Clifford systems, Clifford structures, and their canonical differential forms

  • PDF / 1,535,766 Bytes
  • 15 Pages / 439.37 x 666.142 pts Page_size
  • 111 Downloads / 223 Views

DOWNLOAD

REPORT


Clifford systems, Clifford structures, and their canonical differential forms Kai Brynne M. Boydon1 · Paolo Piccinni2 Received: 28 August 2020 / Accepted: 19 November 2020 © The Author(s) 2020

Abstract A comparison among different constructions in ℍ2 ≅ ℝ8 of the quaternionic 4-form ΦSp(2)Sp(1) and of the Cayley calibration ΦSpin(7) shows that one can start for them from the same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in Spin(7) geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension 16, similar constructions allow to write explicit formulas in ℝ16 for the canonical 4-forms ΦSpin(8) and ΦSpin(7)U(1) , associated with Clifford systems related with the subgroups Spin(8) and Spin(7)U(1) of SO(16) . We characterize the calibrated 4-planes of the 4-forms ΦSpin(8) and ΦSpin(7)U(1) , extending in two different ways the notion of Cayley 4-plane to dimension 16. Keywords  Octonions · Clifford system · Clifford structure · Calibration · Canonical form Mathematics Subject Classification  Primary 53C26 · 53C27 · 53C38

1 Introduction In 1989 R. Bryant and R. Harvey defined the following calibration, of interest in hyperkähler geometry [6]:

1 1 1 ΦK = − 𝜔2R − 𝜔2R + 𝜔2R ∈ Λ4 ℍn . 2 i 2 j 2 k

Communicated by Vicente Cortés. * Paolo Piccinni [email protected] Kai Brynne M. Boydon [email protected] 1

Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines

2

Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy



13

Vol.:(0123456789)



K. B. M. Boydon, P. Piccinni

In this definition, (𝜔Ri , 𝜔Rj , 𝜔Rk ) are the Kähler 2-forms of the hypercomplex structure (Ri , Rj , Rk ) , defined by multiplications on the right by unit quaternions (i, j, k) on the space ℝ4n ≅ ℍn. When n = 2 , the Bryant-Harvey calibration ΦK relates with Spin(7) geometry. This is easily recognized by using the map

L ∶ ℍ2 → 𝕆,

̄ ∈ 𝕆, L(h1 , h2 ) = h1 + (kh2 k)e

from pairs of quaternions to octonions, that yields the identity

L∗ ΦSpin(7) = ΦK .

(1.1)

Here ΦSpin(7) ∈ Λ4 ℝ8 is the Spin(7) 4-form, or Cayley calibration, studied since the R. Harvey and H. B. Lawson’s foundational paper [11], and defined through the scalar product and the double cross product of ℝ8 ≅ 𝕆:

ΦSpin(7) (x, y, z, w) = < x , y × z × w > = < x , y(̄zw) >, assuming here orthogonal y, z, w ∈ 𝕆. The present paper collects some of the results in the first author Ph.D. thesis [3], inspired from viewing formula (1.1) as a way of constructing the Cayley calibration ΦSpin(7) through the 2-forms 𝜔Ri , 𝜔Rj , 𝜔Rk . As well known, by summing the squares of the latter 2-forms one gets another remarkable calibration, namely the quaternionic right 4-form ΩR . Thus 𝜔Ri , 𝜔Rj , 𝜔Rk , somehow building blocks for quaternionic geometry, enter also in Spin(7) geometry. A first result is the following Theorem  1.1, a kind of “other way around” of formula (1.1). To state it, recall that the Cayley calibration ΦSpin(7)