On the convergence of certain indefinite theta series

TL;DR

This paper provides an elementary proof of the convergence of indefinite theta series in signature (n,2) and establishes the necessity of certain incidence conditions for convergence.

Contribution

It offers a new elementary proof for the convergence of indefinite theta series and proves that the incidence conditions are necessary for convergence.

Findings

Elementary proof of convergence for indefinite theta series.

Incidence conditions are necessary for convergence.

Clarifies conditions for convergence in signature (n,2).

Abstract

We give an elementary proof of the convergence of indefinite theta series associated to an inner space of signature $(n,2)$ conjectured in the work of Alexandrov,Banerjee,Manschot and Pioline (2018) and show that the incidence conditions are also necessary for the convergence.

Equations

$$\theta (\tau )=\sum _{x\in L}{q}^{Q(x)},q=e^{-2i\pi \tau }\text{ and }\tau =u+vi\in \mathbb{C}\text{ with }v>0.$$

$${\theta }_{\mu }(\tau ;\{C_1,C_2\})=\sum _{x\in \mu +L}\Phi (x;\{C_1,C_2\})q^{(x,x)/2}$$

$$(C_j,C_j)\le 0,$$

$$\displaystyle(C_{j},C_{j})(C_{j+1},C_{j+1})-(C_{j},C_{j+1})^{2}\begin{cases*}>0,&if $(C_{j},C_{j})(C_{j+1},C_{j+1})>0$,\\ =0,&if $(C_{j},C_{j})(C_{j+1},C_{j+1})=0$,\end{cases*}$$

$$\displaystyle\begin{cases*}(C_{j},C_{j})(C_{j-1},C_{j+1})-(C_{j},C_{j-1})(C_{j},C_{j+1})<0,&if $(C_{j},C_{j})<0$,\\ (C_{j-1},C_{j+1})>0,&if $(C_{j},C_{j})=0$.\end{cases*}$$

$$\mathbf{w}(x;\mathcal{C})=\sum _{j=1}^N\mathrm{sgn}(x,C_j)\mathrm{sgn}(x,C_{j+1}),x\in V.$$

$$\Phi (x;\mathcal{C})=\mathbf{w}(x;\mathcal{C})-\mathbf{w}(v;\mathcal{C}).$$

$${\theta }_{\mu }(\tau )={\theta }_{\mu }(\tau ;\mathcal{C})=\sum _{x\in \mu +L}\Phi (x;\mathcal{C})q^{(x,x)/2}.$$

$${\hat{\theta }}_{\mu }(\tau )=\sum _{x\in \mu +L}\left(-\mathbf{w}(v;\mathcal{C})+\sum _{j=1}^N{E}_2(C_j,C_{j+1};\sqrt{2}x)\right)q^{(x,x)/2},$$

$$E_2(C,C^{\prime };x)={\int }_z{e}^{\pi (y-{\mathrm{pr}}_z(x),y-{\mathrm{pr}}_z(x))}\mathrm{sgn}(y,C)\mathrm{sgn}(y,C^{\prime })dy$$

$$\mathrm{sgn}(v,C_{j-1})\mathrm{sgn}(v,C_{j+1})<0.$$

$$b^2-c^2-d^2-\cdots$$

$$(b^{\prime }{)}^2-(c^{\prime }{)}^2-(d^{\prime }{)}^2-\cdots$$

$$b{b}^{\prime }-c{c}^{\prime }+d{d}^{\prime }+\cdots$$

$$(v,C_{j-1})(v,C_{j+1})=b{b}^{\prime }<0.$$

$$(C_j,C_{j-1})=0=(C_j,C_{j+1}).$$

$$-b^2+(c^2+d^2+...)$$

$$-(b^{\prime }{)}^2+((c^{\prime }{)}^2+(d^{\prime }{)}^2+...)$$

$$-b{b}^{\prime }+(c{c}^{\prime }+d{d}^{\prime }+...)$$

$${\gamma }_{v,w}(s)=(1-s)v+sw,s\in [0,1]$$

$${\gamma }_j(s)={\gamma }_{C_j,C_{j+1}}(s).$$

$$\mathrm{sgn}(\gamma (t),C_{j_r-1})\mathrm{sgn}(\gamma (t),C_{j_r})+\mathrm{sgn}(\gamma (t),C_{j_r})\mathrm{sgn}(\gamma (t),C_{j_r+1})=0.$$

$$\mathrm{sgn}(x_1,C_{j_i})=-\mathrm{sgn}(x_2,C_{j_i})$$

$$\mathrm{sgn}(x_1,C_j)=\mathrm{sgn}(x_2,C_j)$$

$$0$$

$$=\sum _{i=1}^k\mathrm{sgn}(x_1,C_{j_i-1})\mathrm{sgn}(x_1,C_{j_i})+\mathrm{sgn}(x_1,C_{j_i+1})\mathrm{sgn}(x_1,C_{j_i})$$

$$-\mathrm{sgn}(x_2,C_{j_i-1})\mathrm{sgn}(x_2,C_{j_i})-\mathrm{sgn}(x_2,C_{j_i+1})\mathrm{sgn}(x_2,C_{j_i})$$

$$=2\sum _{i=1}^k\mathrm{sgn}(x_1,C_{j_i-1})\mathrm{sgn}(x_1,C_{j_i})+\mathrm{sgn}(x_1,C_{j_i+1})\mathrm{sgn}(x_1,C_{j_i})$$

$$\mathbf{w}(x_1;\mathcal{C})-\mathbf{w}(x;\mathcal{C})$$

$$=$$

$$C_j$$

$$C_{j-1}$$

$$a^2-c^2-d^2-\cdots$$

$$(a^{\prime }{)}^2-(c^{\prime }{)}^2-(d^{\prime }{)}^2-\cdots$$

$$a{a}^{\prime }-c{c}^{\prime }-d{d}^{\prime }-\cdots$$

$$\mathrm{sgn}(x,C_{j-1})+\mathrm{sgn}(x,C_{j+1})=0+0=0.$$

$$(x,x)\ge r_{U,\inf }(x,x{)}_{\mathcal{C}}.$$

Full text

1. 1 Introduction
2. 2 Convergence of indefinite theta series

1. Introduction

Theta series are one of the most fundamental examples in the theory of modular forms. The theta series associated to a lattice $(L,Q)$ with integer valued positive definite quadratic form $Q$ is classically known to converge absolutely termwise and define a holomorphic modular form:

[TABLE]

Write $(-,-)$ for the symmetric bilinear form associated to $Q$ such that $Q(x)=\frac{1}{2}(x,x)$. When the quadratic space is indefinite, the theta series is no longer absolutely convergent (because the isometry group of $(L,Q)$ is infinite) and there are several different approaches to take care of the convergence problem. One approach is to sum over a subset of $L$. In this case the theta series is holomorphic but generally no longer a modular form. However, in some situations the series can be completed to a non-holomorphic modular form. This direction of research began with the work of Göttsche and Zagier [GZ98] and Zwegers ([Zwe02]) for $V$ of signature $(n,1)$: fix two negative vectors $C_{1},C_{2}$ in $V$ (that is, $(C_{1},C_{1}),(C_{2},C_{2})$ are negative) such that $(C_{1},C_{2})<0$ and take $\mu\in L^{\vee}$ (the dual of $L$), then put

[TABLE]

where $\Phi(x;\{C_{1},C_{2}\})=\mathrm{sgn}((x,C_{1}))-\mathrm{sgn}((x,C_{2}))$. Then Zwegers shows that this theta series is termwise absolutely convergent and that $\theta_{\mu}$ be completed to a modular form using certain complementary error function (we refer the reader to[Zag10] for an overview). In [ABMP18a], Alexandrov, Banerjee, Manschot and Pioline established an analogue of $\Phi(x;\{C_{1},C_{2}\})$ for $V$ of signature $(n,2)$ and later on, in [ABMP18b], based on motivations from physics, they consider a variant of $\Phi(x;\{C_{1},C_{2}\})$ for $V$ of signature $(n,2)$ and conjecture the absolute convergence of the corresponding theta series. This is what we will consider in this note.

More precisely, take $N$ non-zero vectors $\mathcal{C}=\{C_{1},\cdots,C_{N}\}$ in $V$ and consider the following incidence conditions on these vectors:

[TABLE]

To make formulas in this note more readable, we set in the following $\mathrm{sgn}(v,w)=\mathrm{sgn}((v,w))$ for $v,w\in V$. Then we put (following the notation in [FK22], up to a sign change)

[TABLE]

Here by convention, $C_{N+1}=C_{1}$. Take a negative vector $v\in V$ and set

[TABLE]

We then define

[TABLE]

The main result of this note is

Theorem 1.1**.**

1. (1)

Suppose $\mathcal{C}$ satisfies (I.1)-(I.3), then $\mathbf{w}(x;\mathcal{C})$ is constant on $V_{<0}$ (the set of negative vectors in $V$) and the indefinite theta series $\theta_{\mu}(\tau)$ is termwise absolutely convergent. 2. (2)

Suppose $\mathcal{C}$ satisfies (I.1)-(1.2) and the indefinite theta series $\theta_{\mu}(\tau)$ is termwise absolutely convergent. If moreover no three consecutive vectors in $\mathcal{C}$ are all null-vectors, then $\mathcal{C}$ also satisfies (I.3).

The value $\mathbf{w}(v;\mathcal{C})$ has the following obvious interpretation: write $v^{\perp}$ for the orthogonal complement of $v$ (a hyperplane in $V$). Then

1. (1)

$(v,C_{j})(v,C_{j+1})\geq 0$ if and only if the straight segment $\gamma_{j}$ in $V$ connecting $C_{j}$ to $C_{j+1}$ passes through the hyperplane $v^{\perp}$; 2. (2)

$(v,C_{j})(v,C_{j+1})<0$ if and only if this segment lies outside this hyperplane.

Write $\gamma(\mathcal{C})$ for the sum $\gamma_{1}+\gamma_{2}+\cdots+\gamma_{N}$, a piecewise smooth loop in $V$. Then the value $\mathbf{w}(v;\mathcal{C})$ measures how many times this loop $\gamma(\mathcal{C})$ passes through the hyperplane $v^{\perp}$, assuming that $(v,C_{j})\neq 0$ for any $j=1,\cdots,N$. If $\gamma(\mathcal{C})$ passes through $r$ times through $v^{\perp}$, then one has $\mathbf{w}(v;\mathcal{C})=N-4r$.

The first part of the theorem is also proved in [FK22] under the further condition that $(C_{j},C_{j})<0$ for all $j=1,\cdots,N$. As explained in loc.cit, (I.3) means that the $2$-dimensional oriented plane $[C_{j},C_{j+1}]$ spanned by $C_{j}$ and $C_{j+1}$ all lie in the same connected component of $D(V)$, the Grassmannian of oriented negative $2$-planes in $V$. This part corrects the conjecture in [ABMP18b, C], which does not include the term $\mathbf{w}(v;\mathcal{C})$ in the definition of $\Phi(x;\mathcal{C})$: as explained in [FK22], for $N=3$, $\mathbf{w}(v;\mathcal{C})$ is non-zero (we can also use the preceding interpretation of $\mathbf{w}(v;\mathcal{C})$) and therefore $\theta_{\mu}(\tau)$ can not be termwise absolutely convergent without this term.

The theta series $\theta_{\mu}$ defines a mock modular form of weight $(n+2)/2$, whose modular completion is given by

[TABLE]

where

[TABLE]

with $z$ the subspace of $V$ generated by $C$ and $C^{\prime}$ and $\mathrm{pr}_{z}(x)$ the projection of $x$ to $z$. We refer the reader to [ABMP18b, C] for more details.

We say a vector $v\in V$ is regular at $\mathcal{C}$ if $(v,C_{j})\neq 0$ for any $j=1,\cdots,N$. Write $\mathrm{Reg}(\mathcal{C})$ for the subset of $V$ consisting of $v$ regular at $\mathcal{C}$, which is an open dense subset of $V$ and also a finite disjoint union of polyhedral cones.

The strategy of proof of 1.1(1) is based on the following simple observation: the function $\mathbf{w}(x;\mathcal{C})$ is locally constant on $\mathrm{Reg}(\mathcal{C})$ and it is constant on $V_{<0}$. So we only need to take care of the connected components of $\mathrm{Reg}(\mathcal{C})$ whose closure intersect with the closure $\overline{V}_{<0}$ only at [math]. On such components, $(x,x)$ takes positive values and in fact is bounded below by some positive definite inner product $(x,x)_{\mathcal{C}}$.

2. Convergence of indefinite theta series

As in the introduction, let $(V,(-,-))$ be an indefinite inner product space over $\mathbb{R}$ of signature $(n,2)$. Take $N$ non-zero vectors $\mathcal{C}=\{C_{1},\cdots,C_{N}\}$ in $V$ satisfying conditions (I.1)-(I.3).

Lemma 2.1**.**

The restriction of the map $\mathbf{w}(-;\mathcal{C})$ to $\mathrm{Reg}(\mathcal{C})$ is locally constant.

Proof.

This is because each $\mathrm{sgn}(x,C_{j})$ is locally constant on $\mathrm{Reg}(\mathcal{C})$. ∎

This is also true without $\mathcal{C}$ satisfying (I.1)-(I.3).

The following three lemmas are the technical heart of this note. The main idea can be explained as follows (see Figure.1 for an illustration): suppose $C_{j-1},C_{j},C_{j+1}$ lie in the same negative plane $V^{\prime}$ (this is possible under the conditions (I.1) and (I.2)), then (I.3) implies that $C_{j-1}$ and $C_{j+1}$ lie in different connected components (half planes) of $V^{\prime}\backslash\mathbb{R}C_{j}$. Therefore, for a negative vector $v$, when $(v,C_{j})$ is very small (compared to the norms of $v,C_{j\pm 1},C_{j},(C_{j},C_{j\pm 1})$), the intersection $V^{\prime}\cap v^{\perp}$ separates the two vectors $C_{j-1},C_{j+1}$. As a result, the value $\mathrm{sgn}(v,C_{j-1})\mathrm{sgn}(v,C_{j+1})$ is negative. This means that when we perturb $v$ slightly such that $(v,C_{j})$ changes signs (assuming the signs $\mathrm{sgn}(v,C_{i})$ remain unchanged for all $i\neq j$), the value $\mathbf{w}(v;\mathcal{C})$ is unchanged.

We fix a basis $(e_{1},e_{2},\cdots,e_{n+2})$ of $V$ such that the bilinear form $(-,-)$ is represented by the diagonal matrix $\mathrm{diag}(-1_{2},1_{n})$.

Lemma 2.2**.**

Fix $j=1,\cdots,N$. For any vector $v\in V_{<0}$ such that $(v,C_{j})=0$, we have

[TABLE]

Proof.

First we assume that $C_{j}$ is not a null vector. We can suppose that the vectors $C_{j}$ and $v$ are of the form: $C_{j}=(1,0,\cdots)$ and $v=(0,1,0,\cdots)$. Then we can write $C_{j-1}=(a,b,c,d,\cdots)$ and $C_{j+1}=(a^{\prime},b^{\prime},c^{\prime},d^{\prime},\cdots)$. (I.2) and (I.3) give

[TABLE]

From this, we get

[TABLE]

Next assume $(C_{j},C_{j})=0$, then (I.2) implies

[TABLE]

As in the above, we can assume $C_{j}=(1,0,1,0,\cdots)$ and $v=(0,1,0,\cdots)$. Then we can write $C_{j-1}=(a,b,a,c,d,\cdots)$ and $C_{j+1}=(a^{\prime},b^{\prime},a^{\prime},c^{\prime},d^{\prime},\cdots)$. (I.1) and (I.3) imply

[TABLE]

We deduce that $bb^{\prime}<0$ and thus $(v,C_{j-1})(v,C_{j+1})=bb^{\prime}<0$. ∎

For two vectors $v,w\in V$, we write

[TABLE]

for the path/segment in $V$ connecting $v$ to $w$. For each $j=1,\cdots,N$, we write

[TABLE]

Then (I.1) and (I.2) imply that $\gamma_{j}(s)$ is a non-positive vector for any $s\in[0,1]$.

Lemma 2.3**.**

Let $\gamma\colon[0,1]\rightarrow V_{<0}$ be a continuous map such that there are only finitely many $t\in[0,1]$ with $\gamma(t)\in V_{<0}\backslash\mathrm{Reg}(\mathcal{C})$. Then $\mathbf{w}(\gamma(t);\mathcal{C})$ is a constant map on $t\in[0,1]$.

Proof.

If $\gamma([0,1])\subset\mathrm{Reg}(\mathcal{C})$, then the conclusion holds thanks to Lemma 2.1. We can then assume there is only one $t_{0}\in[0,1]$ such that $\gamma(t_{0})\notin\mathrm{Reg}(\mathcal{C})$. We can put $t_{0}=1$ and $(\gamma(1),C_{j_{1}})=\cdots=(\gamma(1),C_{j_{p}})=0$ for indices $j_{1}<j_{2}<\cdots<j_{p}$ (we have $j_{r}+1\neq j_{r+1}$ for any $r=1,\cdots,p-1$ by Lemma 2.2) and $(v,C_{j})\neq 0$ for other indices $j$. We can then assume that $\mathrm{sgn}(\gamma(t),C_{j_{r}})<0$ for any $t\in[0,1[$. By Lemma 2.2, we have $\mathrm{sgn}(\gamma(t),C_{j_{r}-1})\mathrm{sgn}(\gamma(t),C_{j_{r}+1})<0$ for any $t\in[0,1]$. As a result,

[TABLE]

For $j\neq j_{r}-1,j_{r}$, the cardinal of the intersection $\gamma(t)^{\perp}\cap\gamma_{j}(]0,1[)$ is constant on $t\in[0,1]$ and therefore $\mathrm{sgn}(\gamma(t),C_{j})\mathrm{sgn}(\gamma(t),C_{j+1})$ is constant on $t\in[0,1]$. So we conclude that $\mathbf{w}(\gamma(t);\mathcal{C})$ is indeed constant on $t\in[0,1]$. ∎

Lemma 2.4**.**

For any negative vectors $v,w\in V_{<0}$, we have $\mathbf{w}(v;\mathcal{C})=\mathbf{w}(w;\mathcal{C})$.

Proof.

It suffices to construct a path $\gamma\colon[0,1]\rightarrow V_{<0}$ as in Lemma 2.3 connecting $v$ and $w$. Since $\mathrm{Reg}(\mathcal{C})\cap V_{<0}$ is open dense in $V_{<0}$, we can find $v_{1},w_{1}\in V_{<0}\cap\mathrm{Reg}(\mathcal{C})$ in sufficiently small open neighborhoods of $v$ and $w$ such that the paths $\gamma_{v,v_{1}}$ and $\gamma_{w_{1},w}$ satisfy Lemma 2.3. Moreover we can always find a negative vector $u\in V_{<0}$ such that $\gamma_{v_{1},u}$ and $\gamma_{u,w_{1}}$ both lie in $V_{<0}$ and satisfy Lemma 2.3: indeed, under a suitable basis of $V$ such that the quadratic form $(-,-)$ is represented by the matrix $\mathrm{diag}(-1_{2},1_{n})$, we can write $v_{1}$ and $w_{1}$ in coordinates as $v_{1}=(1,0,\cdots,0)$ and $w_{1}=(a_{1},a_{2},\cdots,a_{n+2})$ with $(w_{1},w_{1})=-1$. Then we can always find a vector $u=(x_{1},x_{2},0,\cdots,0)\in V$ such that $x_{1}^{2}+x_{2}^{2}=-(u,u)=1$, $x_{1}^{2}=(v_{1},u)^{2}<1$ ($v_{1}$ and $u$ span a negative subspace of dimension $2$) and $(a_{1}x_{1}+a_{2}x_{2})^{2}=(u,w_{1})^{2}<1$ ($u$ and $w_{1}$ span a negative subspace of dimension $2$). Now it is easy to see that the path $\gamma_{v,v_{1}}+\gamma_{v_{1},u}+\gamma_{u,w_{1}}+\gamma_{w_{1},w}$ in $V_{<0}$ satisfies Lemma 2.3. ∎

We write $\mathbf{w}_{\mathcal{C}}=\mathbf{w}(v;\mathcal{C})$ for some $v\in V_{<0}$.

Proof of Theorem1.1(1).

The above proposition gives the first part of Theorem 1.1(1). For the convergence, we proceed by a case-by-case study of the vectors in $V$, the main idea is that, as long as a vector $x$ lies in (the boundary of) a connected component of $\mathrm{Reg}(\mathcal{C})$ which has non-empty intersection with $V_{<0}$, we can approximate $x$ with a vector $x^{\prime}$ in this component and these two values $\mathbf{w}(x;\mathcal{C})$ and $\mathbf{w}(x^{\prime};\mathcal{C})$ should be equal.

It is easy to see that any non-zero vector $x$ in $V$ belongs to one and only one of the following cases:

1. (1)

$x\in U$ for some connected component $U$ (a polyhedral cone) of $\mathrm{Reg}(\mathcal{C})$ such that $U\cap V_{<0}=\emptyset$. Since $\mathbf{w}(x;\mathcal{C})$ is constant on $U$ and also on $V_{<0}$, so $\mathbf{w}(x;\mathcal{C})=\mathbf{w}_{\mathcal{C}}$. 2. (2)

$x$ does not satisfy the preceding case but lies in $\overline{U}_{1}\cap\overline{U}_{2}$ for two distinct connected components $U_{1},U_{2}$ of $\mathrm{Reg}(\mathcal{C})$ such that $U_{1}\cap V_{<0},U_{2}\cap V_{<0}\neq\emptyset$. We can assume that $\overline{U}_{1}\cap\overline{U}_{2}$ is contained in $C_{j_{i}}^{\perp}$ for $j_{1}<j_{2}<\cdots<j_{k}$ and not in any other $C_{i}^{\perp}$. Then these vectors $C_{j_{1}},\cdots,C_{j_{k}}$ are all parallel and it is easy to see $j_{i+1}-j_{i}>1$ for any $i$. Note that $\overline{U}_{1}\cap\overline{U}_{2}$ is a polyhedral cone in a subspace of $V$ of co-dimension $1$. First we assume that $x$ lies in the interior of the polyhedral cone $\overline{U}_{1}\cap\overline{U}_{2}$. So we can choose two vectors $x_{1}\in U_{1}$ and $x_{2}\in U_{2}$ in a sufficiently small open neighborhood of $x$ such that

[TABLE]

By definition of $\mathbf{w}(x_{1};\mathcal{C})$ and $\mathbf{w}(x_{2};\mathcal{C})$, one gets

[TABLE]

On the other hand,

[TABLE]

Thus $\mathbf{w}(x;\mathcal{C})=\mathbf{w}_{\mathcal{C}}$.

If $x$ lies in the boundary of the polyhedral cone $\overline{U}_{1}\cap\overline{U}_{2}$, we can repeat the above argument by choosing points $x_{1}^{\prime},x_{2}^{\prime}$ from $\overline{U}_{1}\backslash U_{1},\overline{U}_{2}\backslash U_{2}$ (we have already shown that $\mathbf{w}(x_{1}^{\prime};\mathcal{C})=\mathbf{w}(x_{2}^{\prime};\mathcal{C})=\mathbf{w}_{\mathcal{C}}$). This process will continue if $x$ lies in the boundary of the boundary of $\overline{U}_{1}\cap\overline{U}_{2}$, etc. It will terminate in finitely many steps as there are only finitely many connected components of $\mathrm{Reg}(\mathcal{C})$. 3. (3)

$x$ does not satisfy the preceding cases but lies in a connected component $U$ of $\mathrm{Reg}(\mathcal{C})$ such that $U\cap V_{<0}=\emptyset$ and $\overline{U}\cap\overline{V}_{<0}\neq\emptyset$. Then there is another connected component $U^{\prime}$ of $\mathrm{Reg}(\mathcal{C})$ such that $U^{\prime}\cap V_{<0}\neq\emptyset$ and $\overline{U}\cap\overline{U^{\prime}}\cap\overline{V}_{<0}\neq\emptyset$. Take $x_{0}\in\overline{U}\cap\overline{U^{\prime}}\cap\overline{V}_{<0}$ and $x^{\prime}\in U^{\prime}$ and assume that $x,x^{\prime}$ are in a sufficiently small open neighborhood of $x_{0}$ (note that $\mathbf{w}(x;\mathcal{C})$ is constant on $U$). Suppose that $\overline{U}\cap\overline{U^{\prime}}$ is contained in $C_{j}^{\perp}$ for some $j$. There are two cases to consider: $(C_{j},C_{j})$ is null or not.

If $(C_{j},C_{j})<0$, we can write

[TABLE]

Conditions (I.2) and (I.3) give

[TABLE]

In particular, we have $aa^{\prime}<0$ and thus $(x_{0},C_{j-1})(x_{0},C_{j+1})=(-a+c)(-a^{\prime}+c^{\prime})<0$. So $\mathrm{sgn}(x_{0},C_{j-1})+\mathrm{sgn}(x,C_{j+1})=0$.

If $(C_{j},C_{j})=0$ and $x_{0}$ is not parallel to $C_{j}$, then we proceed as above and get $\mathrm{sgn}(x_{0},C_{j-1})+\mathrm{sgn}(x_{0},C_{j+1})=0$.

If $(C_{j},C_{j})=0$ and $x_{0}$ is parallel to $C_{j}$, then (I.2) gives

[TABLE]

Since $x,x^{\prime}$ and $x_{0}$ are all sufficiently near to each other, we can use the argument in (2) to conclude that $\mathbf{w}(x;\mathcal{C})=\mathbf{w}(x_{0};\mathcal{C})=\mathbf{w}(x^{\prime};\mathcal{C})=\mathbf{w}_{\mathcal{C}}$. 4. (4)

$x$ does not satisfy the preceding cases but lies in $\overline{U}_{1}\cap\overline{U}_{2}$ for two distinct connected components $U_{1},U_{2}$ of $\mathrm{Reg}(\mathcal{C})$ such that $U_{1}\cap V_{<0}\neq\emptyset$ and $U_{2}\cap V_{<0}=\emptyset$. Note that we have $\overline{U}_{2}\cap\overline{V}_{<0}\neq\emptyset$. From (3), we know that on $U_{1}$ and $U_{2}$, $\mathbf{w}(-;\mathcal{C})$ is equal to $\mathbf{w}_{\mathcal{C}}$. Now we use the argument in (2) to conclude that $\mathbf{w}(x;\mathcal{C})=\mathbf{w}_{\mathcal{C}}$. 5. (5)

$x$ does not satisfy the preceding cases but lies in $U$ for some connected component $U$ of $\mathrm{Reg}(\mathcal{C})$ such that $\overline{U}\cap\overline{V}_{<0}=\{0\}$. In this case we fix a positive definite inner product $(-,-)_{\mathcal{C}}$ on $V$ and consider the quotient $(x,x)/(x,x)_{\mathcal{C}}$, which is a continuous function on $\overline{U}\backslash\{0\}$ and moreover it is homogeneous of degree [math]. Since it is everywhere positive on $\overline{U}\backslash\{0\}$, it admits a positive infimum $r_{U,\mathrm{inf}}$, that is, for any $x\in\overline{U}\backslash\{0\}$,

[TABLE]

We write $\mathcal{U}$ for the set of $U$ as in (5) above, fix the inner product $(-,-)_{\mathcal{C}}$ for all $U\in\mathcal{U}$ and set $r_{\mathrm{inf}}=\min_{U\in\mathcal{U}}r_{U,\mathrm{inf}}>0$. We deduce immediately that if $x$ is not in $U$ for any $U\in\mathcal{U}$, then $\mathbf{w}(x;\mathcal{C})=\mathbf{w}_{\mathcal{C}}$; if $x$ is in $U$ for some $U\in\mathcal{U}$, then $(x,x)\geq r_{\mathrm{inf}}(x,x)_{\mathcal{C}}$. From this one get the absolute convergence of $\theta_{\mu}$:

[TABLE]

∎

We have shown that for any set $\mathcal{C}$ of $N$ non-zero vectors satisfying (I.1)-(I.3), the map $\mathbf{w}(v;\mathcal{C})$ is constant on $v\in V_{<0}$. In fact, we have the following partial converse

Proposition 2.5**.**

For any set of $N$ non-zero vectors $\mathcal{C}=\{C_{1},\cdots,C_{N}\}$ in $V$ satisfying (I.1) and (I.2), if $\mathbf{w}(v;\mathcal{C})$ is constant on $v\in V_{<0}$, then $\mathcal{C}$ also satisfies (I.3).

Proof.

Take $v\in V_{<0}\cap\mathrm{Reg}(\mathcal{C})$. We can choose $N$ non-zero vectors $\mathcal{C}^{\prime}=\{C_{1}^{\prime},\cdots,C_{N}^{\prime}\}$ with $C_{i}^{\prime}$ in a sufficiently small neighborhood of $C_{i}$ for all $i=1,\cdots,N$ such that the following conditions are satisfied

1. (1)

$\mathbf{w}(v;\mathcal{C})=\mathbf{w}(v;\mathcal{C}^{\prime})$, 2. (2)

the vectors $C_{j}^{\prime}$ are not parallel to each other, 3. (3)

$\mathcal{C}$ satisfies (I.1)-(I.3) if and only if $\mathcal{C}^{\prime}$ satisfies (I.1)-(I.3).

Therefore we can assume from the onset that the vectors $C_{j}$ are not parallel to each other.

Fix an index $j=1,\cdots,N$. Take a negative vector $v\in V$ such that $(v,C_{j})=0$ and $(v,C_{i})\neq 0$ for any $i\neq j$ (this is always possible since the $C_{j}$ are not parallel to each other). Then we can find another negative vector $w\in V$ in a sufficiently small open neighborhood of $v$ such that for any $s\in[0,1]$, the following conditions hold

1. (1)

the vector $\gamma_{w,v}(s)$ is a negative vector for any $s\in[0,1]$; 2. (2)

$(\gamma_{w,v}(s),C_{i})\neq 0$ unless $i=j$ and $s=1$; 3. (3)

for any $i\neq j-1,j$, $v^{\perp}\cap\gamma_{i}(]0,1[)=\emptyset$ if and only if $\gamma_{w,v}(s)^{\perp}\cap\gamma_{i}(]0,1[)=\emptyset$ for any $s\in[0,1]$; 4. (4)

$\mathrm{sgn}(v,C_{i})\mathrm{sgn}(\gamma_{w,v}(s),C_{i})=1$ for $i=j-1,j+1$ and for any $s\in[0,1]$.

Thus $\mathrm{sgn}(\gamma_{w,v}(s),C_{i})\mathrm{sgn}(\gamma_{w,v}(s),C_{i+1})$ are all constant on $s\in[0,1]$ (for any $i\neq j-1,j$), we deduce that

[TABLE]

is constant on $s\in[0,1]$. Setting $s=1$, we know that this map is equal to [math]. Now take $s\neq 1$ and we must have $\mathrm{sgn}(\gamma_{w,v}(s),C_{j-1})\mathrm{sgn}(\gamma_{w,v}(s),C_{j+1})=-1$ and therefore $(v,C_{j-1})(v,C_{j+1})<0$.

If $(C_{j},C_{j})<0$, we can write

[TABLE]

(I.2) gives

[TABLE]

Since $(v,C_{j-1})(v,C_{j+1})=aa^{\prime}<0$, we have $aa^{\prime}-cc^{\prime}-dd^{\prime}-\cdots<0$ and thus (I.3) holds for $j$.

If $(C_{j},C_{j})=0$, we can write

[TABLE]

where $D,D^{\prime}\in\mathbb{R}(e_{3},\cdots,e_{n+2})$. (I.1) gives

[TABLE]

Combined with $(v,C_{j-1})(v,C_{j+1})=aa^{\prime}<0$, we get that $(C_{j-1},C_{j+1})=-aa^{\prime}+(D,D^{\prime})\geq 0$, and the equality holds if and only if $a^{2}=(D,D)$, $(a^{\prime})^{2}=(D^{\prime},D^{\prime})$ and $D$ is parallel to $D^{\prime}$. In particular, if the equality holds, we have $(C_{j-1},C_{j-1})=-a^{2}+(D,D)=0$ and similarly $(C_{j+1},C_{j+1})=0$, contradicting the assumption that no three consecutive vectors in $\mathcal{C}$ are all null vectors. Therefore we must have $(C_{j-1},C_{j+1})>0$. ∎

Proof of Theorem 1.1(2).

The absolute convergence of $\theta_{\mu}(\tau)$ implies that the discrete-valued map $\Phi(x;\mathcal{C})$ is identically zero if $(x,x)<0$. Then we apply the above proposition. ∎

Remark 2.6**.**

The strategy of proof of 1.1 can be easily adapted to the case of indefinite theta series $\theta_{\mu}(\tau;\{C_{1},C_{2}\})$ for an integral lattice $(L,(-,-))$ of signature $(n,1)$ mentioned in the introduction. In particular, for two negative vectors $C_{1},C_{2}$ in $V$, if $\theta_{\mu}(\tau;\{C_{1},C_{2}\})$ are termwise absolutely convergent, then $(C_{1},C_{2})<0$.