We now turn to the analytic extension of Selberg-like integrals to dense, open subsets of the space of possible exponents. As discussed in the introduction, the results in this section are apparently sharp for generic Selberg-like integrals, but for e.g. symmetric Selberg-like integrals they are only preliminary. Nevertheless, the results we prove here will be useful in establishing the sharp results later. For our discussion of the symmetric and DF-symmetric cases, it is useful to consider somewhat more general integrals than Eq. (1.2). Let \(\ell ,m,n\in \mathbb \) satisfy \(\ell +m+n=N\in \mathbb ^+\). Fix a finite collection \(\mathcal \) of indexed sets

$$\begin \} \}_ \in \mathcal (K_) } \subseteq \mathbb . \end$$

(3.1)

Define

$$\begin & S_[F](},},}) = \int _}\Big [ \prod _^N |x_i|^|1-x_i|^ \Big ]\nonumber \\ & \quad \Big [ \prod _ (x_k-x_j)^} \Big ] F \,\textrmx_1\cdots \,\textrmx_N, \end$$

(3.2)

for \((},},})\in \Omega _[\mathcal ]\), where

\(\Omega _[\mathcal ]\) denotes the set of \((},},})\in \mathbb ^N_}}\times \mathbb ^N_}}\times \mathbb ^_}}\) such that

$$\begin & \Big [ \prod _^N |x_i|^|1-x_i|^ \Big ] \Big [ \prod _ (x_k-x_j)^} \Big ]\nonumber \\ & \quad \Big [\prod _ \in \mathcal (K_) } x_}^}}\Big ] \in L^1(\triangle _,\,\textrmx_1\cdots \,\textrmx_N ) \end$$

(3.3)

for all \(\} \}_ \in \mathcal (K_) } \in \mathcal \), and

F has the form

$$\begin F= \sum _} \}_ \in \mathcal (K_) }} \Big [ \prod _ \in \mathcal (K_) } x_}^}} \Big ] F_} \}_\in \mathcal (K_)} } \end$$

(3.4)

for some \(F_} \}_\in \mathcal (K_)} } \in C^\infty (K_)\).

We denote the set of such F by \(\mathcal ^}(K_)\). From the definition Eq. (2.6) of \(\triangle _\), the integrand is nonvanishing there, so the absolute values in Eq. (3.2) amount to a choice of branch.

Observe that \(\Omega _[\mathcal ]\) is a nonempty, open, and connected subset of \(\mathbb ^\). In the case \(N=1\), we consider \(\Omega _[\mathcal ]\) as a subset of \(\mathbb ^2_\).

We write \(\Omega [F]\) to denote \(\Omega _[\mathcal ]\) for arbitrary \(\mathcal \) such that \(F\in \mathcal ^}(K_)\). Let

$$\begin \Omega _=\Omega _[\_ \in \mathcal (K_)}\} ]. \end$$

(3.5)

If \(\varphi \in C^\infty (\triangle _)\), then we can consider \(\varphi \) as an element of \(C^\infty (K_)\), so \(S_[\varphi ]:\Omega _\rightarrow \mathbb \) is well-defined, and \(\Omega _[\varphi ]\supseteq \Omega _\). If \(f\in \mathbb [x_1,\ldots ,x_N]\), then the lift of \(f\varphi \) is also a classical symbol on \(K_\) (it is smooth if \(\ell ,n=0\), but not necessarily otherwise), so

$$\begin S_[f\varphi ]:\Omega _[f]\rightarrow \mathbb \end$$

(3.6)

is well-defined, except now we may have \(\Omega _[f\varphi ] \not \subseteq \Omega _\) if \(\ell \ne 0\) or \(n\ne 0\).

In the special case when \(\ell ,n=0\) and \(m=N\), we use the abbreviations \(\Omega _=\Omega _N\), \(\Omega _[\bullet ] = \Omega _N[\bullet ]\), and

$$\begin S_[F](},},}) = S_[F](},},}), \end$$

(3.7)

this being consistent with our earlier notation.

As in the introduction, when \(},},}\) are constant, we just write ‘\(\alpha \)’ in place of ‘\(}\),’ ‘\(\beta \)’ in place of ‘\(}\),’ and ‘\(\gamma \)’ in place of ‘\(}\).’ Let \(U_[\bullet ]\) denote the set of \((\alpha ,\beta ,\gamma )\in \mathbb ^3\) such that \((},},}) \in \Omega _[\bullet ]\) holds when \(}=\alpha \), \(}=\beta \), and \(}=\gamma \).

Similar abbreviations will be used throughout the rest of this paper.

In addition to the general Selberg-like integral above, we have the following general integral of Dotsenko–Fateev type:

$$\begin I_[F](},},}) & = \int _} \Big [ \prod _^N |x_i|^|1-x_i|^ \Big ] \nonumber \\ & \quad \Big [ \prod _ (x_k-x_j+i0)^ } \Big ] F \,\textrmx_1\cdots \,\textrmx_N \end$$

(3.8)

for \((},},}) \in V_[\mathcal ]\), where now \(\mathcal \) denotes a finite collection of indexed sets \(\}\}_ \in \mathcal (A_)}\subseteq \mathbb \),

\(V_[\mathcal ]\) denotes the set of \((},},}) \in \mathbb ^\) for which the integrand in Eq. (3.8) lies in \(L^1(\square _, \,\textrmx_1\cdots \,\textrmx_N)\)—that is the set of \((},},})\) such that

$$\begin & \Big [ \prod _^N |x_i|^|1-x_i|^ \Big ] \Big [ \prod _ |x_k-x_j|^} \Big ]\nonumber \\ & \quad \Big [\prod _ \in \mathcal (K_) } x_}^}}\Big ] \in L^1(\square _,\,\textrmx_1\cdots \,\textrmx_N ) \end$$

(3.9)

for all \(\} \}_ \in \mathcal (A_) } \in \mathcal \), and

F has the form Eq. (3.4) for \(F_}\}_ \in \mathcal (A_) }}\in C^\infty (A_)\).

In Eq. (3.8), \(x^\gamma = e^ e^\) if \(x<0\) and \(x^ = e^\) if \(x>0\). We apply abbreviations for Dotsenko–Fateev-like integrals that are analogous to those used for Selberg-like integrals.

Let \(W_[\bullet ]\) denote the set of \((\alpha ,\beta ,\gamma )\in \mathbb ^3\) such that \((},},}) \in V_[\bullet ]\) holds when \(}=\alpha \), \(}=\beta \), and \(}=\gamma \). Let \(W_^}[F]\) denote the set of \((\alpha _-,\alpha _+,\beta _-,\beta _+,\gamma _-,\gamma _0,\gamma _+) \in \mathbb ^7\) such that \((}^},}^},}^}) \in V_[F]\). Let

$$\begin I^;\texttt}_[F](\alpha _-,\alpha _+,\beta _-,\beta _+,\gamma _-,\gamma _0,\gamma _+) = I_[F](}^},}^},}^}).\nonumber \\ \end$$

(3.10)

This section is split into many short subsections. The general analytic framework in which the extension is performed is discussed in §3.1, and the specific application to Selberg-like integrals is contained in §3.2. We prove a family of identities relating \(I_,I_,I_,\cdots \) in §3.3. As preparation for our discussion of singularity removal in the DF-symmetric case, we discuss in §3.4 an alternative regularization procedure suggested by Dotsenko–Fateev that works for some suboptimal range of parameters (in particular allowing \(\gamma _0=-1\), but not allowing the real parts of \(\alpha _-,\alpha _+,\beta _-,\beta _+\) to be too negative). It should be remarked that this regularization technique can be combined with that in §3.1 to yield proofs of the main theorems without the technicalities associated with needing to understand the analyticity of products of distributions like \((y \pm i0)^\lambda \) in \(\lambda \). As this lacks the purely analytic flavor of the proof in §3.1, it is not the approach we follow here. The \(I_\) are related to the Selberg-like integrals \(S_\) in §3.5. A key lemma used in the removal of singularities is in §3.6. This lemma is a generalization of a result proven by Aomoto [3] and discussed heuristically by Dotsenko–Fateev [8]. For completeness and later convenience, we record in §3.7 the symmetric and DF-symmetric cases of the results in §3.2 regarding the Dotsenko–Fateev integrals.

Let \(\mathfrak _=\mathfrak _\ell \times \mathfrak _m\times \mathfrak _n\), which we consider as the subgroup of \(\mathfrak _N\) leaving each of \(\mathcal _1,\mathcal _2,\mathcal _3\) invariant, where \(\mathcal _1,\mathcal _2,\mathcal _3\) are as in the previous section, a.k.a. the Young subgroup associated with the partition \(\= \mathcal _1\sqcup \mathcal _2 \sqcup \mathcal _3\). Given a permutation \(\sigma \in \mathfrak _\), let

$$\begin I_[F](},},})^\sigma & = \int _} \Big [ \prod _^N |x_i|^|1-x_i|^ \Big ]\nonumber \\ & \quad \Big [ \prod _ (x_-x_+i0)^ } \Big ] F \,\textrmx_1\cdots \,\textrmx_N,\nonumber \\ \end$$

(3.11)

defined for \((},},}) \in V_[F]\). If we define \(}^\sigma ,}^\sigma ,}^\sigma \) by \(\alpha _j^ = \alpha _\), \(\beta _j^\sigma = \beta _\), and \(\gamma _^ = \gamma _\), and

$$\begin F^\sigma (y_1,\ldots ,y_N) = F(y_(1)},\ldots ,y_(N)} ), \end$$

(3.12)

then \(I_[F](},},})^\sigma = I_[F^\sigma ](}^\sigma ,}^\sigma ,}^\sigma )\). This relation will be very useful below. More generally, for any \(\sigma \in \mathfrak _N\), let

$$\begin I_[F](},},})^\sigma&= I_[F^\sigma ](}^\sigma ,}^\sigma ,}^\sigma ) \end$$

(3.13)

$$\begin S_[F](},},})^\sigma&= S_[F^\sigma ](}^\sigma ,}^\sigma ,}^\sigma ), \end$$

(3.14)

defined for \((},},}) \in V_[F]\) in the former case or for

$$\begin & (},},})\in \Omega _[F]^\sigma \nonumber \\ & \quad = \},},})\in \mathbb ^: (}^\sigma ,}^\sigma ,}^\sigma ) \in \Omega _[F^\sigma ] \} \end$$

(3.15)

in the latter case. We will use similar notation for other subsets of \(\mathbb ^\) below, as well as for the meromorphic extensions of \(S_[F]\) and \(I_[F]\).

3.1 Some Generalities

Let \(N\in \mathbb \) be arbitrary. For a Fréchet space \(\mathcal \), let \( \mathscr (\mathbb ^;\mathcal )\) denote the Fréchet space of entire \(\mathcal \)-valued functions on \(\mathbb ^N\), where the topology is that of uniform convergence in compact subsets, as measured with respect to each Fréchet seminorm on \(\mathcal \), and similarly for \(\mathcal \) an LF-space. Let \(\mathcal '(\mathbb ^N)\) denote the LCTVS of compactly supported distributions on \(\mathbb ^N\). By the Schwartz representation theorem,

$$\begin \mathcal '(\mathbb ^N)=\cup _} H_}^(\mathbb ^N), \end$$

(3.16)

where \(H_}^(\mathbb ^N)\) is the set of compactly supported elements of \(H^(\mathbb ^N)\).

Let \(N\in \mathbb ^+\), \(k\in \\), and \(\kappa \in \mathbb \). For any

$$\begin \psi \in C_}^\infty (\mathbb ^k_; \mathcal '(\mathbb ^_,\cdots ,t_N})) = \bigcup _} C_}^\infty (\mathbb ^k_; H_}^(\mathbb ^) )\nonumber \\ \end$$

(3.17)

let, for \(}= (\rho _1,\ldots ,\rho _k)\),

$$\begin I_[\psi ](}) = \int _0^\infty \cdots \int _0^\infty t_1^\cdots t_k^ \langle 1,\psi (t_1,\ldots ,t_k,-) \rangle \,\textrmt_1\cdots \,\textrmt_k,\nonumber \\ \end$$

(3.18)

which we abbreviate as

$$\begin I_[\psi ](}) = \int _^N_k} t_1^\cdots t_k^ \psi (t) \,\textrm^N t. \end$$

(3.19)

Here, \(\mathbb ^N_k = [0,\infty )^k_ \times \mathbb ^_,\cdots ,t_N}\), and \(I_[\psi ](})\) is defined initially for \(\Re \rho _1, \cdots ,\Re \rho _k> -1\), for which the right-hand side of Eq. (3.18) is a well-defined integral.

Let \(H_}^(\mathbb ^N)\) denote the set of compactly supported elements of \(H_}^(\mathbb ^N) = \langle r \rangle ^ H^m(\mathbb ^N)\). Let

$$\begin & \mathscr (\mathbb ^k\times \mathbb ^\kappa ; C_}^\infty (\mathbb ^k_; \mathcal '(\mathbb ^_,\cdots ,t_N})))\nonumber \\ & \quad = \bigcap _ \bigcup _} \mathscr (\Omega ;C_}^\infty (\mathbb ^k_; H_}^(\mathbb ^) )), \end$$

(3.20)

endowed with the strongest topology such that the inclusions

$$\begin \bigcap _ & \mathscr (\Omega ;C_}^\infty (\mathbb ^k_; H_}^(\mathbb ^) ))\nonumber \\ & \quad \hookrightarrow \mathscr (\mathbb ^k\times \mathbb ^\kappa ; C_}^\infty (\mathbb ^k_; \mathcal '(\mathbb ^_,\cdots ,t_N}))) \end$$

(3.21)

are all continuous, where the left-hand side is an LF space. Here, \(\Omega \) is varying over bounded domains in \(\mathbb ^k\times \mathbb ^\kappa \). We are identifying functions on \(\mathbb ^\times \mathbb ^\kappa \) with their restrictions to subdomains. In other words, an element of the space defined by Eq. (3.20) is locally an analytic family of elements of \(C_}^\infty (\mathbb ^k_; H_}^(\mathbb ^) )\) for some \(m,s\in \mathbb \) which are allowed to depend on \(\Omega \).

Proposition 3.1

Suppose that, for each \(}\in \mathbb ^k\) and \(}\in \mathbb ^\kappa \), we are given some \(\psi (-;},})\) as in Eq. (3.17), depending entirely on \(},}\) in the sense that the map

$$\begin \mathbb ^k\times \mathbb ^\kappa \ni (},}) \mapsto \psi \in C_}^\infty (\mathbb ^k;\mathcal '(\mathbb ^)) \end$$

(3.22)

is entire, i.e. lies in \(\mathscr (\mathbb ^k\times \mathbb ^\kappa ; C_}^\infty (\mathbb ^k_; \mathcal '(\mathbb ^_,\cdots ,t_N})))\). Define

$$\begin I_[\psi ](},}) = I_[\psi (},})](}). \end$$

(3.23)

Then, the function \(J_[\psi ]\) defined by

$$\begin I_[\psi ](},}) = \Big [ \prod _^k\Gamma (\rho _j+1) \Big ] J_[\psi ](},}) \end$$

(3.24)

extends to an entire function on \(\mathbb _}}^k\times \mathbb _}}^\kappa \). Moreover, the function

$$\begin & J_[-]:\mathscr (\mathbb ^\times \mathbb ^\kappa ;C^\infty _}( \mathbb ^k_; \mathcal '(\mathbb ^_,\ldots ,t_N} ))) \ni \psi \nonumber \\ & \quad \mapsto J_[\psi ] \in \mathscr (\mathbb ^\times \mathbb ^\kappa ) \end$$

(3.25)

is continuous.

Cf. [19][39, Lemma 10.7.9].

Proof

The \(k=0\) case is essentially tautologous.

We now proceed inductively on k. Let \(k\ge 1\), and assume that we have proven the result for smaller k. Expanding \(\psi \) in Taylor series around \(t_1=0\), there exist

$$\begin \psi ^&\in \mathscr \left( \mathbb ^\times \mathbb ^\kappa ;C^\infty _}\big ( \mathbb ^_; \mathcal '(\mathbb ^_,\ldots ,t_N} )\big )\right) \end$$

(3.26)

$$\begin E^&\in \mathscr \left( \mathbb ^\times \mathbb ^\kappa ;C^\infty \big (\mathbb _;C^\infty _}( \mathbb ^_; \mathcal '(\mathbb ^_,\ldots ,t_N}))\big )\right) , \end$$

(3.27)

which can be regarded as smooth functions (or generalized functions) of \(t_1,\ldots ,t_N\), depending analytically on parameters \(}\in \mathbb ^k\) and \(}\in \mathbb ^\kappa \). such that

$$\begin \psi (t_1,\cdots ,t_N;},}) = \sum _^J t_1^j \psi ^(t_2,\cdots ,t_;},}) + t_1^ E^(t_1,\cdots ,t_N;},})\nonumber \\ \end$$

(3.28)

for all \(J\in \mathbb \). Let \(K \subset \mathbb ^\) be an arbitrary nonempty compact set. There exists some \(T>0\) such that \(} \psi (-;},})\subseteq \\) for all \((},})\in K\). Then, if \(\Re \rho _1,\cdots ,\Re \rho _k>-1\) and \((},})\in K\),

$$\begin I_[\psi ](},})= & \sum _^J \frac[\psi ^](}}}, })} T^\nonumber \\ & + \int _0^T t_1^ I_[E^(t_1,-)](}}},}) \,\textrmt_1,\nonumber \\ \end$$

(3.29)

where \(}}} = (\rho _2,\cdots ,\rho _k)\). We now define \(J_[\psi ](},}): \\times \mathbb ^\kappa _}}\rightarrow \mathbb \) by

$$\begin J_[\psi ](},})= & \frac\sum _^J \frac[\psi ^](}}}, })} T^ \nonumber \\ & + \frac\int _0^T t_1^ J_[E^(t_1,-)](}}},}) \,\textrmt_1.\nonumber \\ \end$$

(3.30)

By construction, Eq. (3.24) holds when \(\Re \rho _1,\cdots ,\Re \rho _k>-1\). By the continuity clause of the inductive hypothesis, the integral in Eq. (3.29) is a well-defined Bochner integral, for each individual \((},}) \in \\times \mathbb ^\kappa \). Moreover, the right-hand side of Eq. (3.30) depends analytically on \((},}) \in \\times \mathbb ^\kappa \). By the inductive hypothesis, this is true for the sum on the first line (multiplied by \(\Gamma (\rho _1+1)^\)), as the simple poles due to the factors of \(1/(\rho _1+j+1)\) cancel with those of \(\Gamma (\rho _1+1)\). So, in order to show that the whole right-hand side of Eq. (3.30) depends analytically on \((},})\) in this domain, we can show it for

$$\begin \int _0^T t_1^ J_[E^(t_1,-)](}}},}) \,\textrmt_1. \end$$

(3.31)

Justifying differentiation under the integral sign, this is a \(C^1\)-function of \((\Re \rho _1,\Im \rho _1) \in \^2,u> -1-J\}\), and it satisfies the Cauchy–Riemann equations, so it follows that the integral in Eq. (3.31) is analytic as a function of \(\rho _1 \in \\), for each fixed \(}}}\in \mathbb ^\) and \(}\in \mathbb ^\kappa \). Adding \(}}},}\)-dependence does not change the argument.

So, the formula Eq. (3.29) yields an analytic extension of \(I_\), and we can take a union over all \(J \in \mathbb \), the various partial extensions agreeing with each other via analyticity. The continuity clause is evident from the formula Eq. (3.30) and the inductive hypothesis. \(\square \)

Consequently, \(I_[\psi ]\) admits an analytic continuation \(}_[\psi ]:\Omega \rightarrow \mathbb \) to the set \(\Omega =(\mathbb ^k_}} \backslash \bigcup _} \^\})\times \mathbb ^\kappa _}}\), and the map

$$\begin }_[-]:\mathscr (\mathbb ^\times \mathbb ^\kappa ;C^\infty _}( \mathbb ^k_; \mathcal '(\mathbb ^_,\ldots ,t_N} ))) \ni \psi \mapsto }_[\psi ] \in \mathscr (\Omega )\nonumber \\ \end$$

(3.32)

is continuous.

If \(\mathcal \) is a consistently orientable collection of codimension-1 interior p-submanifolds on a mwc M, then, letting \(x_}\) for \(\textrm\in \mathcal (M)\) denote a bdf of the face \(\textrm\), it is the case that, for any \(}\in \mathbb ^\mathcal \) and \(}\in \mathbb ^(M)}\), the product

$$\begin \omega (},})= & \prod _\in \mathcal (M)} x_}^}}\prod _} (y_P + i0)^: }^\infty _}(M;\Omega M) \ni \mu \mapsto \lim _ \int _M \prod _\in \mathcal (M)}\nonumber \\ & \prod _} x_}^}} (y_P + i\varepsilon )^ \mu \end$$

(3.33)

is a well-defined classical distribution on M, where \(\_}\) are consistently oriented defining functions. (Here, \(}^\infty _}(M;\Omega M)\) is the set of compactly supported smooth densities on M that are Schwartz at each boundary hypersurface.) That is, \(\omega \) is an extendable distribution on M and defines, for small \(\epsilon >0\), an element of \(C^\infty ([0,\epsilon )_}}; \mathcal '(\textrm))\) for each face \(\textrm\). We write the right-hand side of Eq. (3.33) as \(\int _M \omega (},}) \mu \). More generally, if \(\mu \in C_}^\infty (M;\Omega M)\), then

$$\begin \lim _ \int _M \prod _\in \mathcal (M)}\prod _} x_}^}} (y_P + i\varepsilon )^ \mu = \int _M \omega (},}) \mu \end$$

(3.34)

exists whenever \(\rho _}>-1\) for all \(\textrm\in \mathcal (M)\).

Let \(\varkappa \in \mathbb \). Suppose that we are given some entire family

$$\begin \mu : \mathbb ^(M)}\times \mathbb ^\mathcal \times \mathbb ^\rightarrow C^\infty _}(M;\Omega M) \end$$

(3.35)

of compactly supported smooth densities \(\mu (},},}) \in C^\infty _}(M;\Omega M)\) on M. Consider the function

$$\begin & I[M,\mu ](},},}):\},},}) \in \mathbb ^(M)}\times \mathbb ^\mathcal \times \mathbb ^\varkappa : \rho _}\nonumber \\ & \quad >-1 \text \textrm\in \mathcal (M)\} \rightarrow \mathbb \end$$

(3.36)

defined by

$$\begin I[M,\mu ](},},}) = \int _M \omega (},}) \mu (},},}). \end$$

(3.37)

Proposition 3.2

Suppose that, for some \(N_0\in \mathbb ^+\), we are given an affine map \(L = (L_1,L_2,L_3):\smash ^_}}}\rightarrow \smash ^(M)}_}}}\times \mathbb ^}_}}\times \mathbb ^\varkappa _}}\) such that, for each \(\textrm\in \mathcal (M)\), the affine functional

$$\begin (L\bullet )_}: \mathbb ^\ni }\mapsto (L_1})_} \in \mathbb \end$$

(3.38)

is nonconstant. Then, there exist entire functions \(I_,\textrm}[M,\mu ](L\bullet ): \mathbb ^_}} \rightarrow \mathbb \) associated with the minimal facets \(\textrm\) of M such that

$$\begin I[M,\mu ](L}) = \sum _} \Big [ \prod _\in \mathcal (M), \textrm\supseteq \textrm} \Gamma (1+ (L\varrho )_} ) \Big ] I_,\textrm}[M,\mu ](L}) \end$$

(3.39)

for all \(}\in \mathbb ^\) for which the left-hand side is defined by Eq. (3.37).

Proof

Pass to a partition of unity subordinate to a system of coordinate charts on M and apply Proposition 3.1 locally. \(\square \)

Then, letting \(\mathcal = \}:\textrm\in \mathcal (M)\}\),

$$\begin \Big [ \prod _} \frac}))^} \Big ] I[M,\mu ](L}) \end$$

(3.40)

extends to an entire function \(\mathbb _}}^\rightarrow \mathbb \), where \(\#_\Lambda \in \mathbb ^+\) is the maximum size of any set \(S\subseteq \mathcal (M)\) of faces such that \(\cap _\in S} \textrm \ne \varnothing \) and \((L\bullet )_}=\Lambda \) for all \(\textrm\in S\). Indeed, this follows from the proposition above since, for each facet \(\textrm\),

$$\begin \Big [\prod _} \frac}))^} \Big ] \prod _\in \mathcal (M), \textrm\supseteq \textrm} \Gamma (1+ (L})_} ) \end$$

(3.41)

is entire.

3.2 Specialization to Generic Selberg- and DF-Like Integrals

We now apply the results of the previous section to the specific case of the integrals Eq. (3.2) and Eq. (3.8). Fix \(\ell ,m,n \in \mathbb \) satisfying \(\ell +m+n=N\), \(N\in \mathbb ^+\).

3.2.1 The Selberg Case

Fix \(F\in \mathcal ^\mathcal (K_)\). Let \(\rho _ = \rho _(},},})\) be defined by Eqs. (2.41), (2.42), (2.43), and (2.44). Recalling the definition of \(\texttt(\ell ,m,n)\) given in §2.1:

Proposition 3.3

There exist entire functions

$$\begin S_, \texttt, \} \}_ \in \mathcal (K_) } }[F]: \mathbb ^_},},}} \rightarrow \mathbb , \end$$

(3.42)

associated with pairs of minimal facets \(\textrm\) of \(K_\) and collections \(\} \}_ \in \mathcal (_) }\in \mathcal \) of weights such that

$$\begin S_[F](},},})= & \sum _\in \texttt(\ell ,m,n)} \sum _} \}_} \in \mathcal (K_) \in \mathcal } \Big [\prod _(j,k)\in \texttt}\Gamma (1+\rho _+d__}) \Big ] \nonumber \\ & \times S_, \texttt, \} \}_} \in \mathcal (K_) }[F](},},}) \end$$

(3.43)

for all \((},},}) \in \Omega _[\mathcal ]\).

Proof

This is a corollary of Proposition 2.3 and Proposition 3.2, using the fact that the minimal facets of \(K_\) are in correspondence with the elements of \(\texttt(\ell ,m,n)\) via Eq. (2.50). \(\square \)

Consequently, there exists an analytic extension \(}_[F]:}_[\mathcal ]\rightarrow \mathbb \) of \(S_[F]:\Omega _[\mathcal ]\rightarrow \mathbb \), where

$$\begin }_[\mathcal ]= & \mathbb ^_},},}} \Big \backslash \Big [\bigcup _} \}_ \in \mathcal (K_)} \in \mathcal } \Big (\bigcup _ \in \mathcal _} \nonumber \\ & \ +d__} \in \mathbb ^ \}\Big )\Big ]. \end$$

(3.44)

This is an open and connected subset of full measure; namely, it is the complement of a locally finite collection of complex (affine) hyperplanes in \(\mathbb ^\). In the case \(m=N\), this agrees with Eq. (1.13).

As a corollary of the previous proposition, there exists an entire function

$$\begin S_}[F]:\mathbb ^_},},}}\rightarrow \mathbb \end$$

(3.45)

such that

$$\begin S_[F](},},}) = \Big [\prod _\in \mathcal _ } \Gamma (1+\rho _+d__}^} )\Big ] S_}[F](},},})\nonumber \\ \end$$

(3.46)

holds for all \((},},}) \in \Omega _[\mathcal ]\), where \(d^}_} = \min \}: \_0} \}__0\in \mathcal (K_) } \in \mathcal \}\).

The case of the proposition above where \(m=N\) gives Theorem 1.1. Indeed, if \(F \in C^\infty (\triangle _)\), F lifts to an element of \(C^\infty (K_)\), and the orders of vanishing of F at the relevant facets of \(\triangle _N\) imply the same order of vanishing at the lift in \(K_\).

3.2.2 The Dotsenko–Fateev Case

Fix \(F\in \mathcal ^}(A_)\), where \(\mathcal \) is now a collection of orders for the faces of \(A_\). Recalling the definition of \(\Sigma \texttt(\ell ,m,n)\) given in §2.2:

Proposition 3.4

There exist entire functions

$$\begin I_, \texttt, \} \}_ \in \mathcal (A_) } }[F]: \mathbb ^_},},}} \rightarrow \mathbb \end$$

(3.47)

associated with the \(\texttt\in \Sigma \texttt(\ell ,m,n)\) such that

$$\begin & I_[F](},},})\nonumber \\ & \quad =\sum _\in \Sigma \texttt(\ell ,m,n)} \sum _} \}_ \in \mathcal (A_)} \in \mathcal } \Big ( \Big [ \prod _) \in \texttt} \Gamma (1+\varrho _ + d__} ) \Big ] \nonumber \\ & \qquad \times I_,\texttt \} \}_ \in \mathcal (A_)}}[F](},},})\Big ) \end$$

(3.48)

for all \((},},}) \in V_[\mathcal ]\), where we have abbreviated \(\mathcal _1 \cap \mathcal \), \(\mathcal _2 \cap \mathcal \), and \(\mathcal _3 \cap \mathcal \) as S or Q as appropriate.

Proof

Follows from Proposition 2.7 and Proposition 3.2. \(\square \)

Consequently, \(I_[F]:V_[\mathcal ]\rightarrow \mathbb \) admits an analytic continuation \(}_[F]:}_[\mathcal ]\rightarrow \mathbb \), where

$$\begin }_[\mathcal ] = \mathbb ^_},},}} \Big \backslash \bigcup _}\}_ \in \mathcal (A_)} } \bigcup _}\bigcup _ \ + d__} \in \mathbb ^ \}.\nonumber \\ \end$$

(3.49)

Note that \(}_[F] \supseteq \cap __} }_[F]^\sigma \), as every functional \((},},})\mapsto \varrho _(},},})\) has the form \(\rho _(}^\sigma ,}^\sigma ,}^\sigma )\) for some \(\sigma \in \mathfrak _\) and \(\\in \mathcal _\).

As a corollary of the previous proposition, there exists a function

$$\begin I_}[F]:\mathbb ^_},},}} \rightarrow \mathbb \end$$

(3.50)

such that, for all \((},},}) \in V_[\mathcal ]\),

$$\begin & I_[F](},},})\nonumber \\ & \quad = \Big [ \prod _} \prod _ \Gamma (1+\varrho _ + d__}^} ) \Big ] I_}[F](},},}),\nonumber \\ \end$$

(3.51)

where S, Q vary over subsets of \(\mathcal _1=\\), \(\mathcal _2=\\), and \(\mathcal _3=\\), depending on \(x_0\).

The \(m=N\) case of the previous proposition is Theorem 1.3.

3.3 A Simple Identity

For each permutation \(\sigma \) of \(\\). Let

$$\begin (\ell ',m',n') = (\ell ,m,n) & (\sigma = 1), \\ (n,m,\ell ) & (\sigma = (0\; 1) ), \\ (\ell ,n,m) & (\sigma = (0\; \infty ) ), \\ (m,\ell ,n) & (\sigma = (1\; \infty ) ), \\ (n,\ell ,m) & (\sigma = (0\;1\; \infty ) ), \\ (m,n,\ell ) & (\sigma = (1\;0\; \infty ) ). \end\right. } \end$$

(3.52)

In other words, if the elements of \(\\) label the vertices of a triangle and the edges are labeled accordingly—that is, ‘\(\ell \)’ labels the edge between 0 and \(\infty \), ‘m’ labels the edge between 0 and 1, and ‘n’ labels the edge between 1 and \(\infty \)—then \((\ell ',m',n')\) is the permutation of \((\ell ,m,n)\) resulting from applying \(\sigma \) to the triangle and reading off the new labels.

Let \(}_\sigma :\mathbb P^1\rightarrow \mathbb P^1\) denote the unique automorphism acting on \(\\) via \(\sigma \). These are

$$\begin & }_1(z) =z, \quad }_(z) = 1-z, \quad }_(z) = \frac, \quad }_(z) = -\frac, \end$$

(3.53)

$$\begin & }_(z) = \frac, \qquad }_(z) = \frac. \end$$

(3.54)

Let \(\sigma ^}:\mathbb ^\rightarrow \mathbb ^\) denote the affine map

$$\begin \sigma ^}(},},}) = (},},}) & (\sigma = 1), \\ (},},}) & (\sigma = (0\; 1) ), \\ (-2-}-}-2 }\lrcorner \textbf,},}) & (\sigma = (0\; \infty ) ), \\ (}, -2- }- }-2}\lrcorner \textbf, }) & (\sigma = (1\; \infty ) ), \\ (-2-}-}-2 }\lrcorner \textbf,},}) & (\sigma = (0\;1\; \infty ) ), \\ (},-2-}-}-2 }\lrcorner \textbf,}) & (\sigma = (1\;0\; \infty ) ), \end\right. } \end$$

(3.55)

where \(}\lrcorner \textbf\in \mathbb ^N\) has jth component \(\sum _ \gamma _\). Let \(\textrm\in \mathfrak _\) denote the permutation that reverses the order of the elements in each of the sets \(\\), \(\\), and \(\\). Let \(|\sigma |\) denote the order of \(\sigma \).

Proposition 3.5

If \((},},})\in }_\), then \(\sigma ^}(},},}) \in }_\), and if \((},},})\in }_\), then \(\sigma ^}(},},}) \in }_^^}\), and

$$\begin \begin }_[1](},},})&= }_[1]( \sigma ^} (},},}))^^}, \\ }_[1](},},})&= }_[1]( \sigma ^} (},},}))^^} \end \end$$

(3.56)

for all \((},},})\in }_\).

Proof

It can be checked case-by-case that

$$\begin & \ \circ \sigma ^}: \bullet \in \,S,Q\text \} \nonumber \\ & \quad = \: \bullet \in \,S,Q\text \}, \end$$

(3.57)

where on the left-hand side (S, Q) varies over appropriate pairs of subsets of \(\\), \(\\), and \(\\) and on the right-hand side (S, Q) varies over appropriate pairs of subsets \(\\), \(\\), and \(\\), depending on \(\bullet \). It can be seen from eq. (3.57) that

$$\begin }_ = (\sigma ^})^( }_). \end$$

(3.58)

The case of \(}_\) is similar but more complicated.

Equation (3.56) can be proven for \((},},})\in \Omega _\) by way of a change-of-variables by substituting \(x=}_}(y)\). The full result follows via analytic continuation. \(\square \)

3.4 An Imperfect Alternative

For \(\texttt \in \\) and \(r>0\), let \(\Gamma _,\pm ,r}:(0,1)\rightarrow \mathbb \) be defined by

$$\begin \Gamma _(t) = t\pm irt & (t\in (0,1/3)), \\ t\pm ir/3 & (t\in [1/3,2/3]), \\ t\pm ir/3 \mp ir(t-2/3) & (t\in (2/3,1)), \end\right. } \end$$

(3.59)

\(\Gamma _(t) = \Gamma _(1-t)^\), and \(\Gamma _(t) = 1- \Gamma _(1-t)\). Note that the images of these contours are permuted among themselves by the transformations \(}_\sigma \) above.

Fig. 10

The contours \(\Gamma _\), \(\Gamma _\), \(\Gamma _\), \(\Gamma _\). Cf. [8, Figure 16]. (For our purposes, the contours drawn by Dotsenko & Fateev approach \(\pm \infty \) with imaginary part too small. This is why our \(\Gamma _,\pm ,r}\) look different for \(\texttt\ne [0,1]\)

Suppose that \(F\in \mathbb [x_1,x_1^,\ldots ,x_N,x_N^]\). For any compact \(}\Subset \mathbb \) with nonempty interior, let \(O=O[F,}]\) denote the set, which depends on \(\ell ,m,n\in \mathbb \), though we suppress this dependence notationally, of \((},})\in \mathbb ^\) such that

$$\begin & \int _} \cdots \int _} \Big [ \int _} \cdots \int _} \Big [ \int _} \cdots \int _} \nonumber \\ & \quad \Big ( \prod _^N z_j^(1-z_j)^ \Big ) \prod _ (z_k-z_j)^} F_0 \,\textrmz_N\cdots \,\textrmz_ \Big ] \,\textrmz_\cdots \,\textrmz_ \Big ] \,\textrmz_\ell \cdots \,\textrmz_1 \nonumber \\ \end$$

(3.60)

is an absolutely convergent Lebesgue integral whenever \(\gamma _ \in }\) for all \(j,k \in \\) with \(j<k\), for every monomial \(F_0\) in F. In the definition of the integral above, we are defining the integrand such that the branch cuts are not encountered. For such \((},},})\),

$$\begin (},},})\in }_[F], \end$$

(3.61)

and the integral in Eq. (3.60) is equal to \(}_(},},})[F]\), assuming that we choose our branches appropriately. The latter part of this statement can be proven by checking that the integral defined above depends analytically on its parameters and agrees with \(I_(},},})[F]\) for \((},},})\in V_[F]\), which in turn is proven via a contour deformation argument.

The set O is nonempty, open, and contains an affine cone. If

\(\alpha _j\) has sufficiently large real part for \(j\in \mathcal _1 \cup \mathcal _2\) and sufficiently negative real part for \(j\in \mathcal _3\), and

\(\beta _j\) has sufficiently large real part for \(j\in \mathcal _2\cup \mathcal _3\) and sufficiently negative real part for \(j\in \mathcal _1\),

then \((},})\in O[F,}]\), where what “sufficiently large” means depends on \(}\). Consequently, given any subset \(S\subseteq \mathfrak _\ell \times \mathfrak _m\times \mathfrak _n\), the set \(O^\) defined by

$$\begin O^ = \},}) \in \mathbb ^: (}^\sigma ,}^\sigma )\in O[F^\sigma , }^\sigma ] \text \sigma \in S\} \end$$

(3.62)

is open and nonempty. If \(}\) contains e.g. \(-1\), then \(O[F,}]\) contains some \((},})\) such that \((},},})\notin V_[F]\). So, Eq. (3.60) gives us an alternative definition of \(}_(},},})[F]\) for some range of parameters.

Proposition 3.6

Consider \(\bullet \in \\) and \(j,k\in \mathcal _\bullet \) with \(j<k\) and \(|j-k|=1\). Suppose that \(\gamma _\in \mathbb \). Let \(\tau \in \mathfrak _\) denote the transposition swapping j, k. Then,

$$\begin & I_[F](},},}) - I_[F](},}, })^\tau \nonumber \\ & \quad = \int _} \cdots \in