# Hamiltonian group actions in the context of holomorphic line bundles

When studying Hamiltonian group actions, a very nice set up might be to take the following:

Let $M$ be a compact Kähler manifold with (integral) Kähler form $\omega$, endowed with a Hamiltonian $G$ action with moment map $\mu: M \to \mathfrak g^*$ such that $\iota_{X^\sharp}\omega = d\mu^X$ for $X \in \mathfrak g$, where $X^\sharp$ is the corresponding fundamental vector field, and $\mu^X(p) = \mu(p)(X)$.

However, several recent papers (for example, [1], [2], and [3]) recast this into the language of pre-quantum line bundles. For example,

Let $M$ be a compact Kähler manifold, $L \to M$ a holomorphic, Hermitian line bundle with connection $\nabla$ such that the curvature is $F_\nabla = \omega$. Define the comoment map $\mu^*: \mathfrak g \to C^\infty(M,\mathbb R)$ by $$\mu^*(X) = \frac1{2\pi i} \left[ \mathcal L(X) - \nabla(X^\sharp) \right]$$ where $\mathcal L$ is Lie differentiation of sections $C^\infty(M,L)$ along $X$. In particular, one again has $d\mu^*(X) = \iota_{X^\sharp}\omega$.

The two are used seemingly interchangeably, and I would hazard a guess is that the relationship is related to the notion of quantization. Unfortunately, in my exploration of the literature I rarely see discussions of quantization that also include Hamiltonian group actions, and when it is mentioned the reader is often referred to the tome that is Kostant's Quantization and Unitary Representations, which has neither an index, nor a table of contents, nor is searchable electronically. So my questions are as follows:

1. Does there exist a good reference for the relationship between these two frameworks, specifically in the context of Hamiltonian group actions (and even better would be convexity theorems).
2. Is it clear that the moment map in the first framework is equivalent to the comoment map in the second framework? If so, a few remarks commenting on why this is the case would be greatly appreciated.
3. Precisely when does the datum of a Hamiltonian group action on an integral Kähler manifold have a one-to-one correspondence with the prequantum line bundle set up? Is being integral, compact, and Kähler sufficient?

Edit: It has been pointed out that $\mu^*(X)$ is not a map on $M$. I agree that I cannot see why it should be a function on $M$ rather than a function on sections, so perhaps this is a typo in [2,page 5] and [3,page 3]. All the more reason why a reference would be greatly appreciated.

• Your $\mu ^*(X)$ doesn't look at all like a function on $M$. – abx May 12 '15 at 15:49
• I am certainly inclined to agree: it would seem to act on sections. Yet this is how the map is defined in reference [2, page 641, eq(2)] and [3, page 3], just adding to the confusion. Edit: For the arxiv version, it should be [2, page 5, eq(2)] – Tyler Holden May 12 '15 at 16:00

You may be relieved to hear that Kostant actually started from your first definition of the momentum map, and then used that to construct the lifted group action (denoted $\mathcal{L}(X)$ above). The general picture goes like this:

Construct a principle $U(1)$-bundle $\tau:P\rightarrow M$ and a connection $\alpha$ with curvature $2\pi i\, \omega$, where $\omega$ is the symplectic form. (The line bundle $L$ will be the associated bundle to $P$). We define a the Hamiltonian vector field $X_f$ associated to $f\in C^\infty(M,\mathbb{R})$ by the relation $i_{X_f}\omega = df$. Such fields satisfy $$[X_f,X_g] = X_{-\lbrace f,g\rbrace}.$$ Now using the prequantum data of $U(1)$-bundle and connection $\alpha$, it isn't too difficult to show that the vector fields $$A_f = X_f^h - (f\circ\tau)\left(2\pi i\right)_P$$ on $P$ obey the same relations, namely $$[A_f,A_g] = A_{-\lbrace f,g\rbrace}.$$ Here, $X_f^h$ denotes the horizontal lift of $X_f$ to $P$, and $\left(2\pi i\right)_P$ is the generator along the $U(1)$ fiber in $P$. One motivation for this construction is to produce appropriate quantum operators corresponding to classical observables.

Now, assuming the existence of a comomentum map $\mu^*:\mathfrak{g}\rightarrow C^\infty(M,\mathbb{R})$ that is also a Lie algebra homomorphism $$\lbrace \mu^*(\xi),\mu^*(\zeta)\rbrace = \mu^*([\xi,\zeta]),$$ for $\xi,\zeta\in\mathfrak{g}$, and letting $\mathcal{L}(\xi) = A_{\mu^*(\xi)}$, the above implies $$[\mathcal{L}(\xi),\mathcal{L}(\zeta)] = -\mathcal{L}([\xi,\zeta]).$$ Hence the $\mathcal{L}(\xi)$ form an infinitesimal left action of $\mathfrak{g}$ on $P$. This construction essentially allows Kostant to lift the $G$-action on $M$ to a $G$-action on $P$ (by exponentiation, assuming $G$ is simply connected). So we have $$\mathcal{L}(\xi) = X_{\mu^*(\xi)}^h - (\mu^*(\xi)\circ\tau) (2\pi i)_P = (\xi^\sharp)^h - (\mu^*(\xi)\circ\tau)(2\pi i)_P.$$

Carrying this over to the associated line bundle, this becomes $$\mathcal{L}(\xi) = \nabla_{\xi^\sharp} + \mu^*(\xi) 2\pi i$$ (where this $\mathcal{L}(\xi)$ is the corresponding $\mathfrak{g}$-action on sections of $L$). Rearranging gives your second definition of the comomentum map.

So the only prerequisites are integral symplectic form, and the existence of a momentum/comomentum map with the homomorphism property. Kostant is still the best place to learn this, I'm afraid. You could also try "Moment Maps, Cobordisms, and Hamiltonian Group Actions" by Guillemin, Ginzburg and Karshon.

$\mu^*(X)$ is a function on $M$ since it is a tensorial map on sections of the line bundle, thus a section of $L^*\otimes L$ which is trivial. But the Lie derivative is only defined on sections if $L$ is a $G$-bundle. $X^\sharp$ probably means the infinitesimal action of $X$ on $M$. $\omega$ has to be integral so that there exists a line bundle $L$ whose Chern class is $\omega$; then there exists even a Hermitian connection $\nabla$ on $L$ with curvature $\omega$. Kähler is not necessary.

• MR0464311 (57 #4243) Reviewed Simms, D. J. An outline of geometric quantisation (d'après Kostant). Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), pp. 1–10. Lecture Notes in Math., Vol. 570, Springer, Berlin, 1977.

• D. J. Simms and N. M. J. Woodhouse, Lectures on Geometric Quantization, Lecture Notes in Physics, Vol. 53 (Springer, New York, 1976). MR0672639 (58 #32470)

• J. Sniatycki, Geometric Quantization and Quantum Mechanics (Springer-Verlag, Berlin, 1980). MR0554085 (82b:81001)

• MR2151954 (2006k:81209) Reviewed Ali, S. Twareque(3-CONC-MS); Engliš, Miroslav(CZ-AOS) Quantization methods: a guide for physicists and analysts. (English summary) Rev. Math. Phys. 17 (2005), no. 4, 391–490.

$\mathcal L_X - \nabla_{X^\sharp}$ is a skew Hermitian endomorphism since the Hermitian metric is preserved. So $\mu^*(X) = \frac1{2\pi i}$ of that is a real valued function.
• Sorry if this is a silly question, but then why is it not complex valued functions? If I understand correctly, $\mu^*(X) \in \Gamma(M, End(L))$ and $End(L) \cong L^* \otimes L \cong \mathbb C \times M$. But then shouldn't sections be $\mathbb C$-valued (since we are dealing with holomorphic line bundles)? – Tyler Holden May 12 '15 at 19:55