The Basso-Dixon Formula and Calabi-Yau Geometry

We analyse the family of Calabi-Yau varieties attached to four-point fishnet integrals in two dimensions. We find that the Picard-Fuchs operators for fishnet integrals are exterior powers of the Picard-Fuchs operators for ladder integrals. This implies that the periods of the Calabi-Yau varieties for fishnet integrals can be written as determinants of periods for ladder integrals. The representation theory of the geometric monodromy group plays an important role in this context. We then show how the determinant form of the periods immediately leads to the well-known Basso-Dixon formula for four-point fishnet integrals in two dimensions. Notably, the relation to Calabi-Yau geometry implies that the volume is also expressible via a determinant formula of Basso-Dixon type. Finally, we show how the fishnet integrals can be written in terms of iterated integrals naturally attached to the Calabi-Yau varieties.


Introduction
Over the last decade, it has become clear that the analytic structure of perturbative scattering amplitudes and multi-loop Feynman integrals is tightly related to topics in algebraic geometry.In particular, it is known that Feynman integrals compute (relative) periods in the sense of Kontsevich and Zagier [1,2].As a consequence, understanding the geometry associated to a Feynman integral may inform us about the class of transcendental functions and numbers that appear in the result, and methods for the computation of periods may be adapted to perturbative computations in quantum field theory.Which classes of geometries and functions may arise from Feynman integrals is still an open question.The simplest examples can either be expressed in terms of polylogarithmic functions or involve elliptic or modular curves.The corresponding functions are by now relatively well understood (see, e.g., ref. [3] and references therein for a recent review).It is known that also higher-genus Riemann surfaces may show up [4][5][6], though in that case the relevant functions are still poorly understood [7].In addition, Calabi-Yau varieties may arise, and in this case the relevant class of functions is slowly emerging, see, e.g., refs.[8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].
Particularly interesting representatives of Feynman integrals which involve Calabi-Yau geometries are the ladder, traintrack and fishnet integrals [11-13, 17, 22], because they compute correlators in the so-called fishnet conformal field theories of refs.[23][24][25].Originally, these fishnet theories were discovered in D = 4 spacetime dimensions, where they arise as double-scaling limits of planar gamma-deformed N = 4 Super Yang-Mills theory with gauge group SU(N c ).The gamma-deformed theory depends on three parameters γ 1,2,3 in addition to the Yang-Mills coupling constant g.In its simplest version, the fishnet limit of this model is defined by taking g → 0 and γ 3 → i∞, while ξ 2 := g 2 N c e −iγ 3 is kept fixed and furnishes the new coupling constant of the limit theory.Similar to N = 4 Super Yang-Mills theory, the fishnet theory has an AdS/CFT dual, the so-called fishchain model introduced in refs.[26,27].Different combinations of limits in the above parameters lead to more involved families of fishnet theories, which can in turn be generalised, e.g. to generic spacetime dimensions D. Notably, in the planar limit the fishnet models inherit integrability properties from their AdS/CFT mother theories, which allows one to study Feynman integrals as integrable systems [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44].
Fishnet integrals are also interesting purely from the Feynman integral perspective, because they have a particularly simple analytic structure, which often allows one to obtain analytic expressions even at high loop orders.In D = 4 dimensions, it is known that all four-point ladder integrals can be evaluated in terms of classical polylogarithms [45,46].Analytic results are also known for two-loop traintrack integrals in terms of elliptic polylogarithms [47][48][49].Remarkably, it was conjectured in ref. [29], and later proven in refs.[39,43], that four-point fishnet integrals where the internal vertices are arranged on a square lattice can be expressed as determinants of ladder integrals.This simple analytic structure, known as the Basso-Dixon formula, was also observed for four-point fishnet integrals in D = 2 dimensions in ref. [33].
In ref. [17] we argued that fishnet integrals in two dimensions are tightly related to certain families of Calabi-Yau (CY) varieties, and that the value of the fishnet integral computes the quantum volume of the mirror CY geometry.One of the goals of this paper is to study the relevant Calabi-Yau geometries in the context of four-point fishnet integrals.By conformal invariance, these integrals depend on a single cross ratio, and correspondingly the family of Calabi-Yau varieties is described by a single complex modulus.A lot of information about a Calabi-Yau variety is encoded in its periods.Due to conformal invariance, in the case of four-point fishnet integrals we need to consider a one-parameter family of Calabi-Yau varieties, and the periods are given by solutions of an ordinary differential operator, called the Picard-Fuchs operator.We present a strategy to determine the Picard-Fuchs operator for fishnet integrals.We observe that for ladder integrals, the Picard-Fuchs operators fall into the class of so-called Calabi-Yau operators [50][51][52][53].In the case of a non-ladder integral, the Picard-Fuchs operator is the exterior power of the differential operator for a ladder integral.This implies that the periods can be written as determinants, very reminiscent of the Basso-Dixon formula.Remarkably, this also shows that fishnet integrals and volumes can be written in a Basso-Dixon-like determinantal form.Since the solutions of the ladder Picard-Fuchs operators form irreducible representations of the geometric monodromy group, representation theory determines to a large extent the properties of the solutions for the general fishnet operators.For the ℓ-loop ladder integrals we employ a Picard-Fuchs operator of order ℓ + 1, while the order of the differential operator associated to generic Basso-Dixon graphs is given by b ℓ as specified in table 1 in section 5. 3.This paper is organised as follows: In sections 2, 3 and 4 we give a short review of fishnet integrals, Calabi-Yau geometry and general properties of Picard-Fuchs operators.In particular in section 3.3 we explain how the fishnet integrals are related to the calibrated volume of the Calabi-Yau manifold M or the quantum volume of its mirror W. In section 5 we analyse the Picard-Fuchs operators for the Calabi-Yau varieties obtained from fishnet integrals, and we discuss the properties of their periods.In section 6 we show how to compute the fishnet integrals from these periods, and we demonstrate how the Basso-Dixon formula arises naturally from our geometric considerations.In section 7 we show how the volumes of the Calabi-Yau varieties can naturally be expressed in terms of iterated integrals, which can themselves be cast in the form of a Basso-Dixon-like formula.Finally, in section 8 we present our conclusions.We include several appendices with mathematical proofs omitted throughout the main text.

Fishnet integrals and the Basso-Dixon formula
Throughout this paper, we will consider a class of position space Feynman integrals called fishnet integrals, cf.ref. [28].In particular, we will be interested in the four-point limits of these fishnet integrals, whose corresponding Feynman graphs are shown in figure 1.For M = 1 (or N = 1) we refer to them as ladder integrals.Fishnet integrals can be defined in arbitrary space-time dimensions D, and the external points are labelled by points α i ∈ R D , while the internal points are labelled by ξ i ∈ R D .An edge in the graph connecting two points labelled by a and b represents a propagator [(a − b) 2 ] −D/4 , and we integrate over all internal points.In the following, we denote the four-point integral obtained in this way by I (D) M,N (α), where α = (α 1 , . . ., α 4 ).
Fishnet integrals are interesting for various reasons.In particular, they compute correlation functions in a bi-scalar fishnet theory, whose Lagrangian in D dimensions reads [23,24] It is easy to check that, due to the chiral nature of the interaction, the only Feynman graph contributing to the correlation function ⟨X(α 1 ) M Z(α 2 ) N X(α 3 ) M Z(α 4 ) N ⟩ is the fishnet graph from figure 1.In a more general context it was shown that renormalisation requires additional double-trace couplings to be added to the above fishnet Lagrangian [54,55], and the betafunction has two conformal fixed points [55,56].These additional operators are irrelevant for the correlators considered here, and so we will not discuss them any further.
Fishnet integrals enjoy an enhanced symmetry.In particular, they are invariant under conformal transformations in D dimensions, where the external points carry conformal weights D/4, and they are annihilated by the generators of the Euclidean conformal algebra so(1, D + 1).This implies that we can write where M,N (α) is an algebraic function carrying the conformal weight and ϕ M,N (χ) only depends on conformal cross ratios: In addition, generic fishnet integrals are invariant under the generators of the Yangian over the conformal algebra [30,37] (at least in the case when the external points are not identified, i.e., no two propagators are connected to the same external point, cf.ref. [57]).Since Yangian invariance is not directly relevant to the content of this paper, we will not dwell further on this here.Since the four-point integrals considered in this paper depend on a single holomorphic variable, we can obtain an associated Picard-Fuchs operator by the method explicitly illustrated in section 5. Finally, four-point fishnet integrals exhibit a remarkable simplicity.This was first observed in ref. [29] in D = 4 dimensions, where it was conjectured (and recently proven in refs.[39,43]) that the integrals I (4) M,N can be expressed as determinants of ℓ-loop ladder integrals I (4) 1,ℓ .The latter are known for all values of ℓ in terms of classical polylogarithms [45,46].A similar relation was shown to hold in D = 2 dimensions in ref. [33].One of the aims of this paper is to discuss the relation between geometry and the Basso-Dixon formula in two dimensions.In the remainder of this section, we review some general results about fishnet integrals in two dimensions.

Fishnet integrals in two dimensions
In D = 2 dimensions it turns out to be convenient to package the labels α j , ξ j ∈ R2 into complex variables: It is easy to see that fishnet integrals can be cast in the form1 with a = (a 1 , . . ., a n ), x = (x 1 , . . ., x ℓ ).The integrand depends on the polynomial where the product ranges depend on the graph topology.Up to an algebraic prefactor that carries the conformal weight, the integral then only depends on the cross ratio formed by the four complex variables: The four-point fishnet integrals have been evaluated analytically in ref. [33] in terms of hypergeometric functions.More precisely, we have with θ z := z∂ z and z = − a 12 a 34 a 23 a 14 = 1 z . (2.9) The ladder integrals ϕ W (z) := ϕ 1,W −1 (z) are given in terms of bilinear combinations of hypergeometric functions . (2.10) The hypergeometric functions are (anti-)holomorphic at the origin, W +1 F W (. . .; z) = 1 + O(z), but the derivatives with respect to ϵ introduce logarithmic terms.Equation (2.8) is the two-dimensional analogue of the Basso-Dixon formula of ref. [29] and was first presented in ref. [33].
3 The Calabi-Yau geometries for the fishnet graphs In this section, we define the Calabi-Yau (CY) geometries associated to integrals defined in eq.(2.5).We start in section 3.1 by a description of the singular families of Calabi-Yau varieties defined by multi coverings of P l = × l i=1 P 1 i branched at eq. (2.6), so that the integral becomes a real bilinear of the periods of the Calabi-Yau variety.The periods of the Calabi-Yau varieties associated to the ladder integrals play a special role: we will argue that the periods of more general fishnet geometries can be obtained by anti-symmetric powers of periods of ladder geometries.Moreover, to the latter we can associate generically smooth families of Calabi-Yau manifolds M 1,N , as well as smooth mirror families W 1,N .Using a deep result of Beukers and Heckman on the geometric monodromy group action G N on periods of M 1,N we can understand the periods of general fishnet geometries and their degenerations by geometric representation theory of G N .We provide the necessary information about these building blocks in section 3.2.Finally, in section 3.3, we discuss two different interpretations of fishnet integrals as volumes associated to CY manifolds.

Calabi-Yau varieties associated to ladder integrals in two dimensions
In ref. [17] we argued that fishnet integrals in D = 2 dimensions are closely related to Calabi-Yau (CY) periods on Calabi-Yau varieties.Roughly speaking, a Calabi-Yau manifold M is a Kähler manifold with trivial canonical bundle, K M = 0 (see ref. [58] for the precise definition we use in this paper).CY varieties M include more generally the singular CY geometries in complex deformation families of CY manifolds.As in ref. [16], the restricted physical parameters of Feynman integrals identify them also in the application here with Calabi-Yau periods on these singular loci.Similar to ref. [16], our strategy is again to find the smooth family and to argue that the restriction to the singular loci defines a suitable CY motive.
For the ladder integrals considered here, the smooth motives all turn out to be hypergeometric (Gel ′ fand, Kapranov and Zelevinskȋ) motives associated to complete intersections, cf.
refs.[14,15].The relevant geometries M are d-fold coverings over the base P l defined by an equation of the form The coverings are finitely branched at B = {P (x, a) = 0}.For now, we keep the discussion general, and we consider fishnet graphs with an arbitrary number of external points a, and we specialise below to the four-point graphs in figure 1.The coverings defined by eq.(3.1) are projective and therefore Kähler.By the adjunction formula, the canonical class of Here H i = {x i = 0} is the hyperplane class of the i th P 1 , and ν ∈ N is the common degree of P (x, a) in each x i .The triviality of K M implies the existence of a unique, nowhere vanishing (ℓ, 0)-form.As a section of K M , it is given in the homogeneous coordinates [x i : w i ] of each P 1 by Here we have to make a comment.The polynomial P (x, a) depends on the external points a. Conformal invariance, however, implies that the only non-trivial functional dependence can be in cross ratios z, up to an algebraic function F (a) that carries the conformal weight (cf.eq.(2.7)).Correspondingly, we can define the (ℓ, 0)-form Ω(z) such that it only depends on cross ratios.The function F (a) ∈ Q(a) lives in a simple algebraic extension of Q(a) completely fixed by our normalisation Ω(z), chosen here as in Definition 4.3. of ref. [58]: where M z denotes the fiber over the point z, ω denotes the Kähler form on M z , and K(z) is the Kähler potential on the complex moduli space M cs parametrised by z.Note that ϕ(z) has two interpretations as a volume, as we will further explain in section 3.3.On a family of CY ℓ-folds parametrised by z, we can calculate the monodromy-invariant real quantity ϕ(z) in terms of the periods (see also ref. [59]) Here Π(z) is a vector of period integrals of Ω(z), where the Γ j label an integral basis of the middle homology H ℓ (X, Z), and Σ is the corresponding integral intersection form, which is even for ℓ even and odd for ℓ odd.In particular, for ℓ odd, it can always be chosen to be symplectic.A period of the family M z is a solution of the Calabi-Yau Picard-Fuchs ideal, denoted in the following by CYPFI(M z ).The existence of the CYPFI(M z ) is equivalent to the flatness of the Gauss-Manin connection on the Hodge bundle over the complex moduli space M cs .Every solution vector Π(z) fulfils a set of quadratic relations, which follow from Griffiths transversality of the Gauss-Manin connection (see ref. [16] for a review from the Feynman graph perspective): So far the discussion applies to very general families defined by d-fold branched coverings as in eq.(3.1).We now focus on the special case we are interested in, namely the fishnet graphs from figure 1.In that case, there is a single cross ratio, and so we write simply z instead of z.In particular, the choice (d, ν) = (2, 4) satisfies the condition 2d d−1 = ν implied by eq.(3.2). 2 Writing P = P M,N , as defined in eq.(2.6), we arrive at the singular fishnet geometry M M,N : From now on, we will indicate quantities related to M M,N by an index (M, N ), e.g., we will write F M,N (a), Ω M,N (z), Π M,N (z) and ϕ M,N (z) instead of F (a), Ω(z), Π(z) and ϕ(z).Moreover, z will denote the cross ratio defined as in eq.(2.7).From eq. (3.5) we then see that the value of the fishnet integral ϕ M,N (z) can be computed from the periods Π M,N (z) and the intersection form Σ M,N of M M,N : To each fishnet integral we can associate the Calabi-Yau variety M defined by eq.(3.8) or in case of the ladder integrals alternatively a Calabi-Yau manifold as in eq.(3.13).The geometrisation implies that after absorbing the factor |F (a)| 2 we can geometrise the integral in terms of a real period bilinear in eq.(3.5) that has geometric interpretations in terms of two different volumes of a Calabi-Yau manifold, as we explain in detail in section 3.3.

The Calabi-Yau manifolds associated to the ladder integrals
Let us start from the Picard-Fuchs equation for the N -loop ladder integral.The latter can be derived by the method outlined in section 5 from the general form of the period integral in eq.(3.6) with Ω(z) given in eq. ( 3.3).We note that for the ladder integrals the loop order ℓ, the complex dimension dim C (M 1,N ) and N are equal so that below we can use ℓ, d or N as appropriate for the context.We find that the associated motive is of Gel ′ fand, Kapranov and Zelevinskȋ type before the one-parameter specialisation, and in the one-parameter limit (where the integral only depends on the cross ratio z), it corresponds to the hypergeometric system (see ref. [61] for the conventions): with the Riemann P-Symbol This one parameter hypergeometric motive has been geometrically realised in ref. [62] as the CYPFI for the mirror M 1,N of the complete intersection of N + 1 quadrics in P 2N +1 , abbreviated as Mirror symmetry is a well studied symmetry relating the h 1,1 (W) complexified Kähler structure deformations of a CY N -fold W to the h 1,N −1 (M) complex structure deformations of the mirror CY N -fold M and vice versa, see ref. [16] for a review from the Feynman intregral perspective and more references.Therefore the simple one complex structure parameter families of CY N -folds have been first studied as mirrors of simple complete intersections having only one Kähler structure deformation as in (3.13).In this case the mirror can be constructed as a resolution of an orbifold with respect to the action of a discrete group Γ ⊆ SU(N ) on W 1,N , i.e., it is defined as M 1,N = W 1,N /Γ .The resolution of M 1,3 = W 1,3 /Z3 8 has been constructed in ref. [63] with methods that apply in general.This realises the Calabi-Yau motive as the motive of a smooth Calabi-Yau N -fold M 1,N .
The differential operator L N has a point of maximal unipotent monodromy (MUM) at z = 0.At such a MUM point the local index, here 0, is (N + 1)-times degenerate.That implies that one of the solutions has a log N z behaviour and the monodromy M around z = 0 is maximal unipotent, i.e. (M − 1) k vanishes only for k ≥ N + 1.In fact, the operator L N is a CY operator of degree N + 1 (cf.section 4.2).The fundamental solution that is holomorphic at the MUM-point z = 0 is: Note that b N (M 1,N ) = N + 1 and the Hodge numbers are: We call this a Hodge structure of type (1, 1, . . ., 1, 1).Let J n denote the rank n exchange matrix (J n ) ij = δ i,n−j .We choose the rank N + 1 intersection matrix of M 1,N as This choice is made such that at a given point of maximal unipotent monodromy or MUMpoint (see section 4.1 for details) it induces a pairing between those cycles whose periods degenerate like z λ log N −k z and z λ log k z for k = 0, . . ., d, respectively.The Zariski closure G N of the geometric monodromy group 3 G N is irreducible over C and given by This statement was proven for the hypergeometric motive in eq.(3.11) in Theorem 6.5 of ref. [61] for many more hypergeometric systems, together with the field of definition F that occurs in the monodromy actions for m = 1.Let us note that, in order to evaluate the bilinear in eq.(3.5), it is not mandatory that we have fixed a basis of periods over Z.A G N (R) basis change with G N as in eq.(3.17) is obviously admissible.The geometric monodromies act on the space of solutions, and in particular the vector spaces of the solution to eq. (3.11) form by eq.(3.17) irreducible fundamental representations of the corresponding group G N .We call these irreducible representations V. Representation theory therefore governs the monodromies and the Hodge structures associated to the symmetric and antisymmetric powers discussed in section 4.3.
For N odd, we can make the statement to hold over Z, and for N even with a choice of m.The Γ -class formalism described in ref. [15] promotes the Frobenius basis in eq.(4.5) at the MUM-point z = 0 to an integral basis in eq.(3.6), provided that the intersection numbers between the restriction H of the hyperplane class from P 2N +1 to W 1,N and the Chern classes of the tangent bundle are known.This latter data is defined by where | m means taking the m th coefficient.For the geometrical realisation M 1,N , we checked that eq.(3.5) equals eq.(2.10).This gives the coefficients of the ϵ expansion of eq.(2.10) an interpretation in terms of intersection numbers of W 1,N .For example, the Euler numbers of χ(W 1,N ) = 0, 24, −128, 960, −6912, 51051, . .., for N = 1, 2, 3, 4, 5, 6, . . .that follow from eq. (3.18) describe an admixture of the fundamental solution to the highest log N z logarithmic solution of the form ζ N (2πi) N χ(W 1,N ) etc. Since the periods are solutions of the CYPFI, we review in section 4 general results about CYPFIs for one-parameter families of CY varieties.

Fishnet integrals as Calabi-Yau volumes
There are two relations of the fishnet integrals to volumes of Calabi-Yau varieties, which are both conceptually very interesting and related by mirror symmetry.Let M s be the singular Calabi-Yau variety, which we associated to the fishnet integrals in section 3.1.The definition of the volumes is most easily stated if we have a smooth Calabi-Yau realisation M and a smooth mirror W of it, even though there will be generalisations for generically singular families.For example, for the ladder diagrams the smooth W is given in 3.13, while a smooth Calabi-Yau manifold M was defined as the resolved orbifold M 1,N .Let us assume we have such smooth models.As emphasised in ref. [17] the fishnet integrals defined from periods on M compute the quantum volume of the mirror Calabi-Yau variety W. A second interpretation is as the calibrated volume of M itself.We will give short accounts of booth concepts.
The fishnet integral as the calibrated volume of M Let us go back to the definition of ϕ(z) in eq.(3.5), which up to the absolute value of the factor F (a) ∈ Q(a) is the fishnet integral and encodes its non-trivial dependence on the transcendental periods, which are generalisations of rational or elliptic functions.As explained in [58], see again Definition 4.3., there is a canonical calibration given by which relates Ω(z) to the Kähler form ω. The volume form of M is defined by Then, by integrating the latter over M, we obtain the calibrated volume of M. The calibrated volume is real, positive and monodromy invariant.In contrast to the quantum volume in eq.(3.25) it can be defined classically, given the Kähler form ω defined by the calibration.We related ϕ(z) to the calibrated volume of M.However, its definition depends by eq.(3.19) on the physical parameters encoded in z.
The fishnet integral as the quantum volume of W In ref. [17] we used mirror symmetry and proposed another relation to a volume whose value depends on the physical parameters encoded in z, while its definition stays fixed.This is the quantum volume of the mirror W. To define it, we define a Kähler class on W . At a MUM-point at z = 0, which, e.g., the ladder Calabi-Yau manifolds have, we have a relation t R i (z) = Im t i (z) given by the mirror map, i.e., in terms of the periods (cf.eq.(4.7)) We can then use eq.(3.20) to obtain the classical volume of the mirror W: where the C cl i 1 ,••• ,i ℓ are explicitly-computable integers, namely the (classical) intersection numbers of M, which for the ladder geometries are defined by eq.(3.18), i.e. here we have only the intersection number of ℓ divisors given by . While its definition is fixed, it depends on the values of z via (3.22).Moreover, it gets quantum corrected by world-sheet instanton effects to the quantum volume (q i = exp(2πit i )): It is the key observation of mirror symmetry that the latter can be encoded in where Π are the periods on M and the relation z(q) is given by inverting eq.(3.22).This quantity is also real and positive, but not quite monodromy invariant, because of the normalisation by |Π 0 (z(q))| 2 and the fact that Π 0 (z) undergoes monodromy changes.This normalisation and the absence of instanton corrections in low dimensions dim 2 (W) = 1, 2 is discussed in ref. [17].Restoring the monodromy invariance, we write: That is, the fishnet integrals are proportional to the quantum volume of the associated mirror manifolds W.
In this section we have given two interpretations of eq.(3.5) in terms of Calabi-Yau volumes that apply directly to the ladder graphs where the geometry is smooth.For the general fishnet graphs, we can take eqs.(3.26) and (3.9) as the definition of the volume of the corresponding singular varieties.The calculation of these volumes by iterated integrals in the general case will be discussed in section 7.
4 Picard-Fuchs operators for one-parameter families of Calabi-Yau varieties

Differential operators in one variable
In this section, we briefly review some mathematical concepts related to differential operators in one variable.We follow the review [64] and consider a differential operator of degree n over P 1 : Its n-dimensional C-vector space of solutions is is the least common multiple of the denominators of the a i (z), then the solutions can have singularities only at the components of the discriminant divisor z = z i with z i given by the roots of ∆ L (z) = 0, possibly supplemented by z = ∞ in P where 4 Operators that satisfy the property in eq.(4.3) are called essentially self-adjoint.
A local basis for Sol(L) can be constructed using the Frobenius method.We use the coordinates δ i = z − z i and δ ∞ = 1/z, and we transform the operator to logarithmic derivatives where the p k (δ * ) have no common factor.If we view the L(θ * , δ * ) as polynomials in the formal variables θ * and δ * , then the local exponents5 at z * are given by the roots of the polynomial L(θ * , δ * )| δ * =0 , which is of degree n in θ * .If these roots {λ} are n-fold, i.e. maximally degenerate at z * , the latter point is called a point of maximal unipotent monodromy or MUM-point.Let us assume that at a singularity, which we choose to be at z = 0 to simplify notation, we have k distinct roots λ i with multiplicity m i (with k i=1 m i = n).For simplicity, we first assume that λ i − λ j / ∈ Z. Then the Frobenius method guarantees that, for each λ i , one can construct m i independent solutions of the form these soltuions form the Frobenius basis of Sol(L) in which the local monodromy is lower triangular.We can fix the m i holomorphic power series in z uniquely by demanding ϖ In particular, at a MUM-point the highest logarithmic degeneration is z λ i log n−1 z.If the differences of k indices λ js , s = 1, . . ., k are in Z, the highest logarithmic degeneracy is max{m js − 1|s = 1, . . ., k}, but more logarithmic solutions can appear, as the logarithmic degeneracies of the larger indices can be shifted.One still gets s m js independent solutions involving s m js power series, but the latter need to be indexed differently.
The solutions of the differential operator L generating a CYPFI(M z ) are the periods of M z .Besides eq.(3.7), they obey further restrictions.Part of them are valid for periods on any algebraic variety of dimension ℓ.For example, the non-logarithmic monodromies are finite, and so all local exponents must be rational, λ k ∈ Q.Moreover, Landmann's theorem [66] implies that the highest logarithmic degeneration at singular locii is ∼ δ λ i * log ℓ δ * .The latter implies that a differential operator L describing periods can only have a MUMpoint if its degree is n = ℓ + 1.Others are specific for period degenerations of a CYPFI(M z ), see ref. [15] for examples and references.

Calabi-Yau operators
In this section, we review a class of differential operators in one variable that play a prominent role in the theory of one-parameter families of CY ℓ-folds.Before we state the definition, we need to introduce some concepts.
Throughout this section we assume that L is a differential operator of degree ℓ + 1 with only regular-singular points.We assume that L has a MUM-point at z = 0, and admits a basis of solutions of the form The mirror map is defined by Its exponential is a holomorphic function of z: The periods can be used to define another set of functions that are holomorphic in a neighbourhood of q = 0.They are defined as follows: Let α m (z) = u m,m (z) −1 , where the u m,k (z) are determined recursively: and u 0,k (z) = y k (z) .(4.9) The functions α m (z) are holomorphic at z = 0, and called the structure series of L. In appendix C we present a formula that allows one to compute the structure series as a ratio of determinants of periods.
We define an almost Calabi-Yau operator L of degree ℓ + 1 as a differential operator with regular-singular singularities such that: • (ii) L has a MUM point at z = 0, and there is a local basis of solutions as in eq.(4.6).
In refs.[50][51][52][53]67] a Calabi-Yau operator is required to have the additional property that • (iii) the holomorphic functions y 0 (z), q(z) and α m (z) are N -integral, by which we mean that there is an integer N such that they admit a Taylor expansion of the form ∞ k=0 a k z k , where N k a k is an integer.The latter is expected from the relation of periods to the integrality of BPS states for smooth Calabi-Yau manifolds without torsion from mirror symmetry (see, e.g., refs.[68,69]) and observed in an overwhelming number of cases, but, except for a few cases, not proven.
While many Picard-Fuchs operators describing one-parameter families of CY varieties with Hodge structure (1, 1, . . ., 1, 1) fall into this class, we emphasise that the two concepts are distinct.In particular, for many examples of CY operators of degree four [50,51], it is not known if they are Picard-Fuchs operators of some family of CY three-folds.Oneparameter families whose Hodge structure is not (1, 1, . . ., 1, 1), do not have almost Calabi-Yau operators as Picard-Fuchs operators.For example, the one-parameter families of CY four-folds considered in ref. [70].As we will see, the one-parameter families of Calabi-Yau manifolds M M,N with M ≤ N , M ≥ 2 and N > 2 fulfil neither (i) nor (ii).

Operations on differential operators
In this section, we review standard techniques, such as the Hadamard product, as well as the symmetric and anti-symmetric products.They allow one to start from a given differential system Lf (z) = 0 and its solutions and to construct new ones.These techniques have been studied intensively in the context of differential motives.As we will see in section 5, these products allow us to relate the periods describing the CY varieties M M,N for fishnet integrals for different values of (M, N ).
In the following, it will be useful to consider two differential systems Lf (z) = 0 and Lf (z) = 0 as equivalent, if the dependent function f (z) is changed by an algebraic function α(z) ∈ Q(z). 6In other words, we consider equivalence classes ot differential operators defined as follows: we say that L and L are equivalent, L ∼ L if We start by defining the Hadamard product.Given two functions f and g that are holomorphic at z = 0, their Hadamard product is defined as Moreover, consider the differential operators L f and L g of minimal degree that annihilate f and g, respectively.Their Hadamard product is then defined by Hadamard products have been extensively studied to generate new Calabi-Yau operators [67,50] from old ones.They come geometrically with multi fibre structure [71], whose singularities allow one to predict if the resulting geometry is Calabi-Yau and to calculate their Hodge numbers [72,73] and other topological data.
Let us now define symmetric and anti-symmetric products of differential operators.This is most conveniently done by considering symmetric and anti-symmetric products of the representation of the monodromy group G on Sol(L).If the periods, which are in Sol(L), form an irreducible fundamental representation of G, we call the latter V = V w 1 , following our main reference in representation theory [74], where also the weights w i are defined.The m th symmetric and anti-symmetric power representations will respectively be denoted by Sym m V and ∧ m V.As explained below, the latter are represented by spaces of functions constructed from the solutions of L, and we identify the representations of G with these spaces of functions.The equivalence classes of minimal irreducible operators that annihilate these function spaces are called Sym m L and ∧ m L, respectively.
The solution spaces of Sym m L and ∧ m L can be described very explicitly.Let us start by describing the solution space of the symmetric product.Sym m L is defined to be the operator of minimal degree that annihilates all products of m solutions of L, i.e., it is defined by its solution space This vector space realises in an obvious sense the symmetric power of the fundamental representation of G.If L has degree n, then dim Sol L = n, and we have Next, let us describe the solution space of the anti-symmetric (or exterior) power ∧ m L. We start by noting that z is monodromy invariant, and so V and θ k z V are in the same monodromy representation.Hence, the anti-symmetric power of the fundamental representation can be represented by the set of n m functions represented by , where the j k , k = 1, . . ., m, take ordered, non-repeating values in {0, . . ., n − 1}.Note that this definition does not depend on the choice of J = (j 1 , . . ., j m ), as all choices lead to the same representation of G, and so the corresponding functions can only differ by α(z) ∈ Q(z) and we call the antisymmetric power simply Λ m V. Let A m,n be the set of n m m-tuples J as defined above.For a given choice, e.g.J = (0, . . ., m − 1), we represent the n m functions spanning Λ m V as determinants of m × m minors of the θ-Wronskian W with elements (W ) ij = θ j z y i , i, j = 0, . . ., n − 1 and y i ∈ V. We also define . 7 The m th anti-symmetric power of L is then defined as the irreducible operator of minimal degree with solution space (4.17) Let us make two comments.First, note that we only need to consider m ≤ n 2 because Second, the D I generate the solution space, i.e., every solution can be written as a linear combination of such determinants.In general, however, the determinants do not necessarily form a basis of Sol(∧ m L), but there may be relations among them.If V is in an irreducible representation of the geometric monodromy group G as in (3.17)(5.9),then representation theory decomposes ∧ m V into irreducible representations of G.By identifying one solution in such an irreducible representation, we immediately find the number of independent solutions corresponding to variations of Hodge structure of the geometry that corresponds to ∧ m V and the representation of the monodromy action on the associated periods.We use this fact to analyse the structure of solutions for the general fishnet integrals in eq. ( 5.3).

Period geometry of the fishnet integrals in two dimensions
In section 5.1 we describe how we can compute the Picard-Fuchs operator for the Calabi-Yau varieties given by double coverings like in eq.(3.1), or more specifically eq.(3.8).Using our general results in section 4, we explain in section 5.2 how the M 1,N geometries are connected by symmetric and Hadamard products and finally in section 5.3 how the general fishnet period geometry of M M,N follows form antisymmetric products of the Π 1,N .

The Picard-Fuchs operators for branched covers of fishnet type
In order to study the period geometry of M M,N , it is useful to have the expressions for the Picard-Fuchs operator L M,N .We now describe our method to determine them in some generality.
We know that Ω M,N is the holomorphic (ℓ, 0)-form on M M,N (with ℓ = M N ).The compact Calabi-Yau ℓ-fold (3.9) is given by the double covering of × ℓ i=1 P 1 i branched at the four points in each P 1 at which Ω M,N can develop a residue.We can make a choice of the branch cuts such that circles S 1 in each P 1 lift to non-intersecting circles on the cover to a cycle with topology of an ℓ-dimensional torus T ℓ on M M,N .By taking the residue at say x i = 0, 1 ≤ i ≤ ℓ, it is possible to evaluate an expression for one particular period: (5.1) The relation of the ladder integrals in eq.(2.10) to the geometries in eq.(3.13) can be checked in a stronger way at the level of all periods.Since (3.13) is smooth and the topology is known, we can evaluate eq.(3.9) using the Γ -class and find eq.(2.10), up to the factor |F (a)| 2 .
To compute the special period Π M,N,0 (z), we use a conformal transformation to set a 1 = 1, a 2 = 0, a 4 = ∞ such that we can identify the external point a 3 with the cross ratio z, i.e. a 3 = z.In this specific configuration of the external points, we can compute Π M,N,0 (z) by a systematic expansion for which we will use multiple times the well-known formula of the Taylor expansion of a square root The integration cycle of Π M,N,0 (z) is just given by an ℓ-dimensional torus, which means that we have to compute an ℓ-dimensional residue.This residue can be computed quite easily if we first factor out the product of all the integration variables x i where the polynomial PM,N (x, z) is obtained from P M,N (x, a) in eq.(2.6) after setting the external parameters to the values 0, 1, z, ∞, and we define PM,N (x, z) = PM,N (x, z) ℓ i=1 x −2 i .8Now we can use eq.( 5.2) for every linear factor in the polynomial PM,N (x, z) separately.Then the residue in eq. ( 5.3) only receives contributions from the first term containing the product of all x i variables, if and only if, we pick out in the subsequent expansions the constant term in all x i variables.In this way we obtain a Taylor series representation for the period close to z = 0: (5.4) Let us present some examples to illustrate this procedure.Our first example is the one-loop ladder integral.In this case we find As a second, less trivial example we consider the (2, 2)-fishnet integral.We start by noticing that (5.8) The Picard-Fuchs operator can now be obtained as follows: Consider the differential operator L 0 M,N of minimal degree that annihilates Π M,N,0 (z).We can construct it explicitly by writing down an ansatz for L 0 M,N in the form (4.4) and determine the free coefficients in the ansatz so that L 0 M,N annihilates the Taylor expansion in eq. ( 5.4) up to the order through which we have determined the r (i) M,N .We can check that we obtain the same answer if we compute additional Taylor coefficients.Since the Picard-Fuchs operator L M,N annihilates Π M,N,0 (z), it must lie in the ideal generated by L 0 M,N , i.e., it must be of the form L M,N = L ′ L 0 M,N of some differential operator of degree s.If s > 0, the Picard-Fuchs operator factorises, which implies that the monodromy representation on the periods is reducible.Since CY manifolds are expected to have irreducible monodromy, we conclude that we can pick L M,N = L 0 M,N .Using this approach, we can construct the Picard-Fuchs operators on a case by case basis.For the ladder integrals, the calculation can be done systematically to find that the series Π 1,N,0 is hypergeometric and L 1,N is given in eq.(3.11).By the theorem of Beukers and Heckman, their geometric monodromy is given in eq.(3.17) acting on its periods.This fixes the variation of Hodge structure of the general (M, N ) as antisymmetric powers, as explained in section 5.3.
Let us give finally an example of a non-planar fishnet graph, shown in figure 2. This graph gives rise to a one-parameter variation of a Calabi-Yau six-fold of type (1, . . ., 1) that has been very well studied and is famous in mathematics since the Zariski closure of its geometric monodromy group G is in the Lie group G 2 [75,76] G = G 2 . (5.9) It can also be realized as a singular double covering of × 6 i=1 P 1 i and the analogous equation to eq. (3.8) is now given by [76] which is readily identified with the non-planar fishnet graph depicted in figure 2. Performing the residue integrals in eq. ( 5.1) we get a particular combination of binomials from which we infer that the Picard Fuchs operator reads (2θ z +k) .
( 5.11) This operator has not appeared in the literature before.Note, however, that the Riemann symbol of L 7 given is by and its local monodromy agrees with properties of the monodromy blocks that have been calculated in ref. [75] by analysing the local action on the homology of eq.(5.10) induced by monodromy paths in the z plane around the singular fibres at z = 0, 1, ∞.

Period geometries from symmetric and Hadamard products
We now discuss relations between the Picard-Fuchs operators L N = L 1,N for different values of N .It is easy to see that we have where we defined From the discussion in the previous section, we see that the Picard-Fuchs operators for ladder integrals can be constructed iteratively as Hadamard products.In particular, adding one more integration vertex corresponds to taking the Hadamard product with L 0 : It is known that every CY operator of degree 3 is equivalent to the symmetric square of a CY operator of degree 2 [77,52,53].We find that L 2 = Sym 2 (L 1 ) . (5.16) Since the solutions of CY operators of degree 2 are periods of a family of elliptic curves, we conclude that the periods of M 1,1 and M 1,2 can be expressed in terms of complete elliptic integrals of the first kind.The one-loop ladder integral was discussed in refs.[33,57] in terms of elliptic integrals.Indeed, we find and K(z) denotes the complete elliptic integral of the first kind: For M 1,2 , we find [17]: with and For N > 2 it is not expected that the periods can be expressed in terms of elliptic integrals.However, by mirror symmetry we have interesting integral structures in geometric q expansions of the periods (and the quantum volume in eq.(3.24)) at the points of maximal unipotent monodromy.We discuss the higher dimensional geometries below.

The period geometry for general fishnet integrals
Let us now turn to the genuine fishnet integrals with M > 1.We have computed the periods in eq. ( 5.4) up to (M, N ) = (2, 4) and (3,3).We find that in all cases we can write (cf.eq.(4.16)): where we defined W := M + N − 1 and From this we see that the Picard-Fuchs operator of M M,N is the M th exterior power of L W , where the last equivalence follows from eq. (5.15).Note that this relation remains true for M = 1.We see that the Picard-Fuchs operators for M M,N can be constructed by taking Hadamard and exterior products of the first-order operator L 0 .Equation (5.25) has interesting implications for fishnet graphs.First, eq. ( 4.18) implies that L M,N and L N,M are equivalent: (5.26) This implies that M M,N and M N,M have the same periods, up to multiplication by an algebraic function of z.This is not unexpected from the Feynman diagram perspective, because the corresponding Feynman diagrams are simply related by a permutation of the external points.Another consequence is that we can limit ourselves to the cases M ≤ N .Equation (5.23), or more generally eq. ( 5.25) subject to eq. ( 5.26), implies where without restriction of generality M ≤ N .We can now use the results on the geometric monodromy group of the ladder geometries in eq.(3.17) and the general considerations in sections 4.3 to characterize the solution space V M,N of L M,N completely within the equivalence defined by (4.10) from the solution space V of L 1,M +M using representation theory.The choice of the α(z) ∈ Q(z) must still be made based on physically preferred representatives.Let us start with the dimension of the solution space Sol(∧ M L 1,W ).For W = M + N − 1 even and M < W 2 (or M = W 2 ), it follows from Theorem 19.4 of ref. [74] 9 that ∧ M V = V w M (or ∧ M V = V 2w M ).In particular, this representation is irreducible and of rank r = M +N M .For W = M + N − 1 odd, it follows from Theorem 17.5 of ref. [74] that the largest irreducible sub-representation of We can summarise this by: (5.28) From this we deduce that for M + N odd, the determinants D (W ) I are linearly independent, while for M + N even there must be M +N M −2 relations among them.Indeed, we have already stated that, while the determinants in eq. ( 5.24) generate Sol(L M,N ), they do not necessarily form a basis for it.Thanks to eq. ( 5.26), it is sufficient to discuss the case M ≤ W −1 2 .In appendix B we show that, as a consequence of the anti-symmetry of the intersection form for M + N even and the quadratic relations among periods due to eq. (3.7).We have the following relations for W = M + N − 1 odd and M ≤ W −1 2 : 9 For notations for the weight vectors w k we follow the convention of [74].
For example, for (M, N ) = (2, 2), there are 6 determinants, but we have (with W = 3) and so only five out of six determinants are linearly independent [78].Note that there are precisely M +N M −2 relations of the form (5.29) one can write down.Typically, for a given choice of J, many terms in eq.(5.29) vanish, so one may wonder if some of these relations may be trivially satisfied, e.g., because all determinants in the sum vanish individually.It is easy to see that this will never be the case.Indeed, the determinants would vanish individually if for example k ∈ J, for all 0 ≤ k ≤ W −1 2 .This is impossible, because J contains M ≤ W −1 2 elements, but the sum in eq.(5.29) contains W +1 2 terms.Hence, there is always at least one term in eq. ( 5.29) that is non-zero, and we obtain M +N M −2 relations.Note that by definition dim Sol(L M,N ) equals the rank b M •N of the middle cohomology of M M,N (cf.eq.(3.6)).By comparing the Hodge filtration as well as the Tate filtration of M 1,W with that of M M,N , and using eq.( 5.25) and the irreduciblility of G W for all L W , we can refine eq.(5.28) to give the individual Hodge numbers [79].We present here a more pedestrian argument, using the fact that L W has a MUM-point at z = 0, with the solution structure given in eq.(4.5) with λ = 0 and m = l.Given the fact that eq.(4.5) depends only on the W +1 independent power series ϖ i (z), i = 0, . . ., W , it is straightforward to count the number of independent functions in ∧ M V (with V = Sol(L W )) with leading behaviour log k z as z → 0. These are precisely the Hodge numbers One can also give a generating series for the π M,N (k): Note that we have (see also appendix A) This is precisely the structure expected from the Hodge numbers of a one-parameter family of CY ℓ-folds.For 2 ≤ k ≤ ℓ − 2, we generically have h ℓ−k,k (M M,N ) > 1, and so L M,N is not a CY operator for M > 1.The values of eqs.(5.28) and (5.31) for M + N ≤ 8 are tabulated in Table 1.We see that for (M, N ) = (2, 2), we have We have already seen that the second exterior power of a CY operator of degree 4 is conjectured to be a CY operator of degree 5. Hence, L 2,2 = ∧ 2 L 3 is expected to be a CY operator (cf., e.g., refs.[53,78]).If either M or N is greater than 2, we do not expect that L M,N is a CY operator.
The operators L M,N are polynomials in z and θ z .The degree in θ z is dim Sol(L M,N ), and so it is fixed by eq.(5.28).However, we only have estimates for the degree in z, and this degree also grows fast.For example, the operator ), as well its splitting in Hodge numbers h ℓ−k,k (M M,N ), which correspond to the number of solutions with leading logarithmic divergence is the only operator for M > 1 which is an almost Calabi-Yau operator, and it is quadratic in z, and its discriminant is related to the discriminant of 2 and we checked that it fulfils also the integer properties (iii).On the other hand, the operators are not almost Calabi-Yau operators.They are not essentially self-adjoint and have no MUM point, but they have the expected local exponents at z = 0 as given in Table 1.They have maximal order 10 and 18 in z.From our derivation of eq. ( 5.31), we can read off the degeneracies of all local exponents of L M,N at z = 0, which by construction obey λ i − λ j ∈ Z.The structure of solutions is as described after eq.(4.5).
Since SO(3, 2) ↞ Spin(3, 2) ≃ SL(4, R) one can show that every exterior power of a CY operator of degree 4 is an almost CY operator of degree 5 with Hodge numbers (1, 1, 1, 1, 1) [79] and vice versa.As a consequence, the periods of M 1,4 are found to be 2 × 2 determinants of periods of some one-parameter family of CY three-folds.We find L 4 ∼ ∧ 2 L 3 , with This operator corresponds to the entry 2.33 in the AESZ database of CY operators of degree four [80].This implies that the periods of M 1,4 are 2 × 2 determinants of the solutions of L 3 (and their derivatives).In particular, the solution of L 3 that is holomorphic at z = 0 reads: We have seen that L 4 is the second exterior power of the CY operator L 3 in eq.(5.35).In refs.[78,52,53] it was shown that for every CY operator L of degree 4 one has ∧ 2 ∧ 2 L ∼ Sym 2 L. We can combine this with eq.(5.25) to obtain In other words, the periods of the family M 2,3 of CY six-folds can be expressed in terms of the periods of the same CY three-fold that appeared in the computation of the periods of M 1,4 .
6 The Basso-Dixon formula

The intersection form for fishnet integrals
The Picard-Fuchs operators allow us to compute the periods of M M,N .In fact, it is sufficient to compute the periods of the ladder integrals, because all other cases can be obtained by computing determinants of the periods of ladder integrals.The latter are easy to obtain, because the Picard-Fuchs operators for ladder integrals are hypergeometric differential operators.In order to apply eq.(3.9) to compute the fishnet integrals, we also need the expression for the intersection form Σ M,N for M M,N .Since the Picard-Fuchs operators for ladder integrals are CY operators, the intersection form for the Frobenius basis is fixed by eq.(3.16).For M > 1, the Picard-Fuchs operators are not CY operators (except for (M, N ) = (2, 2)), and a priori it may be a difficult task to write down the intersection form.In the following, we show how one can leverage the knowledge that L M,N = ∧ M L W to relate the intersection forms Σ M,N and Σ W .
We start by reviewing some general facts from linear algebra on exterior products.Consider a vector space V of dimension d + 1 with basis {e i : 0 ≤ i ≤ d}.A basis for ∧ M V is given by the vectors e I := e i 1 ∧ . . .∧ e i M , where Consider an endomorphism M ∈ End(V ) represented in this basis by the matrix with entries M ij .Then this determines an endomorphism M ∈ End ∧ M V represented by the matrix M with entries where M (I, J) = M ij i∈I,j∈J is the matrix obtained from M by only keeping the rows from I and the columns from J. It is easy to check that, if M = M (1) M (2) , then we have IK M (2) Consider a set of d ′ vectors in V : We assume them to be linearly independent, which means the matrix formed by the coefficients c I has full rank d ′ .We denote the d ′ -dimensional vector space they span by Then the endomorphism M ∈ End(V ) determines an endomorphism of the quotient ∧ M V /Q in the following way.Fix a basis { b α } of ∧ M V /Q, e.g., by choosing a maximal linearly independent subset of { e I }. 10 Then there is a d × (d Then M defines the endomorphism of ∧ M V /Q described in this basis by the At this point we make an important observation.We can use M and M to define bilinears on ∧ M V and ∧ M V /Q, respectively, by contracting from the left and right with the appropriate basis vectors.It turns out that the two bilinears agree: We can immediately apply these results to construct the intersection form for M M,N and the bilinear expression in eq.(3.9).Indeed, we simply take V = Sol(L W ), e i = Π W,i (z) and e I = D (W ) I (z), I ∈ T W,M .The intersection form Σ W defines an endomorphism on V = Sol(L W ). If M + N is odd, then the determinants D (W ) I (z) are linearly independent and form a basis of V , and we can construct the intersection form using eq.(6.1).The case M + N even is slightly more complicated, because the determinants D (W ) I (z) are not linearly independent, and we need to quotient by the relations in eq.(5.29).The vectors v (i) are the linear combinations of determinants in the left-hand side of eq.(5.29).For fixed (M, N ), we can solve these relations and choose a basis { b α } of Sol(L M,N ) ≃ ∧ M V /Q, and the intersection form relative to that basis is given by eq.(6.5).However, since we are only interested in the bilinear combination in eq.(3.9), we can use eq.(6.6) to express this bilinear in terms of the overcomplete set of all determinants D (W ) I (z).As a conclusion, we can write down the following formula for the fishnet integrals, valid for all values of M + N : where we defined Equation (6.7) looks very similar to eq. (3.9).Its interpretation, however, is different.For M + N odd, the determinants D (z) are not linearly independent, and so they cannot be identified with a basis Π M,N (z) of Sol(L M,N ).Consequently, Σ M,N cannot be identified with the intersection form Σ M,N on M M,N .Nevertheless, eqs.(6.7) and (3.9) agree in all cases, thanks to eq. (6.6).In particular, we can explicitly check that the bilinear expression in eq.(6.7) is monodromyinvariant for all values of M + N .Let ρ : G W → End(V ) be the monodromy representation on Sol(L W ). It acts on the determinants D (6.9) Under a monodromy transformation, the bilinear expression in eq.(6.7) transforms into: where the third step follows from the monodromy invariance of Σ W . Hence, eq.(6.7) is monodromy-invariant.

The Basso-Dixon formula
Equation (6.7) allows us to write all fishnet integrals as bilinears in the determinants of the periods computed from the ladder integrals.From the BD formula in eq.(2.8), we know that the fishnet integrals can also be expressed as determinants in the ladder integrals, i.e., determinants of bilinears of periods.We now show how the BD formula in D = 2 dimensions arises from the CY geometry.
In the following ε a 1 •••a M denotes the Levi-Civita tensor of rank M , i.e., the totally antisymmetric tensor in M indices.We compute We have, with J = (j 1 , . . ., j M ), and similarly for the anti-holomorphic contribution.Hence,11 To summarize, we see that we have the relation: The previous relation allows one to write, up to an overall constant factor, the fishnet integral ϕ M,N (z) as an M × M determinant involving the derivatives of the ladder integral with W = M + N − 1 loops.It is in fact identical to the BD-formula for 2D fishnet integrals of ref. [33], see eq. (2.8).The only apparent differences come from our normalisation of the ladder integrals and the choice of the conformal cross ratio.Equation (6.14) shows that the BD formula for ϕ M,N (z) in two dimensions is in fact equivalent to the statement that L M,N = ∧ M L W .The latter statement, however, has only been checked explicitly for low loop orders, and it still remains conjectural in the general case.However, since the BD formula was shown to hold independently in ref. [33], this shows that L M,N = ∧ M L W holds for all values of (M, N ).

Volumes and iterated integrals
In this section we argue that our multi-valued definition of the quantum volume in eq.(3.25) has an interesting feature, which makes it particularly appealing in the context of Feynman integrals, namely we expect that Vol q (W M,N ) can be written in terms of iterated integrals with at most logarithmic singularities at the MUM-point. 12Such iterated integrals appear frequently in the context of Feynman integrals, where they are often referred to as pure functions [81,82].While we only show this explicitly for the case of CY operators and their exterior powers (which cover all four-point fishnet integrals considered in this paper), we strongly expect this feature to persist in other cases.
Let us first discuss the case of the ladder integrals ϕ N (z) = ϕ 1,N (z).We have seen that the Picard-Fuchs operator L N is a CY operator of degree N + 1.The mirror map becomes (cf.eq.(3.22)) In ref. [18] it was shown that one can write the (normalised) solutions of a CY operator in terms of iterated integrals (k < N ): 2) where we defined the iterated integral [83]:13 The arguments of the iterated integrals are the Y -invariants attached to the CY operator L [52,53], which can be defined in terms of the structure series α m (z) defined in section 4.2: The Y -invariants satisfy the identity: Inserting eq. ( 7.2) into eq.(3.25), we obtain: where I 1,N (q) is the vector of iterated integrals: where we introduced the derivation14 δ that acts on iterated integrals by clipping off letters from the right: δI(f 1 , . . ., f k ; q) := I(f 1 , . . ., f k−1 ; q) and δ(1) = 0 .(7.8) We see that the quantum volume for a ladder integral can be written in terms of iterated integrals with only logarithmic singularities at the MUM-point.Note that this is not restricted to ladder integrals, but exactly the same argument applies to a quantum volume computed from any CY operator.
Let us now turn to fishnet integrals ϕ M,N (z) with M > 1.We have seen that the solution space of the Picard-Fuchs operator is generated by the determinants in eq.(5.24).Since the determinants also involve the holomorphic period Π W,0 (z) (with W = M + N − 1), it is not entirely obvious that the quantum volume can be written in terms of iterated integrals with only logarithmic singularities at the MUM-point.In appendix C we show that the following identity holds: where we defined: Here ϑ denotes the derivation that clips off letters from the left (cf.eq.(7.8)): ϑI(f 1 , . . ., f k ; q) := I(f 2 , . . ., f k ; q) and ϑ(1) = 0 .
Note that ϑ is related to the ordinary derivative θ q by θ q I(f 1 , . . ., f k ; q) = f 1 (q) ϑI(f 1 , . . ., f k ; q) .(7.12) The argument q of the iterated integrals in eq.(7.10) is the canonical q-coordinate of the CY operator L W , cf. eqs.(7.1) and (7.2).We would expect, however, that the natural variable in the context of the operators L M,N is In appendix A we show that D z).Using eqs.(7.9) and (7.10), we find the following very simple relation between t(z) and t(z): and so q = exp [I(Y W,M −1 ; q)] = q + O(q 2 ) .(7.15) It is then easy to show that we have where we defined Putting everything together, we obtain where Σ M,N was defined in eq.(6.8) and I M,N (q) is the vector of iterated integrals: To conclude, we see that in all cases Vol q (W M,N ) can be expressed as a linear combination of iterated integrals with only logarithmic singularities at the cusp.The arguments of the iterated integrals are the Y -invariants of the W -loop ladder ϕ W (z).
Finally, we can repeat exactly the same steps as in the derivation of the BD-formula in eq.(6.14) (with the derivative θ z replaced by the derivation ϑ) to obtain a variant of the BD-formula directly for the quantum volume of W M,N : where θ denotes the analogue of ϑ, but acting on anti-holomorphic iterated integrals.

Conclusions
In this paper, we have studied the Calabi-Yau geometries M M,N that arise in the context of the four-point fishnet integrals I M,N in D = 2 dimensions, where the internal vertices are arranged on a M × N square lattice.We have focused, in particular, on the Picard-Fuchs operators of M M,N , which encode the information on the periods.We find that, remarkably, these Picard-Fuchs operators can be written as the M th exterior power of the ladder integral with W = M + N − 1 loops.This implies that all periods of M M,N can be written as determinants of periods of M 1,W .We have studied in detail relations among these determinants.The exterior power structure also allows us to determine the intersection form on M M,N .The periods together with the intersection form provide enough information to compute I M,N , and remarkably the solution naturally takes the form of a determinant.This provides a possible geometric origin of the Basso-Dixon formula for fishnet integrals in D = 2 dimensions.We have also studied the quantum volume of the mirror Calabi-Yau, and we find that it can naturally be expressed in terms of iterated integrals.
For the future, it would be interesting to extend our work into two possible directions.First, currently our results only apply to four-point fishnet integrals where all vertices attached to the same side of the square formed by the internal vertices have been identified.Since our results show that the Basso-Dixon formula for these integrals is a consequence of the structure of the Picard-Fuchs operator, it would be interesting to understand if a similar structure is present in fishnet integrals depending on more external points.In this case, the periods are solutions to an ideal of partial differential equations, which complicates the analysis.A set of generators of this ideal is conjectured to be given by the generators of the Yangian over the conformal algebra [17].Second, it would be interesting to understand if also the Basso-Dixon formula in D = 4 dimensions has a similar mathematical origin.While in four dimensions there is no connection to Calabi-Yau geometry, this is not a necessary condition, and it may be sufficient to find a differential operator that can be written as an exterior power.An intermediate step towards these goals could be to extend the results of this paper to the four-point fishnet integrals with more general propagator powers considered in ref. [33].We leave these questions for future work.
• By a very similar argument, we observe that the only solution that behaves like log z M (W −M +1)−1 is For D The term J M,S N y M x S is a sum of all terms that are of the form yx j 1 . . .yx jp = y M x S , products of M factors yx js such that the j s sum to S. Thus, the coefficient J M,S N is the number of sets I = {i 1 , . . ., i M } of {0, . . ., W } such that M k=1 i k = S.For S min = M (M −1) (A.16) The number of distinct I that sum to S min + x and S max − x is equal, which is equivalent to the number of distinct I with D

B Linear relations among determinants of periods
In this section we proof that for W odd and M ≤ W −1 2 , the M × M determinants D (W ) I (z) satisfy the linear relations in eq.(5.29).We follow our notations and conventions for fishnet graphs, but we emphasise that the conclusions hold for an arbitrary CY operator L W of even degree M + N = W + 1.
The periods of a family of CY varieties and their derivatives satisfy the quadratic relations in eq.(3.7).They imply the following relations: If W is odd, then Σ W is antisymmetric, and it is easy to show that we have: where we defined Let J = (j 1 , . . ., j M −2 ).We define The left-hand side of eq.(5.29) can then be written in the form (B.5) We now focus on those terms where the elements of J appear in a fixed position inside (b 1 , . . ., b M ).We only discuss the case (b 3 , . . ., b M ) = (j 1 , . . ., j M −2 ).All other cases are similar.

(C.8)
From this we obtain for the second column of eq.(C.7): θ z I I (q) = 1 α 1 (z) ϑI I (q) , (C.9) where the action of the derivation ϑ on iterated integrals was defined in eq.(7.11), and we extended its action on vectors to be componentwise.For the third column of eq.(C.7), we have: (C.10) The first term is proportional to the second column, so it can be neglected when computing the determinant.Continuing like this, we arrive at D (W ) I (z) = Π W,0 (z) M det I I (q), α 1 (z) −1 ϑI I (q), . . ., α 1 (z ∆ M,N,I (q) .(C.11) It is easy to check that ∆ M,N,I (q) = 1, and so find the explicit expression for the holomorphic solution: in agreement with eq.(7.14).Finally, we note that we can use eq.(C.12) to obtain an explicit expression for the structure series α i (z) in terms of minors of the period matrix:

< l a t e x i t s h a 1 _ b a s e 6 4 =
" j Y o k k e r S / b g I r 4 o r F N 1 3 B 2 w H o n s

Figure 1 :
Figure 1: The four-point graph representing the fishnet integral I (D)

Figure 2 :
Figure 2: Non-planar six-loop non-oriented fishnet graph leading to an irreducible mondromy representation in the Lie group G = G 2 .
can be identified with a basis Π M,N (z) of Sol(L M,N ), and Σ M,N agrees with the intersection form Σ M,N on M M,N .If M + N is even, the determinants D (W ) I via the representation ρ IJ = det ρ(I, J) .

2 J
M )(W −M +2)...W , and so h M (W −M +1)−1 = 1.Finally, we need to show that we have the symmetry property h M (W −M +1)−p = h p .This is equivalent to showing that, for fixed W and M , we get the same number of distinct I with D (W ) I that behave like log n min W,I +x z = log x z and log n max W,I −x .To see that this is indeed true, consider the function J W (y, x) = M,S W y M x S .(A.15)

∼
log n min W,I +x z = log x z and with D (W ) I ∼ log n max W,I −x z being equal.