Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)
Y. Abdelaziz, J.-M. Maillard
TL;DR
This paper generalizes Schwarzian differential equations from hypergeometric functions to higher-order linear differential operators with specific Galois groups, revealing conditions for modular solutions and applications in physics and combinatorics.
Contribution
It introduces a framework for Schwarzian conditions on higher-order operators with selected Galois groups, extending previous work and analyzing modular correspondences and Calabi-Yau operator equivalences.
Findings
Pullback invariance reduces to symmetric powers of order-two operators for orders three and four.
Conditions for modular correspondence solutions of Schwarzian equations are established.
Analysis of hypergeometric examples shows invariance up to operator homomorphisms.
Abstract
We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions, can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We expect this new concept to be well-suited in physics and…
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Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)
Y. Abdelaziz, J.-M. Maillard*†*222Recherche Publique Française.
LPTMC, UMR 7600 CNRS, Université Pierre et Marie Curie, Paris 6, Tour 23-13, 5ème étage, case 121, 4 Place Jussieu, 75252 Paris Cedex 05, France
Abstract
We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We expect this new concept to be well-suited in physics and enumerative combinatorics. We finally consider the more general problem of the equivalence of two different order-four linear differential Calabi-Yau operators up to pullbacks and conjugation, and clarify the cases where they have the same Yukawa couplings.
PACS: 05.50.+q, 05.10.-a, 02.30.Hq, 02.30.Gp, 02.40.Xx
AMS Classification scheme numbers: 34M55, 47E05, 81Qxx, 32G34, 34Lxx, 34Mxx, 14Kxx
Key-words: Malgrange pseudo-group, Galoisian envelope, Schwarzian derivative, infinite order rational symmetries of ODE’s, Fuchsian linear differential equations, Gauss and generalized hypergeometric functions, Heun function, globally nilpotent linear differential operators, homomorphisms of linear differential operators, elliptic functions, isogenies of elliptic curves, modular forms, modular correspondences, modular equations, Landen transformation, Hauptmoduls, mirror symmetries, Calabi-Yau ODEs, Yukawa couplings.
1 Introduction
In a previous paper [1] we focused on identities relating the same hypergeometric function with two different222Beyond the known pullback symmetries of hypergeometric functions. The correspondence between the two pullbacks must be an infinite order rational or algebraic transformation [1, 2]. algebraic pullback transformations. These identities correspond to modular forms, the algebraic transformations being solutions of a (non-linear) differentially algebraic [3, 4] Schwarzian equation, that also emerged in a paper by Casale on Galoisian envelopes [5, 6]. This covariance symmetry of hypergeometric functions could be regarded as one of the simplest illustrations of the concept of symmetries (of the renormalization group type [2, 7]) in physics or enumerative combinatorics, a univariate function being covariant (automorphic) with respect to an infinite set of rational or algebraic transformations. This paper [1] was essentially focused on hypergeometric functions and modular forms actually represented as hypergeometric function with two different algebraic pullback transformations (modular correspondences [1, 8]).
The applications of this Schwarzian equation [1] known to be associated to a quite large mathematical framework111In Casale’s paper [5, 6] the Schwarzian equation is associated with meromorphic functions instead of the rational functions of our paper [1]. See also [9, 10, 11]. (Malgrange’s pseudogroup, Galois groupoid [9, 10, 11, 12, 13, 14, 15]), extend well beyond hypergeometric functions in physics. We have seen, for instance in [1], an example of identity relating the same Heun function with two different pullbacks999This Heun function being not, in general, reducible to a pullbacked hypergeometric function [16].. This Heun example [1] could suggest that such Schwarzian differential equations emerge in physics with holonomic functions having a narrow set of singularities (three for hypergeometric functions, four for Heun functions, …) like the Heun example in [1]. Going further we show, in this paper, that such differentially algebraic [3, 4] Schwarzian equations do emerge in a much more general holonomic framework.
We will show in section 2 that the covariance symmetry condition of general order-two linear differential operators with polynomial coefficients automatically yields this Schwarzian differential equation. We will then show in sections 3 and 4 that the covariance symmetry condition imposed on linear differential operators having order three and order four with respective orthogonal and symplectic differential Galois groups, yield Schwarzian differential equations like the one examined in [1]. When their respective symmetric and exterior powers are of order five (instead of six), one finds that these order-three and order-four operators reduce to symmetric square and symmetric cube of an underlying order-two operator. In section 5 we show that the Schwarzian condition can be derived for linear differential operators of arbitrary order . The reduction of the solutions of this Schwarzian differential equation to only modular correspondences [8] was an open question in [1]: in section 6 a necessary condition to have such modular correspondences [8] is derived. In section 7 generalizations of modular forms provide examples of pullback invariance of an operator, up to operator homomorphism. This invariance should be important to describing the symmetries of linear differential operators and thus, is of relevance to physics. Finally in section 8, we consider the more general problem already addressed in [17] where Schwarzian differential equations also occurred, of the equivalence of two different order-four linear differential Calabi-Yau operators [18] up to pullbacks and conjugation, possibly yielding the same Yukawa couplings [17], and we will generalize it to linear differential operators of arbitrary orders.
2 Beyond hypergeometric and Heun functions: order-two linear differential operators
We will show here that non-linear ODEs involving Schwarzian derivatives (cf. equation (9) below), that we will call “Schwarzian ODEs”222See [1, 19] for a definition. See also [20, 21]., obtained in [1] for hypergeometric and Heun functions [22, 23], can be generalized to arbitrary globally nilpotent [24] linear differential operators having an arbitrary numbers of singularities (as opposed to three and four singularities for hypergeometric and Heun functions).
Let us consider a linear differential operator of order two
[TABLE]
and let us also introduce two other linear differential operators of order two: the operator being the conjugate of (1) by a function , and the pullbacked operator which amounts to changing in (1), the head coefficient being normalized111Throughout the paper we consider, for clarity and simplicity, this normalized form for the linear differential operators. The “true” pullbacked operator which amounts to changing (see the command “dchange” in PDEtools in Maple) is in fact where is given by (4). to . These two linear differential operators read respectively:
[TABLE]
where
[TABLE]
and
[TABLE]
where:
[TABLE]
The identification of these two linear differential operators gives two conditions:
[TABLE]
Since
[TABLE]
one can eliminate the log-derivative between (6) and (7), and obtain the Schwarzian condition previously given in [1]
[TABLE]
where
[TABLE]
and where denotes the Schwarzian derivative [19]:
[TABLE]
Unlike in [1], the number of singularities of the second order operator (1) is arbitrary: it does not need to be three or four like in the hypergeometric or Heun examples in [1]. The second order linear differential operator is a general order-two linear differential operator with polynomial coefficients. Introducing the wronskian of
[TABLE]
we see that the LHS and RHS of the first condition (6) are both log-derivatives. Thus one can immediately integrate the first condition (6) and get (up to a multiplicative factor ) the conjugation function in terms of the wronskian and the pullback function :
[TABLE]
Remark 1: When the wronskian is an -th root of a rational function, the exact expression (12) for the conjugation function , becomes an algebraic function when is an algebraic function. This is actually the case when the order-two linear differential operator is globally nilpotent [24]. In this case the linear differential operator is simply conjugated to its adjoint through its wronskian which is a -th root of a rational function:
[TABLE]
Remark 2: If the linear differential operator is not globally nilpotent [24] the wronskian is not necessarily an algebraic function. Introducing , the log-derivative of the conjugation function , one can rewrite the two conditions (6) and (7) as:
[TABLE]
The elimination of in (14) and (15) gives the Schwarzian condition (9) with (10), however the conjugation function is no longer an algebraic function when is an algebraic function (see (12)): it is a transcendental function, and we certainly move away from a modular correspondence [1, 8] framework111For modular correspondences see also the concept of modular equations [25, 26, 27, 28]..
3 Order-three linear differential operators
3.1 General order-three linear differential operators.
Considering an irreducible order-three linear differential operator
[TABLE]
we introduce two other linear differential operators of order three defined as previously in section 2: the operator conjugated of (16) by a function , namely , and the pullbacked111The coefficient is normalized to . operator which amounts to changing in . These two linear differential operators read respectively
[TABLE]
and:
[TABLE]
The equality of these two order-three linear differential operators gives three conditions , with , corresponding, respectively, to the identification of the coefficients of and . Introducing the wronskian of , the LHS and RHS of condition being, again, log-derivatives, one can easily integrate condition and get the exact expression of the conjugation function in terms of the wronskian of and of the pullback :
[TABLE]
Similarly the elimination of the log-derivative between condition and condition yields the Schwarzian condition
[TABLE]
where this time reads:
[TABLE]
3.2 Symmetric Calabi-Yau condition.
Let us consider the condition corresponding to imposing the symmetric square of to be of order five instead of the generic order six. This (“symmetric” Calabi-Yau [35]) condition reads:
[TABLE]
For a globally nilpotent [24] linear differential operator, this (symmetric Calabi-Yau) condition (3.2) together with (11) yields an order-three linear differential operator (16) simply conjugated to its adjoint:
[TABLE]
where the wronskian is a -th root of a rational function.
Again for a globally nilpotent [24] linear differential operator, the exact expression (19) for the conjugation function , becomes an algebraic function when is an algebraic function.
The symmetric square of an order-two linear differential operator is an order-three linear differential operator (16) with the following coefficients:
[TABLE]
These coefficients (24), (25) automatically verify the (symmetric Calabi-Yau) condition (3.2): the symmetric square of a symmetric square of an order-two linear differential operator is of order five instead of the generic order six. Conversely, the (symmetric Calabi-Yau) condition (3.2) can be parametrized111Note that rewriting the exact expression of given by (21) in terms of and using (24) one recovers (10), and in (10) being now and . by (24) and (25) and amounts to imposing the order-three linear differential operator (16) to be exactly the symmetric square of an order-two operator.
Thus our calculations show that the pullback-compatibility of an order-three linear differential operator is equivalent to saying that this order-three operator reduces to (the symmetric square of) an underlying order-two linear differential operator. The Schwarzian condition (20) with given by (21), is thus inherited from the Schwarzian condition (9) of the underlying order-two linear differential operator.
4 Order-four linear differential operators
Consider the irreducible order-four linear differential operator
[TABLE]
and introduce two other linear differential operators of order four defined as previously in sections 2 and 3.1: the linear differential operator conjugated of (26) by a function and the (normalized) pullbacked operator . These two linear differential operators read respectively
[TABLE]
and:
[TABLE]
The identification of these two order-four linear differential operators and gives this time four conditions , , corresponding, respectively, to the identification of the coefficients of and .
Eliminating once again the log-derivative between and one deduces a Schwarzian condition
[TABLE]
where this time:
[TABLE]
Introducing the wronskian of the order-four linear differential operator with (11), the condition just corresponds to log-derivatives and can be easily integrated giving the exact expression of the conjugation function as:
[TABLE]
The next conditions and yield extremely involved non-linear differential conditions on the miscellaneous derivatives of the various coefficients. It turned out to be very difficult to proceed with such huge expressions. Yet when the linear differential operator has a selected (symplectic) differential Galois group one can go much further in the calculations, as we will see in the coming subsection.
4.1 Calabi-Yau condition (exterior square).
Imposing the Calabi-Yau condition [29, 30] in the case of an order-four linear differential operator gives:
[TABLE]
In this case the exterior square of the order-four operator has order five instead of order six.
When condition (32) is verified, the order-four linear differential operator has a symplectic differential Galois group . Note that if condition (32) is verified, the Calabi-Yau conditions for the pullbacked and conjugated operators and are automatically verified: this is a consequence of the fact that the Calabi-Yau condition (32) is left invariant by conjugation and pullback555To see that the Calabi-Yau condition is preserved by conjugation is straightforward. However, as remarked in [17], to see that the Calabi-Yau condition is preserved by pullback transformations is very hard to see by direct computation, since one gets an enormous fourth-order nonlinear differential equation. . In other words the following identification of the coefficients of and is automatically verified when the Calabi-Yau condition (32) is verified.
Recall that the Calabi-Yau condition (32) together with the globally nilpotent condition [24] automatically yields to be conjugated to its adjoint
[TABLE]
where is a -root of a rational function.
At the last step we consider the identification of the constant terms in in and . After injecting in this “large” non-linear differential equation, equation (11), the Schwarzian condition (29) with given by (30), and the Calabi-Yau condition (32), we eventually find that this last “large” equation becomes independent of the pullback and reduces to a quite simple condition giving as a polynomial expression in the two coefficients and and their derivatives:
[TABLE]
In order to understand what this new condition (4.1) coming on top of the Calabi-Yau condition (32) really means, we calculated, for various MUM111Maximal unipotent monodromy (MUM) linear operators [24, 31]. order-four linear differential operators verifying (32) and (4.1), the corresponding nome and Yukawa couplings [31]. The corresponding Yukawa couplings were actually found to be trivial: !!
This amounts to saying that combining the two conditions (32) and (4.1) corresponds to a drastic reduction: the (irreducible) order-four linear differential operator is not a “true” order-four operator. Typically one can imagine that reduces to an order-two operator, being homomorphic to the symmetric cube of an underlying order-two linear differential operator. In fact it is exactly the symmetric cube of an order-two operator.
Let us consider the symmetric cube of an order-two linear differential operator which is an order-four linear differential (26) with the following coefficients:
[TABLE]
One finds straightforwardly that the coefficients given by (4.1) verify the Calabi-Yau condition (32), as well as the new condition (4.1). In this case the differential Galois group is no longer the symplectic differential Galois group , but actually reduces222When an order-four linear differential operator is the symmetric cube of an underling order-two operator its symmetric square is no longer of order but reduces to order . to the differential Galois group of the underlying order-two linear differential operator, namely . The fact that the Calabi-Yau condition (32) is verified is not a surprise: the exterior square of a symmetric cube is naturally of order less than six. The fact that being the symmetric cube of an underlying order-two operator verifies automatically the new condition (4.1) emerging from a compatibility condition of an order-four linear differential operator by pullback is far less obvious. The “parametrization” (4.1) necessarily yields the Calabi-Yau condition (32) and the new condition (4.1), and, conversely, (32) and (4.1) can be parametrized by (4.1).
Our large calculations thus show that the pullback-compatibility of an order-four linear differential operator which verifies the Calabi-Yau condition (32), amounts to saying that this order-four linear differential operator reduces to (the symmetric cube of) an underlying order-two linear differential operator. The Schwarzian condition (29) with given by (30), is thus inherited from the Schwarzian condition (9) of the underlying order-two linear differential operator.
4.2 Reducible operators
Throughout the paper we make the assumption that the linear differential operators are irreducible. The reduciblility of the linear differential operators is not an academic scenario: it is the situation we encounter (almost) systematically with the linear differential operators emerging in physics, typically in the case of the -fold integral of the two-dimensional Ising model [32, 33, 34]. When the linear differential operators are reducible, it is clear that all the calculations of this paper must be revisited, taking into account the miscellaneous factorization scenarios.
Sketching the kind of situation we may encounter, let us consider an order-four linear differential operator which factorizes into the product of two order-two linear differential operators:
[TABLE]
The simple case where the two operators and are identical is sketched in A. In general the exterior square of is an order-six linear differential operator which is the product of an order-one operator, of the symmetric product of and , and of the order-one linear differential operator . Therefore, this reducible order-four linear differential operator does not verify in general the Calabi-Yau condition (32).
Imposing the (normalized) pullback by of this reducible order-four linear differential operator to be equal to a conjugation by a function of that operator, it is important to remember that a change of variable on a linear differential operator which is the product of two operators, is the product of these two linear differential operators on which this change of variable has been performed. One gets a set of equations where one can disentangle two Schwarzian equations
[TABLE]
where and are the functions (10) already encountered in the analysis of order-two linear differential operators
[TABLE]
corresponding to the Schwarzian conditions written separately on and , together with another relation which couples and :
[TABLE]
Among the four solutions of the order-four operators , the two solutions of the order-two linear differential operator transform nicely under the pullback , provided the Schwarzian condition (37) is satisfied, but this just corresponds to a partial symmetry. In general the set of equations (37), (38), (41) seems to be too rigid to allow solutions other than trivial symmetries or partial symmetries.
It is however worth mentioning a quite curious result. If one imposes the reducible order-four linear differential operator to verify the Calabi-Yau condition (32) (i.e. to be such that the exterior square of that operator is order five instead of order six), one gets a condition that becomes remarkably simple when written in terms of the functions and given by (39) and (40). Introducing the difference , the Calabi-Yau condition (32) simply reads:
[TABLE]
Therefore, if one restricts oneself to which identifies the two Schwarzian conditions (37) and (38), one sees that condition (42) is automatically verified: condition is thus a sufficient condition for the Calabi-Yau condition (32).
The analysis of pullback symmetry on reducible linear differential operators is clearly an interesting and challenging problem in physics. It will require many more calculations to explore the arborescence of these various factorization scenarios.
4.3 Symmetric Calabi-Yau condition
The condition, we called in [35, 36] symmetric Calabi-Yau condition for the order-four linear differential operator (which correspond to impose that its symmetric square is of order less than ), is a huge111This polynomial is the sum of monomials in the coefficients of and their derivatives. polynomial condition on the coefficients of and their derivatives. This condition is invariant by pullback and conjugation. Provided the Schwarzian condition (29) with given by (30) is satisfied, this symmetric Calabi-Yau condition alone is not sufficient to have . Similarly to what we saw with the Calabi-Yau condition (32), would a supplementary condition to the symmetric Calabi-Yau condition be sufficient to have ? Could one also have, in this selected subcase, a reduction of to an underlying order-two operator? This scenario remains open.
Working with such huge polynomials will not get us far. In order to advance, let us introduce a parametrization based on the ideas explained in [36], namely that an order-four linear differential operator , with an orthogonal differential Galois group and such that its symmetric square is of order less than , is necessarily of the form222The differential Galois group with an order- symmetric square situation corresponds to a decomposition , see [36].
[TABLE]
where and are order-one and order-three self-adjoint linear differential operators:
[TABLE]
This yields a parametrization of this huge polynomial differential (symmetric Calabi-Yau) condition:
[TABLE]
One easily verifies that this parametrization (46) … (4.3) is such that the polynomial encoding the symmetric Calabi-Yau condition, is identically equal to zero. Moreover one verifies that the order-four linear differential operator (43), with parametrization (46), (4.3), (4.3), (4.3), is, generically, such that its symmetric square has order (instead of ), its exterior square being of order .
Imposing for an order-four linear differential operator, corresponding to this parametrization (such that it verifies the symmetric Calabi-Yau condition, and such that its symmetric square is of order nine), one naturally finds the Schwarzian condition (29) with (30), as well as the exact expression (31) for the conjugation function . Taking into account the Schwarzian condition (29), the identification of the coefficients of for and yields a relation of the form , where is a rational function. Together with the last condition, this gives an invariance of the form yielding only trivial cases222See [1] for similar calculations. for .
This symmetric Calabi-Yau condition, even if it is invariant by pullback and conjugation, is thus not sufficient to get . We have here a situation similar to the one described in the previous section 4.1, with the emergence of the additional condition (4.1) on top of the Calabi-Yau condition (32). However here the calculations are way too large: finding the additional condition(s) together with the symmetric Calabi-Yau condition yielding , is beyond our reach for now. The case, described in the previous section 4.1, where the order-four operator (43) is the symmetric cube of an underlying order-two operator is also such that the symmetric square of is not of the generic order , but, in fact, of order : in this case the coefficients of verify111This can be verified straightforwardly substituting (4.1) in the monomials symmetric Calabi-Yau condition. the symmetric Calabi-Yau condition. Since the calculations are way too large, it is not possible for now to tell if the additional condition(s) to the symmetric Calabi-Yau condition, also gives eventually a linear differential operator that is the symmetric cube of an order-two operator, as described in the previous section 4.1, or whether it would give something else. This would mean the emergence of the “classic” Calabi-Yau condition (32) combined with the condition (4.1). This remains an open question.
5 Order- linear differential operators
The analysis of irreducible order-five operators is sketched in B. Let us now consider an irreducible order- linear differential operator
[TABLE]
and let us also introduce two other linear differential operators of order : the operator conjugated of (50) by a function , namely , and the (normalized) pullbacked operator which amounts to changing in . The pullbacked operator reads
[TABLE]
and the conjugate of (50) reads:
[TABLE]
We impose the identification of these two order- linear differential operators:
[TABLE]
The identification of the coefficients gives the exact expression of in terms of the wronskian and of the pullback :
[TABLE]
Injecting this exact expression in (52), or eliminating the log-derivative , the identification of the coefficients gives the following Schwarzian equation
[TABLE]
where
[TABLE]
i.e.
[TABLE]
where:
[TABLE]
This is in agreement with the fact that the symmetric -th power of an order-two linear differential operator gives an order- linear differential operator such that
[TABLE]
and thus conversely:
[TABLE]
Injecting (61) in the expression of for an order-two linear differential operator (see (10))
[TABLE]
one gets again the expression (56) for for an order- linear differential operator
Remark: the Schwarzian condition (55) and the associated function given by (56), correspond to an elimination of the conjugation function in (53). If one changes the order- linear differential operator by conjugation, , one gets again (53), being replaced by and being replaced by :
[TABLE]
Consequently one gets again the same Schwarzian condition (55) with the function given by (56), since they are obtained by elimination of the conjugation functions or . Therefore given by (56), which is invariant by the conjugation , is left invariant by:
[TABLE]
Conversely imposing this invariance by conjugation (64), (5), on a function of the form gives (56) up to an overall constant factor.
6 Solutions of the Schwarzian conditions
Let us study the solutions of the Schwarzian equation (55) that emerge for any pullback-symmetry condition of linear differential operators of arbitrary order . This should provide valuable information on the pullbacks that are symmetries of linear differential operators.
6.1 Solutions of the Schwarzian equation that are
diffeomorphisms of the identity: a condition on
The Schwarzian condition (9) has been shown in [1] to be compatible under the composition of the pullback-functions verifying (9). The fact that the composition of two solutions of the Schwarzian condition (9) is also a solution222See Appendix D in [1]. of the Schwarzian condition (9), is crucial to describe the set of solutions of (9). Once a solution of the Schwarzian condition (9) is known, the -th composition , yields automatically a commuting set of solutions555Cum grano salis: when the pullbacks are algebraic functions, they are multivalued functions. The composition of multivalued functions is limited to their analytic series expansions (setting aside Puiseux series). of (9). By obtaining the series expansions of these solutions, one can extend to non integer complex values of , and in order to build a one-parameter family of commuting solution series, consider the infinitesimal composition [2]:
[TABLE]
The one-parameter family of commuting solution series commutes with (66) yielding the functional equations [2]:
[TABLE]
Inserting (66) in the Schwarzian condition (9), one sees that is actually holonomic being solution of the linear differential equation of order-three:
[TABLE]
whose corresponding order-three linear differential operator is exactly the symmetric square of an underlying order-two linear differential operator333The reduction of to a symmetric square (69) does not mean that is solution of a second order linear differential (Liouvillian) equation . :
[TABLE]
Conversely can be expressed in terms of as follows:
[TABLE]
This last result (70) is easily obtained by multiplying the LHS of (68) by and integrating the result. One gets this way111This “gauge” in (70) corresponds to the fact that because of (67) one has which allows to change in the Schwarzian equation (9), as well as in the third order linear differential ODE (68). One easily verifies that inserting (70) in (68) gives an identity. :
[TABLE]
which is (70). Thus, for a given pullback , or for a given one-parameter family of commuting solution series (66), or for a given , one has a one-parameter family (70) of in the Schwarzian equation (9). Conversely, for a given , one has at least a one-parameter family of commuting solution series (66).
6.1.1 Selected subcase of the Schwarzian equation.
Let us consider an order-two linear differential operator (where and are rational functions), such that its corresponding function (see (10)) in the Schwarzian equation (9), is of the form (see subsection 6.2 of [1])
[TABLE]
where is a rational function. Introducing the rational function , the identification of the expression of , namely with (73), gives in terms of and
[TABLE]
which is the condition for the order-two linear differential operator to factorize into two order-one linear differential operators:
[TABLE]
In other words, condition (73) with a rational function, is the condition of factorization of the order-two linear differential operator . In this case, the Schwarzian equation (9) reduces to a simpler second order non-linear differential equation (that was studied extensively in [1, 2]):
[TABLE]
Seeking the following one-parameter solutions (66), , one finds that verifies a linear differential equation of order two [2]
[TABLE]
corresponding to the linear differential operator of order two111In fact the order-two operator is the adjoint of the operator (see [2]). When the linear differential operator is conjugated by the wronskian to the linear differential operator , namely .:
[TABLE]
Introducing the wronskian , reads . Thus the linear differential operator (78) has two solutions: which is the solution of the right factor , and another (transcendental) solution that we denote . The function corresponds to this last (transcendental) solution, and not the solution. Conversely can be expressed222Just integrate the LHS of (77). in terms of as follows:
[TABLE]
One easily verifies that by inserting (79) in (77) ones gets an identity, and that by inserting (79) in (73) one recovers (71) with . Here the term is crucial, because when condition (79) with yield the trivial result, which is different from the transcendental (holonomic) function we are looking for. For instance in the example detailed in [2], we had the condition (79) verified with , namely :
[TABLE]
At first sight one expects the order-two linear differential equation (77) on to be a simple limit of the order-three linear differential equation (68) when the condition (73) is imposed. This reduction is not obvious however and the interested reader can find it explained in C.
Remark: the global nilpotence of the linear differential operators gives an of the form , where the wronskian is an -th root of a rational function [24]. Using , condition (76) can be easily integrated into
[TABLE]
where and are constants of integration.
Now let us describe this one-parameter family of commuting solution series (66) of the Schwarzian equation (9).
6.2 Solutions of the Schwarzian equation that are diffeomorphisms
of the identity: the general formal solution
Let us consider (66) as a series in :
[TABLE]
solution of the functional equation (67). This is sufficient to find, order by order in , the solution (83) of (67) where the are given by
[TABLE]
the most general solution (83) of (67) corresponding to linear combinations of the ’s which amounts to changing in (83) into:
[TABLE]
Note that all the ’s are polynomial expressions of and its derivatives.
The functional equation (67) corresponds to the one-form giving:
[TABLE]
Seeing as a function of , one finds that the series (83) together with the recursion (6.2), gives the well-known Taylor expansion
[TABLE]
meaning that is just a shift in
[TABLE]
corresponding to the integration of the one-form . The two transformations and of the one-parameter family clearly commute222This can also be checked directly using (83) with (6.2) for any rational function .:
[TABLE]
One verifies order by order in , that the one-parameter family of commuting series (83) with (6.2) is solution of the Schwarzian equation
[TABLE]
where is given by (70). In terms of , the expression (70) for can be written using the Schwarzian derivative:
[TABLE]
Recalling the chain rule for the Schwarzian derivative of a composition of functions999Namely . and the fact that , one finds that the Schwarzian condition (90) corresponds to the equality of the two Schwarzian derivatives:
[TABLE]
which is verified since . This is another way to see that the one-parameter family of commuting series (83) (with the ’s given by (6.2)) is solution of the Schwarzian equation.
6.3 A simple modular form example.
We have considered in [1, 29, 30, 31, 37] many examples of modular forms represented as pullbacked hypergeometric functions. Each time the one-parameter commuting series combined with the modular correspondences [8] series yields one-parameter series of the form that are solutions of the Schwarzian equation (90).
In [1] the pullback symmetry of the order-two linear differential operator was given as a covariance of its solution, namely a hypergeometric function with two different555We exclude the trivial well-known changes of variables on hypergeometric functions The transformation must be an infinite order transformation symmetry. pullbacks related by modular equations333 The emergence of a modular form [29, 30, 38] corresponds to the emergence of a selected hypergeometric function having an exact covariance property [39, 40] with respect to an infinite order algebraic transformation (the modular correspondences).
[TABLE]
the pullback being solution of the Schwarzian condition (90).
In this example, the pullback is solution of the Schwarzian solution (90) with and given by111 One can easily check that these expressions (93) for and verify (68).:
[TABLE]
One can also check that these expressions (93) verify (70) with222This selected value of has to be compared with the value in (80). , thus providing a quite non-trivial (non-linear differential) identity between the rational function and the holonomic function .
The one-parameter commuting family (66) solution of the Schwarzian equation (90) can be expressed using the two (mirror maps) differentially algebraic [3, 4] functions and described in [1] and in D, as :
[TABLE]
where .
Besides this one-parameter commuting family (66), the Schwarzian equation (90) has a remarkable (infinite) set of algebraic functions solutions [1] defined by the corresponding modular equations [25, 41, 42, 43, 44, 45]. Their series expansions near read:
[TABLE]
where is an integer These series expansions commute for different values of the integer . This is a consequence of the fact that, up to the previous change of variables , , these modular correspondences (95) correspond to taking the -th power of the nome: (see [1] for more details).
6.3.1 A pre-modular concept.
The composition of the one-parameter series (66) (which corresponds to ) and of the modular correspondences (95), yields an infinite set of one-parameter series for instance [1]:
[TABLE]
These one-parameter series do not commute but verify [1] the simple composition formulae555Consequence of the fact, in the nome, they correspond to the composition of transformations like .:
[TABLE]
When the are arbitrary rational numbers the corresponding series are not globally bounded series [31] in general. Therefore, they cannot be the series expansion of an algebraic function: they are differentially algebraic [3, 4] since they are solutions of the Schwarzian equation (90).
In general, finding the Schwarzian equation (90) is easy, and getting solutions order by order as series expansions is also easy. However finding the selected values of the rational numbers such that the differentially algebraic [3, 4] series are globally bounded and thus can be algebraic functions, and, possibly, modular correspondences, is a quite difficult task222Similar to finding the selected values of the parameters so that a quantum Hamiltonian becomes integrable, or finding modular forms among Beukers’ second order differential equations depending on three parameters [46] (36 cases emerging of a numerical exploration of 10 millions triples). .
We will call “pre-modular888Of course, this “pre-modular” term should not be confused with the term premodular in premodular categories, (ribbon fusion categories). Here we mean prerequisites for the emergence of modular forms.” the existence of an infinite set of one-parameter differentially algebraic series (solution of the Schwarzian equation) of the form which verify (96), but for which one does not know if there exist some selected values of the parameter such that these differentially algebraic series [3, 4] become algebraic functions.
In the next section, we will characterize the Schwarzian equations corresponding to these “pre-modular” structure, thus finding conditions that are necessary for the emergence of modular forms.
6.4 Schwarzian equation: conditions for modular correspondence
In the previous sections it was shown that the pullback symmetry condition of arbitrary order-two linear differential operators yields Schwarzian equation (90). The solutions of these order-two linear differential operators are much more general than hypergeometric functions and Heun functions [1]: they can have an arbitrary number of singularities. Let us see which Schwarzian equation (90), or equivalently, which function gives relations (96) corresponding to rigid constraints necessary to have modular correspondences [1].
Series calculations give the conditions on such that series solutions of the form are solutions of the Schwarzian equation with these ’s verifying relations (96). These constraints are conditions on the Laurent series of . For the solution series of the Schwarzian equation to have the pre-modular structure (96), i.e. the same structure as modular correspondences, the Laurent series of must be of the form:
[TABLE]
One easily verifies that this is the case for the previous modular form example where reads (93), as well as for all the other modular forms emerging in physics or enumerative combinatorics we mentioned in previous papers [29, 30, 31, 35, 37].
Condition (97) is a result whose scope transcends the hypergeometric functions framework. In order to show this, let us apply this result on the open problem of finding Heun functions111Finding the selected values of the parameters of a Heun function [47] (in particular the accessory parameter [48]) such that its series expansion is a series with integer coefficients (or more generally is globally bounded [31]), or such that the corresponding order-two linear differential operator is globally nilpotent [24] is a difficult problem. These classification problems are closely related to finding the Heun functions reducible to pullbacked hypergeometric functions [49], and to modular forms [46]. that could be modular forms [38], or pullbacked functions [16, 50]. The Heun function is solution of a linear differential operator of order two where and read:
[TABLE]
The corresponding function is easily deduced from the formula (10) given by . It has the following Laurent series expansion:
[TABLE]
and has the form (97) given by only when . Thus a general analytical constraint like (97) yields a simple exact constraint on the intriguing problem of the classification of the Heun functions that can be modular forms, and more specifically on the necessary conditions for the Heun functions to have a “pre-modular” structure.
6.4.1 Schwarzian equation for .
In order to understand the Laurent series condition (97), let us try to see what is so “special” in the case where . For
[TABLE]
the most general solutions of corresponding Schwarzian equation read:
[TABLE]
which just amounts to a simple transformation on :
[TABLE]
The solutions of the form are given by and are thus “trivial”: this is the case because the nome222Such that the transformations simply reduce to , see the concept of mirror maps [1]. is nothing but the variable! Similarly, the ratio of periods is just , and thus the condition is a “trivialization” of the mirror map.
6.4.2 Rank-two condition (76) and pre-modular structures.
The factorization of the order-two linear differential operator which corresponds to of the form (73), yields the rank-two non-linear differential equation (76) (see section 6.1.1). We would like to know when the modular correspondences structures (existence of solutions series such that (96), thus requiring ) are compatible with a factorization of the order-two linear differential operator and thus with condition (73). Imposing
[TABLE]
where is a rational function, one finds that must have the following Laurent series expansion:
[TABLE]
This result (105) can be directly obtained by looking for the Laurent series for with a pre-modular structure, i.e. such that the series are solutions of condition (76). As a byproduct, one finds that in the case (105) the solutions are such that (96). In particular the solution is a one-parameter family of commuting series. The case , or , corresponds to the simple order-two linear differential operator where is the homogeneous derivative .
More specifically, if one revisits our Heun classification problems, imposing the factorization condition (see the analysis sketched in E) together with the condition (97) required for the emergence of modular correspondence structure (96), one gets the following Laurent series expansion (see (168) for the definition of the parameters):
[TABLE]
This gives the condition (in agreement with condition (105)) and four other conditions. Excluding the case corresponding to the reduction from the four singularities of the Heun function to three singularities, one gets . The Heun function is a (Liouvillian) solution of a reducible linear differential operator of order two , where then reads:
[TABLE]
The pullbacks are solutions of the rank-two non-linear differential equation (76) which can easily be integrated into (see (81), (82)):
[TABLE]
giving a functional equation on the pullbacks with an Abel integral :
[TABLE]
One has for instance the following one-parameter series solutions for the pullback , which verify (96):
[TABLE]
The fact that solutions of the form occur can be clearly seen on equation (108). Even if the “pre-modular” conditions (96) are verified for this example, this Heun function will not be necessarily a modular form represented as a pullbacked hypergeometric function with more than one pullback for generic parameters111The exponent-differences at the four singularities are: . Introducing ,, the exponents difference of the three singular points of the hypergeometric function each the previous exponent-differences must be a multiple of the ’s..
7 Pullback symmetry of an operator up to equivalence of operators
With the aim of generalizing covariance (92), we introduce the derivative of
[TABLE]
which does not correspond to a modular form, since the derivative of a modular form is not a modular form. A derivative of the simple covariance identity (92) gives
[TABLE]
Using the order-two linear differential equation verified by , one can rewrite the in the RHS of (113), as a linear combination of and its derivative . One then deduces from relation (113) a slightly more general relation than the initial simple covariance (92)
[TABLE]
where and read in this particular example222If instead of the simple derivative (112) we had introduced \,\Phi(x)\,=\,\,$$L_{1}(\,_{2}F_{1}([1/12,5/12],[1],x)) where is an arbitrary order-one linear differential operator, we would have also obtained a relation of the form (114) but where and are much more involved expressions. :
[TABLE]
Recalling two Hauptmoduls and
[TABLE]
one can also write relation (114) in a more “balanced” form (see equation (7) in [2]). Introducing the two algebraic functions and
[TABLE]
one has the (modular form) hypergeometric identity:
[TABLE]
After performing calculations of a similar nature of the ones previously seen, one deduces the balanced relation on :
[TABLE]
which should be viewed as a (rational) parametrization of the relation having the form (114).
The interested reader shall find in F a detailed (and we hope pedagogical) analysis of the more general relation (114) given for a selected hypergeometric function555We thank A.J. Guttmann for showing us this remarkable hypergeometric function emerging in a dual context of combinatorics and random-matrix theory, counting the number of avoiding permutations [51, 52]. solution .
Let us provide an example of the relevance of the relation (114) in the context of integrable models in physics. In the case of the two-dimensional Ising model, the covariance (114) is instantiated on , the simplest of the low-temperature -fold integrals occurring in the decomposition of the susceptibility of the square Ising model [32, 33, 34] (see subsection 5.1 in [54]). When applied to , the Landen transformation , which provides an exact representation of a generator of the renormalization group [2, 7, 53], gives the following covariance relation (see equation222Note a misprint in the expression of the Landen transformation in the unlabelled equation above equation (62) in [54]. (64) in [54]):
[TABLE]
This relation (119) can also be written as
[TABLE]
or introducing the inverse Landen transformation (descending Landen transformation):
[TABLE]
[TABLE]
Remark: Note that the premodular condition (97), has no reason to be verified for such generalizations of modular forms (112), (114). For instance for given by (121), the function (see (10)) has the following Laurent series expansion (here ):
[TABLE]
More generally these (hypergeometric) examples provide simple illustrations of a more general pullback symmetry, where one imposes the pullback of an order linear differential operator to be homomorphic to that operator. In this case there exists two intertwiners (of order in general) and , such that:
[TABLE]
The pullback symmetry up to conjugation studied in sections 2, 3, 4, 5, 6 is appropriate for modular forms [29, 30, 31, 37], but not for derivatives of modular forms that also occur in physics (see for instance the previous relation (119) on the square Ising model). The emergence of such generalized covariance (125) for the representation of the Landen transformation (and more generally the modular correspondences providing exact representations of the generators of the renormalization group) on the other -fold integrals ’s of the susceptibility of the Ising model [32, 33, 34] is a challenging open problem, that will require one to consider reducible operators (see subsection 4.2).
Analyzing these more general constraints (125) will require many additional assumptions (beyond the one of having selected differential Galois group) on the linear differential operator to be able to perform more calculations.
8 Schwarzian conditions for different Calabi-Yau operators with the same Yukawa couplings
In the previous sections we have analyzed the question of the covariance under algebraic pullbacks of a linear differential operator of arbitrary order , i.e. the question of linear differential operators with algebraic pullback symmetries. Let us consider here the more general problem of the equivalence under pullbacks up to conjugations of two different linear differential operators, which is an enlightening sieve when one tries to classify selected linear differential operators in theoretical physics (Calabi-Yau linear differential operators [17, 18]). The interested reader will find in G an illustration of this important question where we revisit in detail some calculations of a paper by Almkvist, van Straten and Zudilin [17]. This calculation reexamines the question of pullback equivalence up to conjugation, of two selected order-four operators and verifying the Calabi-Yau condition:
[TABLE]
One finds that a Schwarzian equation verified by these two order-four linear differential operators and reads:
[TABLE]
where and are given by (30), and where and are the coefficients of and for respectively and , (see (236) and (237) in G).
One sees on this example that the nome and Yukawa couplings, expressed in terms of the variable, are related (see (G.1.1), (G.1.1)) by the pullback transformation. Yet, the Yukawa couplings of the two linear differential operators expressed in term of the nome, are related in an even simpler and “universal” way: , as shown in Appendix E of [31]. For a pullback with a series expansion of the form
[TABLE]
the nome and Yukawa couplings expressed in terms of the variable, of two order-four operators such that
[TABLE]
are simply related through the relations
[TABLE]
The Yukawa couplings expressed in terms of the nome222This function is often viewed as a function of the nome , since its -expansion in the case of degenerating family of Calabi-Yau 3-folds is supposed to encode the counting of rational curves of various degrees on a mirror manifold., are related in an even simpler “universal” way as so:
[TABLE]
The previous example (126) corresponds to and . In the case and , the pullback is a deformation of the identity and the Yukawa couplings expressed in terms of the nome, of the two linear differential operators are equal. Thus one recovers Proposition (6.2) of Almkvist et al. paper [17] where the Yukawa couplings coincide.
Since the Schwarzian equation (128) corresponds to the equivalence of two linear differential operators by pullback with remarkably simple relations (132) on their Yukawa couplings expressed in terms of the nome, the Schwarzian equation (128) can be seen as a condition to have simply related Yukawa couplings. In the case of deformation of the identity pullbacks, it can be seen as a condition of preservation of the Yukawa couplings (seen as functions of the nome). These results are not restricted to order-four linear differential operators (see Appendix E of [31] and G). For instance, one can impose that two different pullbacks of the same order- linear differential operator are homomorphic, i.e. there exist two intertwiners (of order in general) and such that:
[TABLE]
This last generalization turns out to be instructive for physics and enumerative combinatorics.
9 Conclusion
In a previous paper [1] we focused on identities relating the same hypergeometric function with two different algebraic pullback transformations
[TABLE]
along with the existence of analogues of the previous relation. Such remarkable identities correspond to modular forms that emerged in the analysis of multiple integrals related to the square Ising model [29, 30, 31, 35] or in other enumerative combinatorics context [37]. They can be seen as a simple occurence of infinite order999We have for instance in mind to provide exact representations of the renormalization group [2, 7, 53]. covariance symmetries in physics [2] or enumerative combinatorics.
The current paper generalizes these previous results beyond hypergeometric functions555Or even Heun functions, see [1]., analyzing the conditions for order- linear differential operators with an arbitrary number of singularities111Far beyond operators with hypergeometric solutions, or pullbacked hypergeometric solutions. to be pullback invariant up to conjugations:
[TABLE]
One finds that the pullbacks are differentially algebraic [3, 4], being necessarily solutions of the same Schwarzian equations as in [1]
[TABLE]
where the function encoding the Schwarzian equation (136) is a simple expression of the first two coefficients of the linear differential operator (see (56)). For order-two linear differential operators this Schwarzian condition turns out to be sufficient. In the case of linear differential operators with selected differential Galois groups however, we showed, for orders three and four, that the “Calabi-Yau” conditions (see sections 4.1) are rigid enough to force the pullbacked-invariant (up to conjugation) operators (see (135)) to reduce to symmetric powers of an order-two linear differential operator.
The reduction of the solutions of this Schwarzian differential equation to modular correspondences was an open question in [1]. Modular correspondences require the existence, for any integer , of solutions of the Schwarzian equation (136) of the form such that, for any integer and , the following “pre-modular” condition is satisfied:
[TABLE]
We derived in this paper a necessary and sufficient condition to obtain such “pre-modular” solutions for the “Schwarzian condition” (136). This condition turns out to be a simple condition on the Laurent series of encoding the Schwarzian condition:
[TABLE]
In light of what we have discussed so far, the current paper generates more questions than answers that give directions for further research. We have seen for example that (138) is a necessary and sufficient condition for obtaining “pre-modular” solutions for the “Schwarzian condition”, corresponding, in general, to a transcendental222The series (see (137)) are differentially algebraic, but, not necessarily algebraic functions. declination of modular correspondences. To have modular correspondences one needs the existence of selected values of the parameters such that the solution series (see (96)) actually reduce to algebraic functions. Is it only in the case of modular correspondences that such algebraic reductions for selected values take place ?
Then we showed that an order-two linear differential operator emerging in the context of avoiding permutations counting [51, 52], provides a good illustration of a generalization of the pullback-covariance (134) or of the pullback invariance up to conjugation (135): the \,{}_{2}F_{1}\Bigl{(}[-1/4,3/4],[1],x\Bigr{)} that comes up in the context of avoiding permutations counting [51, 52], verify a relation (see (182), (184)), whose general form is given by
[TABLE]
giving a non-trivial explicit example of a pullback invariance of an operator up to operator homomorphisms (see (125))
[TABLE]
Equation (119) providing an exact representation of the Landen transformation (generator of the renormalization group) on , together with the explicit calculations of section 7, make quite clear that conditions like (139) provide a natural and interesting generalization of modular forms, going beyond the Schwarzian equation (136).
At last, we examined the equivalence of two different linear differential operators, under pullback and conjugation, yielding again some Schwarzian condition relating these two linear differential operators (see relation (250)), and we discussed the consequence of such equivalence on the corresponding Yukawa couplings. These results revisiting and complementing the results of [17], provide powerful tools to analyze various symmetry and classification problems of selected linear differential operators, in particular linear differential operators of the Calabi-Yau type [18] (not necessarily of order four [31]).
When dealing with linear differential operators, we have seen the emergence of Schwarzian derivatives, consequence of the fact that the Schwarzian derivative is appropriate for the composition of functions [19] (see the chain rule of the Schwarzian derivative of the composition of function). Do higher order Schwarzian derivatives [55, 56, 57, 58] occur for pullback-symmetries of non-linear ODE’s, or, more generally, for functional equations?
Restraining oneself to the univariate linear differential operators case, let us remark that if condition (134), or (135), describe effectively all the modular forms that often occur in physics [29, 30, 35], or enumerative combinatorics [37], a pullback symmetry up to conjugation constraint like (135) could be restrictive in some sense since it seems to yield systematic reduction555At least in the case where the operators verify Calabi-Yau conditions and thus have selected differential Galois groups. to order-two linear differential operators. In contrast the simple hypergeometric example of section 7 seems to provide a natural generalization of modular forms: the pullback invariance of an operator up to operator homomorphisms condition (140) promises to cover a larger ensemble of exact representations of symmetries in physics or enumerative combinatorics. In particular the emergence of conditions like (139) of higher order, namely generalized covariance (140) for the representation of the Landen transformation111And more generally the modular correspondences providing exact representations of the generators of the renormalization group [2, 53]. on the other -fold ’s of the Ising susceptibility (see [32, 33, 34]), together with their corresponding large order reducible linear differential operators, is a challenging open problem.
Acknowledgments: We would like to thank S. Boukraa, M. van Hoeij and J-A. Weil for very fruitful discussions on differential systems. We thank A.J. Guttmann for providing an interesting pullbacked hypergeometric example. This work has been performed without any ERC, ANR or MAE financial support.
Appendix A A simple reducible linear differential operator of order four
Let us consider an order-four linear differential operator which is the square of an order-two linear differential operator: , where . This reducible order-four linear differential operator is of the form where the two coefficients and read respectively:
[TABLE]
The coefficients of the order-four operator verify the Calabi-Yau condition222 The exterior square of that an order-four operator is of order five instead of order six. This is a general result: the order of the symmetric squares of operators is less than . Such -th powers verify higher order Calabi-Yau conditions. (32). We even have the identity999More generally, the order-one linear differential operator rightdivises the exterior square of the -th power of , for any integer . that the exterior square of is the product of an order-one operator (having the wronskian of as solution), an order-three operator which is the symmetric square of and again the same order-one operator:
[TABLE]
For this reducible order-four linear differential operator the first steps of the calculations give a function given by (30), namely . Using (141) one can rewrite in terms of and . One gets an expression similar to (10) but different, namely , which is exactly (10) but divided by . Therefore the pullback condition on this square operator does not reduce to the pullback condition on the (underlying) .
The change of variable on a linear differential operator which is the product of two operators, is the product of these two linear differential operators on which this change of variable has been performed. More precisely with our normalization of the pullback of a linear differential operator a condition would give the relation
[TABLE]
In other words the pullback of corresponds to a conjugate of another order-four linear differential operator , which is not but is also reducible into two different order-two linear differential operators. Note that the order-two linear differential operator depends on the change of variable .
Appendix B Order-five linear differential operators
Let us consider an irreducible order-five linear differential operator
[TABLE]
and let us also introduce two other linear differential operator of order five, the operator conjugated of (144) by a function , namely , and the pullbacked operator which amounts to changing in . Imposing a generalized (symmetric) Calabi-Yau condition amounts to imposing that the symetric square of (144) is of order less than (the generic order) . Using this (symmetric) Calabi-Yau condition to perform any calculation is a very difficult task since this condition corresponds to a huge polynomial in the coefficients and their derivatives. However, similarly to what we did in section 4.3 we can introduce a parametrization, similar to (46) of this huge (symmetric) Calabi-Yau condition. We saw in [36] that the (symmetric) Calabi-Yau condition for an order-five linear differential operator (which amounts to saying that the symmetric square of is of order less than ), amounts to saying that has the following decomposition
[TABLE]
where and are order-one, order-one, and order-three self-adjoint linear differential operators of the form previously given with (44) and (45), and is another order-one self-adjoint operator:
[TABLE]
It is straightforward to get the coefficients of the order-five operator (144):
[TABLE]
This gives a parametrization of the (symmetric) Calabi-Yau condition and thus a way to perform calculations for an order-five operator that verifies this huge (symmetric) Calabi-Yau condition. Again, one finds that just imposing this (symmetric) Calabi-Yau condition is not sufficient to have .
There is one subcase of that huge polynomial condition that can be written explicitly (in a similar manner we wrote the Calabi-Yau (32) and symmetric Calabi-Yau (3.2) conditions, see (B), (B), (B) below).
Let us consider an order-two linear differential operator , and the symmetric fourth power of , the coefficients of that order-five operator read:
[TABLE]
Conversely this means and
[TABLE]
When the three conditions (B), (B), (B) are verified, the symmetric square of the order-five linear differential operator is of order instead of (and thus its differential Galois group is ). The three conditions (B), (B), (B) are necessary for to be reducible to the symmetric cube of an underlying order-two linear differential operator. If one imposes the three conditions (B), (B), (B), the order-five linear differential operator is simply conjugated to its adjoint:
[TABLE]
where denotes the wronskian of . Recalling that an order-two linear differential operator , having a wronskian is such that , the identity (152) is a simple consequence of the fact that the order-five operator reduces to the symmetric fourth power of an order-two linear differential operator.
Note that by imposing the two conditions111We have the same result imposing the two conditions (B) and (B), or (B) and (B) (B), (B), the symmetric square of the order-five operator becomes of the generic order , yet the symmetric square of does not have a rational solution (the operator and its adjoint are not homomorphic: the differential Galois group of is not equal, or included, in the orthogonal group ).
The identification of these two order-four linear differential operators and gives four conditions , , corresponding respectively to identification of the coefficients of and .
Performing the same pullback-compatibility calculations we did for order-three, and order-four operators for is a tremendously difficult task in a general framework.
The first calculation steps can be performed, giving the exact expression of the conjugation function from as:
[TABLE]
and, eliminating the log-derivative between and , giving the Schwarzian equation
[TABLE]
where, this time:
[TABLE]
Again one finds that the expression of given by (155) gives back the expression (10) when and are deduced from (B) ( and becoming and ).
The condition that we called in [35] symmetric Calabi-Yau condition for the operator (corresponding to impose that its symmetric square is of order less than ) is a huge polynomial condition on the coefficients of and its derivative. Seeing if these pullback-compatibility calculations yield necessarily the huge (symmetric Calabi-Yau222Meaning that the order-five operator has a differential Galois group.) condition and the three conditions (B), (B), (B), or, in other words, that the order-five linear differential operator necessarily reduces again to (a symmetric fourth power of) an underlying order-two linear differential operator, remains an open question.
Appendix C Reduction of the order-three ODE (68) to the order-two ODE (77)
in the rank-two case (73).
The order-three linear differential equation (68) on should reduce to the order-two linear ODE (77) in the rank-two subcase (73). When , the order-three linear differential operator (see (69)) has three solutions:
[TABLE]
This can be seen as a consequence of the fact that the order-two linear differential operator rightdivides the order-three operator :
[TABLE]
In this rank-two subcase (73), the function is and not the third solution which prevails in the general Schwarzian case (see (93)). The form of the last solution can be deduced from the fact that order-three linear differential operator is the symmetric square of an order-two self-adjoint operator (see (69)) which is simply conjugated to the order-two operator given by (78):
[TABLE]
which has clearly the solution as well as the solution , deduced from the solutions of (see (78)). The three solutions (156) correspond to all the products of these two solutions namely the square of and , and their product. Note that the factorization (C) requires condition (73) to be satisfied.
Remark : Recalling (69), (78), (C), one can see that the rightdivision (157) can be seen as a consequence of the identity
[TABLE]
Appendix D Mirror maps for .
The modular correspondences are infinite order algebraic transformations such that
[TABLE]
where is an algebraic function. The modular correspondences are solutions of the Schwarzian condition (90), where simply related to the function (see (68)) are given by equations (93). These modular correspondences have series expansion at of the form
[TABLE]
where and are such that , corresponding to the “simplest” examples of mirror maps [1]. More precisely, the well-known “mirror maps” [61, 62, 63, 64] are often described as series with integer coefficients [65, 66]. These series correspond to a rescaling of and by , namely [1]:
[TABLE]
and:
[TABLE]
The two functions and are differentially algebraic [3, 4], but not holonomic functions. Introducing the function , equation (70) with yields the following Schwarzian relations on
[TABLE]
when the (composition) inverse of verifies the functional equation and Schwarzian equation:
[TABLE]
Note that the one-parameter commuting family (66) solution of the Schwarzian equation (90), can be expressed using these two functions and as where .
Appendix E Selected subcase: Heun function examples.
Since the classification of Heun function is an interesting non trivial problem, let us use the condition (73), , to find the Heun functions corresponding to such factorizations (like the example analysed in detail in [1]). The Heun function is solution of a linear differential operator of order two where and read:
[TABLE]
One thus simply deduces the corresponding function function from the formula (10), namely . At first sight we exclude the values and in order to have Heun functions with four singularities to avoid trivial subcases where the Heun functions could reduce to hypergeometric functions. If one imposes that the function is of the form (73), the rational function must be of the form:
[TABLE]
The identification of given by (73) with of the form (168), with where and are given by (166) and (166), gives a set of five equations in the parameters of the Heun function and in the three coefficients in (168), the simplest one being
[TABLE]
The example analysed in [1] corresponding to the factorization condition (73) corresponds to the following values of these parameters:
[TABLE]
which corresponds to the branch of (168). The analysis of these five equations gives four solutions that we have excluded at first sight because they corresponds to and yield reduction to hypergeometric functions111Like for instance , with . , except when a bunch of conditions occur
[TABLE]
The example analyzed in [1] corresponding to (170) is equivalent to
[TABLE]
Appendix F Pullback invariance up to operator homomorphisms: a simple hypergeometric example.
Let us consider the order-two linear differential operator
[TABLE]
which has the hypergeometric function solution . We have the following homomorphism of the type (133) between pullbacked by two simple different rational functions and :
[TABLE]
Denoting and the two rational pullbacks and in (176) one finds that they are related by the following rational algebraic curve:
[TABLE]
The two Hauptmoduls parametrizing the modular equation222See equation (108) in subsection 5.1 of [1]. corresponding to the representation of , are given as follows:
[TABLE]
Note that we have the following relations between and , and the two Hauptmoduls and :
[TABLE]
which explain the compatibility between the two relations:
[TABLE]
Relation (176) yields the following identity on the hypergeometric function
[TABLE]
[TABLE]
Introducing the order-two linear differential operator annihilating the pullbacked hypergeometric function :
[TABLE]
the compatibility between relation (182) and (184) is a consequence of the identity
[TABLE]
namely that the product is equal to modulo . Of course introducing the order-two linear differential operator annihilating the pullbacked hypergeometric function one also has a very similar identity:
[TABLE]
which means that the product is equal to modulo .
One can get rid of the unpleasant square roots in (183), (185) introducing instead of the pullbacked hypergeometric functions and , the functions
[TABLE]
is a series with integer coefficients
[TABLE]
when is a Laurent series with integer coefficients. These two functions are simply related as follows:
[TABLE]
In fact the function is solution of the order-two linear differential operator
[TABLE]
with a remarkable duality property. It is homomorphic to its pullback by :
[TABLE]
The simple relation (190), which is a rewriting of (182) with the order-one operator being replaced by the order-one operator , is an obvious consequence of the homomorphism (193). Of course we also have the (mirror) relation222Consequence of the (mirror) homomorphism relation: ., compatible with (190), which is a rewriting of (184) with the order-one operator being replaced by the order-one operator
[TABLE]
Note that and given by (191) and (196) are related by the involutive change of variable :
[TABLE]
Denoting the order-two operator annihilating , the compatibility between the relations (190) and (195) corresponds to the relations:
[TABLE]
which should be compared with (188) and (214).
Relations (184), or111Or relations (182) or (190), but in that case the series corresponding to are Puiseux series : (195), can be seen as a particular case of a generalized pullback symmetry condition of the form
[TABLE]
where and are algebraic functions. Identities like (182) can be seen as generalizations of the identities analysed in [1].
F.1 Representation of the composition of the algebraic transformations .
We want to see the algebraic transformations as symmetries. In particular we want to have a representation of the composition of these algebraic transformations, like:
[TABLE]
Let us show here that by building on the previous example we can actually provide identities of the type (200). Introducing
[TABLE]
one has the new pullback symmetry relation similar to (182):
[TABLE]
where:
[TABLE]
One also has the new pullback symmetry relation similar to (184)
[TABLE]
[TABLE]
Let us introduce the order-two linear differential operator annihilating the pullbacked hypergeometric function :
[TABLE]
where
[TABLE]
The compatibility between relation (182) and (184) is a consequence of the identity:
[TABLE]
Of course introducing the order-two linear differential operator annihilating the pullbacked hypergeometric function , one also has a similar identity with the same rational function :
[TABLE]
Again we have that and are obtained from each other by the (involutive) change of variable :
[TABLE]
Note that the two pullbacks and (see (201), (202)) are related to the two previous pullbacks and (see (176)):
[TABLE]
Recalling the new identities (203) and (207) read
[TABLE]
or, introducing :
[TABLE]
Denoting and the two pullbacks in (218), (219),
[TABLE]
one sees that they are related by the simple symmetric algebraic curve:
[TABLE]
Let us consider the algebraic equation (F), that we denote because it is so closely related to the modular equation representing (see their close relation with the Hauptmoduls (179) and (181)). Performing the resultant in of the polynomial with the same one one gets a new algebraic equation . The two pullbacks and are actually a rational parametrization of that new algebraic equation . In other words, if we think identity (184) as a symmetry transformation identity of the type (199), the new identity (203) must be seen as the identity for the iteration of that transformation:
[TABLE]
We are very close to a modular form, the previous algebraic curve (F) playing the role of the modular equation222Given by equation (108) in subsection 5.1.1 in [1]. (see (181)), and the algebraic curve playing the role of the modular equation corresponding to .
Note that if one calculates the function corresponding to the order-two operator , one gets
[TABLE]
which is of the form (in contrast with the result for , see (124)).
Appendix G Schwarzian conditions for different Calabi-Yau operators with related Yukawa couplings
G.1 Revisiting a Calabi-Yau operator in [17]
Following Almkvist, van Straten and Zudilin [17], let us consider the order-four linear differential operator such that its exterior square annihilates555See also [59].
[TABLE]
This order-four linear differential operator such that its exterior square is order-five (it verifies the Calabi-Yau condition (32)) reads
[TABLE]
where and read:
[TABLE]
The other rational functions and are more involved rational functions that will not be given here. The operator can be seen as the “exterior (or antisymmetric) square root111See the concept of Yifan Yang pullback introduced in [60].” of the order-five linear differential operator that annihilates the hypergeometric function (225).
Remark: In [17] the authors introduce a proxy of the exact “exterior square root” namely the so-called Yifan Yang pullback, given in general by the equations in the section “Definition” page 10 of [60]222The author of [60] has benefited from an unpublished result by Yifan Yang. Note that there is a misprint in [60] in the “Definition” of Yifan Yang pullback: on top of page 11, the term should be replaced by . With this correction the exact ‘exterior square root” and the Yifan Yang pullback are related by a simple conjugation , where . and, in this example, by equations (3.11), page 278 in [17], which reads
[TABLE]
where and read:
[TABLE]
the other rational functions and being more involved rational functions that will not be given here. The “Yifan Yang pullback” is related to the exact “exterior square root” by a simple conjugation , with . In general one may prefer to introduce the Yifan Yang pullback defined page 10 and 11 of [60] instead of the exact “exterior square root”, because the corresponding formulae are simpler. It does not make any difference however since the two operators are simply conjugated.
Let us consider the order-four linear differential operator given on page 284 of [17] which annihilates the Hadamard product of two simple hypergeometric functions:
[TABLE]
This order-four operator reads
[TABLE]
where:
[TABLE]
Introducing the pullback and the function
[TABLE]
one has the relation
[TABLE]
and one verifies that a Schwarzian equation (235) is actually verified for (G.1) and (G.1)
[TABLE]
with:
[TABLE]
This Schwarzian equation (235), together with the definitions (236) and (237), are exactly the Schwarzian equation (6.5) together with definition (6.4), page 290 of [17].
G.1.1 Schwarzian conditions for Calabi-Yau operators and Yukawa couplings.
Let us calculate the series expansion of the nome and Yukawa couplings [31] of and . In order to perform the calculations for arbitrary values of and , let us introduce the same variables and as the one introduced by [17]:
[TABLE]
Considering the subcase and , the nome of reads
[TABLE]
while the nome of reads:
[TABLE]
The respective Yukawa couplings of and read:
[TABLE]
[TABLE]
In terms of the nome the Yukawa couplings read:
[TABLE]
and
[TABLE]
On this example we see that the nome and Yukawa couplings expressed in terms of the variable, are simply related (see (G.1.1), (G.1.1)) by the pullback transformation. The Yukawa couplings expressed in term of the nome of the two linear differential operators are related in an even more simple and “universal” way: . This is a general result (see Appendix E of [31]). For a pullback with a series expansion of the form
[TABLE]
the nome and Yukawa couplings expressed in terms of the variable of two order-four linear differential operators such that
[TABLE]
are simply related as follows:
[TABLE]
Their Yukawa couplings, expressed in terms of the nome, are related in an even simpler “universal” way:
[TABLE]
The previous example corresponded to the case and . In the case and , the pullback is a deformation of the identity and the Yukawa couplings expressed in terms of the nome of the two operators are equal. One thus recovers Proposition (6.2) of [17] where the Yukawa couplings coincide.
G.2 Schwarzian conditions for Calabi-Yau operators related by pullback and conjugation.
In fact the Schwarzian condition (235) can be obtained in a totally general framework where two order-four linear differential operators are equal up to pullback and conjugation. Let us consider two order-four operators and such that
[TABLE]
A straightforward calculation similar to the one performed in section 4 yields the Schwarzian relation222This result is the same as the one in [17].
[TABLE]
where the and are given by (30), the and being the ones of the corresponding operators and :
[TABLE]
Remark 1: There is nothing specific with order-four linear differential operators, one has the same result for two operators of arbitrary orders equal up to pullback and conjugation (see (249)): the expressions of and being the ones given in (56), (58). One also has
[TABLE]
Remark 2: The expressions of and are related by (253). Let us assume that is compatible with the modular correspondences structures (existence of solutions of the Schwarzian equations of the form with (96)). One thus has Is this condition automatically satisfied for as a consequence of (253) ? For pullbacks of the form the function deduced from (253), reads:
[TABLE]
The condition (97) for the modular correspondences structures is thus preserved by pullbacks.
G.3 More general framework
For arbitrary orders we observed that the functions that occur in the Schwarzian conditions are left invariant under conjugations of the operators (64) and (5). More generally, one can consider operators that are not conjugated by a function , yet homomorphic, in the sense of the equivalence of operators111Two linear differential operators and of order are homomorphic [35, 36] when there exists operators (intertwiners) of order at most , such that . . For a given operator of order-, one can easily obtain operators homomorphic to . For instance, for an order-two linear differential operator , introducing the order-one operator , an order-two operator homomorphic to is easily obtained performing999In Maple just to rightdivision(LCLM(),). the rightdivision by of the LCLM of and . If one now compares the functions corresponding respectively to and , one sees that they are quite different, except when , in which case one reduces the operator equivalence to a conjugation by a function . The analysis of the conditions for two order- operators and to be homorphic up to pullback
[TABLE]
is a much more general problem corresponding to massive calculations even if one restricts to operators that are homomorphic to their adjoint (thus corresponding to selected, orthogonal or symplectic, differential Galois groups555In that general framework (255), we do not have the Calabi-Yau, or symmetric Calabi-Yau, equations that help us to perform our calculations.). Performing such calculations will require new tools and ideas. This cannot be performed in general (like we did in the first section of this paper) but could be considered on particular problems emerging from physics or enumerative combinatorics, where the operators will be of some “selected” form.
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