The Riemann-Roch Theorem and the Atiyah-Hirzebruch Spectral Sequence

Abstract

We show that under general conditions there is at least one natural inflationary direction for the Kähler moduli of type-IIB flux compactifications. This requires a Calabi-Yau which has h 2,1 > h 1,1 > 2 and for which the structure of the scalar potential is as in the recently found exponentially large volume compactifications. We also need -although these conditions may be relaxed -at least one Kähler modulus whose only non-vanishing tripleintersection is with itself and which appears by itself in the non-perturbative superpotential. Slow-roll inflation then occurs without a fine tuning of parameters, evading the η problem of F-term inflation. In order to obtain COBE-normalised density perturbations, the stabilised volume of the Calabi-Yau must be O(10 5 − 10 7 ) in string units, and the inflationary scale M inf l ∼ 10 13 GeV. We find a robust model independent prediction for the spectral index of 1 − 2 Ne = 0.960 → 0.967, depending on the number of efoldings.

Algebraic $K$-theory and etale cohomology

“…This trivially commutative diagram is the Riemann-Roch theorem. The RiemannRoch theorems of Gillet and of Shekhtman ([42], [106]) for the map from G/J* to etale cohomology may be easily deduced by purely topological calculations of the behavior of Chern classes under topological Gysin maps as in [32], I. D. 2. In reasonable situations where one has an alternative definition of G/^' TOP ( ) and of the Gysin map, [127] shows that they agree with the current one.…”

Algebraic K-theory and the norm-residue homomorphism

“…We now consider the differentials in the B C~ spectral sequence for any smooth quasiprojective variety X. The results presented below are due to Shekhtman [27].…”

“…As above, for any linear Gbundle L on X we denote by ci(L)6 HZ(X, G; JEPl) the element corresponding to L under this i somorphi sm. THEOREM 8.2 (Sherman [71,72], Shekhtman [27], Gillet [48]). Let E be a vector Gbundle over a G-scheme X of rank n, and let P = P(E), L = Op(--l).…”

“…In our computations the Riemanrr-Roch theorem plays an important role [54, 27,48]. We require only the following special case.…”

Determinant bundles and Virasoro algebras

“…The standard GL ί = {(*1 1 )}CGL Π acts on SL n by conjugation, hence on SL n ; put C2 CfL π = SL n . We get a central extension This is just the extension corresponding to the universal Chern class c 2 eH 2 (GL n , JΓ 2 ), [31]. Let us denote by s : GL 1 ->GL Π the canonical section.…”