Question

Based on this mathoverlow question, I would like to have a similar list for the case of string manifolds. An $n$-dim. Riemannian manifold $M$ is said to be string, if the classifying map of its bundle of orthonormal frames $M \to BO(n)$ lifts to a map $M \to BO(n)<8> = BString(n)$, which is the case if and only if the class $\frac{p_1}{2} \in H^4(M, \mathbb{Z})$ vanishes. There are a lot of models that yield geometric realizations of $String(n)$ either as a topological group (see Stolz-Teichner), infinite-dimensional Lie group (see Nikolaus-Sachse-Wockel) or a 2-group (see Schommer-Pries).

What are enlightening examples of string manifolds? What are non-examples? When do you have a geometric interpretation of the obstruction class?

So far, I am aware of the list given at the end of Douglas-Henriques-Hill. What else is out there?

Answer 1

Qingtao Chen and Fei Han have constructed [arxiv:0612055] tons of examples of string manifolds and non-string manifolds as complete intersections in products of complex projective spaces.

For integers $s$ and $n_q$ with $1\leq q \leq s$ consider the product $$ Z:=\mathbb{C}P^{n_1} \times ... \times \mathbb{C}P^{n_s} $$ and the projection $pr_q:Z \to \mathbb{C}P^{n_q}$ to the $q$th factor. For further integers $t$ and $d_{pq}$ with $1 \leq p \leq t$ and $1 \leq q \leq s$, consider for $1 \leq p \leq t$ the line bundle $$ E_p := \bigotimes_{q=1}^s \;\; pr_q^{*}\mathcal{O}^{d_{pq}}_{q} $$ over $Z$, where $\mathcal{O}_q$ is the canonical line bundle over $\mathbb{C}P^{n_q}$. Now let $V_{d_{pq}}$ be the intersection of the zero loci of smooth global sections of $E_1,...,E_t$.

$V_{d_{pq}}$ is always a smooth manifold, and the statement of Proposition 3.1 of the above paper is:

Theorem. Let $m_q$ be the number of non-zero elements in the $q$th column of the matrix $D := (d_{pq})$. Suppose that $m_q +2 \leq n_q$. Then, $V_{d_{pq}}$ is string if and only if $$ D^t D = diag(n_1 + 1,...,n_s+1). $$