Riemann Hypothesis for DAHA superpolynomials and plane curve singularities
Stable Khovanov-Rozansky polynomials of algebraic knots are expected to coincide with certain generating functions, superpolynomials, of nested Hilbert schemes and flagged Jacobian factors of the corresponding plane curve singularities. Also, these 3 families conjecturally match the DAHA superpolynomials. These superpolynomials can be considered as singular counterparts and generalizations of the Hasse-Weil zeta-functions. We conjecture that all -coefficients of the DAHA superpolynomials upon the substitution satisfy the Riemann Hypothesis for sufficiently small for uncolored algebraic knots, presumably for as . This can be partially extended to algebraic links at least for . Colored links are also considered, though mostly for rectangle Young diagrams. Connections with Kapranov’s motivic zeta and the Galkin-Stöhr zeta-functions are discussed.
Dedicated with admiration to Yuri Ivanovich Manin
on the occasion of his 80th birthday
Key words: double affine Hecke algebras; Jones polynomials; HOMFLY-PT polynomials; plane curve singularities; compactified Jacobians; Hilbert scheme; Khovanov-Rozansky homology; iterated torus links; Macdonald polynomial; Hasse-Weil zeta-function; Riemann hypothesis.
MSC (2010): 14H50, 17B22, 17B45, 20C08, 20F36, 33D52, 30F10, 55N10, 57M25
- 1.1 Superpolynomials
- 1.2 DAHA and Weil conjectures
- 1.3 Motivic approach
- 1.4 DAHA approach
- 2 Double Hecke algebras
3 DAHA-Jones theory
- 3.1 Iterated torus knots
- 3.2 From knots to links
- 3.3 Pairs of graphs
- 3.4 DAHA-Jones polynomials
- 3.5 Polynomiality etc
- 4.1 Main theorem
- 4.2 Motivic superpolynomials
- 4.3 Connection conjectures
- 4.4 Riemann Hypothesis
- 4.5 Non-classical features
5 RH numerically
- 5.1 Table organization
- 5.2 On Conjecture 4.5
- 5.3 The table
- 5.4 Brief analysis
- 5.5 Composite theory, etc
- 6 Some formulas, conclusion
The aim of the paper is to approach the Riemann Hypothesis, , for DAHA superpolynomials of algebraic links colored by Young diagrams upon the substitution . The parameter , a counterpart of the cardinality of in the Weil conjectures, is assumed sufficiently small, which is complementary to the classical theory. Then presumably holds for any –coefficients of DAHA superpolynomials of uncolored algebraic knots ; moreover, is sufficient when . For links, stable irregular (non-) zeros appear. For instance, the number of their pairs is conjectured to coincide with the number of components of uncolored algebraic links minus as . We provide tools for finding such bounds for any Young diagrams and arbitrary –coefficients; finding the exact –range of is much more subtle, which is somewhat parallel to the theory of spectral zeta-functions .
Let us try to put this conjecture into perspective and explain the rationale behind it and its possible relation to the Weil conjectures.
1.1.1. Topological and geometric theories
The superpolynomials have several reincarnations in mathematics and physics; the origin is the theory of stable Khovanov-Rozansky polynomials , which are Poincaré polynomials of the HOMFLY-PT triply-graded link homology [Kh, KhR1]. They depend on 3 parameters and are actually infinite series in the unreduced case. Generally, they are difficult to calculate and there are unsettled problems with their definition for links and in the presence of colors. In the unreduced case and for uncolored algebraic links, they are expected to be connected with the ORS superpolynomials , certain generating series for nested Hilbert schemes of the corresponding plane curve singularties [ORS].
These two families in the reduced setting are conjecturally related to the geometric superpolynomials introduced in [ChP1, ChP2]. They were defined there for any algebraic knots colored by columns (wedge powers of the fundamental representation), developing [ChD1, GM, Gor]. Their construction is in terms of the flagged Jacobian factors of unibranch plane curve singularities. Jacobian factors are (indeed) factors of the corresponding compactified Jacobians ; the definition is entirely local. In turn, Jacobian factors are almost directly related to the affine Springer fibers in type (the nil-elliptic case), and therefore to the corresponding –adic orbital integrals; see the end of [ChP1] for some references and discussion.
Inspired by [ORS, GORS], [ChP1, ChP2] and various prior works, especially [Kap, GSh], the ORS superpolynomials and geometric superpolynomials can be considered as “singular” analogs of the Weil polynomials , the numerators of the Hasse-Weil zeta-functions of smooth curves. To analyze this we switch to the th family of superpolynomials, the DAHA ones from [Ch2, Ch3, GN] and further works; the most comprehensive paper on them by now is [ChD2].
1.1.2. DAHA superpolynomials
The DAHA superpolynomials deal with the combinatorial data of iterated torus links and allow any colors (Young diagrams). Importantly, they almost directly reflect the topological type of singularity, in contrast to the ORS construction and the geometric superpolynomials from [ChP1, ChP2]. The connection of the latter with the DAHA superpolynomials seems quite verifiable via the analysis of “iterations”, geometrically and in terms of DAHA. Accordingly, the DAHA superpolynomials are expected to be connected with other theories of superpolynomials too, including physics approach [DGR, AS, GS, DMS, FGS], which is not discussed in this work.
The DAHA superpolynomials are well-established, but they are restricted to iterated torus links . All initial intrinsic conjectures from [Ch2] were proved but the positivity (discussed below) and extended to iterated torus links (not only algebraic). The theory of DAHA-Jones polynomials is very much uniform for any colors (dominant weights) and root systems; the DAHA superpolynomials are in type .
In other approaches, the limitations and practical problems are more significant, especially with links and colors. The ORS polynomials are difficult to calculate since they are based on the weight filtration in cohomology of the corresponding nested Hilbert schemes. The KhR polynomials are known only for simple knots. Though for torus knots, they were recently calculated using Soergel bimodules [Mel2]. Corollary 3.4 there is Conjecture 2.7 (uncolored) from [Ch2]; not all details are provided in [Mel2], but the proof seems essentially a direct identification of Gorsky’s combinatorial formulas with those in [Ch2]. The geometric superpolynomials and flagged Jacobian factors from [ChP1, ChP2] are relatively simple to define, but this is done so far only for algebraic knots and links colored by columns.
1.2. DAHA and Weil conjectures
Let us present the main features of the theory of DAHA superpolynomials as analogs of the Weil conjectures. This connection is mostly heuristic and our mostly serves the sector of , complementary to Weil’s . The key point of the whole work is the identification of in DAHA superpolynomials with in the singular counterpart of the Weil zeta-function.
1.2.1. Polynomiality and super-duality
First of all, the DAHA theory and the geometric construction from [ChP1, ChP2] directly provide superpolynomials, counterparts of the Weil . This is in contrast to the classical theory, where appear due to the rationality theorem.
The coefficients of DAHA superpolynomials are presumably all positive for rectangle Young diagrams and algebraic knots (the positivity conjectures from [Ch2, ChD1]), which is not present in the Weil-Deligne theory [Del1, Del2]. Such a positivity hints at a possible geometric interpretation and “categorification” of these polynomials. However, they can be positive for “very” non-algebraic knots. For links and non-rectangle Young diagrams, the positivity holds only upon the division by some powers of . For instance, is presumably sufficient for uncolored links with components.
To avoid misunderstanding, let us emphasize that the positivity of DAHA superpolynomials for knots (and for links upon the division above) is neither necessary nor sufficient for the validity of the corresponding . However when such positivity holds, one may expect a geometric interpretation of DAHA superpolynomials as in [ChP1, ChP2], which is generally very helpful.
Thus, a counterpart of the first Weil conjecture is the polynomiality for DAHA-Jones polynomials from Theorem 1.2 from [Ch3] and its further generalizations to iterated torus links. The passage to the superpolynomials, Theorem 1.3 there, was announced in [Ch2]; its complete proof was provided in [GN] (using the modified Macdonald polynomials). See [ChD2] for the most general version. Actually, this paper can be seen as its continuation; we provide many examples of algebraic links and the corresponding DAHA procedures.
The super-duality of DAHA superpolynomials is very similar to the second Weil conjecture, the functional equation. It was conjectured in [Ch2] (let us mention prior [GS] in the context of physics superpolynomials) and proved in [GN] on the basis of the –duality of the modified Macdonald polynomials. An alternative approach to the proof (via roots of unity and the generalized level-rank duality) was presented in [Ch3]; it can work for classical root systems and some other families. The proof of the duality from Proposition 3 from [ORS] is parallel to that for the functional equation in the motivic setting [Kap] and also to Section 3 in [Sto] concerning the functional equation for the Galkin-Stöhr zeta-function, related to our geometric superpolynomials.
We note that our parameter and adding colors (numerically, mostly simple rectangle Young diagrams) do not have direct origins in the theory of Hasse-Weil zeta-functions. The parameter is associated with flagged Jacobian factors in [ChP1, ChP2], which is generally compatible with the approach from [Gal],[Sto].
1.2.2. Riemann Hypothesis
For large , corresponding to the cardinality , for our superpolynomials and the zeta-functions from [Sto] generally fails. However the inequality is (surprisingly) sufficient for any uncolored algebraic knots we considered. It becomes more involved for uncolored algebraic links , when the number of pairs of irregular zeros is conjectured to be the number of components minus . Adding colors to knots and links is more subtle, though rectangle Young diagrams satisfy for sufficiently small upon the symmetrization at least for in all examples we considered. We note that can hold for non-algebraic links too, but algebraic links, more generally positively iterated torus links form a major class for this.
Deviations from the classical theory. Focusing on the sector is a significant deviation. Even for close to , irregular zeros generally appear. For instance, one pair (always real) of such zeros is conjectured to occur for uncolored algebraic links with components.
The main change is of course that is arbitrary in the DAHA approach. It is a counterpart of in the Weil conjectures, but this is just a parameter for us. Accordingly, we try to determine minimal such that holds for all . Also, we have colors and one more parameter due to the flagged Jacobian factors (or Hilbert schemes), which is a clear extension of the Weil conjectures.
For instance, the case (“the field with one element”), corresponding to in the DAHA parameters (before the substitution ) and describing the HOMFLY-PT polynomials in topology is of importance. However the bound is mostly beyond and holds only when for sufficiently general knots.
If the geometric superpolynomials are known and coincide with the DAHA superpolynomials (this is conjectured), then the validity of for sufficiently small (sufficiently large ) is relatively straightforward (if true). In the absence of colors and for , the geometric superpolynomials from [ChP1] count –points of the corresponding Jacobian factors over finite fields . This makes them quite close to the Hasse-Weil zeta-function for singular curves and Galkin-Stöhr zeta-functions. Adding colors and is more subtle. The geometric DAHA superpolynomials of algebraic links are expected to exists at least for rectangle Young diagrams; they are defined by now for any columns. DAHA superpolynomials are defined for any Young diagrams, but they are generally very large beyond small columns and the –hook.
1.3. Motivic approach
Let us discuss some algebraic-geometric details. Connecting DAHA superpolynomials with the numerators of the Hasse-Weil zeta-functions seems a priori some stretch, but we think that the following chain of steps provides a sufficiently solid link.
1.3.1. Kapranov’s zeta
The first step is the Kapranov zeta-function of a smooth algebraic curve of genus over a field . It is defined via the classes of of the -fold symmetric products of in the Grothendieck ring of varieties over . The motivic zeta-function of from [Kap] is then a formal series . Here one can replace by for any motivic measure . If and is the number of –points of , then this is the classical presentation of the Hasse-Weil zeta-function.
Theorem 1.1.9 from [Kap] establishes the first two Weil conjectures (the rationality and the functional equation) in the motivic setting. The justification is close to the Artin’s proof in the case of .
One can then extend the definition of motivic zeta to reduced singular curves , replacing by the corresponding Hilbert schemes of points on , subschemes of length to be exact. Let us assume now (and later) that is a rational planar projective reduced curve of arithmetic genus . Then Conjecture 17 from [GSh] states that in :
where . To be exact, this is stated for any reduced curve, not only rational, the total arithmetic genus must be then used in the left-hand side instead of and the resulting expression must be divided by the same series for the normalization of (calculated by Macdonald for smooth ). In the right-hand side, we must set . This substitution is due to the Macdonald formula for .
1.3.2. Nested Hilbert schemes
When the classes are replaced by their (topological) Euler numbers , we arrive at
The rationality here was motivated by Gopakumar and Vafa (via the BPS invariants) and justified in [PaT]. The positivity of was deduced from the approach based on versal deformations: [FGvS] for and then (for arbitrary ) in [Sh].
The OS-conjecture [ObS], (extended and) proved in [Ma], is a geometric interpretation of (1.2) for rational planar curves and their nested Hilbert schemes (pairs of ideals) instead of . It is actually a local formula and one can switch from a rational curve to its germ at the singular point under consideration.
1.3.3. ORS polynomials
Let be an arbitrary plane curve singularity of arithmetic genus (its Serre number); the Hilbert schemes are defined correspondingly. Considering the construction above for and then applying the motivic integration in from Example 1.3.2b from [Kap] is essentially what was suggested in [ORS] (for nested Hilbert schemes). The reduced ORS polynomial is
Here is the Milnor number ( in the unibranch case) and is the weight filtration in the compactly supported cohomology of the corresponding scheme. See the Overview and Section 4 in [ORS]. Proposition 3 there contains the corresponding functional equation. The parameter is associated with this filtration; the factor is some necessary normalization. We put here to distinguish these parameters from the DAHA parameters (below). They are really standard in quite a few topological-geometric papers; also see (1.4.2) below. For , and at the minimal possible degree of , the sum in (1.3) essentially reduces to the right-hand side of (1.2).
Conjecture 2 of [ORS] states that coincides with the reduced stable Khovanov-Rozansky polynomial of the corresponding link, the Poincarè polynomials of the triply-graded HOMFLY-PT homology. Accordingly, its unreduced version is related to the unreduced stable KhR polynomial. The problem with the identification of the ORS and KhR polynomials is that the number of examples is very limited in both theories. See [Mel2] concerning the stable KhR polynomials for torus knots via Soergel bimodules. The weight filtration is known only for the simplest torus knots.
1.4. DAHA approach
The following two families of superpolynomials are much more explicit and calculatable. DAHA superpolynomials are the key for us; their full definition will be provided in this paper. They were defined in [Ch2] for torus knots, triggered by [AS], and extended to any iterated torus links in further papers.
1.4.1. Geometric superpolynomials
The connection of the DAHA superpolynomials to geometry of plane curve singularities goes through geometric superpolynomials , a family of superpolynomials introduced in [ChP1, ChP2], which generalizes [ChD1], [GM] (for and torus knots) and Gorsky’s approach from [Gor] (an efficient combinatorial theory for torus knots and any powers of ). Flagged Jacobian factors were used in [ChP1] instead of the nested Hilbert schemes in [ORS].
Geometric superpolynomials are known by now only for columns. Importantly, their definition does not require root systems at all and the –stabilization with respect to in the DAHA construction. Only iterated torus links are available in the DAHA approach (including all algebraic ones), but the colors can be arbitrary Young diagrams and links are not a problem. Also, DAHA-Jones-WRT polynomials can be uniformly defined for any root systems. Their –stabilization generally requires classical series , but see [ChE] for some superpolynomials for the Deligne-Gross –type family.
The relation between the DAHA superpolynomials and geometric superpolynomials for algebraic knots colored from [ChP1, ChP2] is confirmed in many examples and verifiable theoretically. One can connect the standard (monoidal) transformations of the plane curve singularities with the corresponding recurrence relations in the DAHA theory.
1.4.2. ORS polynomials vs. geometric ones
Our (uncolored) geometric superpolynomials are parallel to from [ORS], though the positivity of the latter is quite non-trivial in contrast to that for the geometric superpolynomials. The latter can be expressed is terms of the weight filtration too via a theorem due to N.Katz; see the end of [ChP1]. This formula and our interpretation of as the number of –points are different from those in [ORS]. However both are expected to match the KhR polynomials, so the DAHA parameters are supposed to be connected with those in (1.3) as follows:
In the geometric superpolynomials , the DAHA parameter (or due to the super-duality) is a counterpart of cardinality of . Accordingly, is the case of the “field with one element”, which leads to the DAHA theory at critical center charge for . When is used, the case is the so-called ”free theory”. In the ORS polynomials, the motivic measure and finite fields vanish at , i.e. at in the DAHA parameters, which is quite different from for the geometric superpolynomials. Also, we heavily use the normalization of singularities and non-standard flagged Jacobian factors to define the geometric superpolynomials. Our construction does not require Hilbert schemes, though see Section 1.4.3 below.
Let us mention here that links, colors and arbitrary root systems (available in the DAHA approach) are generally quite a challenge from the geometric-motivic perspective. However the OS conjecture (the case in DAHA parameters) was established and proved in [Ma] for any colors (Young diagrams) and algebraic links.
1.4.3. Galkin-Stöhr zeta
For , let us take the simplest motivic measure, the count of points over a finite field of cardinality . Then the corresponding version of the (unreduced) ORS construction becomes related to the zeta-function from [Gal, Sto]. With some simplifications, their zeta is defined as the sum of over all ideals for the local ring of a curve singularity. It is assumed Gorenstein to ensure the functional equation for , which is for the substitution . The function is a version of the Weil zeta for singularities. The corresponding generally fails, but if we treat as a variable (it is in the DAHA parameters), then is presumably sufficient in the unibranch case.
The connection with our geometric superpolynomials can be stated as some “combinatorial” identity, which is not straightforward to check because it generally holds only for planar singularities (not for arbitrary Gorenstein ones). Also, the positivity of the coefficients of generally requires the passage to our geometric superpolynomials (and plane curve singularities). The formula from [Sto] has many positive and negative terms canceling each other in a non-trivial way. We provide an example of a non-plane curve singularity where the connection with our (positive) formula can be fixed, but some non-trivial adjustments are needed for this. On the other hand, the functional equation for is not difficult to established (see [Sto]), but we do not see any direct proof of super-duality for our geometric (motivic) superpolynomials, without some passage to ideals.
1.4.4. Modular periods
The construction of DAHA superpolynomials is actually parallel to that for modular periods, a starting point for –adic measures (Mazur, Manin, Katz, eigenvarieties), and for Manin’s zeta-polynomials [OnRS]. Namely, the DAHA-Jones polynomials of torus knots colored by Young diagrams are essentially obtained by applying to the Macdonald polynomials followed by taking the DAHA coinvariant. The superpolynomials are due to the –stabilization for the root systems ; this is important, since the super-duality holds only upon the –stabilization.
This procedure is analogous to taking the integral of a cusp form for multiplied by for certain integers over the paths . Here can be seen (with some stretch) as counterparts of , the integration plays the role of the coinvariant.
The latter obviously has nothing to do with modular forms. However, it is the simplest level-one coinvariant in the family of all DAHA coinvariants of arbitrary levels from [ChM]. They are in one-to-one correspondence with elliptic functions of level (Looijenga functions for any root systems). Using them makes these constructions closer to each other. Vice versa, a challenge is to find modular counterparts of the DAHA-Jones polynomials and superpolynomials for iterated non-torus algebraic knots. It is not impossible that they can be related to Manin’s iterated Shimura integrals [Man].
This is connected with the following (heuristic) interpretation of the Dirichlet –functions of conductor via the families . The sums of DAHA superpolynomials over the knots in such families are supposed to be considered. In a more conceptual way, one applies inside the coinvariant for the corresponding Dirichlet character . The –analogs of zeta and Dirichlet functions from [Ch4] are of this kind, but for a different action of . They are the integrals of for the Gaussian with respect to the Macdonald measure for . See also Section 3 in [ChD2].
2. Double Hecke algebras
2.1. Affine root systems
Let us adjust the standard DAHA definitions to the case of the root systems , which is for the basis , orthonormal with respect to the usual euclidean form . The Weyl group is ; it is generated by the reflections (transpositions) for the set of positive roots ; . The simple roots are The weight lattice is , where are fundamental weights: . Explicitly,
The vectors for form the affine root system , where are identified with . We add to the simple roots for the maximal root . The corresponding set of positive roots is .
2.1.1. Affine Weyl group
Given , let
for . The affine Weyl group is the semidirect product of its subgroups and , where is identified with
The extended Weyl group is , where the action is
It is isomorphic to for . The latter group consists of id and the images of in . Note that is , where is the standard involution of the non-affine Dynkin diagram of , induced by . Generally, we set , where is the longest element in sending to .
The group is naturally identified with the subgroup of of the elements of the length zero; the length is defined as follows:
One has for , where is the element of minimal length such that , equivalently, is of maximal length such that . The elements are very explicit. Let be the longest element in the subgroup of the elements preserving ; this subgroup is generated by simple reflections. One has:
Setting for coincides with the length of any reduced decomposition of in terms of the simple reflections Thus, indeed, is a subgroup of of the elements of length .
2.2. Definition of DAHA
We follow [Ch3, Ch2, Ch1]. Let ; generally it is the least natural number such that The double affine Hecke algebra, DAHA , of type depends on the parameters and will be defined over the ring formed by polynomials in terms of and Note that the coefficients of the Macdonald polynomials will belong to
It is convenient to use the following notation:
We set for and otherwise, generally, when the number of links between in the affine Dynkin diagram is .
For pairwise commutative
For instance, .
The double affine Hecke algebra is generated over by the elements , pairwise commutative satisfying (2.5) and the group where the following relations are imposed:
(i) factors on each side;
(iv) when ;
Given the product
does not depend on the choice of the reduced decomposition of . Moreover,
In particular, we arrive at the pairwise commutative elements
When acting in the polynomial representation (see below), they are called difference Dunkl operators.
2.3. The automorphisms
The following maps can be (uniquely) extended to automorphisms of , where must be added to (see [Ch1], (3.2.10)–(3.2.15)) :
These automorphisms fix and their fractional powers, as well as the following anti-involution :
The following anti-involution results directly from the group nature of the DAHA relations:
To be exact, it is naturally extended to the fractional powers of :
This anti-involution serves the inner product in the theory of the DAHA polynomial representation.
Let us list the matrices corresponding to the automorphisms and anti-automorphisms above upon the natural projection onto , corresponding to . The matrix will then represent the map for . One has:
, , , .
The projective (due to Steinberg) is the group generated by subject to the relation The notation will be ; it is isomorphic to the braid group .
2.3.1. The coinvariant
The projective and the coinvariant , to be defined now, are the main ingredients of our approach.
Any can be uniquely represented in the form
(the DAHA-PBW theorem, see [Ch1]). Using this presentation, the coinvariant is a functional defined as follows:
The main symmetry of the coinvariant is
Also, , where we extend to as follows:
The following interpretation of the coinvariant is important. For any , one has: , where is the standard character (one-dimensional representation) of the affine Hecke algebra , generated by for ; sends and . Therefore acts via the projection of onto the polynomial representation , which is the –module induced from ; see [Ch1, Ch2, Ch3] and the next section.
2.4. Macdonald polynomials
2.4.1. Polynomial representation
It is isomorphic to as a vector space with the action of given by the Demazure-Lusztig operators :
The elements become the multiplication operators and act via the general formula for . Note that naturally acts in the polynomial representation. See formula (1.37) from [Ch3], which is based on the identity
Symmetric Macdonald polynomials. The standard notation is for ; see [Mac, Ch1] (they are due to Kadell for the classical root systems and due to Rogers for ). The usual definition is as follows. Let be such that (it is unique); recall that . For , the following are the defining relations:
Here and further is the constant term of a Laurent series or polynomial ; is considered a Laurent series of with the coefficients expanded in terms of positive powers of . The coefficients of belong to the field . One has (see (3.3.23) from [Ch1]):
Recall that for .
DAHA provides an important alternative (operator) approach to the –polynomials; namely, they satisfy the (defining) relations
for any symmetric (–invariant) polynomial . Here and the coefficient of in is assumed .
Spherical normalization. We call spherical Macdonald polynomials for . One has (the evaluation theorem):
They are necessary below for managing algebraic links (spherical polynomials are sufficient for knots) and are important for the justification of the super-duality .
For , i.e. for a dominant weight with for all , the corresponding Young diagram is as follows:
One has: , . Also, .
Let us calculate the set of all