The singularities of Selberg-- and Dotsenko--Fateev-like integrals

We discuss the meromorphic continuation of certain hypergeometric integrals modeled on the Selberg integral, including the 3-point and 4-point functions of BPZ's minimal models of 2D CFT as described by Felder and Silvotti and Dotsenko and Fateev (the ``Coulomb gas formalism''). This is accomplished via a geometric analysis of the singularities of the integrands. In the case that the integrand is symmetric (as in the Selberg integral itself) or, more generally, what we call ``DF-symmetric,'' we show that a number of apparent singularities are removable, as required for the construction of the minimal models via these methods.


Introduction
Let N = {(x 1 , . . . , x N ) ∈ [0, 1] N : x 1 ≤ · · · ≤ x N } (1) denote the standard N -simplex, which we consider as a subset of C N . We study in this note Selberg-like integrals, by which we mean definite integrals of the form for N ∈ N + , F ∈ C ∞ ( N ), and α = {α j } N j=1 , β = {β j } N j=1 , γ = {γ j,k = γ k,j } 1≤j<k≤N ⊂ C such that the integrand above is absolutely integrable on N . Integrals of this form are relevant to an array of topics in mathematical physics [FW08]. Here, we discuss a particular application to the Coulomb gas formalism (a.k.a. "free field realization," "Feigin-Fuchs representation," etcetera) of 2D CFT [DF84][DF85a] [DF85b] [FS89][PM97, Chp. 9] [FW08]. This approach of Dotsenko-Fateev to the construction of the "minimal models" of Belavin-Polyakov-Zamolodchikov (BPZ) [BPZ84] has been the subject of substantial interest, but it appears that it has not yet been placed on entirely rigorous mathematical footing. The construction in [FS89] [FS92] of the 3-point coefficients of the (1, s)-and (r, 1)-primary fields and their descendants in the minimal models is satisfactorily rigorous, but it has remained an open problem to handle the rest of the primary fields at a similarly satisfactory degree of rigor. From our perspective, the issue is an insufficient treatment of the meromorphic continuation of Selberg-like integrals, which are instead treated somewhat formally in the original works.
The issue is that Dotsenko & Fateev (DF) take some of the γ's to be −1 -see e.g. [DF85a, Appendix A] [FS92,p. 27] [FW08,§2] -and then the integrand above is, say for F = 1, no longer integrable over the integral's domain. As a consequence, the integrals in [DF85a, Appendix A] are formal. One way of making sense of them is via meromorphic continuation in the exponents of the integrand. Dotsenko and Fateev suggest this, but they do not prove that a suitable meromorphic continuation exists, nor do they discuss the singularities of the extension in sufficient detail to justify their manipulations. A construction of Kanie-Tsuchiya [KT86a] [KT86b] rediscovered by Mimachi-Yoshida [MY02][MY03] [Yos03] yields the existence of some meromorphic continuation defined for almost all values of the exponents. But, this extension is not quite sufficient for our purposes: it has removable singularities that, while removable, are nontrivial to actually prove removable. In particular, the Kanie-Tsuchiya construction has an apparent (but isolated, in a suitable sense) singularity at γ = −1 (see [KT86b, §5, above Thm. 5]), along with at a few other problematic affine hyperplanes in the space of possible parameters.
Most of the rigorous work on the analysis of integrals of Dotsenko-Fateev type -see e.g. [FK15a][FK15b] [FK15c][FK15d] [LV19] for some recent work -focuses on screened multipoint correlation functions with at most one screening charge screening per insertion point. Such integrals are related to the N = 1 case of S N (α, β, γ). Not much has been done about the N > 1 case. Moreover, while a fair amount of work has gone into the study of general hypergeometric integrals associated to hyperplane arrangements -the literature on this topic is large, so we just cite [Var95][AK11] -it does not seem possible to deduce the specific, concrete results below from results in the current literature.
Our first goal is to prove that S N [F ] can be analytically continued to a subseṫ Ω N ⊆ C 2N +N (N −1)/2 (9) having full measure in C 2N +N (N −1)/2 . In order to describe precisely the structure of the singularity at C 2N +N (N −1)/2 \Ω N , we introduce some terminology. Let T(N ) denote the collection of maximal families I of consecutive subsets I {0, . . . , N + 1} such that • 2 ≤ |I| ≤ N + 1 for all I ∈ I and • if I, I ∈ I satisfy I ∩ I = ∅, then either I ⊆ I or I ⊆ I. "T" stands either for "tree" in "full binary trees" or "Tamari" in Tamari lattice [Tam62] [Gey94], and the elements of T(N ) can be thought of as specifying the valid ways of adding a maximal number of nonredundant parentheses to a string of N + 2 identical characters. There are #T(N ) = C N +1 such ways, where C N +1 is the (N + 1)st Catalan number. To each I ∈ I, we associate the facet for all (α, β, γ) ∈ Ω N .
Here, Γ : C\{−n : n ∈ N} → C is the Gamma function. As a consequence of the theorem, there exists an entire function S N ;reg [F ] : C 2N +N (N −1)/2 → C such that for all (α, β, γ) ∈ Ω N . The setΩ N contains all elements of C 2N +N (N −1)/2 lying outside of a locally finite arrangement of affine hyperplanes. Consider Example. The simplest case is when N = 1 and F = 1 identically, when the Selberg integral is given by defined initially for α, β > −1 via the definite integral and then extended meromorphically via the formula on the right-hand side above (or via another method). This is Euler's β-function. One method of meromorphic continuation involves the "Pochhammer contour" b −1 a −1 ba ∈ π 1 (C\{0, 1}), where a, b are the generators of π 1 (C\{0, 1}) corresponding to one (say, counterclockwise) circuit around each of 0, 1 respectively.
The proof below is lower-brow than the twisted homological constructions of [KT86a,§5][KT86b] and Aomoto [Aom87], as it is based on the method described in [Var95,Chp. 10]. This involves the geometric analysis of the singularities of the Selberg(-like) integrand. The key observation is that if the N -simplex is blown up to the N -dimensional associahedron [Sta63][MSS02, §1.6] [Pos09] (see Figure 2, Figure 6), then the Selberg integrand -which is not polyhomogeneous on Nbecomes one-step polyhomogeneous (a.k.a "classical") on the resolution. See §2 for details. This observation appears, in an essentially equivalent form (albeit with different terminology), already in [KT86a][KT86b] [MY03], though the term "associahedron" does not appear there. Closely related observations have appeared in the physics literature [Miz17][CKW18][CMT19] [Miz20].
integrand at the corresponding faces. Each I ∈ T(N ) is associated with a minimal facet of the associahedron, and the I ∈ I are associated with the faces containing that facet. Thus, we have a geometric interpretation of each of the terms in eq. (11).
Note thatΩ N does not contain (α, β, γ) with γ j,k = −1 for |j − k| = 1, so the theorem above is insufficient for the construction of the minimal models. Moreover, the theorem cannot be sharpened while maintaining generality. Indeed, the proof of the theorem shows that if F > 0 everywhere in N (including the boundary), then for any (α, β, γ) ∈ R 2N +N (N −1)/2 for which both of • γ j,k, * ∈ Z ≤−(k−j) for precisely one pair of j, k ∈ {0, . . . , N + 1} with j < k, • γ j,k, * > −(k − j) for all other j, k hold, as for such (α, β, γ) the quantity S N ;reg [F ](α, β, γ) is proportional to a convergent integral of a positive integrand over the corresponding face of the associahedron. Consequently, S N [F ] : Ω N → C cannot be analytically continued to the complement of any strictly smaller collection of hyperplanes than that in eq. (13).
In this case, we simply write We now consider F ∈ C[x 1 , . . . , x N ] S N , i.e. symmetric polynomial F . This case includes, of course, Selberg's original example, in which F = 1, as well as the 3-point coefficients of the (1, s)-and (r, 1)-primary fields and their descendants in the BPZ minimal models. The computation of such integrals is listed as an open problem in [KT86a]. Below, we will introduce a more general notion of "DF-symmetric" Selberg-like integrals, this including the other 3-point coefficients. But, for the purposes of an introductory discussion we focus on the -already interesting -symmetric case. The integral eq. (23) is defined initially on the subset U N [F ] ⊂ C 3 α,β,γ given by which contains Example. Consider F = 1, i.e. the Selberg integral. In this case, Selberg proved in [Sel44] that See [FW08] for a review of the history of this result.
The Γ(2+d j +α+β+(N +j−2)γ) term in the denominator of eq. (29) implies thatS N [F ](α, β, γ) = 0 for all When constructing the 3-point coefficients of the BPZ minimal models, this is one mechanism preventing the fusion of (0, s)-primary fields (which are not included in the model) with the primary fields that are included. In BPZ's terminology, this is the truncation of the operator algebras, as originally derived via the constraint of OPE associativity -see [ The proof of the theorem above consists of three steps: (1) The first step is the removal of the fictitious singularities ofṠ N [F ](α, β, γ) only in γ (as required e.g. in the Coulomb gas formalism with both kinds of screening charges). The basic idea is to employ the relation -which can be found in a heuristic form in [DF85a, Ap. A] -between the symmetrization of S N [F ](α, β, γ) and the "DF-like" integral where N = [0, 1] N . We can analytically continue I N [F ] via an argument similar to that used to prove Theorem 1.1. Unlike that of S N [F ](α, β, γ), this extension has no singularities associated with hyperplanes of constant γ. The true singularities of the extension of S N [F ](α, β, γ) associated with hyperplanes of constant γ show up in the relation with the extension of I N [F ](α, β, γ).
(2) The second step removes the other unwanted singularities away from the loci of two or more unwanted singularities, via some identities proven via Aomoto [Aom87] in the F = 1 case (and [DF85a, Ap. A], at a physicist's level of rigor). The use of these identities for computing the original Selberg integral is sketched in [FW08]. It seems there cannot be a similar computation of S N [F ] in the deg F > 1 case, so a statement about the singularities is the best we can do. The simplex N ⊂ R N can be thought of as a subset of (C\{0, 1}) N = (CP 1 \{0, 1, ∞}) N (33) via the embedding R → C → CP 1 , and the rough idea of this step of the proof is to relate the integrals above to the result of replacing N with L N N for L one of the six linear fractional transformations preserving CP 1 \{0, 1, ∞}. Only three of these are essentially different, and one of these three is just the identity and therefore uninteresting. The other two integrals each have meromorphic extensions with different manifest singularities. Using Proposition 4.2, these functions can be related to each other, and this can be used to remove most of the apparent singularities that are not present in all three functions. Some singularities are present in the relations between the integrals, and these cannot be removed.
(3) The third step is the application of Hartog's theorem to remove the remaining removable singularities, which now lie on a codimension two subvariety of C 3 .
This argument is carried out in §4.1. The version more relevant to [DF85a] (with the additional steps needed) is in §4.2. We call I N [F ] a "DF-like" integral because similar integrals appear, albeit at a somewhat formal level, in [DF85a]. A similar construction appears in [Fel89]. Let ΣT(N ) denote the collection of maximal collections I of pairs (x 0 , S) of x 0 ∈ {0, 1} and nonempty subsets S ⊆ {1, . . . , N } such that, given (x 0 , S), (x 0 , Q) ∈ I, either S ⊆ Q or Q ⊆ S.
for all (α, β, γ) for which the left-hand side is a well-defined integral.
If desired, it is possible to replace the sines with Γ-functions with appropriate integral shifts.
Example. When F = 1, Dotsenko and Fateev claim in [DF85a, Eqs. A.8, A.35] 1 that the integral above is given bẏ for each choice of sign.

Associahedra
We use the term 'mwc' to mean manifold-with-corners in the sense of Melrose -e.g. [Mel][HMM97], these possessing C ∞ -structure. In this section, we define the mwcs that will be used to resolve the singularities of Selberg-and Dotsenko-Fateev-like integrands: • in §2.1, we define the associahedra K ,m,n , used to meromorphically continue the Selberg-like integrals, and • in §2.2 we define the mwcs A ,m,n , used to meromorphically continue the DF-like integrals. Since K 0,N,0 is the usual N -dimensional associahedra, we refer to the mwcs defined below as associahedra as well, hence the title of this section. If M is a mwc, we use F(M ) to denote the set of faces of M , where by faces we mean only the boundary hypersurfaces. We use "facet" to refer to the higher codimension boundary components.
It is worth comparing Melrose's notion of mwc to that of polyhedron. A mwc is locally a polyhedron, but the converse is not true, as the basic requirement of M being locally diffeomorphic to a relatively open neighborhood of [0, ∞) N means that every facet f M is the intersection of at most N faces. While the (closed) ball, tetrahedron, cube, and dodecahedron are all mwcs, the octahedron and icosahedron are not. It is critical for the argument in §3 that the associahedra A ,m,n and K ,m,n are not just polyhedra, but rather mwcs. Thus, the notion of "mwc" used here plays a similar role to that of "polyhedra in general position" in [Var95, §10.7], but they are not equivalent. For the purposes of this paper, we find it more natural (and technically simpler, as it avoids the need for polyhedral realizations) to use the language of mwcs.
We keep track of the full C ∞ -structure of these mwcs below. Were it required, we could keep track of C ω -(i.e. real analytic) structure, but since this would require going somewhat beyond the existent literature on mwcs, and since this level of precision is not needed for the rest of the paper, we will restrict ourselves to the smooth category.
where ff = bd −1 (f) is the front face of the blowup. Then, given bdfs x F ∈ C ∞ (M ; R + ) of the faces F ∈ F(M ), we can choose bdfs x [F;f∩F] , x ff of the faces of [M ; f] such that, for each F ∈ F(M ), and, if f ⊆ F, then This follows from the analogous result for blowing up a facet of [0, ∞) N . Note that because M is a mwc and not just a polyhedron, if F 1 , . . . , F d ∈ F(M ) are distinct faces with ∩ δ F δ = ∅, then the connected components of ∩ δ F δ are codimension d facets of M . (The 2D lens is an example of a mwc with two faces whose intersection is disconnected.) If U is an open subset of a mwc, then U can be considered as a mwc in its own right. We will say that some function x ∈ C ∞ (U ; [0, ∞)) is a bdf in U of F ∈ F(M ) if it is a bdf of the face F ∩ U of U , assuming that F ∩ U = ∅, in which case it is automatically a face of U . Let R t = R t ∪ {−∞, +∞} denote the "radial" compactification of R. This is a (C ∞ -)manifold-with-boundary, with 1/t serving as a bdf for {∞} in {t > 0} and −1/t serving as a bdf for {−∞} in {t < 0}.
2.1. The Associahedra K ,m,n . We now define the mwc K ,m,n for , m, n ∈ N not all zero. The blowup procedure below is a generalization of that in [KT86a]. We begin with the set where N = + m + n. This is a compact sub-mwc of R N . Naturally, ,m,n ∼ = ,0,0 × 0,m,0 × 0,0,n .
If , n = 0, in which case m = N , then ,m,n is just the standard N -simplex N .
The dimension of f 0,I is given by For notational simplicity, if I 0 ⊆ I is I with the singletons removed, then we define f I 0 = f 0,I . Thus, f ∅ denotes the "bulk" of ,m,n , and the faces of ,m,n are of the form f {I} for I a consecutive pair. Rephrasing eq. (57), As a bdf of f {I} for I = {k mod Z/(N + 3)Z, k + 1 mod Z/(N + 3)} when k ∈ {1, . . . , N + 1}, we can take In other words, F ,m,n is the set of facets f I for I defining a partition of Z/(N + 3)Z into a single interval of length at least two (not containing any two of 0, + 1, + m + 2) and a number of singletons which are being omitted from the notation.
I.e., we first blow up the elements of the collection F ,m,n;0 (which may be empty, e.g. if , m, n are all nonzero), and then, proceeding from higher to lower codimension, iteratively blow up the lifts of the facets in F ,m,n;d (meaning the closures of the lifts of the interiors). We should check that the blowup eq. (60) is well-defined, which concretely means that, for each d, the blow-ups in the step in which we blow up the lifts of the elements of F ,m,n;d commute. This can be done via a somewhat tedious inductive argument, which we only sketch.
When the time has come to blow up the facets f = f in the lifted F ,m,n;d , their intersection isif nonempty -either a point (which we denote K 0,0,0 ) or else an associahedron K ∩,m∩,n∩ (which will not change upon performing further blowups) of dimension < N , and a neighborhood thereof is diffeomorphic to with f corresponding to {t = 0} and f corresponding to {t = 0}; the blowups of these two faces in the product above commute, with the result being naturally diffeomorphic to In order to prove the claimed decomposition, eq. (61), it is first useful to note when f ∩ f = ∅. If I, I satisfy |I| = N − d + 1 = |I | and I ∩ I = ∅, then the corresponding facets and since this is blown up in an earlier stage of the construction, f and f cannot intersect. So, if our two facets f, f to be blown up have nonempty intersection, then they must be the lifts of f {I} and f {I } for I, I satisfying As seen inductively, the lift of this facet after performing the blow-ups so far is K ∩,m∩,n∩ , although this is not crucial for the proof that the construction is well-defined. Since this has dimension 2d − N , a neighborhood of this facet in our partially blown-up manifold automatically has the form so it just needs to be checked that f, f sit inside of this in the expected way. Indeed, the projections of f, f onto the [0, 1) 2N −2d factor are necessarily transversal. With the fact that they both have dimension d, this implies that we can decompose such that f corresponds to {t = 0} and f corresponds to {t = 0}. This completes our sketch. We now discuss the combinatorial structure of K ,m,n . All of the faces of ,m,n are in F ,m,n;N −1 , so every face of K ,m,n is the front face of one of our blowups. So, the faces of K ,m,n are in bijection with the elements of F ,m,n and thus with I as above. Such a subset is uniquely specified by its endpoints j, k ∈ Z/(N + 3)Z, since only two consecutive subsets of Z/(N + 3)Z have the same endpoints as I, namely I itself and I ∪ {j, k}, and the latter contains two of 0, + 1, + m + 2. Let J ,m,n denote the set of unordered pairs {j, k} arising in this way. For {j, k} ∈ J ,m,n , let I(j, k) = I(k, j) denote the unique consecutive subset of Z/(N + 3)Z having these endpoints and containing at most one member of {0, + 1, + m + 2}. For such j, k, let F j,k = F k,j denote the corresponding face of K ,m,n , and let x F j,k = x F k,j denote a bdf of that face constructed inductively as in the introduction to this section. (Note that these bdfs may depend on the particular order in which the elements of the F ,m,n;d are blown up.) There Example. Consider the case N = 2. Then, up to essential equivalence, the cases to consider are K 1,1,0 and K 0,2,0 . These are depicted in Figure 3. The mwc K 1,1,0 is identical to A 1,1,0 ; in §2.2 we introduce notation for labeling the faces of the A ,m,n , and this notation appears in Figure 4 alongside that used for the K ,m,n . We have introduced an additional notation for the faces of K ,m,n , indicating I in the subscript using the following conventions: • The elements 0, + 1, + m + 2 ∈ Z/5Z are depicted using a '•,' and 0 is omitted if not included in I. • The other elements of Z/5Z are depicted using a '•.' • Except for 0, the elements of Z/5Z are depicted in order. If 0 is to be depicted, it is listed either first or last. The elements included in I are enclosed in parentheses. Figure 3. The associahedra K 1,1,0 (left) and K 0,2,0 (right), realized as polyhedra roughly in accordance with the blowup procedure. In the first figure, the horizontal axis is roughly w 1 = 1/(1 − x 1 ), increasing to the right. In the second figure, it is just (roughly) x 1 . In both figures, the vertical axis is (roughly) x 2 .
Given any face F of K |S|,|Q|,|R| , forg * x F vanishes to first order at each face in forg −1 (F).
This can be proven by inducting on the number of blowups. Figure 5. The mwc K 1,2,0 , with labeled faces, realized as a polyhedron roughly in accordance with the blowup procedure. As above, Figure 6. The mwc K 0,3,0 , with labeled faces, realized as a polyhedron roughly in accordance with the blowup procedure. The faces in the line of sight are Fig. 5.2], where the full blowup procedure is depicted.
Proposition 2.1. Suppose that µ ∈ C ∞ ( ,m,n ; Ω ,m,n ) is a strictly positive smooth density on ,m,n . Then, the lift of µ to K ,m,n has the form for a strictly positive µ ∈ C ∞ (K ,m,n ; ΩK ,m,n ). Here, for j, k ∈ Z/(N + 3)Z, we use the notation In the product, each unordered pair is counted only once.
Proof. We recall the following lemma: The proposition follows from an inductive application of the lemma, once we note that |j − k| is the codimension of F j,k .
Proposition 2.2. The Lebesgue measure on R N , which defines a strictly positive smooth density on • ,m,n , has the form for µ ∈ C ∞ ( ,m,n ; Ω ,m,n ) a strictly positive smooth density on ,m,n .
We now record the results of lifting the factors x i , 1 − x i , and x j − x k comprising the Selberg integrand to K ,m,n . Beginning with the first two cases: If N = 1, then these are all trivial to prove. By applying the universal property of the associahedra, the N ≥ 2 case follows from the N = 1 case. In a similar manner, by working out the case of K 0,2,0 in detail and applying the universal property, we get, for k > j: Indeed, in the case of , n = 0 and m = 2, this says that Indeed, if we construct K 0,2,0 by first blowing up F 1,3 and then blowing up F 2,4 , we get so that x F 1,3 x F 2,3 x F 2,4 = x 2 − x 1 , on the nose. On the other hand, if we reverse the order of the blowups, then we get From this, we can deduce the following.
Proof. Each ρ j,k is an affine function of α, β, γ, so it suffices to check 2N + N (N − 1)/2 + 1 cases, the case when all three of α, β, γ are zero and 2N + N (N − 1)/2 cases where the triple (α, β, γ) ranges over a basis of C 2N +N (N −1)/2 . Write where ρ j,k is the linear part of ρ j,k . Thus, we want to show that, upon lifting to K ,m,n , with it sufficing to check eq. (97) on a basis of C 2N +N (N −1)/2 .
Let T( , m, n) denote the collection of maximal families I of consecutive subsets I Z/(N + 3)Z such that • 2 ≤ |I| ≤ N + 1 for all I ∈ I, • no two of 0, + 1, + m + 2 are in any I ∈ I together, and • if I, I ∈ I satisfy I ∩ I = ∅, then either I ⊆ I or I ⊆ I.
The elements of T( , m, n) can be thought of as specifying valid ways of adding parentheses to group together the elements of Z/(N + 3)Z without grouping any of 0, + 1, + m + 2 together. The minimal facets of K ,m,n are in bijective correspondence with the elements of T( , m, n), with the facet corresponding to I.

The Associahedra
A ,m,n . We now define the mwc A ,m,n for , m, n ∈ N not all zero. We begin with the N = + m + n cube N = [0, 1] N t , which we identify with The facets of N we label by sextuples (S, Q, S , Q , S , Q ) consisting of (possibly empty) The 3-cube 3 and the three blowups A 1,1,1 = K 1,1,1 , A 1,2,0 , A 0,3,0 thereof. The ,m,n (S, S , S ) are eight subcubes corresponding to the eight vertices of 3 . One such cube is depicted in red.
As in the previous section, we should check that, for each d = 1, . . . , N , having already blown up F ,m,n;d 0 for d 0 < d, the blowups of the closures of the lifts of the interiors of all of the F ∈ F ,m,n;d all commute. One way to see this is to split Once we have established that blowing up F ,m,n;0 , · · · , F ,m,n;d−1 is fine, then naturally, with the left-hand side being well-defined if the right-hand side is. Thus, it suffices to check that the blowups [ ,m,n (S, S , S ); F ,m,n;0 ; · · · ; F ,m,n;d ] are all well-defined. To see this, identify ,m,n (S, S , S ) = 0, and note that the blowup prescription is just that of performing the total boundary blowup [HMM97] on each of the three factors. (Note that this is not the same as the total boundary blowup of the product of the factors.) Here, The faces of A ,m,n are in bijection with the elements of F ,m,n . We label the faces of A ,m,n as follows: and given any face F of A |S|,|Q|,|R| , the pullback forg * x F vanishes to first order at each face F 0 (109) This is the "universal property" of the A ,m,n . Via the decomposition in eq. (107), it follows from the corresponding universal property of the total boundary blowup of a product, which is essentially given by Proposition B.2.
Proposition 2.4. Suppose that µ is a strictly positive smooth density on ,m,n . Then, the lift of µ to A ,m,n has the form for a strictly positive smooth density µ ∈ C ∞ (A ,m,n ; ΩA ,m,n ) on A ,m,n .
Proof. Follows via induction on the number of blowups, as in the proof of Proposition 2.1.
Proposition 2.5. The Lebesgue measure on R N , which defines a strictly positive smooth density on • ,m,n , has the form for some strictly positive smooth density µ ∈ C ∞ ( ,m,n ; Ω ,m,n ) on ,m,n . Proof. Consider the neighborhood bd −1 ( ,m,n (S, S , S )) ⊆ A ,m,n . If one of j, k is in S ∪ S ∪ S and the other is not, then the intersection of H j,k with bd −1 ( ,m,n (S, S , S )) is a submanifold disjoint from the boundary and therefore a p-submanifold. It therefore suffices to consider the case when j, k ∈ S ∪ S ∪ S (and the case when neither are in S ∪ S ∪ S is similar). For notational simplicity, we only consider the case when j, k ∈ S . Then, whereH j,k is the closure of {x j = x k } in ([0, 2/3) S × (1/3, 1] S ) tb , which is a p-submanifold [MS08] (this also follows from Proposition B.1). Thus, H j,k ∩ bd −1 ( ,m,n (S, S , S )) is a p-submanifold of bd −1 ( ,m,n (S, S , S )). As the neighborhoods bd −1 ( ,m,n (S, S , S )) cover A ,m,n , the conclusion follows.
This result is illustrated in Figure 9. We now record the results of lifting x i and 1 − x i to A ,m,n , these being derivable via the universal property.
Proof. Follows from the preceding computations, along with Proposition 2.5.
If M is an orientable mwc, we say that a collection P of interior p-submanifolds each of codimension one is consistently orientable if we can choose an orientation on each such that, for any p ∈ M , the subset P ∈P,p∈P does not contain zero, where ++ N * P ⊂ + N * P ⊂ T * M is the induced positively oriented conormal bundle, sans the zero section, and T * M is the extendable cotangent bundle of M . Whether or not this holds does not depend on the choice of orientation of M . Choosing defining functions {y P } P ∈P ⊂ C ∞ (M ; R) for the P ∈ P such that dy P (p) ∈ + N * p P for each p ∈ P , we say that the {y P } P ∈P are consistently oriented defining functions.
Example. In • 0,3,0 = (0, 1) 3 , consider P = {H • 1,2 , H • 2,3 , H • 3,1 }. The functions x 2 − x 1 , x 3 − x 2 , x 1 − x 3 are not consistently oriented defining functions, as but is nonvanishing on ∩ j,k∈S∈P,j<k H j,k . If P consists only of singletons, then this is vacuously true, so it suffices to consider the case when at least one member of P has cardinality > 1. This is certainly true for p ∈ • ,m,n , as dy k,j ∝ dx k − dx j on • ,m,n ∩ H j,k , where the coefficient of proportionality is positive. Indeed, by the results above, for some f j,k ∈ C ∞ (A ,m,n ; R ≥0 ) that is nonvanishing in the interior, so ,m,n , as claimed. This argument does not work for p ∈ ∂A ,m,n , as f j,k may vanish there.
A homogeneity argument can be used to show that, for any p ∈ ∂A ,m,n , there exists a tubular neighborhood T : U → U 0 of a neighborhood U 0 ⊂ F 0 of p in F 0 , where F 0 is the smallest facet containing p, such that the intersections U ∩ P of this neighborhood with the P ∈ P are all vertical subsets, meaning of the form T −1 (B) for some B ⊂ U 0 . This implies that if the 1-form above vanishes at p, then it also vanishes on the fiber of the tubular neighborhood over p and hence somewhere in • ,m,n ∩ H j,k p H j,k .
We illustrate the preceding argument with an example. Consider the case when the only one of , m, n that is nonzero is m, and consider p ∈ ∩ H j,k ∈P H j,k . The set ∩ H j,k ∈P H j,k ⊂ A 0,N,0 (the "small diagonal") is a p-submanifold located away from all but the very first two blowups involved in the construction of A 0,N,0 . Near this p-submanifold, A 0,N,0 is canonically diffeomorphic to and the situation near the opposite corner is similar. In the interior of the front face of that blowup, we can use = x 1 as a bdf and coordinatesx j = x j /x 1 for j = 2, . . . , N as parametrizing the face itself. In terms of these coordinates, locally, and, for 1 ≤ j < k ≤ N , we can write y k,j =ỹ k,j C ∞ (A 0,N,0 ; R + ) forỹ k,j given locally bỹ So, if λ k,j ≥ 0, then 1≤j<k≤N λ j,k dỹ k,j = 0 ⇒ λ j,k = 0 for all k, j. Since the y k,j differ from thẽ y k,j by a (smooth) positive factor, the y k,j have the same property on ∩ H j,k ∈P H j,k . There is a more direct argument using the coordinates in Proposition B.1 (with the decomposition eq. (107)). Namely, using eq. (107), the result follows from the analogous result for [0, 1) N tb . Given any σ ∈ S N , consider the coordinates ,x σ(2) , · · · ,x σ(N ) defined in Proposition B.1, these giving a C ∞ -atlas as σ varies over all permutations. In these coordinate systems, the relevant p-submanifolds are, locally, so have defining functions y k,j = −1 +x j+1 · · ·x k . This satisfies The minimal facets of A ,m,n are in bijective correspondence with the elements of ΣT( , m, n), with the facet corresponding to I.

Meromorphic continuation
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. (2). Let , m, n ∈ N satisfy + m + n = N ∈ N + . Fix a finite collection D of indexed sets for (α, β, γ) ∈ Ω ,m,n [D], where • Ω ,m,n [D] denotes the set of (α, β, γ) for all {d F } F∈F (K ,m,n ) ∈ D, and • F has the form for some F {d F } F∈F (K ,m,n ) ∈ C ∞ (K ,m,n ).
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: • V ,m,n [D] denotes the set of (α, β, γ) ∈ C 2N +N (N −1)/2 for which the integrand in eq. (150) lies in L 1 ( ,m,n , dx 1 · · · dx N ) -that is the set of (α, β, γ) such that for all {d F } F∈F (A ,m,n ) ∈ D, and • F has the form eq. (146) for F {d F } F∈F (A ,m,n ) ∈ C ∞ (A ,m,n ).
In eq. (150), x γ = e πiγ e γ log |x| if x < 0 and x γ = e γ log x if x > 0. We apply abbreviations for Dotsenko-Fateev-like integrals that are analogous to those used for Selberg-like integrals.
Let W ,m,n [•] denote the set of (α, β, γ) ∈ C 3 such that (α, β, γ) ∈ V ,m,n [•] holds when α = α, 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 ,m,n , I ,n,m , I n, ,m , · · · 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 γ 0 = −1, but not allowing the real parts of α − , α + , β − , β + to be too negative). The I ,m,n are related to the Selberg-like integrals S ,m,n 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 [Aom87] and discussed heuristically by Dotsenko-Fateev [DF85a]. 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 S ,m,n = S × S m × S n , which we consider as the subgroup of S N leaving each of I 1 , I 2 , I 3 invariant, where I 1 , I 2 , I 3 are as in the previous section. Given a permutation σ ∈ S ,m,n , let . This relation will be very useful below. More generally, for any σ ∈ S N , let defined for (α, β, γ) ∈ V ,m,n [F ] in the former case or for in the latter case. We will use similar notation for other subsets of C 2N +N (N −1)/2 below, as well as for the meromorphic extensions of S ,m,n [F ] and I ,m,n [F ].
3.1. Some Generalities. Let N ∈ N be arbitrary. For a Fréchet space X , let O(C N ; X ) denote the Fréchet space of entire X -valued functions on C N , where the topology is that of uniform convergence in compact subsets, as measured with respect to each Fréchet seminorm on X , and similarly for X an LF-space. Let E (R N ) denote the LCTVS of compactly supported distributions on R N . By the Schwartz representation theorem, where H m c (R N ) is the set of compactly supported elements of H m (R N ). Let N ∈ N + , k ∈ {0, . . . , N }, and κ ∈ N. For any let, for ρ = (ρ 1 , . . . , ρ k ), which we abbreviate as Here, R N k = [0, ∞) k t 1 ,··· ,t k × R n−k t k+1 ,··· ,t N , and I N,k,κ [ψ](ρ) is defined initially for ρ 1 , · · · , ρ k > −1, for which the right-hand side of eq. (160) is a well-defined integral. Let endowed with the strongest topology such that the inclusions are all continuous.
Proposition 3.1. Suppose that, for each ρ ∈ C k and δ ∈ C κ , we are given some ψ(−; ρ, δ) as in eq. (159), depending entirely on ρ, δ in the sense that the map Then, the function J N,k,κ [ψ] defined by extends to an entire function on C k ρ × C κ δ . Moreover, the function is continuous.
Consequently, I N,k,κ [ψ] admits an analytic continuationİ N,k,κ [ψ] : Ω → C to the set Ω = (C k ρ \ j∈{1,...,k} {ρ j ∈ Z ≤−1 }) × C κ δ , and the maṗ is continuous. If P is a consistently orientable collection of codimension-1 interior p-submanifolds on a mwc M , then, letting x F for F ∈ F(M ) denote a bdf of the face F, it is the case that, for any δ ∈ C P and ρ ∈ C F (M ) , the product exists whenever ρ F > −1 for all F ∈ F(M ). Let κ ∈ N. Suppose that we are given some entire family of compactly supported smooth densities µ(ρ, δ, λ) ∈ C ∞ c (M ; ΩM ) on M . Consider the function defined by Proposition 3.2. Suppose that, for some N 0 ∈ N + , we are given an affine map L = (L 1 , L 2 , L 3 ) : for all ∈ C N 0 for which the left-hand side is defined by eq. (179).
Proof. Pass to a partition of unity subordinate to a system of coordinate charts on M and apply Proposition 3.1 locally.
extends to an entire function C N 0 → C, where # Λ ∈ N + is the maximum size of any set S ⊆ F(M ) of faces such that ∩ F∈S F = ∅ and (L•) F = Λ for all F ∈ S. Indeed, this follows from the proposition above since, for each facet f, is entire.

Specialization to Generic Selberg-and DF-like
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 C 2N +N (N −1)/2 . In the case m = N , this agrees with eq. (13).
In other words, if the elements of {0, 1, ∞} label the vertices of a triangle and the edges are labeled accordingly -that is, ' ' labels the edge between 0 and ∞, 'm' labels the edge between 0 and 1, and 'n' labels the edge between 1 and ∞ -then ( , m , n ) is the permutation of ( , m, n) resulting from applying σ to the triangle and reading off the new labels. Let T σ : CP 1 → CP 1 denote the unique automorphism acting on {0, 1, ∞} via σ. These are Let σ param : C 2N +N (N −1)/2 → C 2N +N (N −1)/2 denote the affine map (t ∈ (0, 1/3)), t ± ir/3 (t ∈ [1/3, 2/3]), t ± ir/3 ∓ ir(t − 2/3) (t ∈ (2/3, 1)), For any compact K C with nonempty interior, let O = O[F, K] denote the set, which depends on , m, n ∈ N, though we suppress this dependence notationally, of (α, β) ∈ C 2N such that is an absolutely convergent Lebesgue integral whenever γ j,k ∈ K for all j, k ∈ {1, . . . , N } 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 (α, β, γ), and the integral in eq. (202) is equal toİ ,m,n (α, β, γ)[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 ,m,n (α, β, γ)[F ] for (α, β, γ) ∈ V ,m,n [F ], which in turn is proven via a contour deformation argument. The set O is nonempty, open, and contains an affine cone. If • α j has sufficiently large real part for j ∈ I 1 ∪ I 2 and sufficiently negative real part for j ∈ I 3 , and • β j has sufficiently large real part for j ∈ I 2 ∪ I 3 and sufficiently negative real part for j ∈ I 1 , where what "sufficiently large" means depends on K. Consequently, given any subset S ⊆ S × S m × S n , the set O S∩ defined by is open and nonempty. If K contains e.g. −1, then O[F, K] contains some (α, β) such that (α, β, γ) / ∈ V ,m,n [F ]. So, eq. (202) gives us an alternative definition ofİ ,m,n (α, β, γ)[F ] for some range of parameters.
Proof. By analyticity, it suffices to prove the result when the quantities above are well-defined Lebesgue integrals. Decomposing ,m,n into !m!n! copies of ,m,n , The right-hand side is which is the right-hand side of eq. (206).
which holds for all N ∈ N and encodes the bijection between S N and the set of possible runs of the bubble sort algorithm. Plugging in ζ = e πiγ , eq. (210) becomes eq. (209).
• Since the prefactor on the right-hand side of eq. by Proposition 3.8. By Proposition 3.11, this extends to an entire function C 3 α,β,γ → C. The product k (γ)(1 − e 2πikγ )(1 − e 2πiγ ) −1 , with its removable singularities removed, vanishes if and only if kγ ∈ N and γ / ∈ N. Thus, S reg ,m,n [F ] extends to an analytic function on where M = max{ , m, n}. Combining these two observations, S reg ,m,n [F ] extends to an analytic function onU ,m, The symmetric case of Proposition 3.9 reads, after multiplying through by 1 − e ±2iγ , Proof. We prove the second claim, and the proof of the first is similar. Suppose that We can apply eq. (249) for = 1 and all pairs of m, n ∈ {0, . . . , N − 1} such that m + n = N − 1.
Combining the plus and minus cases of eq. (249) to eliminate theṠ 1,N −n−1,n [F ] term, We calculate: and where s(t) = sin(πt). So, for (α, β, γ) as above such that none of the trigonometric factors on the right-hand side of eq. (256) vanish, Proof. We begin by defining the following open (and dense) subsets of C 3 : We write By Proposition 4.1, the second line on the right-hand side of eq. (261) defines an entire function. Since Υ 0 extends to an analytic function on U N ;0 ∩ U N ;1 and Υ 1 extends to an analytic function on Q N ;∞ , S N ;Reg [F ] extends to an analytic function on U N ;0 ∩ U N ;1 ∩ Q N ;∞ .
In O N ;0 ∩U 0,N,0 ∩U 0,0,N , Proposition 4.2 gives where This is where • S 1 is the set of hyperplanes that are contained in the complement of one of U N ;0 , U N ;1 , Q N ;∞ , • S 2 is the set of hyperplanes that are contained in the complement of one of U N ;0 , Q N ;1 , U N ;∞ , and • S 3 is the set of hyperplanes that are contained in the complement of one of Q N ;0 , U N ;1 , U N ;∞ .
so that S N ;Reg [F ] defines an analytic function on U = C 3 \∪ H∈H H. Observe that every H ∈ H is an affine subspace of C 3 of complex codimension two or three (since S 1 ∩ S 2 ∩ S 3 = ∅), and the collection H is locally finite. Hartog's theorem therefore implies that S N ;Reg [F ] analytically continues to the entirety of C 3 .
This completes the proof of Theorem 1.2.
It appears that all of the poles that could be present are present. The apparent zeroes of S 2 [F ](α, 1/2, 1/3) in the depicted range of α have been marked with dotted black lines and numerically computed to be ≈ −2.48503 for F = x 2 and ≈ −3.06833, −3.57013, and −4.08562 for F = y 2 .
Proof. A factor of j∈Q∪Q 0 x j appears on the right-hand side of eq. (323) to the power which is, by the binomial theorem, +1 if Q 0 = ∅ and 0 otherwise. Thus, Q 0 ⊆{M +1,...,N } x F Q∪Q 0 ,N = j∈Q x j .