1 Introduction
Let
$\overline {{\mathcal M}}_g$
denote the moduli space of stable curves of genus g, which is a compactification of the moduli space of smooth curves constructed by Deligne and Mumford in [Reference Deligne and Mumford4]. We denote by
$\Delta ^{ns}\subset \overline {{\mathcal M}}_g$
the non-separating node boundary divisor, i.e., the closure of the locus corresponding to irreducible curves with one node.
One records the combinatorial type of a connected nodal curve C by the corresponding dual marked graph
$\Gamma $
: vertices of
$\Gamma $
correspond to irreducible components of C, marked by the genus of the corresponding component, and edges correspond to nodes. Note that multiple edges and loops are both allowed. Stability of C corresponds to the following notion of stability for
$\Gamma $
(see [Reference Arbarello, Cornalba and Griffiths1, Chapter X, Lemma (3.1)]): for every vertex
$v,$
one should have
$2g_v-2+l_v>0$
, where
$g_v$
is the marking of the vertex and
$l_v$
is the number of half-edges emanating from v. The arithmetic genus of C is equal to the genus of
$\Gamma $
computed as
$b(\Gamma )+\sum _v g_v$
, where
$b(\Gamma )$
is the first Betti number of
$\Gamma $
. We denote by
${\mathcal M}_{\Gamma }\subset \overline {{\mathcal M}}_g$
the stratum consisting of curves with a given dual stable graph
$\Gamma $
.
Let
$\varphi $
be a top holomorphic form on
$\overline {{\mathcal M}}_g$
defined in a neighborhood in
$\overline {{\mathcal M}}_g$
of a point
$C_0\in {\mathcal M}_{\Gamma }$
, with a pole of order
$1$
along
$\Delta ^{ns}$
. We are interested in the region of convergence in
$s\in {\mathbb C}$
of the integral
$$ \begin{align*}I_{\Gamma}(\varphi,s)=\int_{B_{\Gamma}}\frac{\varphi\wedge\overline{\varphi}}{|\det(\tau-\overline{\tau})|^s},\end{align*} $$
where
$B_{\Gamma }$
is a sufficiently small ball in an étale neighborhood of
$C_0$
in
$\overline {{\mathcal M}}_g$
, and
$\tau $
is the period matrix.
Recall that the period matrix depends on a local choice of a symplectic basis of
$H_1(C)$
(see Equation (2.1)). This choice changes the function
$I_{\Gamma }(\varphi ,s)$
, but Lemma 2.2 implies that it does not change its region of convergence. Globally,
$\det (\tau -\overline {\tau })$
is related to the natural Hermitian form on the determinant line bundle
$\det H^0(C,\omega _C)$
. The latter line bundle shows up in Mumford’s calculation of the canonical line bundle on
${\mathcal M}_g$
and its superanalog (see [Reference Felder, Kazhdan and Polishchuk5]). For this reason, the integrals of the form
$I_{\Gamma }(\varphi ,s)$
(for
$s=5$
) model local behavior of integrals calculating vacuum amplitudes in superstring theory after integrating out the odd variables and using the GSO projection (see [Reference Witten7]).
The point is that
$\det (\tau -\overline {\tau })$
has a logarithmic growth near
$\Delta ^{ns}$
that offsets the poles of
$\varphi \wedge \overline {\varphi }$
for sufficiently large s. The precise region of convergence of
$I_{\Gamma }(\varphi ,s)$
depends on a graph
$\Gamma $
.
For example, if
$\Gamma $
has a single vertex of genus
$g-1$
with a loop, i.e., we are at the generic point of
$\Delta ^{ns}$
, then
$\det (\tau -\overline {\tau })=u\cdot \log |t|$
, where u is invertible and
$t=0$
is a local equation of
$\Delta ^{ns}$
. So the convergence of our integral for such
$\Gamma $
is the same as for
$\frac {dt\wedge d\overline {t}}{t\cdot \overline {t}\cdot (\log |t|)^s}$
. Thus, in this case, the integral absolutely converges for
$\text {Re}(s)>1$
and diverges for
$s=1$
.
Our main result, Theorem A below, determines the convergence threshold for each
$\Gamma $
with vertices of genus 0. In particular, it shows that for each genus
$\ge 6$
, there exists a stable graph
$\Gamma $
such that
$I_{\Gamma }(\varphi ,5)$
diverges. This means that the definition of superstring vacuum amplitudes for
$g\ge 6$
requires some additional regularization procedure at the non-separating node boundary divisor in addition to the GSO projection.
Using the asymptotics of
$\tau $
near
$C_0$
(controlled by the monodromy around branches of
$\Delta ^{ns}$
), in the case when
$C_0$
has rational components, we relate the integral
$I_{\Gamma }$
to another integral defined in terms of the graph
$\Gamma $
.
Let
$E(\Gamma )$
denote the set of edges of a connected graph
$\Gamma $
. We introduce independent variables
$x_e$
associated with
$e\in E(\Gamma )$
. Let
$b=b(\Gamma )$
denote the 1st Betti number of
$\Gamma $
(which is equal to g, the arithmetic genus of
$C_0$
), and let
$c_1,\ldots ,c_b$
be a collection of simple cycles in
$\Gamma $
giving a basis in
$H_1(\Gamma )$
. For each edge
$e\in E(\Gamma )$
, we define the index
$$ \begin{align*}(c_i,c_j)_e=\begin{cases} 1, &c_i \text{ and } c_j \text{ pass through } e \text{ in the same direction},\\ -1, &c_i \text{ and } c_j \text{ pass through } e \text{ in the opposite directions},\\ 0, &\text{ otherwise},\end{cases}\end{align*} $$
and consider the
$b\times b$
-matrix
$A=(a_{ij})$
with
$$ \begin{align} a_{ij}=\sum_{e\in E(\Gamma)}(c_i,c_j)_e\cdot x_e. \end{align} $$
Let
$\psi _{\Gamma} \in {\mathbb Z}[x_e]_{e\in E(\Gamma )}$
denote the Kirchhoff polynomial (aka the first Symanzik polynomial) of
$\Gamma $
(see [Reference Bloch3, Section 2]), defined as the determinant
$$ \begin{align*}\psi_{\Gamma}=\det(a_{ij}).\end{align*} $$
This polynomial also has an expansion
$$ \begin{align} \psi_{\Gamma}=\sum_T \prod_{e\not\in T} x_e, \end{align} $$
where T runs over all spanning trees of
$\Gamma $
(see [Reference Bloch3, Proposition 3.4] and Prop. 2.1 below). Note that
$\psi _{\Gamma} $
is homogeneous of degree
$b(\Gamma )$
.
Let
$E'(\Gamma )\subset E(\Gamma )$
denote the set of edges which are not bridges. We consider the integral
$$ \begin{align*}J_{\Gamma}(s):=\int_B \frac{\prod_{e\in E(\Gamma)} dz_ed\overline{z}_e}{|\psi_{\Gamma}(\ln |z_\bullet|)|^s\cdot \prod_{e\in E'(\Gamma)} z_e\overline{z}_e},\end{align*} $$
where B is a small ball around
$0$
in
${\mathbb C}^{E(\Gamma )}$
, and
$(z_e)$
are complex coordinates on
${\mathbb C}^{E(\Gamma )}$
.
Given a connected graph
$\Gamma $
without bridges (not necessarily stable), we define a rational constant
$c(\Gamma )>0$
as follows. Consider the vector space
${\mathbb R}^{E(\Gamma )}$
with the basis corresponding to edges of
$\Gamma $
, and let
$v\in {\mathbb R}^{E(\Gamma )}$
denote the sum of all basis vectors. For each spanning tree T, consider the vector
$$ \begin{align*}v_T:=\sum_{e\not\in T} e\in {\mathbb R}^{E(\Gamma)}.\end{align*} $$
Now, we set
$$ \begin{align*}c(\Gamma)=\inf \left\{\sum c_T \ \big{|} \ \sum_T c_Tv_T\geq v, c_T\in {\mathbb R}_{\ge 0}\right\},\end{align*} $$
where
$w \geq v$
means that
$w-v\in {\mathbb R}_{\ge 0}^{E(\Gamma )}$
.
It is easy to see that
$$ \begin{align} c(\Gamma)\ge \frac{e(\Gamma)}{b(\Gamma)}, \end{align} $$
where
$e(\Gamma )=|E(\Gamma )|$
is the number of edges and
$b(\Gamma )$
is the 1st Betti number of
$\Gamma $
. Indeed, let
$\varphi :{\mathbb R}^{E(\Gamma )}\to {\mathbb R}$
denote the map given by the sum of all coordinates. Then,
$\varphi (v)=e(\Gamma )$
and
$\varphi (v_T)=b(\Gamma )$
, so the inequality
$\sum _T c_Tv_T\geq v$
implies that
$b(\Gamma )\cdot (\sum _T c_T)\ge e(\Gamma )$
.
For an arbitrary connected
$\Gamma $
(possibly with bridges), we set
$c(\Gamma ):=c(\Gamma ')$
, where
$\Gamma '$
is obtained from
$\Gamma $
by contracting all the bridges.
Theorem A The integral
$J_{\Gamma }(s)$
converges for
$\text {Re}(s)>c(\Gamma )$
(for a sufficiently small ball around
$0$
) and diverges for
$s=c(\Gamma )$
(and any ball around
$0$
). If
$\Gamma $
is stable and all components of
$C_0$
are rational, the same assertions hold for the integral
$I_{\Gamma }(\varphi ,s)$
.
A simple example is the n-gon graph
$\Gamma =C_n$
. It is easy to see that one has
${c(C_n)=n}$
. Indeed, the vectors
$v_T$
are just the basis vectors
$e_i$
and the condition
$\sum c_ie_i\ge v$
means that
$c_i\ge 1$
, so the minimal
$\sum c_i$
is n. There are similar stable graphs with all vertices of genus
$0$
. For example, if
$\Gamma $
is a
$2n$
-gon with every other side doubled, then
$c(\Gamma )=n$
. Note that the genus of this graph is
$n+1$
. This shows that the boundary
$\text {Re}(s)=c$
of the convergence halfplane for
$I_{\Gamma }(\varphi ,s)$
can have arbitrary large c as the genus grows.
The proof of Theorem A uses some combinatorics of the cographical matroid associated with
$\Gamma $
to reduce to the case of graphs
$\Gamma $
for which inequality (1.3) becomes an equality (recall that the bases of the cographical matroid are complements to spanning trees, see [Reference Oxley6, Section 2.3]). The key combinatorial result needed for Theorem A is that starting from any graph
$\Gamma $
, one can contract some edges in
$\Gamma $
so that the resulting graph
$\overline {\Gamma }$
satisfies
$c(\overline {\Gamma })=e(\overline {\Gamma })/b(\overline {\Gamma })$
(see Corollary 3.8).
Note that the integral
$J_{\Gamma }(s)$
is similar to the Euler–Mellin integrals considered in [Reference Berkesch, Forsgård and Passare2] (but with a different integration domain) and our convergence result is similar to [Reference Berkesch, Forsgård and Passare2, Theorem 2.2] (see Remark 3.4).
Convention. By a graph, we mean a connected undirected multigraph, possibly with multiple edges and loops.
2 Kirchhoff polynomial and the asymptotics of the period matrix
2.1 Kirchhoff polynomial
We are going to give a proof of the determinant identity relating two definitions of
$\psi _{\Gamma} $
, using Cauchy–Binet theorem, and also get a bit more information about the corresponding matrix. There is also a simple recursive proof of this determinant formula in [Reference Bloch3].
Let us consider a more general setup where we are given a surjective morphism of free
${\mathbb Z}$
-modules of finite rank
$$ \begin{align*}{\alpha}:{\mathbb Z}^E\to H,\end{align*} $$
with the property that, for every subset
$S\subset E$
,
$\operatorname {coker}({\alpha }_S)$
is a free
${\mathbb Z}$
-module, where
${\alpha }_S:{\mathbb Z}^S\to H$
is the restriction of A.Footnote
1
We will apply it in the case when
$E=E(\Gamma )$
,
$H=H^1(\Gamma )$
, and
${\alpha }$
is the natural projection. Note that in this case
$\operatorname {coker}({\alpha }_S)$
is canonically identified with
$H^1(\Gamma ^S)$
, where
$\Gamma ^S$
is obtained from
$\Gamma $
by deleting edges in S.
With
${\alpha }$
as above we associate a symmetric bilinear form on
$H^*$
with entries which are linear forms in independent variables
$(x_e)_{e\in E}$
:
$$ \begin{align*}B(\xi_1,\xi_2)=\sum_{e\in E} \xi_1({\alpha}(e))\cdot \xi_2({\alpha}(e))\cdot x_e.\end{align*} $$
Thus, we have a well-defined discriminant
$\det (B)\in R:={\mathbb Z}[x_e \ |\ e\in E]$
.
Let us say that
$S\subset E$
is a basis if
${\alpha }_S:{\mathbb Z}^S\to H$
is an isomorphism. We denote by
${\mathcal B}$
the set of all bases.
Proposition 2.1
(i) One has
$\det (B)=\sum _{S\in {\mathcal B}} \prod _{e\in S} x_e$
.
(ii) Let us view B as a nondegenerate form over the field
$QR$
, the fraction field over R, and let
$B^{-1}$
denote the corresponding bilinear form on the dual space
$H^\vee $
with values in
$QR$
. Then, setting
$x_e=\ln (y_e)$
, where
$y_e>0$
, we have
$$ \begin{align*}\lim_{y\to 0} B^{-1}(\ln(y_e))=0.\end{align*} $$
Proof (i) Let
$f_1,\ldots ,f_h$
be a
${\mathbb Z}$
-basis of H, so we can view
${\alpha }$
as a matrix
$E\times h$
-matrix. Then, the matrix of B is given by
$$ \begin{align*}B={\alpha}\cdot D(x)\cdot {\alpha}^t,\end{align*} $$
where
$D(x)$
is the diagonal
$E\times E$
-matrix with the entries
$(x_e)$
. We can calculate
$\det (B)$
by applying the Cauchy–Binet’s theorem to the decomposition of B in
${\alpha }$
and
$D(x){\alpha }^t$
. In this theorem, we need to sum over subsets
$S\subset E$
, where
$|S|=h$
, such that the corresponding minor of
${\alpha }$
is nonzero. This condition is equivalent to the condition that
${\alpha }_S:{\mathbb Z}^S\to H$
has nonzero determinant. Since
$\operatorname {coker}({\alpha }_S)$
is free by assumption, this is equivalent to S being a basis, in which case the corresponding minor is
$\pm 1$
. Since the corresponding minor of
${\alpha }^t$
is the same sign times
$\prod _{e\in S} x_e$
, the assertion follows.
(ii) In the dual basis
$(f_i^*),$
the matrix of
$B^{-1}$
is the inverse matrix of B. Hence, it is enough to prove that every
$(h-1)\times (h-1)$
-minor M of B satisfies
$$ \begin{align*}\lim_{y\to 0} \frac{M(\ln(y))}{\det B(\ln(y))}=0.\end{align*} $$
Due to the formula for the determinant, it is enough to prove that every monomial appearing in M is a constant multiple of
$\prod _{e\in S} x_e$
, for some
$S\subset S'$
with
$|S|=h-1$
and
$S'$
a basis. Indeed, without loss of generality, we can assume that M corresponds to the rows
$1,\ldots ,h-1$
. Then, by the Cauchy–Binet’s formula, the monomials appearing in M would correspond to some subsets
$S\subset E$
with
$|S|=h-1$
such that the map
${{\mathbb Z}^S\to H/{\mathbb Z}\cdot f_h}$
induced by
${\alpha }_S$
is nondegenerate. But this implies that
${\alpha }_S$
is injective, and so the image of
${\alpha }_S$
is of rank
$h-1$
. Since
${\alpha }$
is surjective, there exists an element
$s\in E\setminus S$
such that
${\alpha }(s)$
is not contained in the image of
${\alpha }_S$
, hence for
$S'=S\cup \{s\}$
, the map
${\alpha }_{S'}$
is nongenerate, i.e.,
$S'$
is a basis.
We are interested in the case of the natural projection
${\alpha }:{\mathbb Z}^{E(\Gamma )}\to H^1(\Gamma )$
for a graph
$\Gamma $
. In this case,
${\mathcal B}$
is the set of bases of the cographical matroid associated with
$\Gamma $
. Choosing a basis of simple cycles
$(c_i)$
, we can view the rows of
${\alpha }$
as coefficients of the edges in
$c_i$
(with respect to a fixed orientation of all edges). Then, the matrix of the symmetric form B will be exactly the matrix (1.1). Hence,
$\det (B)$
gets identified with the polynomial
$\psi _{\Gamma }$
and we derive the expansion (1.2). Note that, in this case, bases are exactly complements to spanning trees. In particular, if an edge e is a bridge, then it is not contained in any bases, so
$\psi _{\Gamma} $
depends only on variables corresponding to non-separating edges (i.e., edges that are not bridges).
2.2 Asymptotics of the period matrix
Assume that
$\Gamma $
is a stable graph of genus g, with all vertices of genus
$0$
, and let
$C_0$
be a stable curve with rational components with the dual graph
$\Gamma $
. It is well known that the arithmetic genus of
$C_0$
is g, so we can view
$C_0$
as a point of
$\overline {{\mathcal M}}_g$
. Then, the set of edges
$E(\Gamma )$
is in bijection with the branches of the boundary divisor through
$C_0$
, and the subset
$E^{ns}(\Gamma )\subset E(\Gamma )$
of non-separating edges is in bijection with the branches of
$\Delta ^{ns}$
through
$C_0$
.
Note that if the graph
$\Gamma $
is trivalent, then the corresponding stratum
${\mathcal M}_{\Gamma }$
is a point and
$e(\Gamma )=3g-3$
. Otherwise, the stratum
${\mathcal M}_{\Gamma }$
has positive dimension.
For every non-separating edge e, there is a vanishing cycle
${\alpha }_e\in H_1(C)$
, where C is a smooth curve close to
$C_0$
, and the corresponding monodromy transformation
$M_e$
on
$H_1(C)$
has form
$$ \begin{align*}M_e(x)=(x\cdot {\alpha}_e){\alpha}_e+x.\end{align*} $$
More precisely, C is obtained from the collection of spheres numbered by the set
$V(\Gamma )$
of vertices of
$\Gamma $
by connecting them with tubes numbered by the set of edges
$E(\Gamma )$
. We fix an orientation of
$\Gamma $
and let
${\beta }_e$
denote a path along the tube corresponding to e going in the direction of e. Then, we define
${\alpha }_e$
as the class of a circle around the tube corresponding to e, so that
$({\beta }_e\cdot {\alpha }_e)=1$
.
The subgroup
$A\subset H_1(C)$
generated by
$({\alpha }_e)$
is maximal isotropic, and we have natural identifications
$$ \begin{align*}H_1(C)/A\simeq H_1(C_0)\simeq H_1(\Gamma)\end{align*} $$
(see [Reference Bloch3, Section 6]).
Let
$({\alpha }_1,\ldots ,{\alpha }_g,{\beta }_1,\ldots ,{\beta }_g)$
be a symplectic basis of
$H_1(C)$
(so
$({\beta }_j\cdot {\alpha }_i)=\delta _{ij}$
,
$({\alpha }_i\cdot {\alpha }_j)=({\beta }_i\cdot {\beta }_j)=0$
) such that
$A={\mathbb Z}{\alpha }_1+\cdots +{\mathbb Z}{\alpha }_g$
. In particular,
$M_e$
does not change
${\alpha }_i$
’s. Let
$\omega _1,\ldots ,\omega _g$
be the basis of
$H^0(C,\omega _C)$
, normalized by
$\int _{{\alpha }_i}\omega _j=\delta _{ij}$
. The period matrix
$\tau =(\tau _{ij})$
is given by
$$ \begin{align} \tau_{ij}=\int_{{\beta}_i}\omega_j. \end{align} $$
The monodromy
$M_e$
preserves
$(\omega _j)$
and acts on the periods by the transformation
$$ \begin{align*}M_e:\tau_{ij}\mapsto \int_{M_e({\beta}_i)}\omega_j=\tau_{ij}+({\beta}_i\cdot {\alpha}_e)\cdot ({\alpha}_e)_j=\tau_{ij}+({\beta}_i\cdot {\alpha}_e)\cdot ({\beta}_j\cdot {\alpha}_e),\end{align*} $$
where the integers
$({\alpha }_e)_j$
are determined from
${\alpha }_e=\sum _j ({\alpha }_e)_j\cdot {\alpha }_j$
.
Let
$z_e$
denote a local equation of the branch of the boundary divisor near
$C_0$
corresponding to the edge e. Then, the monodromy
$M_e$
acts on
$\ln (z_e)/(2\pi i)$
as
$\ln (z_e)/(2\pi i)\mapsto \ln (z_e)/(2\pi i)+1$
. Hence,
$$ \begin{align*}\tau^{\prime}_{ij}:=\tau_{ij}-\sum_{e\in E}\frac{\ln(z_e)}{2\pi i}\cdot ({\beta}_i\cdot {\alpha}_e)\cdot ({\beta}_j\cdot {\alpha}_e)\end{align*} $$
are invariant under all monodromy transformations
$M_e$
. In other words, we have
$$ \begin{align} \tau=\tau_0\left(\frac{\ln(z)}{2\pi i}\right)+\tau', \end{align} $$
where
$\tau =(\tau _{ij})$
,
$\tau '=(\tau ^{\prime }_{ij})$
, and
$\tau _0(x)$
is the matrix with coefficients in
${\mathbb Z}[x_e \ |\ e\in E]$
with the entries
$$ \begin{align*}\tau_0(x)_{ij}=\sum_{e\in E} x_e\cdot ({\beta}_i\cdot {\alpha}_e)\cdot ({\beta}_j\cdot {\alpha}_e).\end{align*} $$
In fact, it follows from the nilpotent orbit theorem (see [Reference Bloch3, Section 9]) that in Equation (2.2) the term
$\tau '$
is regular near
$C_0$
. We will use this to compute the asymptotics for
$\det (\tau -\overline {\tau })$
near
$C_0$
.
Lemma 2.2 Near
$C_0$
, one has
$$ \begin{align*}\det(\tau-\overline{\tau})=\psi_{\Gamma}\left(\frac{\ln(|z|)}{\pi i}\right)\cdot (1+f),\end{align*} $$
where
$z=(z_e)$
and
$f\to 0$
as
$C\to C_0$
.
Proof Let us choose a symplectic basis of
$H_1(C)$
as follows. First, let
$(c_i)_{i=1,\ldots ,g}$
be a basis of simple cycles in
$H_1(\Gamma )$
, and let
$({\alpha }_i)_{i=1,\ldots ,g}$
be the dual basis of A with respect to the pairing between A and
$H_1(C)/A\simeq H_1(\Gamma )$
induced by the intersection pairing. Let
$({\beta }_i)_{i=1,\ldots ,g}$
be a set of mutually orthogonal classes in
$H_1(C)$
projecting to
$(c_i)$
under the projection
$H_1(C)\to H_1(\Gamma )$
(such a set exists since the intersection pairing is perfect). Then,
$({\alpha }_i,{\beta }_i)$
is a standard symplectic basis. Furthermore, due to our definition of
${\alpha }_e$
, the intersection index
$({\beta }_i\cdot {\alpha }_e)$
is
$1$
(resp.
$-1$
) exactly when
$c_i$
passes through e in the direction of the orientation (resp. in the opposite direction).
It follows that
$\tau _0(x)$
coincides with the matrix
$A=(a_{ij})$
given by (1.1). Since
$\tau _0(x)$
depends on
$(x_e)$
linearly with integer coefficients, (2.2) gives
$$ \begin{align*}\tau-\overline{\tau}=A\left(\frac{\ln |z|}{\pi i}\right)+\tau'-\overline{\tau'}=A\left(\frac{\ln |z|}{\pi i}\right)\cdot \left(1+A\left(\frac{\ln |z|}{\pi i}\right)^{-1}\cdot (\tau'-\overline{\tau'})\right),\end{align*} $$
where
$\tau '-\overline {\tau '}$
is regular near
$C_0$
. Now, by Proposition 2.1(ii), we have
$$ \begin{align*}\lim_{C\to C_0}A\left(\frac{\ln |z|}{2\pi i}\right)^{-1}\cdot (\tau'-\overline{\tau'})=0,\end{align*} $$
and the assertion follows.
3 Convergence region
3.1 Elementary observations
Lemma 3.1 Let
$P(x_1,\ldots ,x_n)$
be a (nonzero) polynomial with nonnegative coefficients. Then, for
$s\in {\mathbb C}$
, the integral
$$ \begin{align*}\int_B \frac{\prod_{i=1}^n dz_id\overline{z}_i}{|P(-\ln |z_1|,\ldots,-\ln |z_n|)|^s\cdot \prod_{i=1}^n z_i\overline{z}_i}\end{align*} $$
over a sufficiently small ball B around the origin in
${\mathbb C}^n$
converges if and only if the integral
$$ \begin{align*}\int_{[C,+\infty)^n} \frac{dx_1\ldots dx_n}{P(x_1,\ldots,x_n)^s}\end{align*} $$
converges for sufficiently large C.
Proof This follows from the change of variables
$z_i=e^{-x_i+ i\phi _i}$
, with
$x_i>C$
and
${\phi _i\in (0,2\pi )}$
.
Lemma 3.2 Let
$C>0$
. The integral
$$ \begin{align*}\int_{[C,+\infty)^n}\frac{dx_1\ldots dx_n}{(x_1+\cdots+x_n)^s}\end{align*} $$
converges for
$\text {Re}(s)>n$
and diverges for
$s=n$
.
Proof Convergence for
$\text {Re}(s)>n$
follows from the inequality
$$ \begin{align*}(x_1+\cdots+x_n)^n\ge x_1\ldots x_n.\end{align*} $$
To prove the divergence for
$s=n$
, we will use induction on n. The base case
$n=1$
is well known. Now, let
$n>1$
. It is enough to prove the divergence of the integral
$$ \begin{align*}\int_{x_1>C_1}\ldots \int_{x_n>C_n} \frac{dx_1\ldots dx_n}{(x_1+\cdots+x_n)^n},\end{align*} $$
for any large
$C_1,\ldots ,C_n$
. Performing the integration in
$x_n$
, we get
$$ \begin{align*}\frac{1}{n-1}\int_{x_1>C_1}\ldots \int_{x_{n-1}>C_{n-1}}\frac{dx_1\ldots dx_{n-1}}{(x_1+\cdots+x_{n-1}+C_n)^{n-1}}.\end{align*} $$
The proof now follows from the change of variables
$x_1\mapsto x_1+C_n$
and the inductive hypothesis.
For a vector
$w=(w_1,\ldots ,w_n)\in {\mathbb Z}_{\ge 0}^n$
, we denote
$x^w:=x_1^{w_1}\ldots x_n^{w_n}$
.
Lemma 3.3
(i) Let
$C>0$
. Suppose for a collection of vectors
$v_1,\ldots ,v_N\in {\mathbb Z}_{\ge 0}^n$
and scalars
$c_1,\ldots ,c_N\in {\mathbb R}_{\ge 0}$
, one has
$$ \begin{align*}\sum c_iv_i \ge v:=(1,\ldots,1)\in {\mathbb R}^n.\end{align*} $$
Then, the integral
$$ \begin{align} \int_{[C,+\infty)^n}\frac{dx_1\ldots dx_n}{(x^{v_1}+\cdots+x^{v_N})^s} \end{align} $$
converges for
$\mathrm{Re}(s)>c_1+\cdots +c_N$
. In particular, for
$\Gamma $
without bridges, the integral
$J_{\Gamma }(s)$
converges for
$s>c(\Gamma )$
.
(ii) The integral
$J_{\Gamma }(s)$
diverges for
$s=e(\Gamma )/b(\Gamma )$
.
Proof (i) The first assertion follows from the inequality
$$ \begin{align*}(x^{v_1}+\cdots+x^{v_N})^{c_1+\cdots+c_N}\ge (x^{v_1})^{c_1}\ldots (x^{v_N})^{c_N}\ge x^v=x_1\ldots x_n\end{align*} $$
for
$x_i\ge 1$
. The assertion about
$J_{\Gamma }$
follows from this using Lemma 3.1.
(ii) Set
$n=e(\Gamma )$
and
$b=b(\Gamma )$
. Since
$\psi _{\Gamma }$
has degree b, we have
$$ \begin{align*}\psi_{\Gamma}(x_1,\ldots,x_n)\le (x_1+\cdots+x_n)^b.\end{align*} $$
Now, the divergence for
$s=n/b$
follows immediately from Lemmas 3.1 and 3.2.
Remark 3.4 The convergence statement of Lemma 3.3(i) is similar to the convergence statement [Reference Berkesch, Forsgård and Passare2, Theorem 2.2] about more general Euler–Mellin integrals. Note however that our domain of integration is
$[C,+\infty )^n$
, where
$C>0$
, so our condition on s is different from the one in [Reference Berkesch, Forsgård and Passare2, Theorem 2.2], where the integration is over
$(0,+\infty )^n$
. The converse of Lemma 3.3(i) is false, i.e., the integral (3.1) can converge even when there are no
$c_i\in {\mathbb R}_{\ge 0}$
with
$\sum c_i v_i\ge v$
and
$\text {Re}(s)>\sum c_i$
. For example, consider the integral
$$ \begin{align*}I(s)=\int_{[C,+\infty)}\frac{dx_1dx_2}{(x_1x_2^2+x_1^4x_2^3)^s}.\end{align*} $$
Then, conditions of Lemma 3.3(i) hold only for
$\text {Re}(s)>2/5$
, however, we claim that
$I(s)$
converges for
$\text {Re}(s)>1/3$
. Indeed, changing the variables by
$x_i=y_i+C$
, we get
$$ \begin{align*}I(s)=\int_{[0,+\infty)}\frac{dy_1dy_2}{P(y_1,y_2)^s},\end{align*} $$
where
$P(y_1,y_2)=(C+y_1)(C+y_2)^2+(C+y_1)^4(C+y_2)^3$
. Since the Newton polytope of P is the rectangle with the vertices
$(0,0)$
,
$(4,0)$
,
$(0,3),$
and
$(4,3)$
, convergence for
$\text {Re}(s)>1/3$
follows from [Reference Berkesch, Forsgård and Passare2, Theorem 2.2].
Recall that we have an inequality
$c(\Gamma )\ge e(\Gamma )/b(\Gamma )$
(see (1.3)).
Definition 3.1 We say that
$\Gamma $
optimal if
$c(\Gamma )=e(\Gamma )/b(\Gamma )$
.
Lemma 3.3 proves our assertion about convergence/divergence of
$J_{\Gamma }(s)$
in the case of optimal
$\Gamma $
. Below, we will reduce the case of a general
$\Gamma $
to that of optimal
$\Gamma $
.
3.2 Combinatorial statement
Let M be a loopless matroid on the ground set E. For all
$S\subset E$
, consider the linear functional
$$ \begin{align*}\varphi_S:{\mathbb R}^E\to{\mathbb R}\end{align*} $$
taking an element to the sum of its coordinates in S. The base polytope
$P(M)$
consists of the vectors
$w\in {\mathbb R}^E$
such that
$0\leq \varphi _S(w) \leq \operatorname {rk} S$
for all
$S\subset E$
and
$\varphi _E(w) = \operatorname {rk} E$
.
Let
$$ \begin{align*}m=m(M):= \max\{|S| / \operatorname{rk} S \mid S \neq \emptyset\},\end{align*} $$
and let
$\mathcal {T}_0$
be the (nonempty) collection of subsets of E that attain this maximum.
Lemma 3.5 If
$S,T\in \mathcal {T}_0$
, then
$S\cup T\in \mathcal {T}_0$
. Hence,
$\mathcal {T}_0$
has a maximal element.
Proof For any set
$U\subset E$
of nonzero rank, we have
$|U| \leq m \operatorname {rk} U$
. Applying this inequality to
$U = S \cap T$
, we find that
$$ \begin{align*}|S\cup T| &= |S| + |T| - |S\cap T|\\&\geq m \operatorname{rk} S + m\operatorname{rk} T - m \operatorname{rk} S\cap T\\&= m (\operatorname{rk} S + \operatorname{rk} T - \operatorname{rk} S\cap T)\\&\geq m \operatorname{rk} S\cup T. \end{align*} $$
Applying it next to
$U = S\cup T$
, we find that
$|S\cup T| = m \operatorname {rk} S\cup T$
, thus
$S\cup T \in \mathcal {T}_0$
.
Let
$$ \begin{align*}c=c(M) := \min\{t \mid \text{there exists } w\in t P(M) \text{ with } w_e \geq 1 \text{ for all } e\in E\},\end{align*} $$
and let w be an element of
$c P(M)$
with
$w_e \geq 1$
for all
$e\in E$
(a witness for c).
Proposition 3.6 We have
$c=m$
.
Proof Let
$T_0$
be the maximal element of
$\mathcal {T}_0$
(which exists by Lemma 3.5). Since
$w_e\geq 1$
for all
$e\in E$
and
$w\in c P(M)$
, we have
$$ \begin{align*}|T_0| \leq \varphi_{T_0}(w) \leq c \operatorname{rk} T_0,\end{align*} $$
and therefore
$c \geq |T_0| / \operatorname {rk} T_0 = m$
.
Now, we must prove the opposite inequality. We will do it by constructing an element
$w\in m P(M)$
with
$w_e \geq 1$
for all
$e\in E$
. This construction will proceed in stages.
First, let
$w(0) = (1,\ldots ,1)$
. We then have
$\varphi _S(w(0)) = |S| \leq m \operatorname {rk} S$
for all subsets
$S\subset E$
, with equality if and only if
$S\in \mathcal {T}_0$
. In particular, it is not necessarily the case that
$\varphi _E(w(0)) = m \operatorname {rk} E$
. If
$T_0 = E$
, then
$\varphi _E(w(0)) = m \operatorname {rk} E$
, so
$w(0)\in m P(M)$
and we are done. If not, choose
$e_0\in E\setminus T_0$
. Then, we have
$e_0\notin S$
for all
$S\in \mathcal {T}_0$
. That means that there exists
$\epsilon> 0$
such that
$\varphi _S(w(0) + \epsilon x_e) \leq m \operatorname {rk} S$
for all
$S\subset E$
. Choose the largest such
$\epsilon $
, and let
$w(1) = w(0) + \epsilon x_{e_0}$
.
Now, let
$\mathcal {T}_1$
be the collection of subsets
$S\subset E$
with
$\varphi _S(w(1)) = m \operatorname {rk} S$
. By an argument identical to that of Lemma 3.5, there is a maximal element
$T_1$
of
$\mathcal {T}_1$
. If
$T_1 = E$
, then
$w(1) \in m P(M)$
and we are done. If not, choose
$e_1 \in E \setminus T_1$
, and repeat the procedure to produce a new vector
$w(2)$
.
Eventually, we will have
$T_k = E$
, and this process will terminate with
$w(k) \in m P(M)$
and
$w(k)_e \geq 1$
for all
$e\in E$
.
Proposition 3.7 Given a loopless matroid M and any
$T\in {\mathcal T}_0$
, one has
$$ \begin{align*}c(M) = c(M|_T) = |T| / \operatorname{rk} T.\end{align*} $$
In particular, this is true for the maximal element
$T_0\in \mathcal {T}_0$
.
Proof Since the rank function on
$M|_T$
is the same as that on M, it is clear that
$$ \begin{align*}m(M) = m(M|_T) = |T| / \operatorname{rk} T.\end{align*} $$
It remains to apply Proposition 3.6.
Corollary 3.8 For any connected graph
$\Gamma $
without bridges, there exists a graph
$\overline {\Gamma }$
obtained by contracting some edges in
$\Gamma $
, such that
$$ \begin{align*}c(\Gamma)=c(\overline{\Gamma})=\frac{e(\overline{\Gamma})}{b(\overline{\Gamma})}\end{align*} $$
(the second equality means that
$\overline {\Gamma }$
is optimal).
Proof Let M denote the cographical matroid associated with
$\Gamma $
. Then,
${c(M)=c(\Gamma )}$
. The property that
$\Gamma $
is optimal is the property that
$c(M) = |E| / \operatorname {rk} E$
. Corollary 3.7 says that we can find an optimal deletion of M with the same constant c. That means that we can find an optimal contraction of
$\Gamma $
with the same constant c.
Example 3.9 Let
$\Gamma $
be a
$2n$
-gon with every other side doubled, so
$e(\Gamma )=3n$
. It is easy to see that
$c(\Gamma )=n$
. We can collapse all the doubled sides to get the n-gon
$\overline {\Gamma }$
, which is optimal and has
$c(\overline {\Gamma })=n$
.
3.3 Proof of Theorem A
Lemma 2.2 shows that the case of the integral
$I_{\Gamma }(\varphi ,s)$
(for stable
$\Gamma $
and all components of
$C_0$
rational) reduces to the case of
$J_{\Gamma }(s)$
. Also, since
$\psi _{\Gamma }=\psi _{\Gamma '}$
, where
$\Gamma '$
is obtained by contracting all the bridges, it is enough to consider the case when
$\Gamma $
has no bridges.
Due to Lemma 3.3(i), it remains to prove that
$J_{\Gamma} (s)$
diverges for
$s=c(\Gamma )$
. We define
$S\subset E(\Gamma )$
to be the set of edges that get contracted to get
$\overline {\Gamma }$
. By the Fubini theorem, it is enough to prove that for any fixed values
$z_i=c_i$
with
$i\in S$
, the integral
$$ \begin{align*}\int_{B'} \frac{\prod_{e\not\in S} dz_ed\overline{z}_e}{\psi_{\Gamma}(\ln |z_\bullet|)|_{z_i=c_i, i\in S}^{c(\Gamma)}\cdot \prod_{e\not\in S} z_e\overline{z}_e},\end{align*} $$
where
$B'$
is a small ball around the origin in
${\mathbb C}^{E(\Gamma )\setminus S}$
, diverges. Thus, by Lemma 3.3(ii) applied to
$\overline {\Gamma }$
, it is enough to prove an inequality
$$ \begin{align*}\psi_{\Gamma}(x_\bullet)|_{x_e=a_e, e\in S}\le C\cdot \psi_{\overline{\Gamma}}(x_\bullet),\end{align*} $$
for
$a_e>0$
, with some constant
$C>0$
depending on
$(a_e)$
. Furthermore, it is enough that this inequality holds for
$x_e>C$
. Now, we observe that for every spanning forest
$T\subset \Gamma $
, the intersection
$\overline {T}:=T\cap \overline {\Gamma }$
is also a spanning forest. This implies that for every monomial
$x_T=\prod _{\not \in T} x_e$
of
$\psi _{\Gamma }$
, one has
$$ \begin{align*}x_T|_{x_e=a_e, e\in S}=\prod_{e\in S} a_e\cdot x_{\overline{T}}.\end{align*} $$
This easily leads to the claimed inequality.
Acknowledgements
We thank Erik Panzer for useful discussions.
