On Some Special Cases of Gaiotto's Positivity Conjecture

We prove a conjecture of D. Gaiotto on positivity of inner products arising in studying Landau-Ginzburg boundary conditions in the 1-dimensional case, and in special cases in higher dimensions, for 3d free hypermultiplets.


Introduction
Suppose that V = N ≥0 F N V is a complex vector space with an ascending filtration, such that dim F N V < ∞ for all N .The associated graded space to V is gr where gr N V := F N V /F N −1 V .If V , V ′ are two such filtered vector spaces, then a linear map ϕ : V ′ → V is filtered if ϕ(F N V ′ ) ⊂ F N V .In this case, ϕ gives rise to the associated graded map gr ϕ : gr V ′ → gr V .Note also that gr V may be viewed as a filtered space with F N gr V := ⊕ N j=0 gr j V .Let ( , ) be a bilinear or sesquilinear form on a filtered space V .Let us say that ( , ) is filterednondegenerate if its restriction ( , ) N to the subspace F N V is nondegenerate for every N ≥ 0. In this case we have unique filtered isomorphisms ϕ + , ϕ − : gr V → V with gr ϕ + = gr ϕ − = Id such that for all N , ϕ ± (gr N V ) = (F N −1 V ) ⊥ ± , where for Y ⊂ F N V , Y ⊥ + , Y ⊥ − denote the left, respectively right, orthogonal complement of Y under ( , ) (the Gram-Schmidt biorthogonalization maps).Thus we get a bilinear or sesquilinear form ⟨ , ⟩ on gr V given by ⟨v 1 , v 2 ⟩ := (ϕ + v 1 , ϕ − v 2 ).Moreover, this form is nondegenerate when restricted to each homogeneous component gr N V of gr V , and different homogeneous components are orthogonal.
We will consider this construction when V = C[z 1 , . . ., z n ] and the filtration is given by degree of polynomials, so that F N V = C[z 1 , . . ., z n ] ≤N , the space of polynomials of degree at most N .Namely, let W ∈ C[z 1 , . . ., z n ] be a homogeneous polynomial of degree d (in physical terminology, a Landau-Ginzburg superpotential).Consider the sesquilinear form on C[z 1 , . . ., z n ] defined by the formula It is evident that this form is Z/d-invariant, i.e., (P, Q) W = (P * , Q * ) W , where P * (z) := P e 2πi d z .
This paper is a contribution to the Special Issue on Basic Hypergeometric Series Associated with Root Systems and Applications in honor of Stephen C. Milne's 75th birthday.The full collection is available at https://www.emis.de/journals/SIGMA/Milne.html

P. Etingof
It is clear that the form ( , ) 0 is filtered-nondegenerate, as the monomial basis is orthogonal with respect to ( , ) 0 , with positive norms.Hence ( , ) W is filtered-nondegenerate for sufficiently generic W (outside the zero sets of a countable collection of polynomials on S d C n * regarded as a real vector space).Moreover, while ( , ) W is not, in general, Hermitian-symmetric, we have Hence if for some W the form ( , ) W is filtered-nondegenerate then the resulting homogeneous form ⟨ , ⟩ W on C[z 1 , . . ., z n ] is Hermitian symmetric.
Recently D. Gaiotto, motivated by the study of Landau-Ginzburg boundary conditions for 3d free hypermultiplets, proposed the following conjecture (in the more general setting of quasihomogeneous W ; see [1,Sections 7.6 and 7.7]).
Note that the form ⟨ , ⟩ 0 is clearly positive definite.Thus part (i) of Conjecture 1.1 follows from part (ii) by a deformation argument.In fact, part (i) follows even from a weaker version of part (ii), stating that ( , ) W is filtered-nondegenerate on a dense connected subset of S d C n * .Moreover, we already know that this holds on a dense subset, but it is not clear that this subset is connected, since we removed zero sets of real polynomials.
In particular, for N = 0 Conjecture 1.1 (ii) reduces to Conjecture 1.2.The real number is strictly positive for all W (or, equivalently, nonzero for all W ).
These conjectures are easy to check directly for d ≤ 2 using Gaussian integrals, but already the case d = 3 is not known.
We prove two results in the direction of these conjectures.
Theorem 1.3 is proved in Section 2 using (the "easy direction" of) Schoenberg's theorem on totally positive functions.The relevant inner product in this case is defined by [2, formula (7.34)].
Theorem 1.4 is proved in Section 3; it easily follows from Bochner's theorem on positive definite functions.
(1) It is clear that if Conjecture 1.1 holds for W 1 and W 2 then it holds for (W More generally, by the same argument it holds for in the quasi-homogeneous setting. (2) Let t > 0. Making a change of variable z → tz in the integral, we obtain Thus setting P t (z) := P t −1 z , we may define the inner product If W defines an isolated singularity (i.e., the projective hypersurface W = 0 is smooth), then we can take the limit t → 0 and get the inner product (the integral is conditionally convergent).It is explained in [1, Section 7.7 and Appendices A-C] that this inner product arises in 2d Landau-Ginzburg models, and Conjecture 1.1 holds for it (namely, the positivity is shown by using the Morse flow to deform the contour of integration so that the integrand is manifestly positive).This implies that for any W defining an isolated singularity, Conjecture 1.2 holds for ε −1 W for sufficiently small ε.
2 Proof of Theorem 1.3

Totally positive functions
We start with reviewing the theory of totally positive functions which is used in the proof of Theorem 1.3.More details can be found in [4,5].
Definition 2.1.A function f : R → R is totally positive if for any real we have det 1≤i,j≤N f (x i − y j ) ≥ 0, and is strictly totally positive if this inequality is strict (det 1≤i,j≤N f (x i − y j ) > 0).
We will consider only continuous totally positive functions.For a function f : R → R, denote by f its Fourier transform Lemma 2.2.Let f be a totally positive L 1 -function.Suppose that for some N and some Proof .Let N be the smallest such number.Expanding the determinant, we obtain . Moreover, by assumption there exist Passing to Fourier transforms with respect to x N , we have Thus f = 0, hence f = 0. ■

P. Etingof
Let f : R → R be a bounded measurable function and g : R → R be an L 1 -function.Recall that the convolution f * g is the function
(ii) If moreover f is strictly totally positive, then so is f * g.
(ii) follows from the Cauchy-Binet formula and Lemma 2.2.■ Example 2.4.The determinant det 1≤i,j≤n (e Thus, setting u i := e x i , v i := −e −y i , we get that for any real ,j≤N e −e x i −y j is positive.Hence the function f (x) = e −e x is strictly totally positive (and bounded).

Thus Proposition 2.3 implies
Proposition 2.5.For every totally positive L 1 -function g, the function e −e x * g is strictly totally positive.

Definition 2.6 ([5]
).The function f is said to be extended totally positive if ∆ N (f, x) is strictly positive for all N and all x ∈ R.
Proposition 2.7 ([5, p. 55]; see also [4, p. 41]).If f : R → R is analytic, then it is strictly totally positive if and only if it is extended totally positive.
Theorem 2.8 (Schoenberg [7]; see [3,5] for a review).A non-zero L 1 -function f (x) is totally positive iff its Fourier transform G(s) is of the form 1/Ψ(is), where Ψ is a Laguerre-Polya entire function with Ψ(0) > 0. This means that G(s) admits a Hadamard factorization Example 2.9.The Fourier transform of the (strictly) totally positive function e −e x is 1 √ 2π Γ(is), which has the Hadamard factorization where γ is the Euler constant.This product has a pole at s = 0 since e −e x / ∈ L 1 .However, for any ε > 0 the strictly totally positive function e εx−e x is in L 1 and has the Fourier transform 1 √ 2π Γ(is + ε) with Hadamard factorization exactly in the form of Theorem 2.8.

Oscillating integrals
Fix an integer d > 2. For an integer 0 ≤ p ≤ d − 1 and t ∈ R, let (This function, of course, depends on d, but we will not indicate this dependence in the notation.)Let us make a change of variable w = z d .We then get Thus the function F p is the restriction to the real axis of the Fourier transform of the distribution Recall that the Fourier transform of |w| s |dw| 2 is = O e −(p+1) u d as u → +∞ (the terms with k > 0 are dominated by the term with k = 0 in this limit).
Let us split this integral into two parts, one for Re z d ≥ 0 and one for Re z d ≤ 0, and replace integration from 0 to ∞ by integration from 0 to +i∞ on the first part and to −i∞ on the second part.This contour manipulation is legitimate for the following reason.Suppose Re z d > 0. The difference between the integrals in v from 0 to R and from 0 to iR is the integral over the quarter-circle from R to iR, which we can write as the integral from 0 to π/2 with respect to θ = argv.But this integral goes to zero as R → ∞ since for Im v > 0, |v| = R, we have exp 2iv Re z d = exp −2R sin θ • Re z d , hence the integrand goes to zero as R → ∞ for 0 < θ < π 2 .The case Re z d < 0 is similar.The above change of contours creates factors e ±πs , so we get

P. Etingof
Now the crucial observation is that part of the poles of Γ(p+1−ids) cancel all the zeros of 1 Γ(1−is) to bring the function to the zero-free Laguerre-Polya form.Namely, writing the Euler product, we get G p (s) = Γ(is)H p (s), where By Theorem 2.8, we get as t → +∞, so it is negative for large t, even though F d (0) > 0.

Proof of
| u=w+w .
So D N > 0 by Corollary 2.12.■ Finally, note that Thus from Corollary 2.13, we get the following result.
Proposition 2.14.We have det 0≤k,ℓ≤N −1 (−1) ℓ I k,ℓ,p (t) > 0, where Proposition 2.14 implies that if we define biorthogonal polynomials p k (z), q k (z) using the weight function e −|z| 2 +tz n −tz n as explained in the introduction i.e., by the formulas p where h k > 0. This completes the proof of Theorem 1.3.
3 Proof of Theorem 1.4 Recall that a function ϕ : R → C such that ϕ(−x) = ϕ(x) is positive definite if for distinct x 1 , . . ., x N ∈ R the matrix (ϕ(x i −x j )) is positive definite; equivalently, for distinct x 1 , . . ., x N ∈ R, det ϕ(x i − x j ) > 0. Bochner's theorem [6] states that a continuous function ϕ is positive definite iff it is the Fourier transform of a finite positive measure.Proof .For z ∈ C n with |z| = 1, consider the integral is proportional (with positive coefficient) to So by Example 3.1, G(s) is positive definite for n + ℓ ≤ d (as it is then a product of Gamma functions, namely the Gamma function in the denominator cancels and the resulting function has no zeros).Thus F n+ℓ−1 (t) ≥ 0 for t > 0, i.e., J(z) ≥ 0 (this was also shown using total positivity in Section 2).Also this function is invariant under multiplication by e iθ , so descends to CP n−1 .Now note that I(ρ, W ) = CP n−1 ρ(z)J(z)dzdz.
Thus I(ρ, W ) > 0 (as the function J is real analytic, hence ρJ is a non-negative function which is strictly positive on some subset of positive measure).■ Note that we in fact obtain the following generalization of Theorem 1.4.

Proposition 2 .
10.For p ≤ d − 1, the function F p e u 2 is strictly totally positive.Proof .Let G p (s) be the Fourier transform of F p e u 2 multiplied by √ 2π.Making a change of variable v = e u/2 , we get
−y h p (y)dy, where h p is a totally positive L 1 -function with Fourier transform H p (s).Thus Proposition 2.5 implies that F p e The bound p ≤ d − 1 in Proposition 2.10 is sharp: we have Theorem 1.3For every N and u ∈ R, we have det 0≤k,ℓ≤N −1 (−1) ℓ ∂ k+ℓ u F p e For every N and t ∈ C × , we have det 0≤k,ℓ≤N −1 (−1) ℓ ∂ k t ∂ (t∂ t ) k (t∂ t ) ℓ F p (|t|) . ℓ