2.1 Preliminaries

We begin by reviewing some standard notation and constructions. A good reference is [Reference ShintaniShi75]. Suppose that $L$ , $(\,,\,)$ is a lattice of rank $m=p+q$ with an even integral symmetric bilinear form of signature $(p,q)$ with $p$ , $q>0$ . Let $L^{\vee }\supset L$ be the dual lattice and set $Q(x)=\frac{1}{2}(x,x)$ . Let $V=L\otimes _{\mathbb{Z}}\mathbb{R}$ and let $G=O(V)$ be the orthogonal group of $V$ . Let

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{L}=\{\unicode[STIX]{x1D6FE}\in G\mid \unicode[STIX]{x1D6FE}L=L,~\unicode[STIX]{x1D6FE}|_{L^{\vee }/L}=\text{id}\}.\end{eqnarray}$$

We denote by $\text{Gr}_{q}^{o}(V)$ the Grassmannian of oriented $q$ -planes in $V$ and let

$$\begin{eqnarray}D=D(V)=\{z\in \text{Gr}_{q}^{o}(V)\mid (\,,\,)|_{z}<0\}\end{eqnarray}$$

be the space of oriented negative $q$ -planes in $V$ . For $z\in D$ , the associated Gaussian is

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}(x,z)=e^{-\unicode[STIX]{x1D70B}(x,x)_{z}},\end{eqnarray}$$

where, for $R(x,z)=-(\text{pr}_{z}(x),\text{pr}_{z}(x))$ ,

$$\begin{eqnarray}(x,x)_{z}=(x,x)+2R(x,z)\end{eqnarray}$$

is the majorant determined by $z$ . For fixed $z$ , $\unicode[STIX]{x1D711}_{0}(\cdot ,z)=\unicode[STIX]{x1D711}_{0}(z)\in {\mathcal{S}}(V)$ is a Schwartz function on $V$ , while, for fixed $x\in V$ , $\unicode[STIX]{x1D711}_{0}(x,\cdot )=\unicode[STIX]{x1D711}_{0}(x)\in A^{0}(D)$ is a smooth function on $D$ satisfying the equivariance

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}(gx,gz)=\unicode[STIX]{x1D711}_{0}(x,z)\end{eqnarray}$$

for $g\in G$ , or equivalently

$$\begin{eqnarray}g^{\ast }\unicode[STIX]{x1D711}_{0}(x)=\unicode[STIX]{x1D711}_{0}(g^{-1}x)=:\unicode[STIX]{x1D714}(g)\unicode[STIX]{x1D711}_{0}(x),\end{eqnarray}$$

where $g^{\ast }$ denotes the pullback of functions on $D$ and $\unicode[STIX]{x1D714}(g)$ denotes the action of $g$ on ${\mathcal{S}}(V)$ . Thus

(2.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{0}\in [{\mathcal{S}}(V)\otimes A^{0}(D)]^{G}. & & \displaystyle\end{eqnarray}$$

The action $\unicode[STIX]{x1D714}$ of $G$ on ${\mathcal{S}}(V)$ commutes with the Weil representationFootnote ⁴ action of the two-fold cover $G^{\prime }=\text{Mp}_{2}(\mathbb{R})$ of $\text{SL}_{2}(\mathbb{R})$ on ${\mathcal{S}}(V)$ , and hence there is a representation of $G\times G^{\prime }$ on this space, which we also denote by $\unicode[STIX]{x1D714}$ . Recall that for $b\in \mathbb{R}$ and $a\in \mathbb{R}^{\times }$ , there are elements $n^{\prime }(b)$ , $m^{\prime }(a)$ , and $w^{\prime }$ in $G^{\prime }$ projecting to

$$\begin{eqnarray}n(b)=\left(\begin{array}{@{}cc@{}}1 & u\\ & 1\end{array}\right),\quad m(a)=\left(\begin{array}{@{}cc@{}}a & \\ & a^{-1}\end{array}\right),\quad \text{and}\quad w=\left(\begin{array}{@{}cc@{}} & 1\\ -1 & \end{array}\right)\end{eqnarray}$$

in $\text{SL}_{2}(\mathbb{R})$ whose Weil representation action is given by

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}(n^{\prime }(b))\unicode[STIX]{x1D711}(x) & = & \displaystyle e(bQ(x))\,\unicode[STIX]{x1D711}(x),\nonumber\\ \displaystyle \unicode[STIX]{x1D714}(m^{\prime }(a))\unicode[STIX]{x1D711}(x) & = & \displaystyle |a|^{m/2}\unicode[STIX]{x1D711}(ax),\nonumber\\ \displaystyle \unicode[STIX]{x1D714}(w^{\prime })\unicode[STIX]{x1D711}(x) & = & \displaystyle e\bigg(\frac{p-q}{8}\bigg)\,\hat{\unicode[STIX]{x1D711}}(x)=e\bigg(\frac{p-q}{8}\bigg)\,\int _{V}\unicode[STIX]{x1D711}(y)\,e(-(x,y))\,dy.\nonumber\end{eqnarray}$$

Then, for $\unicode[STIX]{x1D70F}=u+iv\in \mathfrak{H}$ and $g_{\unicode[STIX]{x1D70F}}^{\prime }=n^{\prime }(u)m^{\prime }(v^{1/2})$ , we have

$$\begin{eqnarray}\unicode[STIX]{x1D714}(g_{\unicode[STIX]{x1D70F}}^{\prime })\unicode[STIX]{x1D711}_{0}(x,z)=v^{(p+q)/4}e^{-2\unicode[STIX]{x1D70B}vR(x,z)}\,\mathbf{q}^{Q(x)},\quad \mathbf{q}=e(\unicode[STIX]{x1D70F})=e^{2\unicode[STIX]{x1D70B}i\unicode[STIX]{x1D70F}}.\end{eqnarray}$$

The following invariance property gives rise to the modularity of the theta series. Define a vector-valued tempered distribution

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{L}:{\mathcal{S}}(V)\longrightarrow \mathbb{C}[L^{\vee }/L],\quad \unicode[STIX]{x1D711}\mapsto \unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D711};L)=\mathop{\sum }_{\unicode[STIX]{x1D707}\in L^{\vee }/L}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D711})\,e_{\unicode[STIX]{x1D707}},\end{eqnarray}$$

where $e_{\unicode[STIX]{x1D707}}\in \mathbb{C}[L^{\vee }/L]$ is the characteristic function of the coset $\unicode[STIX]{x1D707}+L$ and

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D711})=\mathop{\sum }_{x\in \unicode[STIX]{x1D707}+L}\unicode[STIX]{x1D711}(x).\end{eqnarray}$$

Let $\unicode[STIX]{x1D6E4}^{\prime }$ be the inverse image of $\text{SL}_{2}(\mathbb{Z})$ in $G^{\prime }$ . Then there is a finite Weil representation $\unicode[STIX]{x1D70C}_{L}$ of $\unicode[STIX]{x1D6E4}^{\prime }$ acting on $\mathbb{C}[L^{\vee }/L]$ , and the theta distribution $\unicode[STIX]{x1D6E9}_{L}$ satisfies

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{L}(\unicode[STIX]{x1D714}(\unicode[STIX]{x1D6FE}^{\prime })\unicode[STIX]{x1D711})=\unicode[STIX]{x1D70C}_{L}(\unicode[STIX]{x1D6FE}^{\prime })\unicode[STIX]{x1D6E9}_{L}(\unicode[STIX]{x1D711}).\end{eqnarray}$$

Let $K^{\prime }$ be the inverse image of $\text{SO}(2)$ in $G^{\prime }$ , and suppose that $\unicode[STIX]{x1D711}$ is an eigenfunction of weight $\ell \in \frac{1}{2}\mathbb{Z}$ for the Weil representation action of $K^{\prime }$ , that is,

$$\begin{eqnarray}\unicode[STIX]{x1D714}(k_{\unicode[STIX]{x1D703}}^{\prime })\unicode[STIX]{x1D711}=e(\ell \unicode[STIX]{x1D703})\,\unicode[STIX]{x1D711},\quad k_{\unicode[STIX]{x1D703}}=\left(\begin{array}{@{}cc@{}}\cos \unicode[STIX]{x1D703} & \sin \unicode[STIX]{x1D703}\\ -\sin \unicode[STIX]{x1D703} & \cos \unicode[STIX]{x1D703}\end{array}\right),\end{eqnarray}$$

for a suitable preimage $k_{\unicode[STIX]{x1D703}}^{\prime }$ of $k_{\unicode[STIX]{x1D703}}$ in $G^{\prime }$ .

Then the invariance of the theta distribution together with a standard calculation [Reference ShintaniShi75, pp. 90–98] implies that the $\mathbb{C}[L^{\vee }/L]$ -valued theta series

$$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D707}\in L^{\vee }/L}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F};\unicode[STIX]{x1D711})\,e_{\unicode[STIX]{x1D707}}=v^{-\ell /2}\,\unicode[STIX]{x1D6E9}_{L}(\unicode[STIX]{x1D714}(g_{\unicode[STIX]{x1D70F}}^{\prime })\unicode[STIX]{x1D711})\end{eqnarray}$$

is a (non-holomorphic) vector-valued modular form of weight $\ell$ and type $(\unicode[STIX]{x1D70C}_{L},\mathbb{C}[L^{\vee }/L])$ .

The Gaussian $\unicode[STIX]{x1D711}_{0}$ is an eigenfunction of $K^{\prime }$ of weight $(p-q)/2$ so that the Siegel theta series $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F},z;\unicode[STIX]{x1D711}_{0})$ are components of vector-valued modular forms and, moreover, via equivariance (2.1), are $\unicode[STIX]{x1D6E4}_{L}$ -invariant as functions of $z$ , that is,

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F},\cdot ;\unicode[STIX]{x1D711}_{0})\in A^{0}(D)^{\unicode[STIX]{x1D6E4}_{L}}.\end{eqnarray}$$

For this semi-classical reformulation of Weil’s construction of theta functions we are following Shintani [Reference ShintaniShi75]; cf. also [Reference Bruinier and FunkeBF04].

2.2 Theta forms

The basic idea is to replace equivariant families of Schwartz functions by equivariant families of Schwartz forms, that is, Schwartz functions valued in differential forms on $D$ . Let $A^{r}(D)$ be the space of smooth $r$ -forms on $D$ . A main result of [Reference Kudla and MillsonKM86, Reference Kudla and MillsonKM87] is the explicit construction of a family of Schwartz forms

(2.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{\text{KM}}\in [{\mathcal{S}}(V)\otimes A^{q}(D)]^{G}. & & \displaystyle\end{eqnarray}$$

Thus, for $x\in V$ and $g\in G$ ,

$$\begin{eqnarray}g^{\ast }\unicode[STIX]{x1D711}_{\text{KM}}(x)=\unicode[STIX]{x1D711}_{\text{KM}}(g^{-1}x)\in A^{q}(D).\end{eqnarray}$$

In particular, for fixed $x\in V$ , $\unicode[STIX]{x1D711}_{\text{KM}}(x)$ is a $G_{x}$ -invariant $q$ -form on $D$ . For example, $\unicode[STIX]{x1D711}_{\text{KM}}(0)$ is a $G$ -invariant form. Under $K^{\prime }$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D714}(k_{\unicode[STIX]{x1D703}}^{\prime })\unicode[STIX]{x1D711}_{\text{KM}}=e\bigg(\frac{p+q}{2}\,\unicode[STIX]{x1D703}\bigg)\unicode[STIX]{x1D711}_{\text{KM}}.\end{eqnarray}$$

Note the shift in weight! Moreover, the $q$ -form $\unicode[STIX]{x1D711}_{\text{KM}}(x)$ is closed,

$$\begin{eqnarray}d\unicode[STIX]{x1D711}_{\text{KM}}=0,\end{eqnarray}$$

where $d:A^{q}(D)\rightarrow A^{q+1}(D)$ is the exterior derivative.

Define the theta form

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F};\unicode[STIX]{x1D711}_{\text{KM}}):=v^{-(p+q)/4}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D714}(g_{\unicode[STIX]{x1D70F}}^{\prime })\unicode[STIX]{x1D711}_{\text{KM}}).\end{eqnarray}$$

Then, by construction, $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F};\unicode[STIX]{x1D711}_{\text{KM}})$ is a closed $\unicode[STIX]{x1D6E4}_{L}$ -invariant $q$ -form on $D$ and hence defines a closed $q$ -form on the (orbifold) quotient $M_{L}=[\unicode[STIX]{x1D6E4}_{L}\backslash D]$ . Moreover, as a function of $\unicode[STIX]{x1D70F}$ , $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F};\unicode[STIX]{x1D711}_{\text{KM}})$ is a component of a (non-holomorphic) modular form of weight $(p+q)/2$ and type $(\unicode[STIX]{x1D70C}_{L},\mathbb{C}[L^{\vee }/L])$ .

2.3 Relation to geodesic cycles

To avoid orientation issues, we take $\unicode[STIX]{x1D6E4}\subset \unicode[STIX]{x1D6E4}_{L}$ to be a neat subgroup of finite index. The theta forms define cohomology classes for the locally symmetric space $M_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}\backslash D$ which are related to totally geodesic cycles. This was the original motivation for their construction. We briefly recall the basic facts. For $x\in V$ with $x\neq 0$ , let $V_{x}=x^{\bot }$ , and let

$$\begin{eqnarray}D_{x}=\{z\in D\mid R(x,z)=0,\text{i.e., }z\subset V_{x}\}.\end{eqnarray}$$

In particular, $D_{x}$ is empty if $Q(x)\leqslant 0$ , and is a totally geodesic sub-symmetric space of codimension $q$ if $Q(x)>0$ .

Let $\text{pr}_{\unicode[STIX]{x1D6E4}}:D\rightarrow \unicode[STIX]{x1D6E4}\backslash D=M_{\unicode[STIX]{x1D6E4}}$ and, for $x$ with $Q(x)>0$ , let

$$\begin{eqnarray}\displaystyle Z(x)=\text{pr}_{\unicode[STIX]{x1D6E4}}(D_{x}), & & \displaystyle \nonumber\end{eqnarray}$$

a totally geodesic codimension $q$ -cycle in $M_{\unicode[STIX]{x1D6E4}}$ with an immersion

$$\begin{eqnarray}\displaystyle i_{x}:\unicode[STIX]{x1D6E4}_{x}\backslash D_{x}\longrightarrow Z(x)\subset \unicode[STIX]{x1D6E4}\backslash D. & & \displaystyle \nonumber\end{eqnarray}$$

Notice that $Z(x)$ depends only on the $\unicode[STIX]{x1D6E4}$ -orbit of $x$ .

The following results are special cases of those obtained in [Reference Kudla and MillsonKM86, Reference Kudla and MillsonKM87, Reference Kudla and MillsonKM90].

1. (i) Suppose that $\unicode[STIX]{x1D702}$ is a closed and compactly supported $(p-1)q$ -form on $M_{\unicode[STIX]{x1D6E4}}$ . Then

   $$\begin{eqnarray}\int _{M_{\unicode[STIX]{x1D6E4}}}\unicode[STIX]{x1D702}\wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F};\unicode[STIX]{x1D711}_{\text{KM}})=\int _{M_{\unicode[STIX]{x1D6E4}}}\unicode[STIX]{x1D702}\wedge \unicode[STIX]{x1D711}_{\text{KM}}(0)+\mathop{\sum }_{\substack{ x\in \unicode[STIX]{x1D707}+L \\ Q(x)>0 \\ \hspace{0.6em}{\rm mod}\hspace{0.2em}\unicode[STIX]{x1D6E4}}}\bigg(\int _{Z(x)}\unicode[STIX]{x1D702}\,\bigg)\mathbf{q}^{Q(x)}.\end{eqnarray}$$

2. (ii) Suppose that $S$ is a compact closed (i.e., $\unicode[STIX]{x2202}S=0$ ) oriented $q$ -cycle on $M_{\unicode[STIX]{x1D6E4}}$ . Then

   $$\begin{eqnarray}\int _{S}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F};\unicode[STIX]{x1D711}_{\text{KM}})=\int _{S}\unicode[STIX]{x1D711}_{\text{KM}}(0)+\mathop{\sum }_{\substack{ x\in \unicode[STIX]{x1D707}+L \\ Q(x)>0 \\ \hspace{0.6em}{\rm mod}\hspace{0.2em}\unicode[STIX]{x1D6E4}}}I(Z(x),S)\,\mathbf{q}^{Q(x)},\end{eqnarray}$$
   where $I(Z(x),S)$ is the intersection number of the cycles $Z(x)$ and $S$ .

In particular, both series are termwise absolutely convergent and define holomorphic modular forms of weight $(p+q)/2$ .

Note that these results exactly fit into the framework of (1.1). Additional discussion is given in [Reference Funke and KudlaFK17]. Many interesting variations are possible! For example, the case of certain non-compact cycles $S$ in $M_{\unicode[STIX]{x1D6E4}}$ is considered in joint work of the first author with John Millson [Reference Funke and MillsonFM02, Reference Funke and MillsonFM11, Reference Funke and MillsonFM14].

2.4 Non-closed compact cycles

Suppose that $S$ is a piecewise smooth compact oriented $q$ -chain in the symmetric space $D$ . Then, from the general machinery sketched in the previous sections, we obtain (non-holomorphic) modular forms, which we will refer to as indefinite theta series, or incomplete theta integrals, as explained in the introduction,

(2.3) $$\begin{eqnarray}\displaystyle I_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F};S):=\int _{S}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F};\unicode[STIX]{x1D711}_{\text{KM}}) & & \displaystyle\end{eqnarray}$$

of weight $m/2$ . Since $S$ is compact, we can compute such integrals termwise. Define an operator

(2.4) $$\begin{eqnarray}\displaystyle I_{S}:{\mathcal{S}}(V)\otimes A^{q}(D)\longrightarrow {\mathcal{S}}(V),\quad \unicode[STIX]{x1D711}\mapsto \int _{S}\unicode[STIX]{x1D711} & & \displaystyle\end{eqnarray}$$

from Schwartz forms to Schwartz functions by integrating out the form part. This operator commutes with the Weil representation action of $G^{\prime }$ . Thus, we have

$$\begin{eqnarray}\displaystyle I_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F};S) & = & \displaystyle \int _{S}v^{-(p+q)/4}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D714}(g_{\unicode[STIX]{x1D70F}}^{\prime })\unicode[STIX]{x1D711}_{\text{KM}})\nonumber\\ \displaystyle & = & \displaystyle v^{-(p+q)/4}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D714}(g_{\unicode[STIX]{x1D70F}}^{\prime })I_{S}(\unicode[STIX]{x1D711}_{\text{KM}})),\nonumber\end{eqnarray}$$

so that the indefinite theta series (2.3) is just the theta series defined by the Schwartz function $I_{S}(\unicode[STIX]{x1D711}_{\text{KM}})$ . We obtain explicit formulas for the modular forms $I_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F};S)$ whenever we can compute the Schwartz function

(2.5) $$\begin{eqnarray}\displaystyle I_{S}(\unicode[STIX]{x1D711}_{\text{KM}})\in {\mathcal{S}}(V) & & \displaystyle\end{eqnarray}$$

for a given $q$ -chain $S$ .

The remainder of this paper is devoted to the explicit computation of the Schwartz functions $I_{S}(\unicode[STIX]{x1D711}_{\text{KM}})$ in two cases: (1) the case of the singular $q$ -cubes defined in the next section; and (2) the case of singular $q$ -simplices defined in § 8.

3.1 The singular $q$ -cube $S({\mathcal{C}})$ and its faces

Let

(3.1) $$\begin{eqnarray}\displaystyle {\mathcal{C}}={\mathcal{C}}^{\Box }=\{\{C_{1},C_{1^{\prime }}\},\{C_{2},C_{2^{\prime }}\},\ldots ,\{C_{q},C_{q^{\prime }}\}\} & & \displaystyle\end{eqnarray}$$

be a collection of $q$ pairs of negative vectors in $V$ . For a subset $I\subset \{1,\ldots ,q\}$ , let $C^{I}$ be the ordered set $\{C_{1}^{I},\ldots ,C_{q}^{I}\}$ of $q$ vectors where we take $C_{j}^{I}=C_{j}$ if $j\notin I$ and $C_{j}^{I}=C_{j^{\prime }}$ if $j\in I$ . The vectors are ordered according to the index $j$ . Thus, $C^{\emptyset }=\{C_{1},\ldots ,C_{q}\}$ , etc. We would like to have the following ‘incidence relations’.

(Inc-1)

Each collection $C^{I}$ spans an oriented negative $q$ -plane

$$\begin{eqnarray}z^{I}=\text{span}{\{C^{I}\}}_{\text{p.o.}}.\end{eqnarray}$$

(Inc-2)

The oriented negative $q$ -planes $z^{I}$ all lie on the same component of $D$ .

These relations, which can be achieved by imposing conditions on the determinants of minors of Gram matrices, should allow us to construct a singular $q$ -cube with the points $z^{I}$ as the vertices. However, as already seen in [Reference KudlaKud18], it will be more convenient to work with the following formalism.

For $s=[s_{1},\ldots ,s_{q}]\in [0,1]^{q}$ , let

$$\begin{eqnarray}B(s)=[B_{1}(s_{1}),\ldots ,B_{q}(s_{q})],\end{eqnarray}$$

where

$$\begin{eqnarray}B_{j}(s_{j})=(1-s_{j})C_{j}+s_{j}C_{j^{\prime }}.\end{eqnarray}$$

Definition 3.1. A collection ${\mathcal{C}}$ is said to be in good position if, for all $s\in [0,1]^{q}$ ,

$$\begin{eqnarray}\text{span}\{B(s)\}_{\text{p.o.}}=\text{span}\{B_{1}(s_{1}),\ldots ,B_{q}(s_{q})\}_{\text{p.o.}}\in D.\end{eqnarray}$$

If ${\mathcal{C}}$ is in good position, then relations (Inc-1) and (Inc-2) hold, and we obtain an oriented singular $q$ -cube

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}:[0,1]^{q}\longrightarrow D,\quad s=[s_{1},\ldots ,s_{q}]\mapsto \text{span}{\{B_{1}(s_{1}),\ldots ,B_{q}(s_{q})\}}_{\text{p.o.}}\in D\end{eqnarray}$$

with the $z^{I}$ as its vertices. Let $S({\mathcal{C}})=\unicode[STIX]{x1D70C}_{{\mathcal{C}}}([0,1]^{q})$ be its image in $D$ . Note that the most degenerate case, in which $C_{j}=C_{j^{\prime }}$ for all $j$ and $S({\mathcal{C}})$ is a point in $D$ , is allowed.

From now on, unless stated otherwise, we assume that ${\mathcal{C}}$ is in good position, so that $\unicode[STIX]{x1D70C}_{{\mathcal{C}}}$ and $S({\mathcal{C}})$ are defined.

As in [Reference MasseyMas91], we define the front $j$ -face

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{j}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}:[0,1]^{q-1}\longrightarrow D,\quad \unicode[STIX]{x1D6FC}_{j}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s_{1},\ldots ,s_{q-1})=\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s_{1},\ldots ,s_{j-1},0,s_{j},\ldots ,s_{q-1}),\end{eqnarray}$$

and back $j$ -face

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{j}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}:[0,1]^{q-1}\longrightarrow D,\quad \unicode[STIX]{x1D6FD}_{j}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s_{1},\ldots ,s_{q-1})=\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s_{1},\ldots ,s_{j-1},1,s_{j},\ldots ,s_{q-1}).\end{eqnarray}$$

We write $\unicode[STIX]{x2202}_{j}^{+}S({\mathcal{C}})$ (respectively, $\unicode[STIX]{x2202}_{j}^{-}S({\mathcal{C}})$ ) for the image of $\unicode[STIX]{x1D6FC}_{j}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}$ (respectively, $\unicode[STIX]{x1D6FD}_{j}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}$ ), viewed as an oriented $(q-1)$ -cube. With this convention, the boundary of the oriented $q$ -cube $S({\mathcal{C}})$ is given by

(3.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}S({\mathcal{C}})=\mathop{\sum }_{j=1}^{q}(-1)^{j}(\unicode[STIX]{x2202}_{j}^{+}S({\mathcal{C}})-\unicode[STIX]{x2202}_{j}^{-}S({\mathcal{C}})). & & \displaystyle\end{eqnarray}$$

Note that, if $y\in V$ with $(y,y)<0$ , then $V_{y}:=y^{\bot }$ has signature $(p,q-1)$ . We write $D(V_{y})$ for the space of oriented negative $(q-1)$ -planes in $V_{y}$ , and we have an embedding

(3.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{y}:D(V_{y})\longrightarrow D,\quad z^{\prime }\mapsto \text{span}\{y,\unicode[STIX]{x1D701}^{\prime }\}_{\text{p.o.}},\quad z^{\prime }=\text{span}{\{\unicode[STIX]{x1D701}^{\prime }\}}_{\text{p.o.}}, & & \displaystyle\end{eqnarray}$$

whose image $D_{y}^{\prime }$ is the space of $z\in D$ such that $y\in z$ .

We define collections

(3.5) $$\begin{eqnarray}\displaystyle {\mathcal{C}}[j]=\{\{C_{1\bot j},C_{1^{\prime }\bot j}\},\ldots ,\widehat{\{C_{j},C_{j^{\prime }}\}},\ldots ,\{C_{q\bot j},C_{q^{\prime }\bot j}\}\} & & \displaystyle\end{eqnarray}$$

and

(3.6) $$\begin{eqnarray}\displaystyle {\mathcal{C}}[j^{\prime }]=\{\{C_{1\bot j^{\prime }},C_{1^{\prime }\bot j^{\prime }}\},\ldots ,\widehat{\{C_{j},C_{j^{\prime }}\}},\ldots ,\{C_{q\bot j^{\prime }},C_{q^{\prime }\bot j^{\prime }}\}\} & & \displaystyle\end{eqnarray}$$

of $(q-1)$ pairs of negative vectors in $V_{j}=C_{j}^{\bot }$ and $V_{j^{\prime }}=C_{j^{\prime }}^{\bot }$ , respectively. Here recall that $C_{i\bot j}$ is the orthogonal projection of $C_{i}$ to $V_{j}$ ; cf. (1.10).

The following easy fact illustrates the advantage of the ‘good position’ formalism.

Proof. Note that, if we set $s^{\prime }=[s_{1},\ldots ,s_{q-1}]\in [0,1]^{q-1}$ and write $\unicode[STIX]{x1D6FC}_{j}s^{\prime }=[s_{1},\ldots ,s_{j-1},0,s_{j},\ldots ,s_{q-1}]$ , then, since ${\mathcal{C}}$ is in good position,

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{j}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s^{\prime }) & = & \displaystyle \unicode[STIX]{x1D70C}_{{\mathcal{C}}}(\unicode[STIX]{x1D6FC}_{j}s^{\prime })\nonumber\\ \displaystyle & = & \displaystyle \text{span}{\{B_{1}(s_{1}^{\prime }),\ldots ,B_{j-1}(s_{j-1}^{\prime }),C_{j},B_{j+1}(s_{j}^{\prime }),\ldots ,B_{q}(s_{q-1}^{\prime })\}}_{\text{p.o.}}\nonumber\\ \displaystyle & = & \displaystyle \text{span}{\{B_{1}(s_{1}^{\prime })_{\bot j},\ldots ,B_{j-1}(s_{j-1}^{\prime })_{\bot j},C_{j},B_{j+1}(s_{j}^{\prime })_{\bot j},\ldots ,B_{q}(s_{q-1}^{\prime })_{\bot j}\}}_{\text{p.o.}}\in D,\nonumber\end{eqnarray}$$

which implies that ${\mathcal{C}}[j]$ is in good position for $V_{j}$ and $D(V_{j})$ . Similarly for ${\mathcal{C}}[j^{\prime }]$ .◻

We write $S({\mathcal{C}}[j])$ and $S({\mathcal{C}}[j^{\prime }])$ for the corresponding oriented singular $(q-1)$ -cubes in $D(V_{j})$ and $D(V_{j^{\prime }})$ with parametrizations analogous to (3.2),

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{{\mathcal{C}}[j]}:[0,1]^{q-1}\longrightarrow D(V_{j})\end{eqnarray}$$

and

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{{\mathcal{C}}[j^{\prime }]}:[0,1]^{q-1}\longrightarrow D(V_{j^{\prime }}).\end{eqnarray}$$

We let $\unicode[STIX]{x1D705}_{j}=\unicode[STIX]{x1D705}_{(-1)^{j-1}C_{j}}$ and $\unicode[STIX]{x1D705}_{j^{\prime }}=\unicode[STIX]{x1D705}_{(-1)^{j-1}C_{j^{\prime }}}$ so that

(3.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{j}\circ \unicode[STIX]{x1D70C}_{{\mathcal{C}}[j]}=\unicode[STIX]{x1D6FC}_{j}\unicode[STIX]{x1D70C}_{{\mathcal{C}}} & & \displaystyle\end{eqnarray}$$

and

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{j^{\prime }}\circ \unicode[STIX]{x1D70C}_{{\mathcal{C}}[j^{\prime }]}=\unicode[STIX]{x1D6FD}_{j}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}. & & \displaystyle\end{eqnarray}$$

Here the key point to note is that

$$\begin{eqnarray}\displaystyle & & \displaystyle \text{span}{\{B_{1}(s_{1}),\ldots ,B_{q}(s_{q})\}}_{\text{p.o.}}|_{s_{j}=0}\nonumber\\ \displaystyle & & \displaystyle \quad =\text{span}{\{B_{1\bot j}(s_{1}),\ldots ,B_{(j-1)\bot j}(s_{j-1}),\text{}\underline{C}_{j},B_{(j+1)\bot j}(s_{j+1}),\ldots ,B_{q\bot j}(s_{q})\}}_{\text{p.o.}}\nonumber\\ \displaystyle & & \displaystyle \quad =\text{span}{\{(-1)^{j-1}\text{}\underline{C}_{j},B_{1\bot j}(s_{1}),\ldots ,B_{(j-1)\bot j}(s_{j-1}),B_{(j+1)\bot j}(s_{j+1}),\ldots ,B_{q\bot j}(s_{q})\}}_{\text{p.o.}}\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D705}_{j}\circ \unicode[STIX]{x1D70C}_{{\mathcal{C}}[j]}(s_{1},\ldots ,\widehat{s_{j}},\ldots ,s_{q}),\nonumber\end{eqnarray}$$

where, for example,

$$\begin{eqnarray}B_{1\bot j}(s_{1})=(1-s_{1})C_{1\bot j}+s_{1}C_{1^{\prime }\bot j}.\end{eqnarray}$$

3.2 The regular case

Recall from [Reference KudlaKud18] that a vector $x\in V$ is said to be regular with respect to ${\mathcal{C}}$ if $(x,C)\neq 0$ for all $C\in {\mathcal{C}}$ . Parts (i) and (ii) of the following are an analogue of [Reference KudlaKud18, Lemma 4.2] and the proofs given there extend immediately to the general case. Part (iii) will be proved in the Appendix, where the definition of the local intersection number will also be reviewed.

5.1 Local formulas

We fix a base point $z_{0}\in D$ and an orthonormal basis $\{e_{1},\ldots ,e_{m}\}$ , $m=p+q$ , $(e_{r},e_{s})=\unicode[STIX]{x1D716}_{r}\unicode[STIX]{x1D6FF}_{rs}$ , $\unicode[STIX]{x1D716}_{r}=+1$ for $1\leqslant r\leqslant p$ and $\unicode[STIX]{x1D716}_{r}=-1$ for $r>p$ , with

$$\begin{eqnarray}z_{0}=\text{span}{\{e_{p+1},\ldots ,e_{m}\}}_{\text{p.o.}}.\end{eqnarray}$$

In particular $V\simeq \mathbb{R}^{m}$ , and the associated to $z_{0}$ Gaussian is given by

(5.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{0}(x)=\unicode[STIX]{x1D711}_{0}(x,z_{0})=e^{-\unicode[STIX]{x1D70B}\mathop{\sum }_{j}x_{j}^{2}}\in {\mathcal{S}}(V),\quad x=\mathop{\sum }_{i}x_{i}e_{i}. & & \displaystyle\end{eqnarray}$$

Let $K$ be the stabilizer of $z_{0}$ in $G$ and write $\mathfrak{g}_{o}=\text{Lie}(G)=\mathfrak{k}_{o}+\mathfrak{p}_{o}$ where $\mathfrak{k}_{o}=\text{Lie}(K)$ and $\mathfrak{p}_{o}$ are the $+1$ and $-1$ eigenspace for the Cartan involution at $z_{0}$ . There is a canonical isomorphism $T_{z_{0}}(D)\simeq \mathfrak{p}_{o}$ . Under the identification

$$\begin{eqnarray}V\otimes V~\overset{{\sim}}{\longrightarrow }~\text{End}(V),\quad (v_{1}\otimes v_{2})(v)=(v_{2},v)v_{1},\end{eqnarray}$$

a basis for $\mathfrak{p}_{o}$ is given by

$$\begin{eqnarray}X_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D707}}=e_{\unicode[STIX]{x1D6FC}}\otimes e_{\unicode[STIX]{x1D707}}+e_{\unicode[STIX]{x1D707}}\otimes e_{\unicode[STIX]{x1D6FC}},\quad 1\leqslant \unicode[STIX]{x1D6FC}\leqslant p<\unicode[STIX]{x1D707}\leqslant p+q.\end{eqnarray}$$

Let $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D707}}$ be the dual basis for $\mathfrak{p}_{o}^{\ast }$ .

By the equivariance property (2.2), $\unicode[STIX]{x1D711}_{\text{KM}}(x)$ is determined by the element of the complexFootnote ⁶

$$\begin{eqnarray}\left[{\mathcal{S}}(V)\otimes \bigwedge ^{\bullet }(\mathfrak{p}_{o}^{\ast })\right]^{K}\end{eqnarray}$$

obtained by restriction to the point $z_{0}$ .

For $1\leqslant s,t\leqslant q$ , and for $x=\sum _{j}x_{j}e_{j}\in V$ , let

$$\begin{eqnarray}\unicode[STIX]{x1D714}(s)=\unicode[STIX]{x1D714}(x;s)=\mathop{\sum }_{j=1}^{p}x_{j}\,\unicode[STIX]{x1D714}_{j,p+s}\in \mathfrak{p}_{o}^{\ast }\end{eqnarray}$$

and

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}(s,t)=\mathop{\sum }_{j=1}^{p}\unicode[STIX]{x1D714}_{j,p+s}\wedge \unicode[STIX]{x1D714}_{j,p+t}\in \bigwedge ^{2}(\mathfrak{p}_{o}^{\ast }).\end{eqnarray}$$

For $\unicode[STIX]{x1D706}$ with $0\leqslant \unicode[STIX]{x1D706}\leqslant [q/2]$ , we define $q$ -forms

(5.2) $$\begin{eqnarray}\displaystyle \mathbf{AO}_{\unicode[STIX]{x1D706}}(q)=A[\unicode[STIX]{x1D714}(1)\wedge \cdots \wedge \unicode[STIX]{x1D714}(q-2\unicode[STIX]{x1D706})\wedge \unicode[STIX]{x1D6FA}(q-2\unicode[STIX]{x1D706}+1,q-2\unicode[STIX]{x1D706}+2)\wedge \cdots \wedge \unicode[STIX]{x1D6FA}(q-1,q)], & & \displaystyle\end{eqnarray}$$

where $A$ is the alternation

(5.3) $$\begin{eqnarray}\displaystyle A[\unicode[STIX]{x1D714}(1)\wedge \cdots \wedge \unicode[STIX]{x1D6FA}(t-1,t)]=\frac{1}{t!}\mathop{\sum }_{\unicode[STIX]{x1D70E}\in S_{t}}\text{sgn}(\unicode[STIX]{x1D70E})\,\unicode[STIX]{x1D714}(\unicode[STIX]{x1D70E}(1))\wedge \cdots \wedge \unicode[STIX]{x1D6FA}(\unicode[STIX]{x1D70E}(t-1),\unicode[STIX]{x1D70E}(t)). & & \displaystyle\end{eqnarray}$$

Note that these are homogeneous of degree $q-2\unicode[STIX]{x1D706}$ in $x$ , and it will sometimes be useful to write $\mathbf{AO}_{\unicode[STIX]{x1D706}}(q)(x)$ to indicate this dependence. With this notation, we have the following formula for the restriction of $\unicode[STIX]{x1D711}_{\text{KM}}(x)$ at the point $z_{0}$ (cf. [Reference Kudla and MillsonKM86, p. 371]):

(5.4) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{\text{KM}}(x)=2^{q/2}\mathop{\sum }_{\unicode[STIX]{x1D706}=0}^{[q/2]}C(q,\unicode[STIX]{x1D706})\,\mathbf{AO}_{\unicode[STIX]{x1D706}}(q)(x)\,\unicode[STIX]{x1D711}_{0}(x),\end{eqnarray}$$

where

(5.5) $$\begin{eqnarray}\displaystyle C(t,\unicode[STIX]{x1D706})=\bigg(-\frac{1}{4\unicode[STIX]{x1D70B}}\bigg)^{\unicode[STIX]{x1D706}}\frac{t!}{2^{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D706}!(t-2\unicode[STIX]{x1D706})!}. & & \displaystyle\end{eqnarray}$$

There are two auxiliary $(q-1)$ -forms associated to $\unicode[STIX]{x1D711}_{\text{KM}}(x)$ that will play a fundamental role in our calculations. We will recall their relation to $\unicode[STIX]{x1D711}_{\text{KM}}(x)$ in a moment. The first of these is given by

(5.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{\text{KM}}(x)=2^{q/2-1}\mathop{\sum }_{\unicode[STIX]{x1D706}=0}^{[(q-1)/2]}\mathop{\sum }_{s=1}^{q}(-1)^{s}x_{p+s}\,C(q-1,\unicode[STIX]{x1D706})\,\mathbf{AO}_{\unicode[STIX]{x1D706}}(q;s)(x)\,\unicode[STIX]{x1D711}_{0}(x), & & \displaystyle\end{eqnarray}$$

where the $(q-1)$ -form $\mathbf{AO}_{\unicode[STIX]{x1D706}}(q;s)$ is defined by the alternation analogous to $\mathbf{AO}_{\unicode[STIX]{x1D706}}(q-1)$ but for the index set $\{1,\ldots ,{\hat{s}},\ldots ,q\}$ replacing $\{1,\ldots ,q-1\}$ . For example, $\mathbf{AO}_{\unicode[STIX]{x1D706}}(q;q)=\mathbf{AO}_{\unicode[STIX]{x1D706}}(q-1)$ .

Now we include the parameter $\unicode[STIX]{x1D70F}=u+iv$ . Writing

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{\text{KM}}(x) & = & \displaystyle \unicode[STIX]{x1D711}_{\text{KM}}^{0}(x)\,e^{-\unicode[STIX]{x1D70B}(x,x)},\nonumber\\ \displaystyle \unicode[STIX]{x1D713}_{\text{KM}}(x) & = & \displaystyle \unicode[STIX]{x1D713}_{\text{KM}}^{0}(x)\,e^{-\unicode[STIX]{x1D70B}(x,x)},\nonumber\end{eqnarray}$$

we have, for $\mathbf{q}=e(\unicode[STIX]{x1D70F})$ and $Q(x)=\frac{1}{2}(x,x)$ ,

(5.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{\text{KM}}(\unicode[STIX]{x1D70F},x)=\unicode[STIX]{x1D711}_{\text{KM}}^{0}(v^{1/2}x)\,\mathbf{q}^{Q(x)}=v^{-(p+q)/4}\unicode[STIX]{x1D714}(g_{\unicode[STIX]{x1D70F}}^{\prime })\unicode[STIX]{x1D711}_{\text{KM}}(x) & & \displaystyle\end{eqnarray}$$

and

(5.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{\text{KM}}(\unicode[STIX]{x1D70F},x)=v\,\unicode[STIX]{x1D713}_{\text{KM}}^{0}(v^{1/2}x)\,\mathbf{q}^{Q(x)}. & & \displaystyle\end{eqnarray}$$

Note that

$$\begin{eqnarray}-2i\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70F}}}\unicode[STIX]{x1D711}_{\text{KM}}(\unicode[STIX]{x1D70F},x)=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}v}\{\unicode[STIX]{x1D711}_{\text{KM}}^{0}(v^{1/2}x)\}\,\mathbf{q}^{Q(x)}.\end{eqnarray}$$

On the set of $x$ such that $R(x,z_{0})>0$ , let

(5.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x)=-\int _{1}^{\infty }\unicode[STIX]{x1D713}^{0}(t^{1/2}x)\,t^{-1}\,dt. & & \displaystyle\end{eqnarray}$$

The point here is that

$$\begin{eqnarray}\unicode[STIX]{x1D713}^{0}(t^{1/2}x)=(\text{form valued polynomial in }t^{1/2}x)\cdot e^{-2\unicode[STIX]{x1D70B}tR(x,z_{0})},\end{eqnarray}$$

so that the integral only makes sense when $R(x,z_{0})>0$ . For $x$ with $R(x,z_{0})>0$ , let

(5.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}_{\text{KM}}(\unicode[STIX]{x1D70F},x):=\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(v^{1/2}x)\,\mathbf{q}^{Q(x)}=-\int _{v}^{\infty }\unicode[STIX]{x1D713}^{0}(t^{1/2}x)\,t^{-1}\,dt\,\mathbf{q}^{Q(x)}. & & \displaystyle\end{eqnarray}$$

The following basic relations between the primitives $\unicode[STIX]{x1D713}_{\text{KM}}(\unicode[STIX]{x1D70F},x)$ , $\unicode[STIX]{x1D6F9}_{\text{KM}}(\unicode[STIX]{x1D70F},x)$ and the form $\unicode[STIX]{x1D711}_{\text{KM}}(\unicode[STIX]{x1D70F},x)$ are given in [Reference Funke and KudlaFK17, § 3, Proposition 3.2]; cf. also [Reference Kudla and MillsonKM90, § 8].

Taking homogeneity in $x$ of various terms into account and writing $R=R(x,z_{0})$ , we have the explicit formulas

(5.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}_{\text{KM}}(\unicode[STIX]{x1D70F},x) & = & \displaystyle 2^{q/2-1}\mathop{\sum }_{\unicode[STIX]{x1D706}=0}^{[(q-1)/2]}\mathop{\sum }_{s=1}^{q}(-1)^{s-1}x_{p+s}\,C(q-1,\unicode[STIX]{x1D706})\,\mathbf{AO}_{\unicode[STIX]{x1D706}}(q;s)\nonumber\\ \displaystyle & & \displaystyle \times \,(2\unicode[STIX]{x1D70B}R)^{-(q-2\unicode[STIX]{x1D706})/2}\,\unicode[STIX]{x1D6E4}\bigg(\frac{1}{2}(q-2\unicode[STIX]{x1D706}),2\unicode[STIX]{x1D70B}Rv\bigg)\,\mathbf{q}^{Q(x)}\end{eqnarray}$$

and

(5.12) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{\text{KM}}(\unicode[STIX]{x1D70F},x)=2^{q/2}\mathop{\sum }_{\unicode[STIX]{x1D706}=0}^{[q/2]}C(q,\unicode[STIX]{x1D706})\,\mathbf{AO}_{\unicode[STIX]{x1D706}}(q)\,v^{(q-2\unicode[STIX]{x1D706})/2}\,e^{-2\unicode[STIX]{x1D70B}vR}\,\mathbf{q}^{Q(x)}.\end{eqnarray}$$

Here $\unicode[STIX]{x1D6E4}(s,a)=\int _{a}^{\infty }e^{-t}\,t^{s-1}\,dt$ is the incomplete $\unicode[STIX]{x1D6E4}$ -function.

5.2 Global formulas

We now explain how the formulas of the previous section define global differential forms on $D$ . We will use the notation and conventions explained in [Reference KudlaKud18], especially the Appendix, which we now briefly recall.

Let

$$\begin{eqnarray}\text{FD}=\{\unicode[STIX]{x1D701}=[\unicode[STIX]{x1D701}_{1},\ldots ,\unicode[STIX]{x1D701}_{q}]\in V^{q}\mid (\unicode[STIX]{x1D701},\unicode[STIX]{x1D701}):=((\unicode[STIX]{x1D701}_{i},\unicode[STIX]{x1D701}_{j}))<0\}\end{eqnarray}$$

be the bundle of oriented negative frames, and let

$$\begin{eqnarray}\text{OFD}=\{\unicode[STIX]{x1D701}=[\unicode[STIX]{x1D701}_{1},\ldots ,\unicode[STIX]{x1D701}_{q}]\in V^{q}\mid (\unicode[STIX]{x1D701},\unicode[STIX]{x1D701})=-1_{q}\}\end{eqnarray}$$

be the bundle of oriented orthonormal negative frames. Let $\unicode[STIX]{x1D70B}:\text{FD}\rightarrow D$ be the natural projection, taking $\unicode[STIX]{x1D701}$ to its oriented span. Then, for $\unicode[STIX]{x1D701}\in \text{OFD}$ , we have an identification of tangent spaces

$$\begin{eqnarray}V^{q}\simeq T_{\unicode[STIX]{x1D701}}(\text{FD})\supset T_{\unicode[STIX]{x1D701}}(\text{OFD})=\{\unicode[STIX]{x1D702}=[\unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{q}]\in V^{q}\mid (\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})+(\unicode[STIX]{x1D701},\unicode[STIX]{x1D702})=0\}.\end{eqnarray}$$

For $z\in D$ , we let $U(z)=z^{\bot }$ . Then the ‘horizontal’ subspace $U(z)^{q}\subset T_{\unicode[STIX]{x1D701}}(\text{OFD})$ is identified with $T_{z}(D)$ under $d\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D701}}$ . Note that, while the space $U(z)^{q}$ depends only on $z$ , the identification with $T_{z}(D)$ depends on $\unicode[STIX]{x1D701}$ . The identifications for different choices of $\unicode[STIX]{x1D701}$ differ by the action of $\text{SO}(q)$ .

A priori, the expressions given in (5.12) and (5.11) are elements of $S(V)\otimes \bigwedge ^{r}(\mathfrak{p}_{o}^{\ast })$ with $r=q$ and $q-1$ respectively, where $\mathfrak{p}_{o}$ is identified with the tangent space to $D$ at the base point

$$\begin{eqnarray}z_{0}=\text{span}{\{e_{p+1},\ldots ,e_{p+q}\}}_{\text{p.o.}}\in D\end{eqnarray}$$

determined by our chosen orthonormal basis. They yield global formulas as follows. For any $\unicode[STIX]{x1D701}\in \text{OFD}$ , the function $R(x,z)$ is defined by $R(x,z)=(x,\unicode[STIX]{x1D701})(\unicode[STIX]{x1D701},x)$ . For vectors $\unicode[STIX]{x1D702}=[\unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{q}]$ and $\unicode[STIX]{x1D707}=[\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{q}]$ in $U(z)^{q}$ , define

(5.13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}(s)(\unicode[STIX]{x1D702})=(x,\unicode[STIX]{x1D702}_{s}),\quad \unicode[STIX]{x1D6FA}(s,t)(\unicode[STIX]{x1D702},\unicode[STIX]{x1D707})=(\unicode[STIX]{x1D702}_{s},\unicode[STIX]{x1D707}_{t})-(\unicode[STIX]{x1D702}_{t},\unicode[STIX]{x1D707}_{s}). & & \displaystyle\end{eqnarray}$$

Also note that, in the global version of (5.11),

(5.14) $$\begin{eqnarray}\displaystyle x_{p+s}=-(x,\unicode[STIX]{x1D701}_{s}). & & \displaystyle\end{eqnarray}$$

Proof. We observe that for some non-zero constant $c$ ,

(5.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathbf{AO}_{\unicode[STIX]{x1D706}}(q)(\unicode[STIX]{x1D702}^{1},\ldots ,\unicode[STIX]{x1D702}^{q})\nonumber\\ \displaystyle & & \displaystyle \quad =c\,\frac{1}{q!}\mathop{\sum }_{\unicode[STIX]{x1D70E}\in S_{q}}\text{sgn}(\unicode[STIX]{x1D70E})\,\det \left(\begin{array}{@{}cccccc@{}}(x,\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}(1)}^{1}) & \cdots \, & (x,\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}(q-2\unicode[STIX]{x1D706})}^{1}) & \unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}(q-2\unicode[STIX]{x1D706}+1)}^{1} & \cdots \, & \unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}(q)}^{1}\\ \vdots & & \vdots & \vdots & & \vdots \\ (x,\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}(1)}^{q}) & \cdots \, & (x,\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}(q-2\unicode[STIX]{x1D706})}^{q}) & \unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}(q-2\unicode[STIX]{x1D706}+1)}^{q} & \cdots \, & \unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}(q)}^{q}\end{array}\right),\end{eqnarray}$$

where, in expanding the determinant, the product of vectors is taken using $(\,,\,)$ .◻

Thus (5.12) (respectively, (5.11)) defines a global $q$ -form $\unicode[STIX]{x1D711}_{\text{KM}}(\unicode[STIX]{x1D70F},x)$ on $D$ (respectively, a global $(q-1)$ -form $\unicode[STIX]{x1D6F9}_{\text{KM}}(\unicode[STIX]{x1D70F},x)$ on $D-D_{x}$ ) and these forms satisfy

$$\begin{eqnarray}d\unicode[STIX]{x1D6F9}_{\text{KM}}(\unicode[STIX]{x1D70F},x)=\unicode[STIX]{x1D711}_{\text{KM}}(\unicode[STIX]{x1D70F},x)\end{eqnarray}$$

on $D-D_{x}$ .

7.1 The regular case

Suppose that $x$ is regular with respect to ${\mathcal{C}}$ , so that, by Lemma 3.3, the intersection $D_{x}~\cap ~S({\mathcal{C}})$ is either empty or consists of a single interior point $\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s(x))$ depending on whether $\unicode[STIX]{x1D6F7}_{q}(x;{\mathcal{C}})$ vanishes or not. If $\unicode[STIX]{x1D6F7}_{q}(x;{\mathcal{C}})\neq 0$ and for $\unicode[STIX]{x1D716}>0$ sufficiently small, define a collection

$$\begin{eqnarray}{\mathcal{C}}^{\unicode[STIX]{x1D716}}(x)=\{\{B_{1}(s(x)_{1}-\unicode[STIX]{x1D716}),B_{1}(s(x)_{1}+\unicode[STIX]{x1D716})\},\ldots ,\{B_{q}(s(x)_{q}-\unicode[STIX]{x1D716}),B_{q}(s(x)_{q}+\unicode[STIX]{x1D716})\}\}.\end{eqnarray}$$

For simplicity, we will abbreviate this as

$$\begin{eqnarray}{\mathcal{C}}^{\unicode[STIX]{x1D716}}={\mathcal{C}}^{\unicode[STIX]{x1D716}}(x)=\{\{C_{1}^{\unicode[STIX]{x1D716}},C_{1^{\prime }}^{\unicode[STIX]{x1D716}}\},\ldots ,\{C_{1}^{\unicode[STIX]{x1D716}},C_{q^{\prime }}^{\unicode[STIX]{x1D716}}\}\}.\end{eqnarray}$$

Proof. We note that, for $t\in [0,1]$ ,

$$\begin{eqnarray}(1-t)C_{j}^{\unicode[STIX]{x1D716}}+tC_{j^{\prime }}^{e}=(1-s(x)_{j}+\unicode[STIX]{x1D716}-2t\unicode[STIX]{x1D716})C_{j}+(s(x)_{j}-\unicode[STIX]{x1D716}+2t\unicode[STIX]{x1D716})C_{j^{\prime }}\end{eqnarray}$$

so that, for $t\in [0,1]^{q}$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{{\mathcal{C}}^{\unicode[STIX]{x1D716}}(x)}(t)=\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s(x)-\unicode[STIX]{x1D716}+2\unicode[STIX]{x1D716}t)\in D,\end{eqnarray}$$

that is, ${\mathcal{C}}^{\unicode[STIX]{x1D716}}(x)$ is in good position.◻

By construction, the singular $q$ -cube $S({\mathcal{C}}^{\unicode[STIX]{x1D716}}(x))$ contains the point $D_{x}~\cap ~S({\mathcal{C}})$ . For $x$ regular with respect to ${\mathcal{C}}$ and $\unicode[STIX]{x1D6F7}_{q}(x;{\mathcal{C}})=0$ , we let $S({\mathcal{C}}^{\unicode[STIX]{x1D716}}(x))$ be the empty set. In general, we let

$$\begin{eqnarray}S^{\unicode[STIX]{x1D716}}(x;{\mathcal{C}})=S({\mathcal{C}})-\text{int}~S({\mathcal{C}}^{\unicode[STIX]{x1D716}}).\end{eqnarray}$$

Then Stokes’s theorem and the inductive relation of Corollary 6.3 imply the following inductive formula.

Proposition 7.2. Suppose that $x$ is regular with respect to ${\mathcal{C}}$ . Then the set $D_{x}$ does not meet $\unicode[STIX]{x2202}S({\mathcal{C}})$ , the integral

$$\begin{eqnarray}I^{00}(x;{\mathcal{C}}):=\int _{\unicode[STIX]{x2202}S({\mathcal{C}})}\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x)\end{eqnarray}$$

is well defined, and

$$\begin{eqnarray}I^{0}(x;{\mathcal{C}})=I^{00}(x;{\mathcal{C}})-\lim _{\unicode[STIX]{x1D716}\downarrow 0}\,I^{00}(x;{\mathcal{C}}^{\unicode[STIX]{x1D716}}(x)).\end{eqnarray}$$

Moreover,

(7.1) $$\begin{eqnarray}\displaystyle I^{00}(x;{\mathcal{C}}) & = & \displaystyle 2^{1/2}\,\mathop{\sum }_{j=1}^{q}(x,\text{}\underline{C}_{j})\bigg(\int _{1}^{\infty }e^{-2\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j})^{2}}\,I^{0}(tx_{\bot j};{\mathcal{C}}[j])\,dt\bigg)\nonumber\\ \displaystyle & & \displaystyle -\,(x,\text{}\underline{C}_{j^{\prime }})\bigg(\int _{1}^{\infty }e^{-2\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j^{\prime }})^{2}}\,I^{0}(tx_{\bot j^{\prime }};{\mathcal{C}}[j^{\prime }])\,dt\bigg),\end{eqnarray}$$

where ${\mathcal{C}}[j]$ and ${\mathcal{C}}[j^{\prime }]$ are given by (3.5) and (3.6).

Proof. Combining (3.3), (3.7), (3.8), and Corollary 6.4, we obtain

$$\begin{eqnarray}\displaystyle I^{00}(x;{\mathcal{C}}) & = & \displaystyle \mathop{\sum }_{j=1}^{q}(-1)^{j}\bigg(\int _{\unicode[STIX]{x2202}_{j}^{+}S({\mathcal{C}})}\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x)-\int _{\unicode[STIX]{x2202}_{j}^{-}S({\mathcal{C}})}\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x)\bigg)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{j=1}^{q}(-1)^{j}\bigg(\int _{S({\mathcal{C}}[j])}\unicode[STIX]{x1D705}_{j}^{\ast }\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x)-\int _{S({\mathcal{C}}[j^{\prime }])}\unicode[STIX]{x1D705}_{j^{\prime }}^{\ast }\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x)\bigg)\nonumber\\ \displaystyle & = & \displaystyle 2^{1/2}\mathop{\sum }_{j=1}^{q}(x,\text{}\underline{C}_{j})\bigg(\int _{1}^{\infty }e^{-2\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j})^{2}}\,I^{0}(tx_{\bot j};{\mathcal{C}}[j])\,dt\bigg)\nonumber\\ \displaystyle & & \displaystyle -\,(x,\text{}\underline{C}_{j^{\prime }})\bigg(\int _{1}^{\infty }e^{-2\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j^{\prime }})^{2}}\,I^{0}(tx_{\bot j^{\prime }};{\mathcal{C}}[j^{\prime }])\,dt\bigg),\nonumber\end{eqnarray}$$

as claimed.◻

7.2 The case $q=1$

As a basis for the inductive proof of Theorem 4.1, we first suppose that $q=1$ , so that $\text{sig}(V)=(m-1,1)$ . This case is discussed in several places, among them [Reference KudlaKud13, Reference Funke and KudlaFK17, Reference LivinskyiLiv16], but we give the calculation for convenient reference. We have

$$\begin{eqnarray}D\simeq \{\unicode[STIX]{x1D701}\in V\mid Q(\unicode[STIX]{x1D701})=-1\},\quad z=\text{span}\{\unicode[STIX]{x1D701}\}_{\text{p.o.}},\end{eqnarray}$$

and the tangent space at $z\in D$ is

$$\begin{eqnarray}T_{z}(D)\simeq U(z):=z^{\bot }.\end{eqnarray}$$

For any $x\in V$ , the $1$ -form $\unicode[STIX]{x1D714}(1)$ on $D$ is defined by

$$\begin{eqnarray}\unicode[STIX]{x1D714}(1)_{z}(\unicode[STIX]{x1D702})=(x,\unicode[STIX]{x1D702}),\quad \unicode[STIX]{x1D702}\in U(z)\simeq T_{z}(D),\end{eqnarray}$$

and the Schwartz form is given by

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{\text{KM}}^{0}(x)=2^{1/2}\,\unicode[STIX]{x1D714}(1)\,e^{-2\unicode[STIX]{x1D70B}R(x,z)},\end{eqnarray}$$

with $R(x,z)=(x,\unicode[STIX]{x1D701})^{2}$ . Take $C$ , $C^{\prime }\in V$ such that

$$\begin{eqnarray}Q(C)<0,\quad Q(C^{\prime })<0,\quad (C,C^{\prime })<0,\end{eqnarray}$$

where the third condition ensures that

$$\begin{eqnarray}\{C\}_{\text{p.o.}}\simeq \text{}\underline{C}=C\,|(C,C)|^{-1/2},\quad {\{C^{\prime }\}}_{\text{p.o.}}\simeq \text{}\underline{C}^{\prime }\end{eqnarray}$$

lie on the same component of $D$ . For $s\in [0,1]$ , we define

$$\begin{eqnarray}B(s)=(1-s)C+sC^{\prime },\end{eqnarray}$$

and note that

$$\begin{eqnarray}(B(s),B(s))=(1-s)^{2}(C,C)+2s(1-s)(C,C^{\prime })+s^{2}(C^{\prime },C^{\prime })<0,\end{eqnarray}$$

so that the collection ${\mathcal{C}}=\{\{C,C^{\prime }\}\}$ is in good position. Writing

$$\begin{eqnarray}\unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}(s)=B(s)|(B(s),B(s))|^{-1/2},\end{eqnarray}$$

we obtain a geodesic curve

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}:[0,1]\longrightarrow D,\quad s\mapsto \{B(s)\}_{\text{p.o.}}\simeq \unicode[STIX]{x1D701}(s)\end{eqnarray}$$

joining $\text{}\underline{C}$ and $\text{}\underline{C}^{\prime }$ . The tangent vector to this curve will be $\dot{\unicode[STIX]{x1D701}}=(d/ds)\unicode[STIX]{x1D701}$ , and

$$\begin{eqnarray}\displaystyle I^{0}(x;{\mathcal{C}}) & = & \displaystyle 2^{1/2}\,\int _{0}^{1}(x,\dot{\unicode[STIX]{x1D701}}(s))\,e^{-2\unicode[STIX]{x1D70B}(x,\unicode[STIX]{x1D701}(s))^{2}}\,ds\nonumber\\ \displaystyle & = & \displaystyle 2^{1/2}\,\int _{0}^{1}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}\bigg(-\int _{(x,\unicode[STIX]{x1D701}(s))}^{\infty }e^{-2\unicode[STIX]{x1D70B}t^{2}}\,dt\bigg)\,ds\nonumber\\ \displaystyle & = & \displaystyle 2^{1/2}\,\bigg(\int _{(x,\text{}\underline{C})}^{\infty }e^{-2\unicode[STIX]{x1D70B}t^{2}}\,dt-\int _{(x,\text{}\underline{C}^{\prime })}^{\infty }e^{-2\unicode[STIX]{x1D70B}t^{2}}\,dt\bigg).\nonumber\end{eqnarray}$$

Since

$$\begin{eqnarray}\int _{u}^{\infty }e^{-2\unicode[STIX]{x1D70B}t^{2}}\,dt=2^{-3/2}(1-E(u\sqrt{2})),\end{eqnarray}$$

for

$$\begin{eqnarray}E(u)=2\int _{0}^{u}e^{-\unicode[STIX]{x1D70B}t^{2}}\,dt=2\,\text{sgn}(u)\int _{0}^{|u|}e^{-\unicode[STIX]{x1D70B}t^{2}}\,dt,\end{eqnarray}$$

as in [Reference ZagierZag10], we obtain the expression

$$\begin{eqnarray}\displaystyle I^{0}(x;{\mathcal{C}}) & = & \displaystyle {\textstyle \frac{1}{2}}(E((x,\text{}\underline{C}^{\prime })\sqrt{2})-E((x,\text{}\underline{C})\sqrt{2}))\nonumber\\ \displaystyle & = & \displaystyle {\textstyle \frac{1}{2}}(E_{1}(x\sqrt{2};C^{\prime })-E_{1}(x\sqrt{2};C)),\nonumber\end{eqnarray}$$

which is the $q=1$ case of Theorem 4.1. Here we use the fact that, for $C\in V$ with $Q(C)<0$ , a simple calculation shows that $E_{1}(x;C)=E((x,\text{}\underline{C}))$ . Note that in this calculation we have not used the Stokes’s theorem argument. However, it is instructive to note that

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{\text{KM}}^{0}(x)=2^{-1/2}\,(x,\unicode[STIX]{x1D701})\,e^{-2\unicode[STIX]{x1D70B}(x,\unicode[STIX]{x1D701})^{2}},\end{eqnarray}$$

so that, for $z=\text{span}\{C\}_{\text{p.o.}}\in D-D_{x}$ , the primitive is given by

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x) & = & \displaystyle -2^{-1/2}\,(x,\text{}\underline{C})\int _{1}^{\infty }e^{-2\unicode[STIX]{x1D70B}t(x,\text{}\underline{C})^{2}}\,t^{-1/2}\,dt\nonumber\\ \displaystyle & & \displaystyle =-2^{1/2}\,(x,\text{}\underline{C})\int _{1}^{\infty }e^{-2\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C})^{2}}\,dt\nonumber\\ \displaystyle & & \displaystyle =-\text{sgn}(x,\text{}\underline{C})\int _{\sqrt{2}|(x,\text{}\underline{C})|}^{\infty }e^{-\unicode[STIX]{x1D70B}t^{2}}\,dt\nonumber\\ \displaystyle & & \displaystyle =\frac{1}{2}\,\text{sgn}(x,\text{}\underline{C})\bigg(2\int _{0}^{\sqrt{2}|(x,\text{}\underline{C})|}e^{-\unicode[STIX]{x1D70B}t^{2}}\,dt-1\bigg)\nonumber\\ \displaystyle & & \displaystyle =\frac{1}{2}(E_{1}(x\sqrt{2};C)-\text{sgn}(x,C)).\nonumber\end{eqnarray}$$

Thus the Stokes’ theorem calculation gives

$$\begin{eqnarray}I^{00}(x;{\mathcal{C}})=\int _{\unicode[STIX]{x2202}S({\mathcal{C}})}\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x)=\frac{1}{2}(E_{1}(x\sqrt{2};C_{1^{\prime }})-\text{sgn}(x,C_{1^{\prime }})-E_{1}(x\sqrt{2};C_{1};x)+\text{sgn}(x,C_{1})),\end{eqnarray}$$

so that the basis for Zwegers’s ‘completion’ construction emerges.

7.3 Induction

Next we consider the inductive step. Note that we are assuming that $x$ is regular with respect to ${\mathcal{C}}$ so that (7.1) holds, and we suppose that the identity (4.1) holds for all $q^{\prime }<q$ and all ${\mathcal{C}}^{\prime }$ in good position. Let $I[j]$ and $I[j^{\prime }]$ be subsets of $\{1,\ldots ,\widehat{j},\ldots ,q\}$ and let $C[j]^{I[j]}$ (respectively, $C[j^{\prime }]^{I[j^{\prime }]}$ ) be obtained by the recipe defining $C^{I}$ in Theorem 4.1, starting with the set ${\mathcal{C}}[j]$ defined in (3.5) (respectively, the set ${\mathcal{C}}[j^{\prime }]$ defined in (3.6)). Then (7.1) becomes

(7.2) $$\begin{eqnarray}\displaystyle I^{00}(x/\sqrt{2};{\mathcal{C}}) & = & \displaystyle \mathop{\sum }_{j=1}^{q}(x,\text{}\underline{C}_{j})\bigg(\int _{1}^{\infty }e^{-\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j})^{2}}\,I^{0}(tx_{\bot j}/\sqrt{2};{\mathcal{C}}[j])\,dt\bigg)\nonumber\\ \displaystyle & & \displaystyle -\,(x,\text{}\underline{C}_{j^{\prime }})\bigg(\int _{1}^{\infty }e^{-\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j^{\prime }})^{2}}\,I^{0}(tx_{\bot j^{\prime }}/\sqrt{2};{\mathcal{C}}[j^{\prime }])\,dt\bigg)\nonumber\\ \displaystyle & = & \displaystyle (-1)^{q-1}2^{1-q}\mathop{\sum }_{j=1}^{q}\bigg(\mathop{\sum }_{I[j]}(-1)^{|I[j]|}\bigg((x,\text{}\underline{C}_{j})\int _{1}^{\infty }e^{-\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j})^{2}}\,E_{q-1}(tx_{\bot j};C[j]^{I[j]})\,dt\bigg)\nonumber\\ \displaystyle & & \displaystyle -\,\mathop{\sum }_{I[j^{\prime }]}(-1)^{|I[j^{\prime }]|}\bigg((x,\text{}\underline{C}_{j^{\prime }})\int _{1}^{\infty }e^{-\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j^{\prime }})^{2}}\,E_{q-1}(tx_{\bot j^{\prime }};C[j^{\prime }]^{I[j^{\prime }]})\,dt\bigg)\bigg).\end{eqnarray}$$

We want to compare this to the expression

$$\begin{eqnarray}-2^{-q}\mathop{\sum }_{I}(-1)^{|I|}\,E_{q}(x;C^{I}).\end{eqnarray}$$

The key is to relate the individual quantities $E_{q}(x;C^{I})$ in this sum and the terms on the right-hand side of (7.2) where $I=I[j]$ or $I=\{j\}\cup I[j^{\prime }]$ . Note that if $I=I[j]$ then the collection $C[j]^{I[j]}$ spans a negative ( $q-1$ )-plane in $V_{j}$ which maps to $z^{I}$ under $\unicode[STIX]{x1D705}_{j}$ . Similarly, if $I=\{j\}\cup I[j^{\prime }]$ then the collection $C[j^{\prime }]^{I[j^{\prime }]}$ spans a negative ( $q-1$ )-plane in $V_{j^{\prime }}$ which maps to $z^{I}$ under $\unicode[STIX]{x1D705}_{j^{\prime }}$ . Thus, we are collecting all of the terms which ‘correspond to’ a given vertex of the $q$ -cube $S({\mathcal{C}})$ . The required identities are all consequences of that for $I=\emptyset$ , and thus the main identity needed is as given in the following proposition.

Proposition 7.3. Suppose that the set of vectors $\boldsymbol{c}=\{c_{1},\ldots ,c_{r}\}$ spans an oriented negative $r$ -plane in $V$ and that $x$ is regular with respect to $\boldsymbol{c}$ . Then

(7.3) $$\begin{eqnarray}\displaystyle E_{r}(x;\boldsymbol{c})-\text{sgn}(x;\boldsymbol{c})=-2\mathop{\sum }_{j=1}^{r}(x,\text{}\underline{c}_{j})\int _{1}^{\infty }e^{-\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{c_{j}})^{2}}\,E_{r-1}(C[j];tx_{\bot j})\,dt, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{c}[j]=\{c_{1\bot j},\ldots ,\widehat{c_{j}},\ldots ,c_{r\bot j}\}$ , and $\text{sgn}(x;\boldsymbol{c})$ is defined in (1.6).

Proof. Let $z$ be the negative $r$ -plane spanned by $\boldsymbol{c}$ , and, for $y$ , $y^{\prime }\in z$ , let $(\!(y,y^{\prime })\!)=-(y,y^{\prime })$ . We also suppose that $x=\text{pr}_{z}(x)$ . If $f$ is a smooth function on $z$ , then

(7.4) $$\begin{eqnarray}\displaystyle -\int _{1}^{\infty }(\!(\unicode[STIX]{x1D6FB}f(t\,x),x)\!)\,dt & = & \displaystyle -\int _{1}^{\infty }\frac{d}{dt}\{f(tx)\}\,dt\nonumber\\ \displaystyle & = & \displaystyle f(x)-\lim _{t\rightarrow \infty }f(tx).\end{eqnarray}$$

Here $\unicode[STIX]{x1D6FB}$ is the gradient operator and we assume that the radial limit of $f$ exists. On the other hand, by equation (25) in [Reference NazarogluNaz18, Proposition 3.6],

(7.5) $$\begin{eqnarray}\displaystyle -(\!(\unicode[STIX]{x1D6FB}E_{r}(x;\boldsymbol{c}),x)\!)=2\mathop{\sum }_{j=1}^{r}(\!(x,\text{}\underline{c_{j}})\!)\,e^{-\unicode[STIX]{x1D70B}\,(\!(x,\text{}\underline{C}_{j})\!)^{2}}\,E_{r-1}(x_{\bot j};\boldsymbol{c}[j]). & & \displaystyle\end{eqnarray}$$

Moreover, for $x$ regular with respect to $\boldsymbol{c}$ , we have, [Reference Alexandrov, Banerjee, Manschot and PiolineABMP18, (6.2)] and [Reference NazarogluNaz18], that

(7.6) $$\begin{eqnarray}\displaystyle \lim _{t\rightarrow \infty }E_{r}(tx;\boldsymbol{c})=\text{sgn}(x;\boldsymbol{c}). & & \displaystyle\end{eqnarray}$$

For convenience, we will give the proof of (7.5) in the Appendix. Combining them and noting that the identity (7.4) is valid for the function $f(x)=E_{q}(x;\boldsymbol{c})$ when $x$ is regular with respect to $\boldsymbol{c}$ , we have

$$\begin{eqnarray}E_{r}(x;\boldsymbol{c})-\text{sgn}(x;\boldsymbol{c})=-2\mathop{\sum }_{j}(x,\text{}\underline{c_{j}})\,\int _{1}^{\infty }e^{-\unicode[STIX]{x1D70B}\,t^{2}(x,\text{}\underline{c_{j}})^{2}}\,E_{r-1}(tx_{\bot j};\boldsymbol{c}[j])\,dt,\end{eqnarray}$$

as required.◻

Corollary 7.5.

$$\begin{eqnarray}I^{00}(x;{\mathcal{C}})=(-1)^{q}2^{-q}\mathop{\sum }_{I}(-1)^{|I|}(E_{q}(x\sqrt{2};C^{I})-\text{sgn}(x;C^{I}))\end{eqnarray}$$

and

$$\begin{eqnarray}I^{0}(x;{\mathcal{C}})=I^{00}(x;{\mathcal{C}})+(-1)^{q}\unicode[STIX]{x1D6F7}_{q}(x;{\mathcal{C}})=(-1)^{q}2^{-q}\mathop{\sum }_{I}(-1)^{|I|}E_{q}(x\sqrt{2};C^{I}).\end{eqnarray}$$

Note that the second identity in Corollary 7.5 follows from Proposition 7.2, since the first identity implies that

$$\begin{eqnarray}\lim _{\unicode[STIX]{x1D716}\downarrow 0}\,I^{00}(x;{\mathcal{C}}^{\unicode[STIX]{x1D716}}(x))=-(-1)^{q}\unicode[STIX]{x1D6F7}_{q}(x;{\mathcal{C}}).\end{eqnarray}$$

The identity of Theorem 4.1 follows immediately from this and the continuity of $E(x;C^{I})$ with respect to $C^{I}$ .

Summarizing, if we include the parameter $\unicode[STIX]{x1D70F}$ by using (5.7), we have now established the basic formula

(7.7) $$\begin{eqnarray}\displaystyle I(\unicode[STIX]{x1D70F},x;{\mathcal{C}}) & := & \displaystyle \int _{S({\mathcal{C}}^{\Box })}\unicode[STIX]{x1D711}_{\text{KM}}(\unicode[STIX]{x1D70F},x)\nonumber\\ \displaystyle & = & \displaystyle (-1)^{q}\,2^{-q}\mathop{\sum }_{I}(-1)^{|I|}E_{q}(x\sqrt{2v};C^{I})\,\mathbf{q}^{Q(x)}.\end{eqnarray}$$

8.1 Some geometry

For $V$ of signature $(p,q)$ , we consider a collection of vectors

$$\begin{eqnarray}{\mathcal{C}}={\mathcal{C}}^{\triangle }=[C_{0},\ldots ,C_{q}],\end{eqnarray}$$

$C_{i}\in V$ with $(C_{i},C_{i})<0$ . We suppose that, for all $j$ ,

$$\begin{eqnarray}P_{j}=\text{span}\{C_{0},\ldots ,\widehat{C_{j}},\ldots ,C_{q}\}\end{eqnarray}$$

is a negative $q$ -plane. We assume that the collection ${\mathcal{C}}$ is linearly independent and let $U=\text{span}({\mathcal{C}})$ . Note that $\text{sig}(U)=(1,q)$ , and let $D(U)$ be the space of oriented negative $q$ -planes in $U$ .

Let

$$\begin{eqnarray}{\mathcal{C}}^{\vee }=[C_{0}^{\vee },\ldots ,C_{q}^{\vee }]={\mathcal{C}}\,({\mathcal{C}},{\mathcal{C}})^{-1}\end{eqnarray}$$

be the dual basis of $U$ with respect to $(\,,\,)$ . Since $C_{j}^{\vee }$ then spans $P_{j}^{\bot }$ , we have $(C_{j}^{\vee },C_{j}^{\vee })>0$ . Let

and, for $s\in \unicode[STIX]{x1D6E5}_{q}$ , let

$$\begin{eqnarray}C^{\vee }(s)=\mathop{\sum }_{j=0}^{q}s_{j}C_{j}^{\vee }={{\mathcal{C}}^{\vee }\,}^{t}s.\end{eqnarray}$$

Note that $s_{j}=(C^{\vee }(s),C_{j})$ . We say that ${\mathcal{C}}$ is in good position if

$$\begin{eqnarray}0<(C^{\vee }(s),C^{\vee }(s))=s({\mathcal{C}}^{\vee },{\mathcal{C}}^{\vee })^{t}s=s{({\mathcal{C}},{\mathcal{C}})^{-1}}^{t}s\end{eqnarray}$$

for all $s\in \unicode[STIX]{x1D6E5}_{q}$ . For example, if all entries of $({\mathcal{C}}^{\vee },{\mathcal{C}}^{\vee })=({\mathcal{C}},{\mathcal{C}})^{-1}$ are non-negative, then ${\mathcal{C}}$ is in good position.

Given ${\mathcal{C}}$ in good position, we define

$$\begin{eqnarray}z(s)=C^{\vee }(s)^{\bot }\in D,\end{eqnarray}$$

with orientation $\unicode[STIX]{x1D708}_{z(s)}\in \bigwedge ^{q}z(s)$ defined by

(8.1) $$\begin{eqnarray}\displaystyle C^{\vee }(s)\wedge \unicode[STIX]{x1D708}_{z(s)}=\unicode[STIX]{x1D708}_{U}, & & \displaystyle\end{eqnarray}$$

where we have fixed an orientation

$$\begin{eqnarray}\unicode[STIX]{x1D708}_{U}=C_{0}\wedge C_{1}\wedge \cdots \wedge C_{q}\end{eqnarray}$$

in $\bigwedge ^{q+1}U$ . For example,

$$\begin{eqnarray}z_{j}=z(0,\ldots ,1,\ldots ,0)=(C_{j}^{\vee })^{\bot }=\text{span}\{C_{0},\ldots ,\widehat{C_{j}},\ldots ,C_{q}\}\end{eqnarray}$$

with orientation given as follows. Let $R_{j}$ be the $j$ th column of the matrix $(C,C)^{-1}$ , so that

(8.2) $$\begin{eqnarray}\displaystyle C_{j}^{\vee }={\mathcal{C}}\,R_{j}=\mathop{\sum }_{i=0}^{q}R_{ij}C_{i}. & & \displaystyle\end{eqnarray}$$

Then

$$\begin{eqnarray}C_{j}^{\vee }\wedge C_{0}\wedge \cdots \wedge \widehat{C_{j}}\wedge \cdots \wedge C_{q}=(-1)^{j}R_{jj}\,C_{0}\wedge C_{1}\wedge \cdots \wedge C_{q}.\end{eqnarray}$$

Since $R_{jj}=(C_{j}^{\vee },C_{j}^{\vee })>0$ ,

(8.3) $$\begin{eqnarray}\displaystyle z_{j}=\text{span}\{C_{0},\ldots ,\widehat{C_{j}},\ldots ,C_{q}\}[j] & & \displaystyle\end{eqnarray}$$

where the ‘twist’ $[j]$ indicates that the given basis gives $(-1)^{j}\unicode[STIX]{x1D708}_{z(s)}$ .

For example, for $q=1$ we have

(8.4) $$\begin{eqnarray}\displaystyle z_{0}=\text{span}{\{C_{1}\}}_{\text{p.o.}},\quad z_{1}=\text{span}\{-C_{0}\}_{\text{p.o.}}. & & \displaystyle\end{eqnarray}$$

In particular, good position requires $(C_{0},C_{1})>0$ in this case! For $q=2$ , we have

(8.5) $$\begin{eqnarray}\displaystyle z_{0}=\text{span}{\{C_{1},C_{2}\}}_{\text{p.o.}},\quad z_{1}=\text{span}\{-C_{0},C_{2}\}_{\text{p.o.}},\quad z_{2}=\text{span}{\{C_{0},C_{1}\}}_{\text{p.o.}}. & & \displaystyle\end{eqnarray}$$

By construction, all the $z_{j}$ lie in the same component of $D$ and, by linear independence, the map

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{{\mathcal{C}}}:\unicode[STIX]{x1D6E5}_{q}\longrightarrow D,\quad s\mapsto z(s)\end{eqnarray}$$

is an embedding. Let $S({\mathcal{C}})=\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(\unicode[STIX]{x1D6E5}_{q})$ be its image. The $j$ th face of this tetrahedron is given by restricting to the subset of $s$ with $s_{j}=0$ , so that it is given as

$$\begin{eqnarray}\{z\in S({\mathcal{C}})\mid (C^{\vee }(s),C_{j})=0\}=\{z\in S\mid C_{j}\in z\}.\end{eqnarray}$$

Moreover, in the image $U_{j}$ of $U$ under the projection to $V_{j}=C_{j}^{\bot }$ , we have that

$$\begin{eqnarray}[C_{0}^{\vee },\ldots ,\widehat{C_{j}^{\vee }},\ldots ,C_{q}^{\vee }]\end{eqnarray}$$

is the dual basis to

$$\begin{eqnarray}{\mathcal{C}}_{\bot j}:=[C_{0\bot j},\ldots ,C_{q\bot j}].\end{eqnarray}$$

Thus, up to orientation, to be discussed in a moment, the restriction of $\unicode[STIX]{x1D70C}_{{\mathcal{C}}}$ to a face of $\unicode[STIX]{x1D6E5}_{q}$ is again a simplex $\unicode[STIX]{x1D70C}_{{\mathcal{C}}_{\bot j}}$ in $D(V_{j})$ ! Note that, in particular, ${\mathcal{C}}$ in good position implies that ${\mathcal{C}}_{\bot j}$ is in good position for all $j$ .

Next consider $D_{x}~\cap ~S({\mathcal{C}})$ . This set depends only on $\text{pr}_{U}(x)$ and is given by

$$\begin{eqnarray}D_{x}\cap S({\mathcal{C}})=\left\{\begin{array}{@{}ll@{}}D(U)_{\text{pr}_{U}(x)}\cap S({\mathcal{C}})\quad & \text{if }Q(\text{pr}_{U}(x))>0,\\ \emptyset \quad & \text{if }\text{pr}_{U}(x)\neq 0\text{ and }Q(\text{pr}_{U}(x))\leqslant 0,\\ S({\mathcal{C}})\quad & \text{if }\text{pr}_{U}(x)=0.\end{array}\right.\end{eqnarray}$$

Here, when $Q(\text{pr}_{U}(x))>0$ so that $\text{pr}_{U}(x)$ is a positive vector in $U$ , $D(U)_{\text{pr}_{U}(x)}$ is a pair of oriented negative $q$ -planes in $U$ given by the orthogonal complement to $\text{pr}_{U}(x)$ with its two orientations. One of these has orientation determined by $\text{pr}_{U}(x)$ by the analogue of the recipe (8.1). Then $D_{x}~\cap ~S({\mathcal{C}})=\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s(x))$ is the same $q$ -plane with orientation shifted by

$$\begin{eqnarray}\text{sgn}(\text{pr}_{U}(x),C^{\vee }(s(x))^{q}=\text{sgn}(x,C^{\vee }(s(x)))^{q}.\end{eqnarray}$$

To determine $s(x)$ , we solve

$$\begin{eqnarray}\text{pr}_{U}(x)=\unicode[STIX]{x1D706}\,C^{\vee }(s),\quad s\in \unicode[STIX]{x1D6E5}_{q},~\unicode[STIX]{x1D706}\in \mathbb{R}^{\times },\end{eqnarray}$$

that is,

$$\begin{eqnarray}(x,C_{j})=\unicode[STIX]{x1D706}\,s_{j},\quad 0\leqslant j\leqslant q.\end{eqnarray}$$

The existence of a solution implies that $\text{sgn}(x,C_{j})$ , if non-zero, is independent of $j$ and that

(8.6) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{j=0}^{q}(x,C_{j})=\unicode[STIX]{x1D706}. & & \displaystyle\end{eqnarray}$$

Thus we have the following simple description.

Lemma 8.1. Suppose that $Q(\text{pr}_{U}(x))>0$ . If $\text{sgn}(x,C_{j})$ is independent of $j$ when it is non-zero, then

$$\begin{eqnarray}D_{x}\cap S({\mathcal{C}})=\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s(x)),\end{eqnarray}$$

where

$$\begin{eqnarray}s(x)_{j}=(x,C_{j})\unicode[STIX]{x1D706}(x;{\mathcal{C}})^{-1}\end{eqnarray}$$

with

$$\begin{eqnarray}\unicode[STIX]{x1D706}(x,{\mathcal{C}})=\mathop{\sum }_{j}(x,C_{j}).\end{eqnarray}$$

Otherwise $D_{x}~\cap ~S({\mathcal{C}})$ is empty.

When $D_{x}~\cap ~S({\mathcal{C}})$ is non-empty, we determine the intersection number of the oriented $q$ -simplex $S({\mathcal{C}})$ with the oriented codimension $q$ cycle $D_{x}$ . The claim is that this is determined by the sign of the inner product of $\text{pr}_{U}(x)$ with $C^{\vee }(s(x))$ .

Proposition 8.2. Let $\unicode[STIX]{x1D6F7}_{q}^{\triangle }(x;{\mathcal{C}})$ be as in (1.3). Then, if $x$ is regular with respect to ${\mathcal{C}}$ ,

(8.7) $$\begin{eqnarray}\displaystyle I(D_{x},S({\mathcal{C}}))=\unicode[STIX]{x1D6F7}_{q}^{\triangle }(x;{\mathcal{C}}). & & \displaystyle\end{eqnarray}$$

Suppose that $\text{pr}_{U}(x)\neq 0$ . Then $\unicode[STIX]{x1D6F7}_{q}^{\triangle }(x;{\mathcal{C}})$ is non-zero precisely when all of the non-zero $\text{sgn}(x,C_{i})$ coincide. Suppose further that $s(x)$ lies on $r$ ‘walls’, that is, that exactly $r$ of the inner products $(x,C_{j})$ vanish. Then

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{q}^{\triangle }(x;{\mathcal{C}})=2^{-r}\,(-1)^{q}\text{sgn}(\unicode[STIX]{x1D706}(x,{\mathcal{C}}))^{q}.\end{eqnarray}$$

When $\text{pr}_{U}(x)=0$ , we have that $\unicode[STIX]{x1D6F7}_{q}^{\triangle }(x;{\mathcal{C}})=2^{-q}$ for $q$ even and vanishes for $q$ odd. Note that if $x$ is not regular with respect to ${\mathcal{C}}$ , then the intersection number is not defined.

Proof. Recall that if $\unicode[STIX]{x1D701}\in \text{OFD}$ is a properly oriented $q$ -frame projecting to $z\in D$ , then $T_{z}(D)\simeq U(z)^{q}$ , where $U(z)=z^{\bot }$ in $V$ . Also note that, under this isomorphism, the natural metric on $T_{z}(D)$ is given by $(\!(\unicode[STIX]{x1D702},\unicode[STIX]{x1D702}^{\prime })\!)=-\text{tr}((\unicode[STIX]{x1D702}_{i},\unicode[STIX]{x1D702}_{j}^{\prime }))$ where $\unicode[STIX]{x1D702}=[\unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{q}]$ and $\unicode[STIX]{x1D702}^{\prime }=[\unicode[STIX]{x1D702}_{1}^{\prime },\ldots ,\unicode[STIX]{x1D702}_{q}^{\prime }]$ . For our fixed collection ${\mathcal{C}}$ with $U=\text{span}\{{\mathcal{C}}\}$ , we have an embedding $D(U)\longrightarrow D$ , where $D(U)$ is the space of oriented negative $q$ -planes in $U$ . Recall that $\text{sig}(U)=(1,q)$ . For $z\in D(U)$ , write $W(z)$ for its orthogonal complement in $U$ . Again supposing that $\unicode[STIX]{x1D701}\in \text{OFD}$ with projection $z$ is given, we have

$$\begin{eqnarray}T_{z}(D(U))\simeq W(z)^{q}.\end{eqnarray}$$

Note that $\dim W(z)=1$ , and suppose that $w=w(z)$ is a properly oriented basis vector. Then $T_{z}(D(U))$ is spanned by the vectors $\unicode[STIX]{x1D70F}_{1}(w)=[w,0,\ldots ,0]$ , $\unicode[STIX]{x1D70F}_{2}(w)=[0,w,0,\ldots ,0]$ , etc. Similarly, if $z\in D_{x}$ , then the normal subspace to $T_{z}(D_{x})$ is spanned by the vectors $\unicode[STIX]{x1D70F}_{i}(x)$ , $1\leqslant i\leqslant q$ . For $z=\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s(x))$ , we have $w=C^{\vee }(s(x))$ , and the intersection number of these two cycles is then given by

$$\begin{eqnarray}\displaystyle \text{sgn}(\!(\unicode[STIX]{x1D70F}_{1}(x)\wedge \cdots \wedge \unicode[STIX]{x1D70F}_{q}(x),\unicode[STIX]{x1D70F}_{1}(w)\wedge \cdots \wedge \unicode[STIX]{x1D70F}_{q}(w))\!) & = & \displaystyle (-1)^{q}\det ((\unicode[STIX]{x1D70F}_{i}(x),\unicode[STIX]{x1D70F}_{j}(w)))\nonumber\\ \displaystyle & = & \displaystyle (-1)^{q}\text{sgn}(x,C^{\vee }(s(x)))^{q}.\nonumber\end{eqnarray}$$

But now

$$\begin{eqnarray}C^{\vee }(s(x))=\unicode[STIX]{x1D706}(x;{\mathcal{C}})^{-1}\mathop{\sum }_{j}(x,C_{j})\,C_{j}^{\vee },\end{eqnarray}$$

and, recalling (8.2),

$$\begin{eqnarray}(x,C^{\vee }(s(x)))=\unicode[STIX]{x1D706}(x;{\mathcal{C}})^{-1}\mathop{\sum }_{j}(x,C_{j})(x,C_{j}^{\vee })=\unicode[STIX]{x1D706}(x;{\mathcal{C}})^{-1}\mathop{\sum }_{i,j}(x,C_{j})R_{i,j}(x,C_{i}).\end{eqnarray}$$

If we assume that all of the non-zero $(x,C_{i})$ have the same sign, and recalling that $R_{i,j}\geqslant 0$ , we see that

$$\begin{eqnarray}\text{sgn}(x,C^{\vee }(s(x)))=\text{sgn}(\unicode[STIX]{x1D706}(x;{\mathcal{C}})).\Box\end{eqnarray}$$

For $q=1$ , and $x$ regular with respect to ${\mathcal{C}}$ ,

$$\begin{eqnarray}I(D_{x},S({\mathcal{C}}))=-{\textstyle \frac{1}{2}}(\text{sgn}(x,C_{0})+\text{sgn}(x,C_{1})).\end{eqnarray}$$

Note that, due to the ‘twist’ occurring in (8.3), our negative lines are $z_{0}=\text{span}{\{C_{1}\}}_{\text{p.o.}}$ and $z_{1}=\text{span}\{-C_{0}\}_{\text{p.o.}}$ Thus the ‘cubical’ data is ${\mathcal{C}}^{\Box }=\{C_{1},-C_{0}\}$ , and $I(D_{x},S({\mathcal{C}}))$ coincides with

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{1}^{\Box }(x;{\mathcal{C}}^{\Box })={\textstyle \frac{1}{2}}(\text{sgn}(x,-C_{0})-\text{sgn}(x,C_{1})).\end{eqnarray}$$

8.2 The integral of the theta form

We now compute

$$\begin{eqnarray}I^{0}(x;{\mathcal{C}}^{\triangle })=\int _{S({\mathcal{C}})}\unicode[STIX]{x1D711}_{\text{KM}}^{0}(x).\end{eqnarray}$$

The case $q=1$ coincides with the Zwegers case for ${\mathcal{C}}^{\Box }=\{C_{1},-C_{0}\}$ , and we have

(8.8) $$\begin{eqnarray}\displaystyle I^{0}(x/\sqrt{2};{\mathcal{C}})=-{\textstyle \frac{1}{2}}(E_{1}(x;C_{0})+E_{1}(x;C_{1})). & & \displaystyle\end{eqnarray}$$

As a check on signs, note that, since

$$\begin{eqnarray}\lim _{t\rightarrow \infty }E_{1}(tx;C)=\text{sgn}(x,C),\end{eqnarray}$$

this is consistent with the value of $I(D_{x},S({\mathcal{C}}))$ for $q=1$ above.

For the general case, we suppose that $x$ is regular with respect to ${\mathcal{C}}$ and proceed by induction. Due to regularity, $D_{x}~\cap ~S({\mathcal{C}})$ either is empty or is a single point $\unicode[STIX]{x1D70C}_{{\mathcal{C}}}(s(x))$ on the interior of $S({\mathcal{C}})$ . Recall that

(8.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}S({\mathcal{C}})=\mathop{\sum }_{j=0}^{q}(-1)^{j}S({\mathcal{C}}_{\bot j}). & & \displaystyle\end{eqnarray}$$

Then, by [Reference Funke and KudlaFK17, Remark 3.4], we have

(8.10) $$\begin{eqnarray}\displaystyle I^{0}(x;{\mathcal{C}})=\int _{S({\mathcal{C}})}\unicode[STIX]{x1D711}_{\text{KM}}^{0}(x)=I(D_{x},S({\mathcal{C}}))+\int _{\unicode[STIX]{x2202}S({\mathcal{C}})}\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x). & & \displaystyle\end{eqnarray}$$

Since $\lim _{t\rightarrow \infty }\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(tx)=0$ , this identity gives the limiting value

$$\begin{eqnarray}\lim _{t\rightarrow \infty }I^{0}(tx;{\mathcal{C}})=\lim _{t\rightarrow \infty }\int _{S({\mathcal{C}})}\unicode[STIX]{x1D711}_{\text{KM}}^{0}(tx)=I(D_{x},S({\mathcal{C}})).\end{eqnarray}$$

Now using Corollary 6.3, we have the inductive formula

(8.11) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x2202}S({\mathcal{C}})}\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x) & = & \displaystyle \mathop{\sum }_{j=0}^{q}(-1)^{j}\int _{S({\mathcal{C}}_{\bot j})}\unicode[STIX]{x1D705}_{j}^{\ast }\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{j=0}^{q}2^{1/2}\,(x,\text{}\underline{C}_{j})\int _{1}^{\infty }e^{-2\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j})^{2}}\,I^{0}(tx_{\bot j};{\mathcal{C}}_{\bot j})\,dt.\end{eqnarray}$$

Using this, we obtain the following explicit formula.

Theorem 8.3. For a subset $I\subset \{0,1,\ldots ,q\}$ , let ${\mathcal{C}}^{(I)}$ be the collection of $q+1-|I|$ elements where the $C_{i}$ with $i\in I$ have been omitted. Then

$$\begin{eqnarray}I^{0}(x/\sqrt{2};{\mathcal{C}})=(-1)^{q}2^{-q}\mathop{\sum }_{r=0}^{[q/2]}\mathop{\sum }_{\substack{ I \\ |I|=2r+1}}E_{q-2r}(x;{\mathcal{C}}^{(I)}).\end{eqnarray}$$

Here $E_{0}(\cdots )=1$ .

Remark 8.4. (i) Note that if this formula is proved for $x$ regular, then it holds for all $x$ by continuity.

(ii) Substituting $tx$ for $x$ and letting $t$ go to infinity, we obtain the ‘holomorphic’ part,

(8.12) $$\begin{eqnarray}\displaystyle (-1)^{q}2^{-q}\mathop{\sum }_{r=0}^{[q/2]}\mathop{\sum }_{\substack{ I \\ |I|=2r+1}}\text{sgn}(x;{\mathcal{C}}^{I}), & & \displaystyle\end{eqnarray}$$

where $\text{sgn}(x;\emptyset )=1$ . In the case of $x$ regular, (8.10) implies that this must coincide with $I(D_{x},S({\mathcal{C}}))$ . In fact, it is easily checked that (8.12) is equal to $\unicode[STIX]{x1D6F7}_{q}^{\triangle }(x;{\mathcal{C}})$ for all $x$ . Thus our theta integral is the non-holomorphic completion of the series

$$\begin{eqnarray}\mathop{\sum }_{x\in \unicode[STIX]{x1D707}+L}\unicode[STIX]{x1D6F7}_{q}^{\triangle }(x;{\mathcal{C}})\,\mathbf{q}^{Q(x)}.\end{eqnarray}$$

Proof. The case $q=1$ is (8.8). In the induction, we use the notation

$$\begin{eqnarray}{\mathcal{C}}[j]=[C_{0\bot j},\ldots ,C_{j-1\bot j},C_{j+1\bot j},\ldots ,C_{q\bot j}].\end{eqnarray}$$

Let $A=\{0,1,\ldots ,q\}$ and for a subset $I\subset A$ , let ${\mathcal{C}}^{I}$ be the collection of $q+1-|I|$ vectors obtained by omitting the $C_{i}$ with $i\in I$ . Also denote by $I[j]$ a subset of $A[j]:=\{0,1,\ldots ,{\hat{j}},\ldots ,q\}$ .

We have

$$\begin{eqnarray}\displaystyle & & \displaystyle I^{0}(x/\sqrt{2};{\mathcal{C}})-I(D_{x},S({\mathcal{C}}))\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{\unicode[STIX]{x2202}S({\mathcal{C}})}\unicode[STIX]{x1D6F9}_{\text{KM}}^{0}(x/\sqrt{2})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{j=0}^{q}(x,\text{}\underline{C}_{j})\int _{1}^{\infty }e^{-\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j})^{2}}\,I^{0}(tx_{\bot j}/\sqrt{2};{\mathcal{C}}[j])\,dt\nonumber\\ \displaystyle & & \displaystyle \quad =(-1)^{q}2^{-q}\mathop{\sum }_{j=0}^{q}-2(x,\text{}\underline{C}_{j})\int _{1}^{\infty }e^{-\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j})^{2}}\,\mathop{\sum }_{r=0}^{[(q-1)/2]}\mathop{\sum }_{\substack{ I[j]\subset A[j] \\ |I[j]|=2r+1}}E_{q-1-2r}(tx_{\bot j};{\mathcal{C}}[j]^{I[j]})\,dt\nonumber\\ \displaystyle & & \displaystyle \quad =(-1)^{q}2^{-q}\mathop{\sum }_{r=0}^{[(q-1)/2]}\mathop{\sum }_{j\in A}\mathop{\sum }_{\substack{ I\subset A \\ |I|=2r+1 \\ j\notin I}}-2(x,\text{}\underline{C}_{j})\int _{1}^{\infty }e^{-\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j})^{2}}\,E_{q-1-2r}(tx_{\bot j};{\mathcal{C}}[j]^{I})\,dt\nonumber\\ \displaystyle & & \displaystyle \quad =(-1)^{q}2^{-q}\mathop{\sum }_{r=0}^{[(q-1)/2]}\mathop{\sum }_{\substack{ I\subset A \\ |I|=2r+1}}\mathop{\sum }_{\substack{ j\in A \\ j\notin I}}-2(x,\text{}\underline{C}_{j})\int _{1}^{\infty }e^{-\unicode[STIX]{x1D70B}t^{2}(x,\text{}\underline{C}_{j})^{2}}\,E_{q-1-2r}(tx_{\bot j};({\mathcal{C}}^{I})[j])\,dt\nonumber\\ \displaystyle & & \displaystyle \quad =(-1)^{q}2^{-q}\mathop{\sum }_{r=0}^{[(q-1)/2]}\mathop{\sum }_{\substack{ I\subset A \\ |I|=2r+1}}(E_{q-2r}(x;{\mathcal{C}}^{I})-\text{sgn}(x;{\mathcal{C}}^{I}))\nonumber\\ \displaystyle & & \displaystyle \quad =(-1)^{q}2^{-q}\mathop{\sum }_{r=0}^{[q/2]}\mathop{\sum }_{\substack{ I\subset A \\ |I|=2r+1}}E_{q-2r}(x;{\mathcal{C}}^{I})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,(-1)^{q}2^{-q}\mathop{\sum }_{r=0}^{[(q-1)/2]}\mathop{\sum }_{\substack{ I\subset A \\ |I|=2r+1}}\text{sgn}(x;{\mathcal{C}}^{I})-(-1)^{q}2^{-q}\,\unicode[STIX]{x1D6FF}_{q,\text{even}}.\nonumber\end{eqnarray}$$

Thus, to finish the proof, we note that

(8.13) $$\begin{eqnarray}\displaystyle I(D_{x},S({\mathcal{C}}))=(-1)^{q}2^{-q}\mathop{\sum }_{r=0}^{[q/2]}\mathop{\sum }_{\substack{ I\subset A \\ |I|=2r+1}}\text{sgn}(x;{\mathcal{C}}^{I}), & & \displaystyle\end{eqnarray}$$

where we use the convention that $\text{sgn}(x;\emptyset )=1$ . Here recall that we are assuming that $x$ is regular with respect to ${\mathcal{C}}$ . To check this, observe that

$$\begin{eqnarray}\displaystyle (-1)^{q}2^{-q}\mathop{\sum }_{r=0}^{[q/2]}\mathop{\sum }_{\substack{ I\subset A \\ |I|=2r+1}}\text{sgn}(x;{\mathcal{C}}^{I}) & = & \displaystyle (-1)^{q}2^{-q}\mathop{\sum }_{\substack{ J\subset A \\ |J|\equiv q(2)}}\mathop{\prod }_{j\in J}\unicode[STIX]{x1D70E}_{j}\nonumber\\ \displaystyle & = & \displaystyle (-1)^{q}2^{-q-1}\bigg(\mathop{\prod }_{j\in A}(1+\unicode[STIX]{x1D70E}_{j})+(-1)^{q}\mathop{\prod }_{j\in A}(1-\unicode[STIX]{x1D70E}_{j})\bigg).\Box \nonumber\end{eqnarray}$$