{∞−p} . (31)

7. Construction of Clifford bundle

To define the Clifford bundle, we modify R(e^{∞}) to the following operator E^{∞} = E_{u}^{∞}

E^{∞} =

( \begin{matrix}0 & (−1)^{\frac{ν(ν+1)}{2} − \frac{ν_{−}}{2} +p}D_{u}^{2k}detD_{u}^{−k−d/2} \\ (−1)^{\frac{ν(ν+1)}{2} +ν_{−}+νp}detD_{u}^{k+d/2}G_{Au}^{2k} & 0 \end{matrix} ) (32)

acting on the space ∧^{p}(W^{−k}(X)) ⊕ ∧^{p}(W^{k}(X)) ⊗ τ, where we put D_{u} = D + A_{u}. Here we assume ν and ν_{−} both integers.

By definitions, we have

(E^{∞})^{2}u^{p} = (−1)^{p(ν−p)+ν_{−}}u^{p}, (E^{∞})^{2}f^{∞−p} = (−1)^{(p−ν)p+ν_{−}}f^{∞−p} . (33)

That is, we may consider the action of E^{∞} to be the Hodge operation.

Since (E^{∞})^{−1} is equal to (−1)^{(p−ν)p+ν_{−}}E^{∞} on the space ∧^{p}W^{−k}(X)⊕∧^{p}W^{k}(X)⊗ τ, we have

E_{u}^{∞}(E_{v}^{∞})^{−1} =

(34) ( \begin{matrix}(D + A_{u})^{2k}detD_{u}^{−k−d/2}detD_{u}^{k+d/2}G_{Av}^{2k} & 0 \\ 0 & detD_{u}^{k+d/2}G_{Au}^{2k} D_{v}^{2k} detD_{v}^{−k−d/2} \end{matrix} ) .

The Clifford bundle Cliff on Map(X, M) is the bundle with the transition {E_{u}^{∞}(E_{v}^{∞})^{−1}} and the fibre ∧(W^{−k}(X)) ⊕ ∧(W^{k}(X)) ⊗ τ = C(W^{k})[e^{∞}]. We set

Y = {x ∈ Map(X, M)|ker(D + A_{u}(x)) ≠ {0}} ,

(35) Y_{r} = {x ∈ Map(X, M)| dim ker(D + A_{u}(x)) = r} ,

and assume {Y_{r}} is a non singular submanifold of Map(X, M). Originally, E_{u} is defined only on E_{u} − Y.

Let x be an element of Y_{r}, {e_{λ}}, the proper function of D − A_{u}(x) such that

(D − A_{u}(x))e_{λ_{1}} = ⋯ = (D − A_{u}(x))e_{λ_{r}} = 0 . (36)

Then Y_{r} is defined by e_{λ_{1}} = ⋯ = e_{λ_{r}} = 0 near x. Hence we may identify ker(D + A_{u}(x)) and the normal space of Y_{r} at x. In other word, the normal bundle N(Y_{r}) of Y_{r} is given by

N(Y_{r}) = ∪_{x∈Y_{r}}ker(D + A_{u}(x)) . (37)

Therefore, if N(Y_{r}) is trivial, E_{u}^{∞}(E_{v}^{∞})^{−1} is defined on Y_{r}.

Then applying results in [1] and [6], we have

Theorem 7.1. The Clifford bundle associated to the tangent bundle of ΩM = Map(S^{1}, M) is defined on \overline{Y}_{r} if the string classes S^{1}(T), ⋯, S^{[(r+1)/2]}(T) of T vanish.

8. Parity exchange and Dirac-Kähler type operator

C(W^{−k}) is the direct sum of modules of even spinors and odd spinors. Corresponnding subbundles of Cliff are denoted by S^{0} and S^{1}. We also define the subbundle S^{0}(+), S^{0}(−), S^{1}(+) and S^{1}(−) of Cliff by

S^{0}(±) = {u ± ∗u|u ∈ S^{0}} ,

S^{1}(±) = {u ± ∗u|u ∈ S^{1}} . (38)

Here ∗u means E^{∞} · u. By definition, we have

Cliff = S^{0}(+) ⊕ S^{0}(−) ⊕ S^{1}(+) ⊕ S^{1}(−) . (39)

We also set

S(+) = S^{0}(+) ⊕ S^{1}(+) , S(−) = S^{0}(−) ⊕ S^{1}(−) . (40)

The formal adjoint δ of d is defined by

δu^{p} = (−1)^{p} ∗^{−1} d ∗ u^{p} , δf^{∞−p} = (−1)^{p} ∗^{ν−p} d ∗ df^{∞−p} . (41)

We note δ is not nilpotent on the space of finite degree forms, that is δ^{2} ≠ 0 on the space of sections of C(W^{−k}), because d is not nilpotent on the space of (∞ − p)–forms.

We define the operators D(+) and D(−) by

D(+)u^{p} = (d + δ)u^{p}, D(+) ∗ u^{p} = (−1)^{p−1}(d − δ) ∗ u^{p} ,

D(−)u^{p} = (−1)^{p−1}(d − δ)u^{p}, D(−) ∗ u^{p} = (d + δ) ∗ u^{p} , (42)

then D(+) and D(−) both map the space of sections of S(±) to the space of sections of S(∓) (cf. [13]).

Since d and δ are not nilpotent, (d + δ)^{2} is not equal to Δ = dδ + δd. But we have

Δ = \frac{1}{2}(D(+)^{2} + D(−)^{2}) . (43)

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Department of Mathematics, Faculty of Science, Shinshu University Matumoto, 390-8621, JAPAN E-mail: asada@math.shinshu-u.ac.jp