3.8 More applications of moduli space integrals [04CF]

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3.8 More applications of moduli space integrals

We collect a number of further topics involving the moduli space integral technique. Sections 3.8.2 and 3.8.3 are applications of the moduli integral formula (39) for the Solomon functional.

3.8.1 Lotay-Pacini convexity

Lotay and Pacini proved the convexity of their $J$ -functional (cf. Prop. 2.13) through rather heavy calculations, so it is instructive to see that in the Calabi-Yau case, this result has a much simpler conceptual argument.

We interpret their ‘geodesic’ as a bordism current $\mathcal{C}$ between two Lagrangians $L,L^{\prime}$ , constructed from universal families of holomorphic curves, such that automatic transversality and the positivity condition hold. In their highly idealized setting, only holomorphic strips $\Sigma\simeq\mathbb{R}_{s}\times[0,1]_{t}$ appear in the construction of $\mathcal{C}$ . We define the holomorphic function $F$ as usual. The 1-parameter family of totally real submanifolds is given by the constant $t$ -coordinate slices $\mathcal{C}_{t}$ of $\mathcal{C}$ , whose $J$ -volume functional is expressible through moduli space integrals

\text{Vol}_{J}(\mathcal{C}_{t})=\int_{\mathcal{C}_{t}}|\Omega|=\int_{\mathcal{M}}\int_{\mathbb{R}_{s}\times\{t\}}|\frac{\partial F}{\partial s}|ds.

Since $F$ is holomorphic, so is $\frac{\partial F}{\partial s}$ , whence $|\frac{\partial F}{\partial s}|$ is subharmonic, which combined with the exponential decay at $s\to\pm\infty$ implies the convexity of the function in $t$

\int_{\mathbb{R}_{s}\times\{t\}}|\frac{\partial F}{\partial s}|ds.

Thus $\text{Vol}_{J}(\mathcal{C}_{t})$ is convex as a function of $t$ , as Lotay and Pacini observed.

3.8.2 Lower bound of the Solomon functional

The theme of Chapter 5 will be on the variational approach to find special Lagrangians by minimizing the Solomon functional in a fixed derived category class. As an important motivation, special Lagrangians are formal local minimizers of the Solomon functional under Hamiltonian deformations (cf. section 2.8). In fact we can do better under the automatic transversality and the positivity condition:

Proposition 3.40.

(‘special Lagrangians are minimizers’) Suppose $L_{0}$ is an exact immersed special Lagrangian of phase $\hat{\theta}\in(-\frac{\pi}{2},\frac{\pi}{2})$ , with unobstructed bounding cochain structure. Let $L$ be an almost calibrated, exact, immersed Lagrangian in the same $D^{b}Fuk(X)$ class, which intersects $L_{0}$ transversely. Suppose the bordism current $\mathcal{C}$ with $\partial\mathcal{C}=L-L_{0}$ satisfies automatic transversality and the positivity condition. Then $\mathcal{S}(L)\geq\mathcal{S}(L_{0})$ .

Proof.

The incline angle of the tangent vector to $F(\partial\Sigma)\subset\mathbb{C}$ is equal to the Lagrangian angle modulo $\pi\mathbb{Z}$ . Since $L_{0}$ is a special Lagrangian, along the $L_{0}$ boundary portion $\arg F=\hat{\theta}$ . Thus $\text{Im}(e^{-i\hat{\theta}}F)=0$ at $p,q$ and the self intersections on $L_{0}$ . The Solomon functional integrand simplifies to

\text{Im}\int_{\Sigma}e^{-i\hat{\theta}}F\omega+\sum_{\text{$L$-self intersections on $\partial\Sigma$}}\text{Im}(e^{-i\hat{\theta}}F)f_{L}|^{+}_{-}.

By the almost calibrated assumption on $L,L_{0}$ , and the positivity condition, we obtain Claim 3.23, namely $F(\Sigma)$ lies above its $L_{0}$ boundary,

\text{Im}(e^{-i\hat{\theta}}F)\geq 0\quad\text{on }\Sigma.

Morever, the Novikov positivity requirement for the bounding cochain on $L$ says that $f_{L}|^{+}_{-}\geq 0$ at the degree one self intersections on $\partial\Sigma\cap L$ . Thus the Solomon functional integrand is nonnegative, which implies $\mathcal{S}(L)\geq 0=\mathcal{S}(L_{0})$ . ∎

Remark 3.20.

Suppose we drop the positivity condition, then the key step $\text{Im}(e^{-i\hat{\theta}}F)\leq 0$ would break down, so the above proof of Prop. 3.40 would be invalidated. However, the conclusion may still be true (cf. section 5.7.1).

3.8.3 Bounded part of the Solomon functional

Let $L,L_{0}$ be both exact, immersed Lagrangians with unobstructed bounding cochain structures, lying in the same $D^{b}Fuk(X)$ class, such that all intersections are transverse. Assume the bordism current $\mathcal{C}$ with $\partial\mathcal{C}=L-L_{0}$ satisfies automatic transversality and the positivity condition. We consider $L_{0}$ as a fixed reference Lagrangian, while $L$ can vary. We wish to find uniform a priori bound on certain parts of the Solomon functional, under natural conditions on $L$ .

We shall assume:

•

(Quantitative almost calibratedness) Both $L$ and $L_{0}$ have Lagrangian phase angles within $[-\frac{\pi}{2}+\epsilon,\frac{\pi}{2}-\epsilon]$ for some fixed small constant $\epsilon$ .
•

(Potential clustering, cf. Lemma 6.3) The immersed Lagrangian $L$ can be represented by a twisted complex (17) built from the immersed Lagrangians $L_{1},\ldots L_{N}$ , such that the oscillation of the Lagrangian potentials have uniform bounds

$\sup_{L_{i}}f_{L_{i}}-\inf_{L_{i}}f_{L_{i}}\leq A,$

while for any $i>j$ ,

$\sup_{L_{j}}f_{L_{j}}\leq\inf_{L_{i}}f_{L_{i}}.$

Without loss of generality, we also assume $\sup_{L_{0}}f_{L_{0}}-\inf_{L_{0}}f_{L_{0}}\leq A$ for the fixed Lagrangian $L_{0}$ .

Proposition 3.41.

(Uniform energy bound) Under the potential clustering assumption, all holomorphic polygons $u:\Sigma\to X$ with boundary on $L$ and $L_{0}$ contributing to $\mathcal{C}$ have uniformly bounded energy independent of $L$ :

E(u)=\int_{\Sigma}u^{*}\omega\leq A(N+1),

and along $\partial\Sigma$ the degree one self intersections of $L_{0},L_{1},\ldots L_{N}$ arising from the bounding cochains satisfy a uniform bound

\sum_{i}\sum_{b_{i}}f_{L_{i}}|^{+}_{-}(b_{i})\leq A(N+1).

Proof.

We consider holomorphic polygons whose boundary $\partial\Sigma$ encounters in the clockwise order intersections in $p_{N-1}\in CF^{1}(L_{N},L_{N-1}),\ldots$ , $p_{1}\in CF^{1}(L_{2},L_{1})$ , $p_{0}\in CF^{0}(L_{1},L_{0})$ , $p_{N}\in CF^{0}(L_{0},L_{N})$ , juxaposed possibly by more degree one self intersections $b_{i}$ of $L_{i}$ . The notation here does not constrain the number of self intersections of $L_{i}$ that can occur on $\partial\Sigma$ . The topological energy formula (66) expresses $E(u)$ in terms of the Lagrangian potentials at the intersections

\begin{split}&E(u)+\sum_{i}\sum_{b_{i}}f_{L_{i}}|^{+}_{-}(b_{i})\\ &=f_{L_{N}}(p_{N})-f_{L_{0}}(p_{N})+\sum_{i=0}^{N-1}(f_{L_{i}}-f_{L_{i+1}})(p_{i})\\ &=f_{L_{0}}(p_{0})-f_{L_{0}}(p_{N})+\sum_{i=1}^{N}(f_{L_{i}}(p_{i})-f_{L_{i}}(p_{i-1}))\\ &\leq(N+1)A.\end{split}

By the Novikov positivity requirement of the bounding cochains $f_{L_{i}}|^{+}_{-}(b_{i})\geq 0$ , and the energy of the holomorphic curve is also positive, so they are individually bounded.

More generally, the polygons may miss some of the Lagrangians in $L_{1},\ldots L_{N}$ , but cannot reverse the order of the Lagrangians. This amounts to using a smaller effective value $N$ , and the same argument implies the energy bound. ∎

We now consider the holomorphic function $F$ as before. Recall by Claim 3.22 we have $0\leq\text{Re }(F)\leq\text{Re }F(p)$ , where $p$ is the intersection point in $CF^{0}(L,L_{0})$ . In fact $F(\Sigma)$ must be contained in a triangular region determined by $\text{Re }F(p)$ :

Lemma 3.42.

(Wedge region bound) Under the quantitative almost calibrated hypothesis, we have $|\arg F|\leq\frac{\pi}{2}-\epsilon$ , or equivalently $|\text{Im}(F)|\leq(\cot\epsilon)\text{Re }(F).$ In particular $|F|\leq\frac{1}{\sin\epsilon}\text{Re }F\leq\frac{1}{\sin\epsilon}\text{Re }F(p).$

Proof.

The incline angle of the tangent vector of $F(\partial\Sigma)$ is equal to the Lagrangian angle mod $\pi\mathbb{Z}$ . Together with Claim 3.22 this implies $|\arg F|\leq\frac{\pi}{2}-\epsilon$ on the $\partial\Sigma$ , whence the same bound holds on $\Sigma$ by the maximum principle for holomorphic functions. ∎

We now introduce an elementary functional

\bar{\mathcal{S}}(L)=\text{Im}\left(\sum_{1}^{N}(\sup_{L_{i}}f_{L_{i}})e^{-i\hat{\theta}}\int_{L_{i}}\Omega\right)-(\sup_{L_{0}}f_{L_{0}})\text{Im}(e^{-i\hat{\theta}}\int_{L_{0}}\Omega).

(43)

As in section 3.5, we introduce complex valued volume form $\tilde{\Omega}_{L_{i}}$ on the $(n-1)$ -dimensional moduli spaces of holomorphic curves, whose core properties are

\text{Re }\tilde{\Omega}_{L_{i}}\geq 0,\quad\int_{\mathcal{M}}\tilde{\Omega}_{L_{i}}=\int_{L_{i}}\Omega,\quad i=0,1,\ldots N.

(44)

Thus the elementary functional is also a moduli space integral, with integrand

\text{Im}\sum_{1}^{N}(e^{-i\hat{\theta}}\tilde{\Omega}_{L_{j}})\sup f_{L_{j}}-\text{Im}(e^{-i\hat{\theta}}\tilde{\Omega}_{L_{0}})\sup f_{L_{0}}.

(45)

We decompose the Solomon functional into $\bar{\mathcal{S}}(L)$ and $\mathcal{S}(L)-\bar{\mathcal{S}}(L)$ .

Theorem 3.43.

(Bounded part of the Solomon functional) Under the quantitative almost calibratedness and the potential clustering assumption, and all the standing assumptions of this section, there is a uniform a priori bound independent of $L$ ,

|\mathcal{S}(L)-\bar{\mathcal{S}}(L)|\leq\frac{A(4N+2)}{\sin\epsilon}\int_{L_{0}}\text{Re }\Omega.

Proof.

We analyze the moduli space integrand (39) of the Solomon functional. Applying the uniform energy bound and the wedge region bound, the first term is bounded by

|\text{Im}\int_{\Sigma}e^{-i\hat{\theta}}F\omega|\leq\int_{\Sigma}|F|\omega\leq\frac{\text{Re}(F(p))}{\sin\epsilon}\int_{\Sigma}\omega\leq\frac{\text{Re}(F(p))}{\sin\epsilon}A(N+1).

More intrinsically $F(p)$ defines the complex valued volume form $\tilde{\Omega}_{L_{0}}$ on the moduli space, hence

|\text{Im}\int_{\Sigma}e^{-i\hat{\theta}}F\omega|\leq\frac{1}{\sin\epsilon}A(N+1)\text{Re}(\tilde{\Omega}_{L_{0}}).

(46)

The other two terms in (39) are rewritten as a sum of contributions from intersection points in (38). As in Prop. 3.41, we consider holomorphic polygons whose boundary $\partial\Sigma$ encounters in the clockwise order $p_{N-1}\in CF^{1}(L_{N},L_{N-1}),\ldots$ , $p_{1}\in CF^{1}(L_{2},L_{1})$ , $p=p_{0}\in CF^{0}(L_{1},L_{0})$ , $q=p_{N}\in CF^{0}(L_{0},L_{N})$ juxaposed possibly by more degree one self intersections $b_{i}$ of $L_{i}$ . (The other cases, where $\partial\Sigma$ misses some Lagrangians, can be handled completely similarly.) We first deal with these extra self intersections. Using the wedge region bound, and the Novikov positivity requirement,

|\text{Im}\sum_{i}\sum_{b_{i}}e^{-i\hat{\theta}}Ff_{L_{i}}|^{+}_{-}(b_{i})|\leq\sum_{i}\sum_{b_{i}}|F(b_{i})|f_{L_{i}}|^{+}_{-}(b_{i})\leq\frac{\text{Re}(F(p))}{\sin\epsilon}\sum_{i}\sum_{b_{i}}f_{L_{i}}|^{+}_{-}(b_{i}).

By Lemma 3.41, we have

|\text{Im}\sum_{i}\sum_{b_{i}}e^{-i\hat{\theta}}Ff_{L_{i}}|^{+}_{-}(b_{i})|\leq\frac{1}{\sin\epsilon}A(N+1)\text{Re}(\tilde{\Omega}_{L_{0}}).

(47)

We are left with the contributions of $p_{0},p_{1},\ldots p_{N}$ to (38):

\text{Im}\sum_{0}^{N-1}e^{-i\hat{\theta}}F(f_{L_{j+1}}-f_{L_{j}})(p_{j}).

If we replace $f_{L_{i}}$ by its supremum value $\sup_{L_{i}}f_{L_{i}}$ for all $i=0,1,\ldots N$ , the new expression would be

\begin{split}&\text{Im}\sum_{0}^{N-1}e^{-i\hat{\theta}}F(p_{j})(\sup f_{L_{j+1}}-\sup f_{L_{j}})\\ =&\text{Im}\sum_{1}^{N}e^{-i\hat{\theta}}\sup f_{L_{j}}(F(p_{j-1})-F(p_{j}))-\text{Im}(e^{-i\hat{\theta}}F(p)\sup f_{L_{0}})\end{split}

which is more intrinsically the integrand (45) of the elementary functional. Using the potential clustering assumption and the wedge region bound lemma, the error of replacing the potentials by $\sup_{L_{j}}f_{L_{j}}$ can be bounded by

2A\sum_{1}^{N}|F(p_{j})|\leq 2AN\frac{\text{Re}(F(p))}{\sin\epsilon}=\frac{2AN}{\sin\epsilon}\text{Re}(\tilde{\Omega}_{L_{0}}).

(48)

Now (46)(47)(48) are upper bounds on the three contributions to the difference between the Solomon functional integrand (39) and the elementary functional integrand. Their sum is bounded by

\frac{A(4N+2)}{\sin\epsilon}\text{Re}(\tilde{\Omega}_{L_{0}}),

so after integration on the moduli space,

|\mathcal{S}(L)-\bar{\mathcal{S}}(L)|\leq\frac{A(4N+2)}{\sin\epsilon}\int_{L_{0}}\text{Re }\Omega

as required. ∎

Remark 3.21.

In section 5.2 below we will deduce the potential clustering and an upper bound on $N$ as consequences of almost quantitative calibratedness, and very mild conditions on the ambient manifold $X$ . In section 5.5 the boundedness of $|\mathcal{S}-\bar{\mathcal{S}}|$ will be essential for relating the asymptote of the Solomon functional to stability conditions.

Remark 3.22.

In Kähler geometry, it is often useful to decompose natural functionals into two parts. For instance, the K-energy functional can be decomposed into an entropy part and a pluripotential part [17, section 2.4], which is important in the study of constant scalar curvature Kähler metrics.

Remark 3.23.

We suggested in section 2.10 that the Solomon functional is essentially the logarithm of the tunneling amplitude between Lagrangian branes. Pushing forth with this physics analogy, we may regard the elementary functional as a semiclassical approximation,⁴⁴⁴⁴ 44 The elementary functional is proportional to the period integrals over the cycles $L_{i}$ , which may be regarded as coming from integration over the moduli of constant maps. Such integrals are regarded as more classical then those involving nontrivial holomorphic curves. and $\mathcal{S}-\bar{\mathcal{S}}$ as quantum fluctuation effects. Our main assertion then becomes that quantitative almost calibratedness with some extra hypotheses imply the a priori bound on the quantum fluctuation effects. The author is not aware of previous suggestions in the physics literature, but Jake Solomon’s formal Riemannian picture in section 2.8 may offer partial explanations for the relevance of the almost calibrated condition.

What if we relax the positivity condition?

Suppose we drop the positivity condition on the bordism current, but keep all the other assumptions. Then the key difference is that for holomorphic curves contributing negatively to $\partial\mathcal{C}$ , we need to replace Claim 3.22 by Claim 3.29, and Claim 3.23 by Claim 3.30. Correspondingly, all appearance of $\text{Re }F(p)$ is replaced by its absolute value. Then the conclusion in Theorem 3.43 is replaced by

|\mathcal{S}(L)-\bar{S}(L)|\leq\frac{A(4N+2)}{\sin\epsilon}\int_{\mathcal{M}}|\text{Re }\tilde{\Omega}_{L_{0}}|.

(49)

The problem is that the RHS is no longer a manefestedly a priori bounded quantity.