Bundles and Rigidity

In this paper, we establish rigidity and vanishing theorems for Dirac operators twisted by E8 bundles. Introduction Let X be a closed smooth connected manifold which admits a nontrivial S1 action. Let P be an elliptic differential operator on X commuting with the S1 action. Then the kernel and cokernel of P are finite dimensional representation of S1. The equivariant index of P is the virtual character of S1 defined by (0.1) Ind(g, P ) = tr|gkerP − tr|gcokerP, for g ∈ S1. We call that P is rigid with respect to this circle action if Ind(g, P ) is independent of g. It is well known that classical operators: the signature operator for oriented manifolds, the Dolbeault operator for almost complex manifolds and the Dirac operator for spin manifolds are rigid [2]. In [30], Witten considered the indices of Dirac-like operators on the free loop space LX. The Landweber-Stong-Ochanine elliptic genus ([20], [28]) is just the index of one of these operators. Witten conjectured that these elliptic operators should be rigid. See [19] for a brief early history of the subject. Witten’s conjecture were first proved by Taubes [29] and Bott-Taubes [4]. Hirzebruch [13] and Krichever [18] proved Witten’s conjecture for almost complex manifold case. Various aspects of mathematics are involved in these proofs. Taubes used analysis of Fredholm operators, Krichever used cobordism, Bott-Taubes and Hirzebruch used Lefschetz fixed point formula. In [22, 23], using modularity, Liu gives simple and unified proof as well as various generalizations of the Witten conjecture. Several new vanishing theorems are also found in [22, 23]. Liu-Ma [24, 25] and Liu-Ma-Zhang [26, 27] established family versions of rigidity and vanishing theorems. In this paper, we study rigidity and vanishing properties for Dirac operators twisted by E8 bundles. Let X be an even dimensional closed spin manifold and D the Dirac operator on X. Let P be an (compact-)E8 principal bundle over X. Let W be the vector bundle over X associated to the complex adjoint representation ρ of E8. The twisted Dirac operator D W plays a prominent role in string theory and M theory. In [31], the index of such twisted operator is discovered as part of the phase of the M -theory 1 2 FEI HAN, KEFENG LIU, AND WEIPING ZHANG action. In [8], the partition function in M -theory, involving the index theory of an E8 bundle, is compared with the partition function in type IIA string theory described by K-theory to test M -theory/Type IIA duality. In this paper, we are interested in the equivariant index of the operator DW and establish rigidity and vanishing theorems for this operator. More precisely, let X be a 2k dimensional closed spin manifold, which admits a nontrivial S1 action. Let P be an (compact-)E8 principal bundle over X such that the S1 action on X can be lifted to P as a left action which commutes with the free action of E8 on P . Let W be the complex vector bundle associated to the complex adjoint representation of E8 mentioned above. Then the S1 action on P naturally induces an action on W by g · [s, v] = [g · s, v], where [s, v] with s ∈ P, v ∈ C248, is the equivalent classes defining the elements in W by the equivalent relations (s, v) ∼ (s ·h, ρ(h−1) · v) for h ∈ E8. Let XS 1 be the fixed point manifold and π be the projection from XS 1 to a point pt. Let u be a fixed generator of H2(BS1,Z). We have the following theorem: Theorem 0.1. Assume the action only has isolated fixed points and the restriction of the equivariant characteristic class 1 30c2(W )S1 − p1(TX)S1 to XS 1 is equal to n · π∗u2 for some integer n. (i) If n < 0, then Ind(g,DW ) is independent of g and equal to −Ind(DTCX), minus the index of the Rarita-Schwinger operator. In particular, one has IndDW=−IndDTCX and when k is odd, i.e. dim X ≡ 2 (mod 4), one has Ind(g,DW ) ≡ 0. (ii) If n = 0, then Ind(g,DW ) is independent of g. Moreover, when k is odd, one has Ind(g,DW ) ≡ 0. (iii) If n = 2 and k is odd, then Ind(g,DW ) ≡ 0. Actually we have established rigidity and vanishing results in more general settings concerning the twisted spinc Dirac operators. See Theorem 2.1 and Theorem 2.2 for details. The above theorem is a corollary of Theorem 2.1. We prove our theorems by studying the modularity of Lefschetz numbers of certain elliptic operators involving the basic representation of the affine Kac-Moody algebra of E8. In the rest of the paper, we will first briefly review the Jacobi theta functions and the basic representation for the affine E8 by following [16] (see also [17]) as the preliminary knowledge in Section 1 and then state our theorems as well as give their proofs in Section 2. 1. Preliminaries 1.1. Jacobi theta functions. The four Jacobi theta-functions are defined as follows (cf. [5]), (1.1) θ(z, τ) = 2q sin(πz) ∞ ∏ j=1 [(1− q)(1− e √ −1zqj)(1− e−2π √ −1zqj)], E8 BUNDLES AND RIGIDITY 3 (1.2) θ1(z, τ) = 2q 1/8 cos(πz) ∞ ∏ j=1 [(1− q)(1 + e √ −1zqj)(1 + e−2π √ −1zqj)], (1.3) θ2(z, τ) = ∞ ∏ j=1 [(1 − q)(1− e √ −1zqj−1/2)(1 − e−2π √ −1zqj−1/2)], (1.4) θ3(z, τ) = ∞ ∏ j=1 [ (1− q)(1 + e √ −1zqj−1/2)(1 + e−2π √ −1zqj−1/2) ] , where q = e2π √ −1τ , τ ∈ H, the upper half plane. They are all holomorphic functions for (z, τ) ∈ C×H, where C is the complex plane. Let θ′(0, τ) = ∂ ∂z θ(z, τ)|z=0. One has the following Jacobi identity (c.f. [5]), (1.5) θ′(0, τ) = πθ1(0, τ)θ2(0, τ)θ3(0, τ). Let SL(2,Z) := {( a1 a2 a3 a4 )∣∣∣∣ a1, a2, a3, a4 ∈ Z, a1a4 − a2a3 = 1 } be the modular group. Let S = ( 0 −1 1 0 ) , T = ( 1 1 0 1 ) be the two generators of SL(2,Z). Their actions on H are given by S : τ 7→ − τ , T : τ 7→ τ + 1. The actions on theta-functions by S and T are given by the following transformation formulas (cf. [5]), (1.6) θ(z, τ+1) = e π √ −1 4 θ(z, τ), θ (z,−1/τ) = 1 √ −1 ( τ √ −1 )1/2 e √ −1τz2θ (τz, τ) ;