Brane Quantization of Toric Poisson Varieties

Abstract

In this paper we propose a noncommutative generalization of the relationship between compact Kähler manifolds and complex projective algebraic varieties. Beginning with a prequantized Kähler structure, we use a holomorphic Poisson tensor to deform the underlying complex structure into a generalized complex structure, such that the prequantum line bundle and its tensor powers deform to a sequence of generalized complex branes. Taking homomorphisms between the resulting branes, we obtain a noncommutative deformation of the homogeneous coordinate ring. As a proof of concept, this is implemented for all compact toric Kähler manifolds equipped with an R-matrix holomorphic Poisson structure, resulting in what could be called noncommutative toric varieties. To define the homomorphisms between generalized complex branes, we propose a method which involves lifting each pair of generalized complex branes to a single coisotropic A-brane in the real symplectic groupoid of the underlying Poisson structure, and compute morphisms in the A-model between the Lagrangian identity bisection and the lifted coisotropic brane. This is done with the use of a multiplicative holomorphic Lagrangian polarization of the groupoid.

Comparison of Groupoids

In this appendix we compare the two constructions of prequantized (graded) holomorphic symplectic groupoids integrating the toric Poisson structure $\sigma _{C}$: the groupoid whose construction was sketched at the end of Sect. 3.4, and the groupoid constructed in Theorem 3.4.

1.1 Magnetic deformation construction

We first carefully flesh out the construction of the groupoid outlined in Sect. 3.4. The starting data is the (prequantized) source simply connected holomorphic symplectic groupoid ${\mathcal {G}}_{C}$ integrating $\sigma _{C}$. Recall that as a holomorphic symplectic manifold, it is given by the holomorphic cotangent bundle $(T^{*}M, \Omega _{0})$, where $\Omega _{0}$ is the holomorphic symplectic form whose imaginary part is the real canonical symplectic form. The target and source maps are given, respectively, by

$$\begin{aligned} t(z) = e^{\frac{1}{2}CJ_{0}(z)} \pi (z), \qquad s(z) = e^{-\frac{1}{2}CJ_{0}(z)} \pi (z), \end{aligned}$$

and the deformed multiplication is given by

$$\begin{aligned} m(x,y) = e^{- \frac{1}{2}CJ_{0}(y)} x + e^{\frac{1}{2}CJ_{0}(x)} y, \end{aligned}$$

where $J_{0} : T^{*}M \rightarrow {\mathfrak {t}}_{{\mathbb {C}}}^{*}$ is the holomorphic moment map generating the cotangent lift of the action of ${\mathbb {T}}_{{\mathbb {C}}}$. It is defined, for $\alpha \in T^{*}M$ and $u \in {\mathfrak {t}}_{{\mathbb {C}}}$ by

$$\begin{aligned} \langle J_{0}(\alpha ), u \rangle = i \langle \alpha , V_{u} \rangle , \end{aligned}$$

where $V_{u}$ is the toric vector field corresponding to u.

This symplectic groupoid has a multiplicative prequantization given by the trivial line bundle with connection $d - 2 \pi i \alpha _{0}$, where $\alpha _{0}$ is the canonical primitive of $\Omega _{0}$, and mutliplicative cocycle defined on the space of composable pairs of arrows by

$$\begin{aligned} \theta _{C}(x,y) = e^{i \pi C(J_{0}(x), J_{0}(y))}. \end{aligned}$$

Now consider the autoequivalence $T_{C}$ and the sequence of branes

$$\begin{aligned} B_{n} :=T^{n}_{C}(B_{0}) = (L^{\otimes n}, \overline{ \nabla ^{\otimes n}}), \end{aligned}$$

constructed in Sect. 3.3. The curvature of $\overline{ \nabla ^{\otimes n}}$ is given by $-2\pi i F_{n}$. Using this data, we will construct a graded prequantized holomorphic symplectic groupoid. The following is based on the constructions of [5,6,7].

• 1. Consider the open cover of M consisting of a ${\mathbb {Z}}$-indexed family of copies of M: $\{ M_{i} \}_{i \in {\mathbb {Z}}}$. We localize the groupoid ${\mathcal {G}}_{C}$ to this cover. The result is a groupoid

  $$\begin{aligned} {\mathcal {G}}_{C,{\mathbb {Z}}} = \bigsqcup _{(i,j) \in {\mathbb {Z}}^2} {\mathcal {G}}_{C, i,j}, \end{aligned}$$

  defined over the disjoint union $M_{{\mathbb {Z}}} = \sqcup _{i \in {\mathbb {Z}}} M_{i}$. The groupoid structures are defined as above, but the arrows in ${\mathcal {G}}_{C, i,j}$ go from $M_{j}$ to $M_{i}$. As a result, it is clear that ${\mathcal {G}}_{C,{\mathbb {Z}}}$ is a prequantized holomorphic symplectic groupoid integrating the holomorphic Poisson structure consisting of $\sigma _{C}$ on each copy $M_{i}$.

• 2. Consider the real closed 2-form $F_{{\mathbb {Z}}}$ on $M_{{\mathbb {Z}}}$ defined to be $F_{i}$ on $M_{i}$. Recall that this form satisfies Eq. 12. Therefore, by [5, Proposition 6.3], the deformed complex form $\Omega _{0} + t^{*}F_{{\mathbb {Z}}} - s^{*}F_{{\mathbb {Z}}}$ defines a new holomorphic symplectic structure on ${\mathcal {G}}_{C,{\mathbb {Z}}}$, such that it becomes a holomorphic symplectic groupoid over the manifold $M_{{\mathbb {Z}}}$, now equipped with the holomorphic Poisson structure $(I_{i}, \sigma _{i})$ on $M_{i}$. Note that the holomorphic symplectic form on ${\mathcal {G}}_{C, i,j}$ is given by

  $$\begin{aligned} \Omega _{i,j} = \Omega _{0} + t^{*}F_{i} - s^{*}F_{j}. \end{aligned}$$

  Denote the new groupoid ${\mathcal {Z}}_{C, {\mathbb {Z}}} = \sqcup _{(i,j) \in {\mathbb {Z}}^2} Z_{i,j}$.

• 3. The 2-form $F_{{\mathbb {Z}}}$ is prequantized by the line bundle with connection $(A, \nabla _{A})$ which consists of $(L^{\otimes i}, \overline{ \nabla ^{\otimes i}})$ on $M_{i}$. By [6, Theorem 4.4.5], this leads to a holomorphic multiplicative prequantization of ${\mathcal {Z}}_{C, {\mathbb {Z}}}$ given by

  $$\begin{aligned} ({\mathcal {U}}_{C, {\mathbb {Z}}}, D_{C, {\mathbb {Z}}}) = t^{*}(A, \nabla _{A}) \otimes s^{*}(A, \nabla _{A})^{*} \otimes ({\mathcal {O}}, d - 2\pi i \alpha _{0}). \end{aligned}$$

  Indeed, the component over $Z_{i,j}$ is given by

  $$\begin{aligned} (U_{i,j}, D_{i,j}) = (t^{*}(L^{\otimes i})\otimes s^{*}(L^{\otimes j})^{*}, t^{*}(\overline{ \nabla ^{\otimes i}}) \otimes s^{*}(\overline{ \nabla ^{\otimes j}})^{*} - 2\pi i \alpha _{0}), \end{aligned}$$

  which is a complex line bundle with connection whose curvature equals $-2\pi i \Omega _{i,j}$. As a result, $D_{i,j}^{(0,1)}$ defines a holomorphic structure with respect to which $(U_{i,j}, D_{i,j})$ defines a holomorphic prequantization of $(Z_{i,j}, \Omega _{i,j})$. In order to construct a multiplicative structure for $({\mathcal {U}}_{C, {\mathbb {Z}}}, D_{C, {\mathbb {Z}}})$, we need a flat $\delta$-closed trivialization of $\delta ({\mathcal {U}}_{C, {\mathbb {Z}}}, D_{C, {\mathbb {Z}}})$ over the space of composable pairs of arrows ${\mathcal {Z}}_{C, {\mathbb {Z}}}^{(2)}$. But we have the canonical identification

  $$\begin{aligned} \delta ({\mathcal {U}}_{C, {\mathbb {Z}}}, D_{C, {\mathbb {Z}}})&= \delta (t^{*}(A, \nabla _{A}) \otimes s^{*}(A, \nabla _{A})^{*}) \otimes ({\mathcal {O}}, d - 2\pi i \delta (\alpha _{0})) \\&= ({\mathcal {O}}, d - 2\pi i \delta (\alpha _{0})), \end{aligned}$$

  so that a flat multiplicative prequantization is provided by $\theta _{C}$. We may view the complement of the zero section in ${\mathcal {U}}_{C, {\mathbb {Z}}}$ as a groupoid over $M_{{\mathbb {Z}}}$. It is a central extension of ${\mathcal {Z}}_{C, {\mathbb {Z}}}$.

• 4. Let ${\mathcal {L}} \subset {\mathcal {G}}_{C}$ be the identity bisection, and let ${\mathcal {L}}_{i,j}$ denote this bisection when it is viewed as a (non-holomorphic) submanifold of $Z_{i,j}$. Then ${\mathcal {L}}_{i,j}$ is a section of both the source and target, it defines the identification of $M_{i}$ and $M_{j}$ as smooth manifolds, and we have

  $$\begin{aligned} \Omega _{i,j}|_{{\mathcal {L}}_{i,j}} = F_{i} - F_{j}. \end{aligned}$$

  Furthermore, ${\mathcal {L}}_{i,j} *{\mathcal {L}}_{j,k} = {\mathcal {L}}_{i,k}$.

• 5. Consider the Q-Poisson diffeomorphism $\varphi _{1}$ underlying $T_{C}$. By Eq. 15, $\varphi _{1}$ defines a holomorphic Poisson isomorphism from $(M, I_{i+1}, \sigma _{i+1})$ to $(M, I_{i}, \sigma _{i})$. We thus view it as a holomorphic Poisson automorphism of $M_{{\mathbb {Z}}}$. As explained in Sect. 3.4, $\varphi _{1}$ integrates to a symplectic groupoid automorphism $\Phi$ of the groupoid $(T^{*}M, {\mathrm {Im}}(\Omega _{0}))$ integrating Q. We may view it as a (at the moment only $C^{\infty }$) groupoid automomorphism of ${\mathcal {G}}_{C, {\mathbb {Z}}}$ covering $\varphi _{1}$. It is defined by mapping ${\mathcal {G}}_{C, i,j}$ to ${\mathcal {G}}_{C, i-1,j-1}$. But now recall from Lemma 3.3 that $\Phi ^{*}(\Omega _{i,j}) = \Omega _{i+1, j+1}$. Hence, $\Phi$ defines a holomorphic symplectic groupoid automorphism of ${\mathcal {Z}}_{C, {\mathbb {Z}}}$. This map satisfies $\Phi ({\mathcal {L}}_{i,j}) = {\mathcal {L}}_{i-1,j-1}$.

• 6. The automorphism $\Phi$ may be lifted to a flat multiplicative automorphism $\hat{\Phi }$ of ${\mathcal {U}}_{C, {\mathbb {Z}}}$. First, for every $(i,j) \in {\mathbb {Z}}^2$, we need a flat isomorphism

  $$\begin{aligned} \hat{\Phi }_{i,j} : (U_{i+1, j+1}, D_{i+1, j+1}) \rightarrow \Phi ^{*}(U_{i,j}, D_{i,j}), \end{aligned}$$

  over $Z_{i+1, j+1}$. Since both connections have the same curvature we may first construct the isomorphism along ${\mathcal {L}}_{i+1, j+1}$, and then integrate it to all $Z_{i+1, j+1}$. Restricting the line bundles, we get

  $$\begin{aligned} (U_{i+1, j+1}, D_{i+1, j+1})|_{{\mathcal {L}}_{i+1, j+1}}&= (L^{\otimes i+1} \otimes (L^{\otimes j+1})^{*}, \overline{ \nabla ^{\otimes i+1}} \otimes (\overline{ \nabla ^{\otimes j+1}})^{*}) \end{aligned}$$

  (26)
  $$\begin{aligned} \Phi ^{*}(U_{i,j}, D_{i,j})|_{{\mathcal {L}}_{i+1, j+1}}&= \varphi _{1}^{*}(L^{\otimes i} \otimes (L^{\otimes j})^{*}, \overline{ \nabla ^{\otimes i}} \otimes (\overline{ \nabla ^{\otimes j}})^{*}). \end{aligned}$$

  (27)

  Recall from Sect. 3.3 that

  $$\begin{aligned} (L^{\otimes n+1}, \overline{ \nabla ^{\otimes n+1}}) \cong T_{C}(L^{\otimes n}, \overline{ \nabla ^{\otimes n}}) = (L, \overline{ \nabla }) \otimes \varphi _{1}^{*}(L^{\otimes n}, \overline{ \nabla ^{\otimes n}}). \end{aligned}$$

  To be more precise, the isomorphism from the left hand side to the right hand side is given by $id \otimes \hat{\varphi }_{1}^{\otimes n}$, where $\hat{\varphi }_{1}$ is the time-1 flow of $\hat{W}$ on L. Therefore, we see that $(id \otimes \hat{\varphi }_{1}^{\otimes i}) \otimes (id \otimes \hat{\varphi }_{1}^{\otimes j})^{*}$ takes us from (26) to (27). On the level of line bundles, this is the morphism

  $$\begin{aligned} \hat{\varphi }_{1}^{\otimes (i-j)} : L^{\otimes i-j} \rightarrow \varphi _{1}^{*}(L^{\otimes i-j}). \end{aligned}$$

  Extending this to a flat isomorphism over $Z_{i+1, j+1}$ we obtain $\hat{\Phi }_{i,j}$. In order to check that the resulting map is a homomorphism, we need to establish that $\hat{\Phi }^{*}(\theta _{C}) = \theta _{C}$. Because the cocycle is flat, it suffices to check equality on ${\mathcal {L}}_{i+1,j+1} \times _{M} {\mathcal {L}}_{j+1, k+1}$ (for all $(i,j,k) \in {\mathbb {Z}}^3$). The product on this submanifold is given by the pairing of dual line bundles. For example,

  $$\begin{aligned} (L^{\otimes i+1} \otimes (L^{\otimes j+1})^{*} )\otimes (L^{\otimes j+1} \otimes (L^{\otimes k+1})^{*} ) \rightarrow (L^{\otimes i+1} \otimes (L^{\otimes k+1})^{*} ). \end{aligned}$$

  It is now straightforward to check that $\hat{\Phi }$ is multiplicative.

• 7. Finally, we define the structure of a prequantized holomorphic symplectic groupoid on the disjoint union

  $$\begin{aligned} \bigsqcup _{i \in {\mathbb {Z}}} (Z_{i,0}, \Omega _{i,0}, U_{i,0}, D_{i,0}). \end{aligned}$$

  The source map is defined as before, and is a holomorphic Poisson morphism to $-\sigma _{C}$. We modify the target map as follows. On $Z_{i,0}$ we define the new target map to be $\varphi _{1}^{i} \circ t$. Since t is holomorphic Poisson to $\sigma _{i}$, the new target map is holomorphic Poisson to $\sigma _{0}$. We represent a point of $Z_{i,0}$ as (i, g). The product on this groupoid is defined by

  $$\begin{aligned} \tilde{m}( (j,g), (i,h) ) = (j+ i, m(\Phi ^{-i}(g), h)). \end{aligned}$$

  It is straightforward to check, using the properties established above of ${\mathcal {Z}}_{C,{\mathbb {Z}}}$ and $\Phi$, that this defines the structure of a holomorphic symplectic groupoid over $(M, \sigma _{C})$. A similar formula involving $\hat{\Phi }$ defines a multiplicative prequantization. Finally, the submanifolds ${\mathcal {L}}_{i,0}$ satisfy ${\mathcal {L}}_{i,0} *{\mathcal {L}}_{j,0} = {\mathcal {L}}_{i+j,0}$ with respect to the new multiplication.

1.2 Comparison with reduction construction

In this section we construct an isomorphism between the groupoid constructed in A.1 and the one constructed in Theorem 3.4. Recall that the latter groupoid has the decomposition

$$\begin{aligned} ({\mathcal {Z}}, \Omega ) := \sqcup _{n \in \mathbb {{\mathbb {Z}}}} (Z_{n}, \Omega _{n}), \end{aligned}$$

and each component contains a bisection ${\mathcal {L}}_{n} \subset Z_{n}$, which we view as a section of the source map. The groupoid in ‘degree 0’ is isomorphic to the symplectic groupoid of $\sigma _{C}$

$$\begin{aligned} (Z_{0}, \Omega _{0}) \cong ({\mathcal {G}}_{C}, \Omega _{0}). \end{aligned}$$

Therefore, the groupoid multiplication defines an action of $({\mathcal {G}}_{C}, \Omega _{0})$ on each $(Z_{n}, \Omega _{n})$. We now use this action and the bisection ${\mathcal {L}}_{n}$ to define a diffeomorphism

$$\begin{aligned} T_{n} : {\mathcal {G}}_{C} \rightarrow Z_{n}, \qquad g \mapsto m({\mathcal {L}}_{n}(t(g)), g). \end{aligned}$$

Lemma A.1

The map $T_{n}$ satisfies the following equations

$$\begin{aligned} t \circ T_{n} = \varphi _{1}^n \circ t, \qquad s \circ T_{n} = s, \qquad T_{n}^{*}\Omega _{n} = \Omega _{0} + t^{*}F_{n}. \end{aligned}$$

Proof

The formulas follow from Theorem 3.4. First, for $g \in {\mathcal {G}}_{C}$ we have

$$\begin{aligned} s(T_{n}(g))&= s(m({\mathcal {L}}_{n}(t(g)), g)) = s(g), \\ t(T_{n}(g))&= t(m({\mathcal {L}}_{n} (t(g)), g)) = (t\circ {\mathcal {L}}_{n}) (t(g)) = \varphi _{1}^n (t(g)). \end{aligned}$$

For the formula involving the symplectic form, we use the fact that the graph of the action is a holomorphic Lagrangian submanifold of

$$\begin{aligned} (Z_{n}, -\Omega _{n}) \times ({\mathcal {G}}_{C}, -\Omega _{0}) \times (Z_{n}, \Omega _{n}). \end{aligned}$$

Consider the map from ${\mathcal {G}}_{C}$ into the graph given by

$$\begin{aligned} g \mapsto ({\mathcal {L}}_{n} (t(g)), g, T_{n}(g)). \end{aligned}$$

The symplectic form pulls back to 0, and hence

$$\begin{aligned} T_{n}^{*}\Omega _{n} = \Omega _{0} + t^{*}{\mathcal {L}}_{n}^{*}\Omega _{n} = \Omega _{0} + t^{*}F_{n}. \end{aligned}$$

$\square$

As a result of this lemma, $T_{n}$ is a holomorphic symplectic isomorphism $Z_{n,0} \rightarrow Z_{n}$ which intertwines the source and target maps. Putting all the maps together, we obtain the map

$$\begin{aligned} T : \bigsqcup _{i \in {\mathbb {Z}}} (Z_{i,0}, \Omega _{i,0}) \rightarrow ({\mathcal {Z}}, \Omega ). \end{aligned}$$

Proposition A.2

The map T is a holomorphic symplectic groupoid isomorphism.

Proof

It suffices to check that T is multiplicative. Define a map $\Psi _{n} : {\mathcal {G}}_{C} \rightarrow {\mathcal {G}}_{C}$ by the equation

$$\begin{aligned} m({\mathcal {L}}_{n}(t(g)), g) = m(\Psi _{n}(g), {\mathcal {L}}_{n}(s(g))). \end{aligned}$$

Since ${\mathcal {L}}_{n}$ is a bisection, this map is a groupoid automorphism, since ${\mathcal {L}}_{n}$ is Lagrangian with respect to ${\mathrm {Im}}(\Omega _{n})$, $\Psi _{n}$ is a symplectomorphism of $({\mathcal {G}}_{C}, {\mathrm {Im}}(\Omega _{0}))$, and finally, because $t \circ {\mathcal {L}}_{n} = \varphi _{1}^{n}$, the automorphism $\Psi _{n}$ covers $\varphi _{1}^n$. But this implies that we must have $\Psi _{n} = \Phi ^{n}$, where $\Phi$ is the automorphism considered in Sect. A.1. Stated otherwise, the bisection ${\mathcal {L}}_{n}$ satisfies

$$\begin{aligned} m({\mathcal {L}}_{n}(t(g)), g) = m(\Phi ^n(g), {\mathcal {L}}_{n}(s(g))). \end{aligned}$$

Now let $g \in Z_{i,0}$ and $h \in Z_{j,0}$ satisfy $s(g) = \varphi _{1}^{j}\circ t(h)$. Using the above equation, along with the multiplicativity of the ${\mathcal {L}}_{n}$, we obtain (suppressing the multiplication from the notation in the calculation)

$$\begin{aligned} m(T_{i}(g), T_{j}(h))&= \big ({\mathcal {L}}_{i} (t(g)) \ g\big ) \big ({\mathcal {L}}_{j}(t(h)) \ h\big ) \\&= {\mathcal {L}}_{i} (t(g)) \ \big ({\mathcal {L}}_{j}(t (\Phi ^{-j}(g))) \ \Phi ^{-j}(g) \big ) \ h \\&= {\mathcal {L}}_{i + j} (t (\Phi ^{-j}(g))) \ \big (\Phi ^{-j}(g) h \big ) \\&= T_{i+j}(\tilde{m}(g,h)). \end{aligned}$$

$\square$

We now lift T to an isomorphism $\hat{T}$ between the multiplicative prequantizations. First, to construct the lift $\hat{T}_{n}$ of $T_{n}$ consider the following map which factors through the graph of the multiplication

$$\begin{aligned} {\mathcal {G}}_{C} \rightarrow Z_{n} \times _{M} {\mathcal {G}}_{C} \rightarrow Z_{n} \times {\mathcal {G}}_{C} \times Z_{n}, \qquad g \mapsto ({\mathcal {L}}_{n} \circ t(g), g, T_{n}(g)). \end{aligned}$$

Pulling back the bundle $(U_{n}, D_{n})^{*} \boxtimes (U_{0}, D_{0})^{*} \boxtimes (U_{n}, D_{n})$ we obtain the following bundle over ${\mathcal {G}}_{C}$:

$$\begin{aligned} t^{*}{\mathcal {L}}_{n}^{*}(U_{n}, D_{n})^{*} \otimes (U_{0}, D_{0})^{*} \otimes T_{n}^{*}(U_{n}, D_{n}). \end{aligned}$$

The multiplicative cocycle $\Theta _{C}$, which is defined over $Z_{n} \times _{M} {\mathcal {G}}_{C}$, may be pulled back to ${\mathcal {G}}_{C}$, yielding a flat isomorphism

$$\begin{aligned} \Theta _{C,n}: t^{*}{\mathcal {L}}_{n}^{*}(U_{n}, D_{n}) \otimes (U_{0}, D_{0}) \rightarrow T_{n}^{*}(U_{n}, D_{n}). \end{aligned}$$

Finally, combining this with the isomorphism constructed in Theorem 3.4, we arrive at our definition of $\hat{T}_{n}$:

$$\begin{aligned} \hat{T}_{n} = \Theta _{C,n} \circ \big (t^{*}( {\mathcal {L}}_{n}^{*}(\phi _{n})^{-1} \circ e^{-R_{n}}) \otimes id \big ) : t^{*}(L^{n}, \overline{\nabla ^{\otimes n}}) \otimes (U_{0}, D_{0}) \rightarrow T_{n}^{*}(U_{n}, D_{n}).\nonumber \\ \end{aligned}$$

(28)

Putting all the maps together, we get

$$\begin{aligned} \hat{T} : \bigsqcup _{i \in {\mathbb {Z}}} (U_{i,0}, D_{i,0}) \rightarrow ({\mathcal {U}}, D). \end{aligned}$$

In the following, we will find it useful to decompose $\Theta _{C}$ as the a product of two contributions

$$\begin{aligned} \Theta _{C} = e^{i \pi C(J(x), J(y))} E. \end{aligned}$$

Lemma A.3

The multiplication E defines a (non-flat) multiplicative cocycle, which satisfies the following equation involving the maps $\phi _{n}$ of Theorem 3.4

$$\begin{aligned} e^{-CJ_{j}(y)}\phi _{i}|_{x} \otimes \phi _{j}|_{y} = \phi _{i+j}|_{m(x,y)} E(x,y), \end{aligned}$$

(29)

where $x \in Z_{i}$ and $y \in Z_{j}$.

Proof

The multiplicative structure E is obtained by reducing the ‘trivial’ multiplicative structure on $T^{*}P_{L}$, and $\phi _{n}$ are obtained by reducing the morphisms on $T^{*}P_{L}$ obtained from the canonical sections $s_{n}$ (see the proofs of Theorems 2.4 and 3.4). Working with these structures on $T^{*}P_{L}$ the result follows. $\square$

Proposition A.4

The map $\hat{T}$ is an isomorphism of multiplicative line bundles with connection.

Proof

It suffices to check that $\hat{T}$ is multiplicative, and this may be checked along the submanifolds ${\mathcal {L}}_{i,0} \times _{M} {\mathcal {L}}_{j,0}$. Since $\hat{T}$ factors as the composition of two maps (28), and so we check the multiplicativity of each separately.

The multiplicativity of the term involving $\Theta _{C,n}$ follows from the associativity of the product (i.e. $\delta \Theta _{C} = 1$). More precisely, the induced product on $t^{*}{\mathcal {L}}_{n}^{*}(U_{n}, D_{n}) \otimes (U_{0}, D_{0})$ is given by two applications of $\Theta _{C}$. Choosing a point $x \in M$, so that $({\mathcal {L}}_{i,0}(\varphi _{j}(x)), {\mathcal {L}}_{j,0}(x)) \in {\mathcal {L}}_{i,0} \times _{M} {\mathcal {L}}_{j,0}$, we may write this product as follows:

$$\begin{aligned}&\Theta _{C}\otimes \Theta _C : (U_{i}|_{{\mathcal {L}}_{i}\varphi _{j}(x)} \otimes U_{0}|_{{\mathcal {L}}_{0}\varphi _{j}(x)}) \otimes (U_{j}|_{{\mathcal {L}}_{j}(x)} \otimes U_{0}|_{{\mathcal {L}}_{0}(x)}) \\&\quad \rightarrow (U_{i+j}|_{{\mathcal {L}}_{i+j}(x)} \otimes U_{0}|_{{\mathcal {L}}_{0}(x)}). \end{aligned}$$

Now note that $U_{0}$ is the trivial bundle with the trivial equivariant structure. Hence, for $a \in U_{i}|_{{\mathcal {L}}_{i}\varphi _{j}(x)}$ and $b \in U_{0}|_{{\mathcal {L}}_{0}\varphi _{j}(x)}$ we get

$$\begin{aligned} \Theta _{C}(a \otimes b)&= e^{i \pi C(J_{i}{\mathcal {L}}_{i}\varphi _{j}(x), J_{0}{\mathcal {L}}_{0}\varphi _{j}(x))} \Theta (e^{- \frac{1}{2}CJ_{0}{\mathcal {L}}_{0}\varphi _{j}(x)} a, e^{\frac{1}{2}CJ_{i}{\mathcal {L}}_{i}\varphi _{j}(x)} b) \\&= \Theta (a, e^{\frac{1}{2}CJ_{i}{\mathcal {L}}_{i}\varphi _{j}(x)} b) \\&= \Theta (a, b) \\&= ab. \end{aligned}$$

The second line follows because ${\mathcal {L}}_{0}^{*}J_{0} = 0$, the third line follows because the equivariant structure on $U_{0}$ is trivial, and the final line follows because $\Theta$ is a tensor product, which reduces to scalar multiplication when one of the terms is trivial. As a result of this, the product reduces to

$$\begin{aligned} \Theta _{C} : U_{i}|_{{\mathcal {L}}_{i}\varphi _{j}(x)} \otimes U_{j}|_{{\mathcal {L}}_{j}(x)} \rightarrow U_{i+j}|_{{\mathcal {L}}_{i+j}(x)}. \end{aligned}$$

The product on $U_{n,0}$, restricted to the point $({\mathcal {L}}_{i,0}(\varphi _{j}(x)), {\mathcal {L}}_{j,0}(x))$, is given by

$$\begin{aligned} \hat{\varphi }_{-j} \otimes id : L^{\otimes i}|_{\varphi _{j}(x)} \otimes L^{j}|_{x} \rightarrow L^{\otimes (i+j)}|_{x}. \end{aligned}$$

Therefore, we need to check that the following diagram commutes.

We can decompose the commutativity into two parts. First, in order to match the exponential pre-factors we must show that

$$\begin{aligned} R_{i+j}(x) = R_{i}(\varphi _{j}(x)) + R_{j}(x) - i \pi C(J_{i}{\mathcal {L}}_{i}\varphi _{j}(x), J_{j}{\mathcal {L}}_{j}(x)). \end{aligned}$$

Using the expressions for these functions from Theorem 3.4, the right hand side is given by

$$\begin{aligned}&i\pi \big ( \int _{0}^{i}\int _{0}^{t} C(\varphi _{t+j}^{*}\mu , \varphi _{s+j}^{*}\mu ) ds dt + \int _{0}^{j}\int _{0}^{t}C(\varphi _{t}^{*}\mu , \varphi _{s}^{*}\mu ) ds dt \\&\quad - i^2 C( \int _{0}^{i} \varphi _{t+j}^{*}\mu dt, \int _{0}^{j} \varphi _{s}^{*}\mu ) ds \big ), \end{aligned}$$

which is equal to $R_{i+j}$.

Second, applying Eq. 29, we obtain the equality

$$\begin{aligned} e^{-CJ_{j}{\mathcal {L}}_{j}(x)}\phi _{i}|_{{\mathcal {L}}_{i}\varphi _{j}(x)} \otimes \phi _{j}|_{{\mathcal {L}}_{j}(x)} = \phi _{i+j}|_{{\mathcal {L}}_{i+j}(x)} E({\mathcal {L}}_{i}\varphi _{j}(x),{\mathcal {L}}_{j}(x)). \end{aligned}$$

It therefore remains to show that the flow $\hat{\varphi }_{-j}$ is obtained by acting by the element

$$\begin{aligned} \exp (-CJ_{j}{\mathcal {L}}_{j}(x)) = \exp (iC \int _{0}^{j} \mu \varphi _{s}(x) ds). \end{aligned}$$

This follows from the integral equation defining the flow of $\hat{W}$. $\square$