Generalized Theta Functions, Strange Duality, and Odd Orthogonal Bundles on Curves

Abstract

This paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial Stiefel–Whitney class and the associated space of twisted spin bundles. In particular, we prove a Verlinde type formula and a dimension equality that was conjectured by Oxbury–Wilson. Modifying Hitchin’s argument, we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection. We furthermore address the issue of strange duality for odd orthogonal bundles, and we demonstrate that the naive conjecture fails in general. A consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the Hitchin connection for these moduli spaces. Finally, we answer a question of Nakanishi–Tsuchiya about rank-level duality for conformal blocks on the pointed projective line with spin weights.

Appendix A: Computations in the Clifford algebra

In this section, we compute some vectors in the highest weight modules as explicit elements in the infinite dimensional Clifford algebra.

1.1 A.1. Action of $L(B^{i}_j)$

Consider the rectangle $r\times s$ as a Young diagram Y where the rows are indexed by integer in $\{1,\ldots , r\}$ and the columns by $\{-s,\ldots ,-1\}$. Let (i, j) be the coordinates of Y and let $v:= \bigwedge _{\tilde{Y}_{i,j}=\,\blacksquare }\phi _{i,j}$ (cf. Sect. 8.4).

Proposition A.1

Let v as before be the highest weight vector of the component with highest weight $(\omega _r, (2r+1)\omega _s)$. Let $0\le k\le m\le r$, then

$$\begin{aligned}&L(B^{-k}_{(k+1)})L(B^{-(k+2)}_{k+3})\cdots L(B^{-(m-3)}_{m-2})L(B^{-(m-1)}_{m})\cdot v\\&\quad =\phi _{k,0}\wedge \phi _{k+1,0}\wedge \cdots \wedge \phi _{m,0}\wedge v\ . \end{aligned}$$

1.2 A.2 Action of $R^k(B^0_1)$

Let $v_k=\phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )\cdots \phi ^{k,1}(-\tfrac{1}{2} )\cdot 1$. In this section, we want to give explicit expressions for $R^k(B^i_j)v_k$. First, consider the case when $k=1$.

Lemma A.2

Consider the highest weight vector $\phi ^{1,1}(-\tfrac{1}{2} )\cdot 1$ of the component with highest weight $(\omega _1,\omega _1)$. Then $R(B^0_1)\phi ^{1,1}(-\tfrac{1}{2} )\cdot 1=\phi ^{1,0}(-\tfrac{1}{2} )$.

Now we want to compute $R^2(B^0_1)v_2$. We first have the following lemma

Lemma A.3

For the component $(\omega _2,2\omega _1)$, $\phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,2}(-\tfrac{1}{2} )\cdot 1$, is a highest weight vector. Moreover,

$$\begin{aligned} R(B^0_1)v_2= \phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )+\phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,0}(-\tfrac{1}{2} )\ . \end{aligned}$$

(A.1)

Proof

The proof is by a direct computation. As before, we know that $R(B^0_1)$ acts as on the infinite dimensional Clifford algebra for $\widehat{\mathfrak {so}}(2d+1)$. Hence,

$$\begin{aligned}&R(B^{0}_{1})\phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} ) \\&\quad = \sum _{-r \le q\le r}\bigl [\phi ^{q,0}(-\tfrac{1}{2} )\phi _{q,1}(\tfrac{1}{2} )-\phi _{q,1}(-\tfrac{1}{2} )\phi ^{q,0}(\tfrac{1}{2} )\bigr ]\phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )\\&\quad = \sum _{-r \le q\le r}\bigl [\phi ^{q,0}(-\tfrac{1}{2} )\phi _{q,1}(\tfrac{1}{2} )\bigr ] \phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )\\&\quad = \phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )-\phi ^{2,0}(-\tfrac{1}{2} )\phi ^{1,1}(-\tfrac{1}{2} )\ . \end{aligned}$$

$\square$

Proposition A.4

We have:

$$\begin{aligned} R^2(B^0_1)v_2=2\left[ \phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,0}(-\tfrac{1}{2} )\right) -\left( \phi ^{1,-1}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )+\phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,-1}(-\tfrac{1}{2} )\right] \ . \end{aligned}$$

Proof

Compute using (A.1),

$$\begin{aligned}&R(B^{0}_{1})\phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )\\&\quad = \sum _{-r \le q\le r}\bigl [\phi ^{q,0}(-\tfrac{1}{2} )\phi _{q,1}(\tfrac{1}{2} )-\phi _{q,1}(-\tfrac{1}{2} )\phi ^{q,0}(\tfrac{1}{2} )\bigr ]\phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )\\&\quad =-\phi ^{2,0}(-\tfrac{1}{2} )\phi ^{1,0}(-\tfrac{1}{2} )-\phi _{-1,1}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )\\&\quad = \phi ^{1,0}(-\tfrac{1}{2} )\phi _{-2,0}(-\tfrac{1}{2} )+\phi ^{2,1}(-\tfrac{1}{2} )\phi _{-1,1}(-\tfrac{1}{2} )\\&R(B^{0}_{1})\phi ^{2,0}(-\tfrac{1}{2} )\phi ^{1,1}(-\tfrac{1}{2} )\\&\quad = \sum _{-r \le q\le r}\bigl [\phi ^{q,0}(-\tfrac{1}{2} )\phi _{q,1}(\tfrac{1}{2} )-\phi _{q,1}(-\tfrac{1}{2} )\phi ^{q,0}(\tfrac{1}{2} )\bigr ]\phi ^{2,0}(-\tfrac{1}{2} )\phi ^{1,1}(-\tfrac{1}{2} )\\&\quad =-\phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,0}(-\tfrac{1}{2} )-\phi _{-2,1}(-\tfrac{1}{2} )\phi ^{1,1}(-\tfrac{1}{2} )\\&\quad = -\phi ^{1,0}(-\tfrac{1}{2} )\phi _{-2,0}(-\tfrac{1}{2} )-\phi ^{2,-1}(-\tfrac{1}{2} )\phi _{-1,-1}(-\tfrac{1}{2} )\ .\\ \end{aligned}$$

$\square$

We use the following calculation in the proof of strange duality for the pair $(\omega _2,\omega _r,\omega _r)$ and $(2\omega _1,(2r+1)\omega _s, (2r+1)\omega _s)$.

Lemma A.5

Let $w=\phi _{1,0}\wedge \phi _{2,0}\wedge \bigwedge _{ 1\le i \le r, -s\le j\le -1}\phi _{i,j}$. Then the following hold in $\mathcal {H}_{\omega _r}(\mathfrak {so}(2r+1))\otimes \mathcal {H}_{(2s+1)\omega _s}(\mathfrak {so}(2s+1))$:

$$\begin{aligned} B^{2,1}_{-1,1}w=B^{2,-1}_{-1,-1}w=0\ ;\ B^{1,0}_{-2,0}w=\bigwedge _{ 1\le i \le r, -s\le j\le -1}\phi _{i,j}\ . \end{aligned}$$

Next we compute $R^3(B^0_1)v_3$. Our strategy is same as the previous steps.

Proposition A.6

We have:

$$\begin{aligned} R^3(B^0_1)v_3&=6\left[ \phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,0}(-\tfrac{1}{2} )\phi ^{3,0}(-\tfrac{1}{2} )\right] -3\bigl [ \phi ^{1,-1}(-\tfrac{1}{2} )\phi ^{2,0}(-\tfrac{1}{2} )\phi ^{3,1}(-\tfrac{1}{2} )\\&\quad + \,\phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,-1}(-\tfrac{1}{2} )\phi ^{3,1}(-\tfrac{1}{2} )+ \phi ^{1,-1}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )\phi ^{3,0}(-\tfrac{1}{2} )\\&\quad +\, \phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )\phi ^{3,-1}(-\tfrac{1}{2} ) + \phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,-1}(-\tfrac{1}{2} )\phi ^{3,0}(-\tfrac{1}{2} )\\&\quad +\, \phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,0}(-\tfrac{1}{2} )\phi ^{3,-1}(-\tfrac{1}{2} )\bigr ]\ . \end{aligned}$$

Proof

The proof follows by applying the expression for $R(B^0_1)$ successively:

$$\begin{aligned} R(B^0_1)v_3&=\phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )\phi ^{3,1}(-\tfrac{1}{2} )+ \phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,0}(-\tfrac{1}{2} )\phi ^{3,1}(-\tfrac{1}{2} )\\&\qquad + \phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,1}(-\tfrac{1}{2} )\phi ^{3,0}(-\tfrac{1}{2} )\ . \\ R^2(B^0_1)v_3&= 2\bigl (\phi ^{1,0}(-\tfrac{1}{2} )\phi ^{2,0}(-\tfrac{1}{2} )\phi ^{3,1}(-\tfrac{1}{2} )+\phi ^{1,0}(-\tfrac{1}{2} \phi ^{2,1}(-\tfrac{1}{2} )\phi ^{3,0}(-\tfrac{1}{2} )\\&\qquad +\phi ^{1,1}(-\tfrac{1}{2} \phi ^{2,0}(-\tfrac{1}{2} )\phi ^{3,0}(-\tfrac{1}{2} )\bigr )\\&\qquad -\bigl (\phi ^{1,1}(-\tfrac{1}{2} \phi ^{2,1}(-\tfrac{1}{2} )\phi ^{3,-1}(-\tfrac{1}{2} )+\phi ^{1,1}(-\tfrac{1}{2} )\phi ^{2,-1}(-\tfrac{1}{2} )\phi ^{3,1}(-\tfrac{1}{2} )\\&\qquad +\phi ^{1,-1}(-\tfrac{1}{2} \phi ^{2,1}(-\tfrac{1}{2} )\phi ^{3,1}(-\tfrac{1}{2} )\bigr ). \end{aligned}$$

and acting once more by $R(B^0_1)$. $\quad \square$

We now gather these calculations into the following algorithm:

• If $v_k=\phi ^{1,1}(-\tfrac{1}{2} )\cdots \phi ^{k,1}(-\tfrac{1}{2} )$, then the $\mathfrak {h}_2$-weight of $R^k(B^0_1)v_k$ is zero, where $\mathfrak {h}_2$ is the Cartan subalgebra of $\mathfrak {so}(2s+1)$.

• The expression for $R(B^0_1)$, viewed as an operator on the Clifford module for $\widehat{\mathfrak {so}}(2d+1)$, implies that

  • if $v =\phi ^{1,a_1}(-\tfrac{1}{2} )\cdots \phi ^{k,a_k}(-\tfrac{1}{2} )$, where $0\le a_1+\cdots +a_k\le k$, and each $a_i\in \{-1,0,1\}$, then the action of $R({B^0_1})$ on v is a sum of expressions of the form $\phi ^{1,b_1}(-\tfrac{1}{2} )\cdots \phi ^{k,b_k}(-\tfrac{1}{2} )$, where exactly one of the $b_i$’s is different from $a_i$;

  • the operator $R(B^0_1)$ can change an $a_i=1$ to $b_i=0$, or $a_i=0$ to $b_i=-1$. In the latter case, this introduces a minus sign in front of the new expression. In particular for each expression $\phi ^{1,b_1}(-\tfrac{1}{2} )\cdots \phi ^{k,b_k}(-\tfrac{1}{2} )$ appearing in $R(B^0_1)v$, we get $b_1+\cdots + b_k+1=a_1+\cdots +a_k$. For examples, see the previous lemmas.

• Thus, applying the operator $R(B^0_1)$ to $v_k$, k-times, we get an expression which is a sum of terms of the form $(-1)^m\phi ^{1,c_1}(-\tfrac{1}{2} )\cdots \phi ^{k,c_k}(-\tfrac{1}{2} )$, with multiplicities, where $c_1+\cdots +c_k=0$, and each $-1\le c_i\le 1$, and m is the number of $(-1)$’s appearing among the $c_i$’s.

• The multiplicity of the expression $\phi ^{1,0}(-\tfrac{1}{2} )\cdots \phi ^{k,0}(-\tfrac{1}{2} )$ is k!.

To summarize, we have the following.

Proposition A.7

As an element of $\mathcal {H}_{\omega _k}(\mathfrak {so}(2r+1))\otimes \mathcal {H}_{k\omega _1}(\mathfrak {so}(2s+1))$, the vector $R^k(B^0_1)v_k$ is of the form $k!\phi ^{1,0}(-\tfrac{1}{2} )\cdots \phi ^{k,0}(-\tfrac{1}{2} )$, plus a sum of terms of the form $B^{i,a}_{-j,b}(-1)w$, where $i\ne j$ are positive integers and a, b are nonzero.