Periods of Feynman Diagrams Citation Nasrollahpoursamami, Emad (2017) Periods of Feynman Diagrams. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9GX48MR. https://resolver.caltech.edu/CaltechTHESIS:05052017-144643636 Abstract We study differential equations for Feynman amplitudes and show that the corresponding D-module is isomorphic to a GKZ D-modules. We show that the sheaf of solutions to the D-module is isomorphic to a certain relative homology and that the amplitudes are periods of a relative motive. Using these ideas, we develop a method of regularization which specializes to dimensional regularization and analytic regularization. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Feynman Diagrams; D-modules; A-Hypergeometric; Regularization Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Marcolli, Matilde Thesis Committee: Marcolli, Matilde (chair) Rains, Eric M. Zhu, Xinwen Graber, Thomas B. Defense Date: 28 April 2017 Record Number: CaltechTHESIS:05052017-144643636 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05052017-144643636 DOI: 10.7907/Z9GX48MR ORCID: Author ORCID Nasrollahpoursamami, Emad 0000-0002-9658-1529 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10156 Collection: CaltechTHESIS Deposited By: Emad Nasrollahpoursamami Deposited On: 23 May 2017 23:14 Last Modified: 28 Oct 2021 19:00 Thesis Files Preview PDF - Final Version See Usage Policy. 406kB Repository Staff Only: item control page

Periods of Feynman Diagrams Thesis by Emad Nasrollahpoursamami In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2017 Defended April 28, 2017 ii © 2017 Emad Nasrollahpoursamami ORCID: 0000-0002-9658-1529 All rights reserved iii ACKNOWLEDGEMENTS I would like to thank all the individuals who have advised and helped me in the preparation of this thesis, especially the following: my thesis advisor, Matilde Marcolli, for introducing the problem discussed in this work to me and her continuous encouragement and advice during the completion of the thesis, and Xinwen Zhu for introducing me to his work which is the basis for the second chapter of this thesis, as well as for his invaluable advice throughout my time at Caltech. I would also like to acknowledge and thank Tom Graber and Majid Hadian for their many insightful discussions with me. I am grateful to my wife, Taban, for her love and encouragement. I am also thankful to my parents for their continuous support all the way. iv ABSTRACT We study differential equations for Feynman amplitudes and show that the corresponding D-module is isomorphic to a GKZ D-modules. We show that the sheaf of solutions to the D-module is isomorphic to a certain relative homology and that the amplitudes are periods of a relative motive. Using these ideas, we develop a method of regularization which specializes to dimensional regularization and analytic regularization. v TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . Chapter II: Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Parametric Representation of Feynman Amplitudes . . . . . . . . . . 2.2 Amplituded As a Function of Momenta . . . . . . . . . . . . . . . . Chapter III: GKZ A-Hypergeometric Differential Equations . . . . . . . . . . 3.1 Geometric Origin of GKZ D-modules . . . . . . . . . . . . . . . . . 3.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter IV: Amplitudes and Regularization . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv v 1 6 8 8 15 26 26 41 46 57 1 Chapter 1 INTRODUCTION In perturbative quantum field theory the scattering amplitudes, which are the probabilities of physical processes, can be approximated by sums over Feynman diagrams. Feynman diagrams are graphs corresponding to certain integrals. The integral corresponding to each Feynman diagram is a function of parameters called external momenta. The resulting functions are called amplitudes of Feynman diagrams. To compute the actual scattering amplitudes one needs to add these functions, and thus understanding the properties of these functions is necessary for both experimental and theoretical physics. In this thesis, we restrict our attention to the case of a scalar field theory. This means that the external momenta are just vectors in RD , with D the dimension of the theory. Suitable generalizations exist for arbitrary quantum field theories. A Feynman diagram in D-dimensional scalar field theory is a graph (possibly with multiple edges and self edges) together with an assignment of vectors (momenta) in RD to the vertices of the graph such that they add up to zero, and an assignment of positive real numbers (masses) to the edges. To compute the corresponding integral, we first choose an orientation for the edges of the graph and integrate a product of propagators on the space of flows to the graph. By a flow we mean an assignment of vectors in RD to edges such that at each vertex sum of incoming vectors minus sum of outgoing vectors is equal to the vector at that vertex. The propagator corresponding to an edge with assigned vector p and positive real number m is defined by the following formula: 1 . 2 p + m2 See chapter 2.1 for a detailed explanation. We compute this integral for an example. Consider the following graph on vertices 1, 2, 3 with momenta P1, P2, P3 and masses m1, m2, m3 for edges 12, 23, 31, respectively. 2 P1 1 (Q−P1 )2 +m32 1 P2 Q2 +m12 1 (Q+P3 )2 +m22 P3 The space of flows through this diagram is D dimensional and can be parametrized by vector corresponding to the edge 12 which we denote by Q. Propagators corresponding to edges are shown in the figure above. The corresponding integral is ∫ d DQ .    2 + m2 2 + m2 2 + m2 D (Q − P ) (Q + P ) Q 1 3 R 3 2 1 Note that we have the condition P1 + P2 + p3 = 0 otherwise the space of flows is empty. This integral does not necessary converge (depending on D). Since the early years of quantum field theory physicists have used a different form of this integral which is called the parametric form. The parametric form that we use here was first introduced by Symanzik in [Sym58]. For graph Γ, we denote the set of edges by E, and for each e ∈ E we define a new variable te . The form is defined in section 2.1 as π D`/2 ∫ |E | e − Í P te me2 − ΨΓ R+ Γ 1 Ö ΨΓD/2 e dte, where ΨΓ and PΓ are polynomials in te ’s, which are called first and second Symanzik polynomials and are defined in definitions 2.5 and 2.4. ` is equal to the number of independent cycles in the graph. For the diagram above we have PΓ = P12 t1 t2 + P22 t2 t3 + P32 t3 t1 ΨΓ = t1 + t2 + t3 ` = 1. The integral above does not always converge. To extract a meaningful function from the integral, physicists think of D as a complex number and consider the Laurent expansion of the integral in D. As we show in this thesis for values of D in an open subset of the complex plane, the integral is well defined and can be extended to the 3 complex plane as a meromorphic function of D. For the diagram above this can be seen by first integrating radially. ∫ exp − P12 t1 t2 + P22 t2 t3 + P32 t3 t1 + (t1 + t2 + t3 )(m1 t1 + m2 t2 + m3 t3 ) t1 + t2 + t3 R3+ = Γ(3 − D/2) ∫ ∆2 ! dt1 dt2 dt3 (t1 + t2 + t3 )D/2 P12 t1 t2 + P22 t2 t3 + P32 t3 t1 + (t1 + t2 + t3 )(m1 t1 + m2 t2 + m3 t3 ) (t1 + t2 + t3 )−3+D
−3+D/2 Ω, Í where Ω = i (−1)i ti dt1 ∧ · · · dtˆ i · · · ∧ dtn and ∆2 is the standard embedding of 2-simplex in R3 . We see that in dimension 6 the integral has a simple pole in D. An example of a convergent integral is when D = 4 and we get ∫ 1 Γ(1) Ω. 2 2 2 ∆2 P1 t1 t2 + P2 t2 t3 + P3 t3 t1 + (t1 + t2 + t3 )(m1 t1 + m2 t2 + m3 t3 ) The integral converges and is a function of momenta and masses. Note that masses and momenta appear as coefficients of the polynomial in the integrand. This particular example is considered in [DD98]. Authors compute the integral in dimension four as a linear combination of Di-logs. An interesting type of integral corresponds to primitive log divergent graphs with D = 4. A Feynman diagram is called a primitive log divergent in D = 4, if for all subgraphs (subset of edges), the number of edges is strictly greater than twice the number of loops in the subgraph. In this case the integral has a simple gamma function pole in D and the interesting function is the coefficient of the pole, which is constant and depends on masses and momenta. In [BK95] authors considered this type of integrals with the degree of the nodes restricted to be less than 4 (φ4 theory). They showed that for all graphs of up to 6 loops one gets a linear combination of multiple zeta values. Multiple zeta values are a generalization of values of Riemann zeta function at integers and their relation to mixed Tate motives has been studied extensively. In [GM04] authors show that all multiple zeta values appear as a period of mixed Tate motives. In [Bro12] the author shows that any period of a mixed Tate motive over Z is a linear combination of multiple zeta values. Mixed Tate motives are a quotient of a certain type of varieties with an equivalence relation. Two varieties with the same image in the category of mixed Tate motives have same counting function. All varieties which map to the category of mixed Tate motives have polynomial counting functions, i.e. the number of their points in Fq are polynomials in q. 4 In the case of primitive log divergent graphs one gets the integral ∫ 1 Ω. 2 ∆n ΨΓ (t1, . . . , tn ) One can think of the integrand as a differential form defined on the complement of zeros of ΨΓ and ∆n as a relative cycle in the pair (Pn \ V(ΨΓ ), Σ), where Σ is union of coordinate hyperplanes (see the next paragraph). The work of [BK95] and the relation between mixed Tate motives and counting function led Kontsevich to informally conjecture that zeros of the first Symanzik polynomial (for a general graph) in Pn have polynomial counting functions. The conjecture was verified for all graphs of up to 12 edges in [Ste98]. But the conjecture was later shown wrong in [BB03]. Authors showed that the counting functions of zeros of second Symanzik polynomial are very general. But the question remained to identify the graphs for which one gets a mixed Tate motive. This direction has been studied in [Sta98], [AM09], [Blo10], and [BS12]. The picture above is not completely accurate since zeros of ΨΓ might intersect Σ. Later it was shown in [BEK06] that the integral for a primitive log divergent case is in fact a period in the sense of algebraic geometry, i.e. a pairing between a relative homology class and algebraic differential form both defined over Q in a pair of varieties defined over Q. To construct this pair, authors use a sequence of blowups along the intersection of coordinate hyperplanes to separate the integration cycle from the the poles of the integrand. They also use techniques from homotopy theory to show that for an infinite class of graphs, the pair is of the mixed Tate type. Later it was shown in [BK10] that for the triangle graph the pair of varieties is of mixed Tate type. Note that the triangle graph is not primitive log divergent. As we saw in the examples above, one gets interesting functions or numbers out of these integrals either as the coefficients of the poles or when the integral converges. Two important questions that arise here are the following: what is the analytic structure of the integral in D, and what type of functions do we get after removing the poles? We show that the answers to these questions are related. We construct a toric variety and interpret the integral as a pairing between cohomology and homology classes on it. It turns out that for each codimension one component of the boundary of the toric variety, one gets an arithmetic progression of poles in D. These toric varieties appear when we study the differential equations corresponding to the integral as masses and momenta change. 5 As we have seen in the example, amplitudes of Feynman diagrams are functions of external momenta and masses. We map this space to the vector space constructed by coefficients of the first and second Symanzik polynomial, denoted by VΓ so that the amplitude is the pull back of a (multi-valued)function on VΓ . On this new vector space, we construct a holonomic regular D-module, of which the function we are considering is a solution. As a result, we show that the Feynman amplitude satisfies a holonomic regular system of differential equations. The differential equations on VΓ are a special case of GKZ or A-Hypergeometric system of differential equations introduced in [GZK89] and [GKZ90]. It follows from the results in these references that the corresponding D-module is holonomic and regular. It is well known that these D-modules come from twisted Gauss-Manin connections on toric varieties. Recently it is shown in [Hua+15] that, in the CalabiYau case, the relative homology computes the sheaf of solutions. Using results of [AB01] we show that their construction can be generalized to a non-Calabi-Yau case which includes Feynman diagrams. We construct a toric variety and a family of hypersurfaces in it parametrized by VΓ . The integral becomes a pairing between a differential form defined on the complement of the hypersurface paired with the positive real points of the toric variety considered as a cycles. We show that the variation of the hypersurface corresponds to the differential equations. Since the construction is explicit we can compute the cohomology using generators and relations. Using the description of cohomology with generators and relations, we show how one can define the integral for the divergent case as a Laurent expansion. In particular, we prove the following theorem. Theorem 1.1. Given a graph Γ with n edges and first Symanzik polynomial Ψ and second Symanzik polynomial Q (including mass terms), the amplitude in dimensional regularization, up to a constant, can be computed by the following integral: ∫ Õ 1  i Ai (D/2). c0 A(D/2 + ) = e−Q/Ψ D/2+ = n Ψ R+ i≥−n The left hand side is meromorphic and poles can be described in the following way. For a 2-connected subgraph γ ⊂ Γ, let `γ be the dimension of the first homology of 6 γ. |E(γ)| is the number of edges of γ. A(D/2) has a pole at D/2 ∈ C iff D/2 `γ − |E(γ)| ∈ Z≥0 for a 2-connected subgraph γ. Ai ’s are coefficients of the Laurent expansion of the left hand side. Furthermore, the lowest coefficient, A−n (D/2), at integers comes from a pairing between an algebraic relative cohomology class and a Betti homology class explicitly constructed in section 3.1. 1.1 Structure of the Thesis In section 2.1 we define the Feynman amplitude for a Feynman diagram and show how one can present it in the parametric form. The new result is that a product of a power of the first Symanzik polynomial and the second Symanzik polynomial is the determinant of a matrix. We also show that the coefficients of the first and second Symanzik polynomials are norms of Plücker coordinates for a Grassmannian naturally defined by the graph. In section 2.2 we study how the integral changes as we vary the coefficients of the first and second Symanzik polynomials. In the convergent case, we find a set of linear PDEs satisfied by the integral. Using analytic continuation we define the integral for the divergent case but the proof is not constructive. We first show that the analytic continuation exists and, using that, we find the PDEs satisfied by coefficients in the Laurent expansion. This method is based on [BW09]. It turns out that the set of differential equations is a special case of the GKZ differential equations for the convergent case. For the divergent case, the coefficients of the Laurent expansion are solutions to iterated extension of the GKZ differential equations. In section 3.1 we study the GKZ differential equations as a D-module on a vector space V. Given a polynomial f in n variables with Newton polytope A such that the dimension of V is the number of points in A, and a vector β ∈ k n+1 , we consider the corresponding GKZ D-module H1×A(β). We construct a projective toric variety PΣ together with a line bundle on it. The vector space of global sections of the line bundle is isomorphic to V. Let D be the complement of the torus T in PΣ . Let U be the complement of the zeros of f in V × PΣ , where V parametrizes the coefficients of the polynomial f . Given v ∈ V we show that the algebraic relative cohomology of the pair (Uv, D ∩ Uv ) with the Gauss-Manin connection is isomorphic to HA(β) as a D-module, where Uv is the fiber of U over v. Using the Riemann-Hilbert 7 correspondence, we deduce that the cycle to period map gives us a complete set of solutions. In section 3.2 we study integrals over positive real points of the toric variety, considered as a chain in relative homology. We show that a condition necessary and sufficient in order for an integral of the type ∫ f β0 dt1 . . . dtn β β R+n t 1 . . . tn n 1 to converge is that β is semi non-resonant, as defined in Definition 3.15. Using relations in the cohomology ring, we develop a method to define this integral for any value of β by meromorphic continuation. We show that the poles of this function appear in translates of the faces of a cone in Rn+1 . This cone is the cone over the Newton polytope of f . In chapter 4 we apply the methods developed in the section 3.2 to amplitudes and we explicitly construct a motive such that its periods give us the amplitude. We completely describe the Newton polytope in this case and show that its facets correspond to the product of subgraphs and quotient graphs. Using regularization methods, we define the  expansion of the amplitude for divergent graphs. 8 Chapter 2 FEYNMAN DIAGRAMS 2.1 Parametric Representation of Feynman Amplitudes Feynman diagrams (or Feynman graphs) are one-dimensional simplicial complexes with half edges attached to some of the vertices. These half edges are called external edges, while all other one-dimensional simplices are called internal edges. In the physics literature, for each external edge, it is common to fix a vector in RD . These vectors are called external momenta. They are subject to a momentum conservation law, given by the requirement that the sum of all external momenta of the graph is zero. Since the amplitude only depends on the sum of the external momenta at each vertex, we can equivalently assign a momentum vector to each vertex of the graph and forget about external edges. Namely, we assign to a vertex the sum of the external momenta of all the external edges attached to that vertex, or zero if there are no external edges at that vertex. So in the following external momenta will always be assigned to vertices. Let E be the set of edges of the graph and let V be the set of vertices. We have an exact sequence of free Z modules η0 0 → H1 (Γ, Z) −−−−−−−→ Z|E | −−−−−−−→ Z|V |−1 → 0, (2.1) where H1 (Γ, Z) is the first homology of the graph Γ with coefficients in Z, that is, the free Z module generated by loops. The morphism on the right is the boundary map. Note that, to define this map, we need to fix an orientation on the edges of the graph, but the final result is independent of this choice. Taking the tensor product of the sequence above with RD gives the exact sequence η β 0 → H1 (Γ, RD ) −−−−−−→ RD|E | −−−−−−−→ RD(|V |−1) → 0. Note that the choice of external momenta {pv ∈ RD |v ∈ V, choice of a vector a in RD(|V |−1) . Í (2.2) v∈V pv = 0} is just a Define Q e = Pe2 + me2 : R|E |D → R, where Pe2 is given by first projecting onto the D coordinates corresponding to e and then taking the sum of the squares of these D 9 coordinates. Let ã ∈ RD|E | be a lift of a, under the map β of 2.2. Definition 2.1. The amplitude of a Feynman graph Γ with external momenta a ∈ RD(|V |−1) is given by the integral ∫ Ö 1 A(Γ, a, me ) : = (η + ã)∗ (dµ) β−1 (a) e Q e ∫ (2.3) Ö 1 ∗ )dµ, = (η + ã) ( Qe H1 (Γ,RD ) e where η is as in 2.2 and dµ is the tensor product of the standard measure on RD with the measure on H1 (Γ, R) induced by the morphism H1 (Γ, Z) → H1 (Γ, R). This is the unique positive translation invariant measure on H1 (Γ, R), with the property that a basis of H1 (Γ, Z) generates a parallelogram of measure 1. Note that, with this definition, the amplitude is a function on RD(|V |−1) . ∫∞ The Schwinger trick simply consists of using the identity 0 e−ax dx = a1 in order to rewrite the amplitude in “parametric form". For each edge e we introduce a new variable te . Definition 2.2. A subset of edges S ⊂ E is called a spanning tree if the subgraph with edges in S is a tree and is maximal in the sense that, if we add any of the remaining edges to it, it will contain a loop. Denote the set of spanning trees by Span. Definition 2.3. A subset of edges C ⊂ E is called a cut if it has the following properties. 1. When we remove these edges, the graph becomes a disconnected union of trees. 2. The set C is minimal in the sense that, if we add back any edges to the remaining graph, it will either have a loop or become connected. Since cuts are minimal, they divide vertices into disjoint sets VC and VCc . For a cut C we denote by PC the norm of the sum of momenta in either component, Õ Õ PC = ( pv )2 = ( pv )2 . v∈VC v∈VCc 10 Denote the set of cuts by Cut. Definition 2.4. The first Symanzik (or Kirchhoff) polynomial of a Feynman graph is given by Õ Ö ΨΓ (t1, ..., t|E | ) := te . S∈Span e<S Definition 2.5. The second Symanzik polynomial of a Feynman graph is given by Õ Ö PΓ (t1, ..., t|E |, PC ) := PC te . C∈Cut e∈C Given a Feynman graph Γ, we enumerate edges by 1, ..., |E |. We define p p √ √ √ T := diag( t1, t2, ..., t|E | ) ⊗ IdD×D, and Tred := diag( t1, ..., t|E | ). (2.4) Let P® be a vector in RD×|E | , where the coordinates are ordered in the same way as the variables ti . Note that, for each edge, we have D coefficients. Let H denote the image of H1 (Γ, R) in R|E | . Lemma 2.6. The measure dν on T β−1 (a) induced by the standard measure on RD|E | satisfies dν = ΨΓ (t1, ..., t|E | )D/2 (Tη + T ã)∗ (dµ), (2.5) with ΨΓ the Kirchhoff polynomial, T defined as in 2.4, and η as in 2.2. Proof. Since Tη + T ã is linear, (Tη + T ã)∗ (dµ) is a constant multiple of the measure on T β−1 (a) induced from RD|E | . In order to compute this constant, we can compare the volume of the image of the standard cube in these two measures. We have (Tη + T ã)∗ (dµ) (Tη(Cube) + T ã) = dµ(Cube) = 1, by definition. On the other hand, we have dν (Tη(Cube) + T ã) = dν (Tη(Cube)) , since the standard measure on RD|E | is translation invariant. We choose bases {v1, ...v` } for H1 (Γ, Z) and {A1, ..., AD } for RD . With η as in 2.2, η0 as in 2.1, and Tred as in 2.4, the volume of the image of the standard cube is then given by q

D/2 det Tη(vi ⊗ a j ) · Tη(vk ⊗ al ) = det(Tred η0vi · Tred η0vk ) , since the volume form corresponding to a metric g is det(g)1/2 . Consider Tred η0 : H1 (Γ, R) → R|E | . We have ∧` Tred η0 : ∧` H1 (Γ, R) → ∧` R|E |, 11 where ` is the dimension of H1 (Γ, R). The determinant above is the norm square of ∧` Tred η0(v1 ∧ v2 ∧ ... ∧ v` ) in the induced metric; thus it can be computed as the sum of the squares of the coefficients in an orthonormal basis. Let {w1, ..., w|E | } be the standard basis for R|E | . Then {∧i∈I wi }|I |=`,I ⊂{1,..,|E |} is an orthonormal basis for ∧` R|E | . We have ∧` Tred η0(v1 ∧ v2 ∧ ... ∧ v` ) = (Tred η0(v1 ) ∧ Tred η0(v2 ) ∧ ... ∧ Tred η0(v` )) =( |E | Õ η10 j |E | |E | Õ Õ p p p 0 tj wj ∧ η2 j t j w j ∧ ... ∧ η`0 j t j w j )). j=1 j=1 j=1 Note that, since ηi,0 j = ±1, the coefficient of the term wi1 ∧ wi2 ∧ ... ∧ wi` is either zero Î √ or equal to ± k ti k . On the other hand, the coefficient of wi1 ∧ wi2 ∧ ... ∧ wi` is nonzero iff the orthogonal projection onto the subspace W = span(wi1, wi2, ..., wi` ) is an isomorphism when we restrict it to the image Tred η0. Since T 0 fixes the coordinate subspaces, this map is an isomorphim iff Im(η0) ∩ W = 0, and that happens iff the subgraph with edges i 1, ..., i ` does not have a loop, which means it is the complement of a spanning tree. To summarize, we have Õ Ö
D/2
D/2 det(Tred η0vi · Tred η0vk ) = te . S∈Span e<S  Proposition 2.7. When the integral 2.3 converges, it is equal to ∫ ∫ Ö Í 1 ® ® − e te me2 e−P. P dν, e dte D/2 |E | ΨΓ (t1, ..., t|E | ) T β−1 (a) R+ e∈E where dν is the measure on T β−1 (a) induced by the standard measure on RD|E | . ∫∞ Proof. Using the Schwinger trick 0 e−ax dx = a1 we write ∫ β−1 (a) Ö 1 e Qe (η + ã)∗ (dµ) = ∫ β−1 (a) ! ∫ |E | R+ e− Í e te Q e Ö dte (η + ã)∗ (dµ). e∈E By the definition of Q e , this is equal to ! ∫ ∫ Ö Í Í 2 2 ( e− e te me − e te Pe dte (η + ã)∗ (dµ). β−1 (a) |E | R+ e∈E In vector form, with T as in 2.4, this can be written equivalently as ! ∫ ∫ Ö Í 2 ® ® e− e te me −T P·T P dte (η + ã)∗ (dµ). β−1 (a) |E | R+ e∈E 12 Since all functions are positive, convergence is the same as absolute convergence and we can switch integrals. This gives ∫  ∫ Ö Í ® P® − e te me2 −T P·T e dte e (η + ã)∗ (dµ) . |E | R+ β−1 (a) e∈E ® ® ® ® Using the fact that (η + ã)∗ (e−T P·T P ) = (Tη + T ã)∗ (e−P·P ), we rewrite the above as ∫  ∫ Ö Í ® P® − e te me2 −P· e dte e (Tη + T ã)∗ (dµ) . |E | R+ T β−1 (a) e∈E Then applying the result of Lemma 2.6 we obtain ∫ ∫ Ö Í ® ® − e te me2 e dte e−P·P |E | R+ e∈E T β−1 (a)  dν . ΨΓ (t1, ..., t|E | )D/2  A standard computation shows the following simple facts. Lemma 2.8. Let H be a d-dimensional affine linear subspace in Rn and let L be the distance of the affine subspace H from the origin. Then the integral of a Gaussian 2 function on H with the induced measure is equal to π d/2 e−L . Lemma 2.9. Let v1, v2, ..., vk be vectors in Rn . One can compute the volume squared of the parallelogram generated by these vectors in the induced metric on the subspace they generate, in the form |v1 ∧ v2 ∧ ... ∧ vk | 2 = det(vi · v j ). The next statement then follows easily. Lemma 2.10. Suppose given a vector subspace V of Rn with a basis v1, ..., vk , and a vector a ∈ Rn . Let P(w1, ..., wm ) denote the parallelogram generated by w1, ..., wm . The distance of an affine subspace a + V from the origin is equal to Vol(P(a, v1, ..., vk )) |a ∧ v1 ∧ ... ∧ vk | = . Vol(P(v1, ..., vk )) |v1 ∧ ... ∧ vk | Proof. Volume is defined by the metric, and thus the distance times V ol(P(v1, ..., vk )) is the volume of P(a, v1, ..., vk ).  13 ® ® Proposition 2.11. The Gaussian integral of e−P·P , with respect to the measure dν on T β−1 (a) defined as above, is given by ∫ ® ® e−P·P dν = π D`/2 e−PΓ (t)/ΨΓ (t,a), T β−1 (a) where ΨΓ and PΓ are the two Symanzik polynomials. Proof. By lemma 2.8, it is enough to show that the distance squared of T β−1 (a) from the origin is PΓ (t)/ΨΓ (t, a). Let v1, v2, ..., v` be a basis of H1 (Γ, Z). We denote the image of these vectors in H1 (Γ, Z) ⊗ R by the same notation. Note that the affine subspace over which we are integrating the Gaussian is parallel to the space generated by Tη v1 ⊗ e1, v1 ⊗
e2, ..., v1 ⊗ e D, v2 ⊗ e1, ..., v2 ⊗ e D, ..., v` ⊗ e D , where {e1, ..., e D } is the standard basis for RD . As we have shown before we then have Tη(v1 ⊗ e1 ) ∧ ... ∧ Tη(v` ⊗ e D ) = ! ! Õ Ö√ Õ Ö√ (± (± te ) ∧e<S we ⊗ e1 ∧ te ) ∧e<S we ⊗ e2 ∧ S∈Span S∈Span e<S ... ∧ Õ S∈Span e<S ! Ö√ (± te ) ∧e<S we ⊗ e D , e<S where {we }e∈E is a basis for R|E | . We can write ã = we see that the vector Í e,i Pe,i we ⊗ ei . Then, for i , j, T(we ⊗ ei ) ∧ Tη(v1 ⊗ e1 ) ∧ ... ∧ Tη(v` ⊗ e D ) is orthogonal to T(we 0 ⊗ e j ) ∧ Tη(v1 ⊗ e1 ) ∧ ... ∧ Tη(v` ⊗ e D ). One can see this from the expansion in the standard basis: all terms in the first expression have ` + 1 terms with ei , while the second one has ` terms with ei . Thus, one can compute the norm squared of Õ T( Pe,i we ⊗ ei ) ∧ Tη(v1 ⊗ e1 ) ∧ ... ∧ Tη(v` ⊗ e D ) e for different i’s and add them up to get the squared norm of T(ã) ∧ Tη(v1 ⊗ e1 ) ∧ ... ∧ Tη(v` ⊗ e D ). 14 The vector above is in ∧D`+1 R|E | ⊗ RD . We can identify R|E | ⊗ RD with D copies of R|E | . The norm squared of this vector is equal to the volume squared of the parallelogram generated by the vectors. Since we have ` + 1 vectors in one of the copies and ` vectors in the other copies, we can compute the volume of each of them and multiply them together. For j , i we have v1 ⊗ e j , ..., v` ⊗ e j , all of which have the same volume squared, equal to Ψ. To compute the volume of the copy with ` + 1 vectors, it is enough to compute Õ Tred ( Pe,i we ) ∧ Tred η0(v1 ) ∧ ... ∧ Tred η0(v` ) e = Õ e ! √ Pe,i te we ∧ Õ ! Ö√ (± te ) ∧e∈s we . S∈Span e∈s The terms that appear in the coefficients in the standard basis {wei1 ∧ ... ∧ wei`+1 }i1 <...<i`+1 are sums of Pe,i . Nonzero terms correspond to ` + 1 edges that are a complement of a spanning tree plus one extra edge. Note that, if we remove these edges from the graph, it becomes disconnected and, if we add any of these edges to the graph, it becomes a spanning tree. So the term wei1 ∧ ... ∧ wei`+1 appears ` + 1 times, once Î p Í of each of its edges. Thus, the coefficient is j te j j Pe j ,i . Since Pe j ,i is a lift of a, this sum is equal to the sum of momenta in one of the connected components of the graph. We get the norm as Õ Ö PΓ (t, PC ) = PC te . C∈Cut e∈C The original norm squared we wanted to compute is then PΓ ΨΓD−1 and, by Lemma 2.10, the distance squared is PΓ ΨΓD−1 ΨΓD = PΓ . ΨΓ  The following result then follows from Proposition 2.11 and Proposition 2.7. 15 Proposition 2.12. When the integral 2.3 converges, it is equal to ∫ Í Ö P − te me2 − ΨΓ 1 D`/2 Γ A(Γ, a) = π e dte |E | R+ ΨΓD/2 e Remark 2.13. The coefficients of the second Symanzik polynomials are always positive. According to the computation in Proposition 2.11, they correspond to squared norms of certain differential forms. We will use this property in chapter 4. 2.2 Amplituded As a Function of Momenta The amplitude is defined as an integral which depends on the external momenta. This integral does not always converge. For some graphs that are called ultraviolet divergent, the integral diverges for any value of external momenta, while for some graphs the divergences happen only for special values of the external momenta. The most widely used method in physics for treating these divergences is called dimensional regularization. Within this method, a regularization of divergent integrals is achieved by formally computing the integral for D a complex variable in a neighborhood of the integer spacetime dimension in the complex plane. For a detailed explanation of this method see [CM08]. In this chapter, we define the integral for any D and find differential equations satisfied by it. The integral depends on a parameter in CD(|V |−1) and masses of edges. We map this vector space into the vector space VΓ , which parametrizes the coefficients of the first and second Symanzik polynomials so that it agrees with the amplitude on the image. We generalize the integral to an integral which has VΓ as its parameter space. Note that all coefficients in the second Symanzik polynomial are equal to 1. We consider general coefficients for these terms and look at the integral as we vary them. Over VΓ the differential equation satisfied by the new integral is geometric in nature and can be solved using series. One can identify VΓ with the parameter space of a family of hypersurfaces in toric varieties. The value of the integral for integer D is a period of a relative motive defined by the complement of this hypersurface. From now on we consider the following polynomials: ΨΓ (t, PS ) := PΓ (t, PC ) := Õ PS Ö S∈Span e<S Õ PC Ö C∈Cut e∈C te te 16 Õ Õ Ö Õ Q Γ (t, PC, me ) = PΓ (t, PC ) + ( te me2 ) te = QT t T e S∈Span e<S (2.6) T Here T ranges over all monomials appearing in Q Γ and by t T we mean the monomial corresponding to T. The amplitude is ∫ 1 Ö D`/2 dte, A(Γ, a, me ) = π e−Q Γ /ΨΓ D/2 |E | R+ ΨΓ e where coefficients of ΨΓ are 1 and coefficients of Q Γ come from equation (2.6) and are sums of masses and momentum variables PC . Definition 2.14. (Parameter Space) Let VΓ denote the parameter space for QT and PS . It is a complex vector space of dimension equal to the number of monomials in Q Γ plus |Span|. Definition 2.15. (Generalized Amplitude I) Given (c1, c2, v®) ∈ Cn+2 , let I(c1, c2, QT , PS, v®) := ∫ |E | e −Q Γ /ΨΓ R+ Q cΓ1 v® Ö t dte, ΨΓc2 e (2.7) where t v® means t1v1 t2v2 ...tnvn and n = |E |. Remark 2.16. Note that this integral is not well defined for all values of the parameters and is a (multi-valued) function on a dense domain in Cn+2 × VΓ . To define this (multi-valued) function first we define it for some open subset and then we take the analytic continuation. Lemma 2.17. When the generalized amplitude I converges we have: I(c1, c2, QT , PS, v®) = Γ(n + |®v | + c1 (` + 1) − c2 `) where |®v | = ∫ v |+(c2 −c1 )` Q−n−|® Γ −n−|®v |+(c2 −c1 )(`+1) ∆n−1 Ψ Γ t v® Ω, Í i vi , Ω= n Õ (−1)i+1 ti dt1 ∧ ... ∧ dtˆ i ∧ ... ∧ dtn, i=1 and ∆n−1 is the standard n − 1 simplex embedded in Rn . Proof. To show this we parametrize R+|E | with ∆n−1 × R+ . Consider the map φ : ∆n−1 × R+ → R+|E | given by φ(x, s) = sx. 17 Note that φ∗ (dt1 dt2 ...dtn ) = Ω|∆n−1 sn−1 ds, Q Γ (sx) = s`+1 Q Γ (x), ΨΓ (sx) = s` ΨΓ (x) and (sx)v® = s |®v | x v® . Pulling back the integrand to ∆n−1 × R+ we have: I(c1, c2, QT , PS, v®) = ∫ e −sQ Γ (x)/ΨΓ (x) ∆n−1 ×R+ Q cΓ1 (x) v® Ω c2 x = ΨΓ (x) ∆n−1 ∫ ∫ s c1 (`+1) Q cΓ1 (x) |®v | v® n−1 s x Ωs ds s c2 ` ΨΓc2 (x) (2.8) s c1 (`+1)−c2 `+|®v |+n−1 e−sQ Γ (x)/ΨΓ (x) ds. R+ (2.9) ∫ The fact that R e−sλ s x ds = λ−x−1 Γ(x + 1) then implies the lemma.  + Definition 2.18. (Generalized Amplitude II) Given (c, d, v®) ∈ Cn+2 , let ∫ Q cΓ v® J(c, d, QT , PS, v®) := t Ω. d ∆n−1 ΨΓ (2.10) Remark 2.19. By Lemma 2.17, the generalized amplitudes I and II of Definitions 2.15 and 2.18 are related by I(c1, c2, QT , PS, v®) = Γ(n + |®v | + c1 (` + 1) − c2 `)× J(−n − |®v | + (c2 − c1 )`, −n − |®v | + (c2 − c1 )(` + 1), QT , PS, v®). (2.11) Note that this is a function of PS and QT . For S ∈ Span, let S® be the vector in Zn with 1 for edges that are not in S and zero in the other coefficients. For a monomial T = t1α1 · · · tnαn , let T® be the vector (α1, · · · , αn ). We have ∂ ® I(c1, c2, QT , PS, v®) = c1 I(c1 − 1, c2, QT , PS, v® + T) ∂QT ® −I(c1, c2 + 1, QT , PS, v® + T) (2.12) ∂ ® I(c1, c2, QT , PS, v®) = I(c1 + 1, c2 + 2, QT , PS, v® + S) ∂PS ® −c2 I(c1, c2 + 1, QT , PS, v® + S). (2.13) 18 Let A ⊂ Zn+1 be the set containing the following points. For each monomial T in ® and, for any Spanning tree S, consider the lattice Q Γ , consider the lattice point (0, T) ® point (1, S). Denote the subset of A of lattice points corresponding to spanning trees by AS and the subset of lattice points that correspond to monomials by AC = A \ AS . For ® ∈ AS , Pa refers to PS and for a = (0, T) ® ∈ AC , Pa refers to QT . a = (1, S) Since Q Γ is of degree ` + 1 and ΨΓ is of degree `, all the lattice points lie on the affine hyperplane where the sum of the coordinates is ` + 1. Let φ : Zn+1 → Z be the function that computes the sum of the coordinates and let p0 : Zn+1 → Z be the projection onto the first coordinate and p1 : Zn+1 → Zn the projection onto the last Í n coordinates. For any integer relation a∈A na a® = 0 among lattice points in the set A, we have ! Õ Õ Õ na φ(a®) = (` + 1) na 0 = φ(0) = φ na a® = a∈A a∈A a∈A ! 0 = p0 (0) = p0 Õ a∈A na a® = Õ a∈A na p0 (a®) = Õ na . a∈AS Combining these two we have Í Í    a∈AS,na >0 na = a∈AS,na <0 −na  Í Í   a∈AC,na >0 na = a∈AC,na <0 −na .  (2.14) For the relation (na )a∈A, we consider the following differential operator: Ö Ö ∂
na ∂
−na − . ∂Pa ∂Pa a:n >0 a:n <0 a a Proposition 2.20. Let A and Pa be as above. For any Z-linear relation we have ! Ö Ö ∂
na ∂
−na − J(c, d, Pa, v®) = 0 ∂P ∂P a a a:na >0 a:na <0 ! Ö Ö ∂
na ∂
−na − I(c1, c2, Pa, v®) = 0. ∂Pa ∂Pa a:n >0 a:n <0 a a Í ®=0 a∈A na a 19 Proof. Consider the set Z = {(c1, c2 ) + Z2 } ⊂ C2 . Each time we apply a derivation 2.12 or 2.13 to I(c1, c2, QT , PS, v®), we get a weighted sum of two I(c10 , c20 , QT , PS, v® + p1 (a)) where (c10 , c20 ) ∈ Z. If we apply the positive part of the differential operator Í above, we get a weighted sum of I(c10 , c20 , QT , PS, v® + p1 ( a∈A,a>0 na a)). If we apply the negative part of differential operator above, we get a weighted sum of Í I(c10 , c20 , QT , PS, v® + p1 ( a∈A,a<0 −na a)). To show that the generalized amplitude goes to zero under this differential operator, it is enough to show that the weights are the same for positive and negative parts. Note that 2.14 implies that we apply each type (derivation with respect to QT or PS ) of derivation the same number of times on both sides. Therefore it is enough to show that δ(c1,c2 ) 7→ c1 δ(c1 −1,c2 ) − δ(c1,c2 +1) commutes with δ(c1,c2 ) 7→ δ(c1 +1,c2 +2) − c2 δ(c1,c2 +1) , which can be verified by direct inspection. A similar argument works for J(c, d, Pa, v®).  Proposition 2.21. For each i = 0, ..., n and for a ∈ A, let ai be the i-th coefficient of a. Assume that the generalized amplitude I converges. We have !   Õ i , 0 (−1 − v®i )I(c1, c2, Pa, v®)  ∂ ai Pa I(c1, c2, Pa, v®) = Í  ∂Pa i = 0 ((c1 − c2 )(` + 1) + i v®i + n)I(c1, c2, Pa, v®). a∈A  Proof. For i , 0 consider the action (α, Pa ) 7→ α ai Pa of Gm on the Pa ’s. We want to see how the integral changes under this action. We have ∫ c1 T Q (αTi Q ,t ,...,t ) − Γ S T 1 n Q Γ (α i QT , t1, ..., tn ) v® ΨΓ (α i PS ,t1,...,tn ) e t dt1 ...dtn = |E | ΨΓc2 (α Si PS, t1, ..., tn ) R+ ∫ c1 Q (Q ,t ,...,αt ,...,t ) − ΨΓ (PT,t 1,...,αt i,...,tnn) Q Γ (QT , t1, ..., αti, ..., tn ) v® e Γ S1 i t dt1 ...dtn = |E | ΨΓc2 (PS, t1, ..., αti, ..., tn ) R+ ∫ c1 Q 1 v®1 − ΨΓ Q Γ e Γ c2 t1 ...(αti )v®i ...tnv®n dt1 ...d(αti )...dtn . ® v +1 |E | i Ψ R+ Γ α The last line is valid for multiplication by α real and positive which does not change the integration cycle hence it is also valid for all α. As a result we have I(c1, c2, α ai Pa, v®) = α−1−®vi I(c1, c2, Pa, v®). Taking the derivative with respect to α, evaluated at α = 1, we have ! Õ ∂ ∂ ai Pa I(c1, c2, Pa, v®) = I(c1, c2, α ai Pa, v®)|α=1 ∂P ∂α a a∈A = (−1 − v®i )I(c1, c2, Pa, v®). 20 The other case we consider is when i = 0. Note that a0 is nonzero iff a corresponds to a spanning tree. We scale all terms by (α, PS ) 7→ α a0 PS and we obtain c1 Q (Q ,t ,...,t ) − Ψ Γ(αPT ,t1 ,...,tnn ) Q Γ (QT , t1, ..., tn ) v® Γ S 1 e t dt1 ...dtn = |E | ΨΓc2 (αPS, t1, ..., tn ) R+ ∫ c1 Q (Q ,t ,...,t ) − αΨΓ (PT ,t1 ,...,tnn ) Q Γ (QT , t1, ..., tn ) v® Γ S 1 e t .dt1 ...dtn = . |E | α c2 ΨΓc2 (PS, t1, ..., tn ) R+ ∫ After setting si = ti /α we obtain ∫ − QΓ (QT ,s1,...,sn ) α c1 (`+1)−c2 ` Q cΓ1 (QT , s1, ..., sn ) Í ®i v® n iv α s α ds1 ...dsn = c 2 c |E | 2 α ΨΓ (PS, s1, ..., sn ) R+ ∫ c Q (Q ,s ,...,sn ) Q 1 (Q , s , ..., s ) Í T 1 n v® − Γ T 1 (c1 −c2 )(`+1)+ i v®i +n α e ΨΓ (PS,s1,...,sn ) Γc2 s ds1 ...dsn . |E | ΨΓ (PS, s1, ..., sn ) R+ e ΨΓ (PS,s1,...,sn ) As a result we have I(c1, c2, α a0 Pa, v®) = α(c1 −c2 )(`+1)+ i v®i +n I(c1, c2, Pa, v®). Í Taking the derivative with respect to α, evaluated at α = 1, we then have ! Õ ∂ ∂ a0 Pa I(c1, c2, Pa, v®) = I(c1, c2, α a0 Pa, v®)|α=1 ∂P ∂α a a∈A Õ v®i + n)I(c1, c2, Pa, v®). = ((c1 − c2 )(` + 1) + i  Proposition 2.22. For each i = 0, ..., n and for a ∈ A, let ai be the i-th coefficient of a. Assume that c is positive, d is negative and that all coefficients of v® are positive. We have ! Õ ∂ ai Pa J(c, d, Pa, v®) ∂Pa a∈A   i , 0 (−vi − 1)J(c, d, Pa, v®) + (c(` + 1) − d` + |v| + n)J(c, d, Pa, v® + ei )  =  i = 0 −dJ(c, d, Pa, v®).  Proof. The case i = 0 can be dealt with easily by applying the scaling argument of the previous lemma. For other values of i, the scaling argument does not work, since scaling changes the integration cycle. Let θ = dt1 ∧ dt2 ... ∧ dtn and let v be the 21 vector field ti ∂t∂i . Then it is not hard to see that Ω = ιv θ. Since v − ∂t∂i is tangent to ∆n−1 , we have ιv θ|∆n−1 = ι ∂ θ|∆n−1 . We then obtain Í ∂ti Q cΓ (αPa ) v® t Ω d ∆ n−1 ΨΓ (αPa ) ! ∫ c (P ) Q ∂ a Γ = |α=1 α∗ t v® α−vi Ω ∂α ΨΓd (Pa ) ∆ n−1 ! ∫ Q cΓ (Pa ) v® ∂ ∗ = |α=1 α t Ω α−vi −1 d (P ) ∂α Ψ ∆ n−1 a Γ ! ∫ Q cΓ (Pa ) v® ∂ ∗ = (−vi − 1)J(c, d, Pa, v®) + t Ω |α=1 α ΨΓd (Pa ) ∆ n−1 ∂α ! ! Q cΓ (Pa ) v® Q cΓ (Pa ) v® t Ω = L ti ∂ t Ω ∂t i ΨΓd (Pa ) ΨΓd (Pa ) ! Q cΓ (Pa ) v® = (ιti ∂ ◦ d + d ◦ ιti ∂ ) t Ω ∂t i ∂t i ΨΓd (Pa ) ! ! Q cΓ (Pa ) v® Q cΓ (Pa ) v® = (ιti ∂ ◦ d) ◦ ιv t θ + d ι ti ∂ d t Ω ∂t i ∂t i Ψ (P ) ΨΓd (Pa ) a Γ ! ! Q cΓ (Pa ) v® Q cΓ (Pa ) v® = ι ti ∂ L v t θ + d ι ti ∂ d t Ω ∂t i ∂t i Ψ (P ) ΨΓd (Pa ) a Γ ! ! ! Q cΓ (Pa ) v® Q cΓ (Pa ) v® Q cΓ (Pa ) v® = deg t θ ι ti ∂ t θ + d ι ti ∂ d t Ω . ∂t i ∂t i Ψ (P ) ΨΓd (Pa ) ΨΓd (Pa ) a Γ ∂ ∂ |α=1 J(c, d, αPa, v®) = |α=1 ∂α ∂α ∂ |α=1 α∗ ∂α ∫ Let Σ be the union of the coordinate hyperplanes. We have Q cΓ (Pa ) v® t Ω|Σ = 0, ΨΓd (Pa ) and hence Q cΓ (Pa ) v® ιti ∂ ( d t Ω)|Σ = 0. ∂ti Ψ (P ) Γ a Here we are using the fact that the vector field ti ∂t∂i is tangent to Σ, so that we can first restrict to Σ and then perform the contraction. Since the boundary of ∆n−1 lies on Σ, the integral of the second term vanishes. 22 Using ιv θ|∆n−1 = ι ∂ θ|∆n−1 , we can compute the integral of the first term as ∂ti ! ∫ ! ! c (P ) c (P ) c (P ) Q Q Q ∂ a a a Γ Γ Γ |α=1 α∗ deg t v® Ω = t v® θ ti ι ∂ t v® θ d d d ∂t ∂α i ΨΓ (Pa ) ΨΓ (Pa ) ΨΓ (Pa ) ∆n−1 ∆n−1 !∫ ! Q cΓ (Pa ) v®+ei Q cΓ (Pa ) v® = deg t θ t Ω d ΨΓd (Pa ) ∆n−1 ΨΓ (Pa ) ∫ = (c(` + 1) − d` + |v| + n)J(c, d, Pa, v® + ei ). Summing up, we obtain ∂ |α=1 J(c, d, αPa, v®) = (−vi −1)J(c, d, Pa, v®)+(c(`+1)−d`+|v|+n)J(c, d, Pa, v® +ei ). ∂α  Lemma 2.23. The generalized amplitude J(c, d, Pa, v®), which is holomorphic for <(c) > 0, <(d) < 0 and <(v) > 0, has an analytic continuation which is meromorphic on Cn+2 . Proof. This basically follows from resolution of singularity and the following fact. Let Pi (x) be polynomials in n variables which are bounded away from zero on the hypercube [0, 1]n . One needs to show that the integral ∫ P1 (t)c P2 (t)d t1a1 t2a2 · · · tnan dt1 · · · dtn, [0,1]n which is defined and holomorphic in {<(ai ) > 0}, has an analytic continuation to Cn+2 . We prove this by induction on n. The base case is the observation that C c D d has analytic continuation for C and D nonzero, which is clearly true. Note that we have ∫   (−1)t1 +1 P1 (t)c P2 (t)d t1a1 +1 t2a2 · · · tnan dt2 · · · dtn [0,1]n ∩t1 ={0,1} ∫  ∂  an c d a1 +1 a2 P1 (t) P2 (t) t1 t2 · · · tn dt1 · · · dtn = [0,1]n ∂t1 ∫  ∂  c d a1 +1 a2 = P1 (t) P2 (t) t1 t2 · · · tnan dt1 · · · dtn n ∂t1 ∫ [0,1] + P1 (t)c P2 (t)d (a1 + 1)t1a1 t2a2 · · · tnan dt1 · · · dtn [0,1]n   ∫ c−1 ∂P1 d c ∂P2 d−1 a1 +1 a2 P2 (t) + dP1 (t) P2 (t) t1 t2 · · · tnan dt1 · · · dtn = cP1 (t) n ∂t ∂t 1 1 ∫ [0,1] + P1 (t)c P2 (t)d (a1 + 1)t1a1 t2a2 · · · tnan dt1 · · · dtn . [0,1]n 23 Assume we have the analytic continuation for the region <(ai ) > mi . Note that we also have it for mi = 0, since the Pi ’s are nonzero on the hypercube. From the computation above we have ∫ P1 (t)c P2 (t)d t1a1 t2a2 · · · tnan dt1 · · · dtn n [0,1] ∫   1 an t1 +1 c d a1 +1 a2 = (−1) P1 (t) P2 (t) t1 t2 · · · tn dt2 · · · dtn a1 + 1 [0,1]n ∩t1 ={0,1} ∫ ∂P1 a1 +1 a2 1 cP1 (t)c−1 P2 (t)d t1 t2 · · · tnan dt1 · · · dtn + a1 + 1 [0,1]n ∂t1 ∫ 1 ∂P2 a1 +1 a2 + dP1 (t)c P2 (t)d−1 t1 t2 · · · tnan dt1 · · · dtn . a1 + 1 [0,1]n ∂t1 By induction, the first term is meromorphic. The second and third terms have analytic continuation to the region <(a1 ) + 1 > m1 and <(ai ) ≥ mi . Thus, we have analytic continuation to the region <(a1 ) > m1 − 1. We can continue this for all mi and prove it for any value of ai . Now, using a special case of resolution of singularities, we can rewrite any integral over ∆n−1 as a sum of integrals of the form above. This is the same argument that is used in [BW09] Theorem 2, so we do not repeat it explicitly here.  By Remark 2.11, we can define I(Pa ) using the corresponding value of J(Pa ), i.e. I(c1, c2, QT , PS, v®) = C J(−n − |®v | + (c2 − c1 )`, −n − |®v | + (c2 − c1 )(` + 1), QT , PS, v®), where C is constant. Therefore we see that the second term in the case i , 0 of proposition 2.22 vanishes. c = −n − |®v | + (c2 − c1 )` d = −n − |®v | + (c2 − c1 )(` + 1) c(` + 1) − d` + |®v | + n = 0 Theorem 2.24. Let w0 = (c1, c2, v®) and w = (x, y, u®) be any vectors in Cn+2 . One can pull back I(Pa, w0 +  w) to a neighborhood of  = 0 and take the Laurent expansion. Assume that the Laurent expansion has the following form: Õ I(Pa, w0 +  w) =  i Ii (Pa ). i≥−n 24 Then we have ! Ö ∂
na ∂
−na − Ii (Pa ) = 0 ∂P ∂P a a a:n >0 a:n <0 Ö a a and ! ∂ Ii (Pa ) = ((` + 1, −` − 1, 1, · · · , 1) · w0 + n)Ii (Pa ) a0 Pa ∂Pa a∈A Õ + (` + 1, −` − 1, 1, · · · , 1) · w Ii−1 (Pa ) and for k , 0 ! ∂ a k Pa Ii (Pa ) = (−1 − e k · v®)Ii (Pa ) − e k · u® Ii−1 (Pa ). ∂P a a∈A Õ Proof. Our definition of the integral is by analytic continuation of J(c, d, Pa, v®). Note that for any differential operator L, L(J) is meromorphic. Since the differential equation is satisfied for an open subset of Cn+2 (<(c) > 0,<(d) < 0 and <(®v ) > 0), it is valid for all values of c, d and v®. Note that I(Pa, w0 +  w) for all values of  satisfies the first equation since J has the same property. By the calculation above we see that the second term in the case k , 0 vanishes and we have ! Õ ∂ I(Pa, w0 +  w) = (−1 − e k .(®v +  u®)) I(Pa, w0 +  w) a k Pa ∂Pa a∈A = (−1 − e k .®v )I(Pa, w0 +  w) −  e k .u® I(Pa, w0 +  w). In the case k = 0 we have ! ∂ a0 Pa I(Pa, w0 +  w) ∂Pa a∈A Õ = (n + |®v +  u®| − (c2 +  y − c1 −  x)(` + 1)) I(Pa, w0 +  w) = ((` + 1, −` − 1, 1, · · · , 1).w0 + n)I(Pa, w0 +  w) +  (` + 1, −` − 1, 1, · · · , 1).w I(Pa, w0 +  w). The theorem follows from expanding I(Pa, w0 +  w) and comparing terms with different powers of .  25 Remark 2.25. In standard dimensional regularization for the amplitude, assuming that we take the expansion with respect to  in D/2 + , i.e. ∫ Õ 1 i  Ii (Pa ) = e−Q Γ /ΨΓ D/2+ , |E | R+ ΨΓ i≥−n ® and w = (0, 1, 0), ® and thus the amplitude satisfies the we have w0 = (0, D/2, 0) differential equations ! Ö Ö ∂
−na ∂
na − Ii (Pa ) = 0 ∂Pa ∂Pa a:n >0 a:n <0 a a ! ∂ Ii (Pa ) = (n − (` + 1)D/2)Ii (Pa ) − (` + 1)Ii−1 (Pa ) a0 Pa ∂P a a∈A Õ and for k , 0 ! ∂ a k Pa Ii (Pa ) = −Ii (Pa ) . ∂Pa a∈A Õ In particular the lowest coefficient satisfies the so called GKZ hypergeometric differential equation, which we consider in the next chapter. 26 Chapter 3 GKZ A-HYPERGEOMETRIC DIFFERENTIAL EQUATIONS 3.1 Geometric Origin of GKZ D-modules Definition 3.1. Given Zn , the n-dimensional lattice, we fix a basis and denote an element as n-tuple of integers. We define φi as the map from Zn to Z, which gives us the i-th coordinate in the fixed basis. Let A = {a1, · · · , aN } ⊂ Zn be a set of lattice points such that they all lie in the hyperplane φ1 = 1 and generate the lattice as a Z module. For a tuple of integers r = (na : a ∈ A) consider the relation among the points of A of the form Õ na a = 0. a∈A Denote the set of relations by R. For each r ∈ R, we consider a corresponding differential operator Ö  ∂  na Ö  ∂  −na − (3.1) r := ∂pa ∂pa a∈A a∈A na <0 na >0 and for i = 1, · · · , n we define Zi := Õ φi (a)pa a∈A ∂ . ∂pa (3.2) On V = CN , with coordinates p1, ..., pN , consider the differential equations r φ = 0 . For β = (β1, ..., βn ) ∈ k n ⊂ Cn , consider the differential equations (Zi − βi )φ = 0 . We want to find solutions to these differential equations. We denote by W the Weyl algebra W = k[pa, ∂ ∂ ∂ ∂ : a ∈ A]/([ , pb ] = δba, [pa, pb ] = 0, [ , ] = 0), ∂pa ∂pa ∂pb ∂pa 27 where k is a sub-field of C. Then the GKZ left W-module is defined by Õ Õ HA(β) = W/ W(Zi − βi ) + W r . r i Remark 3.2. One can consider HA(β) as a DV -module, i.e. as a sheaf of modules over the sheaf of differential operators on V. W is the ring of global sections of DV and H A(β) is the space of global section of the corresponding sheaf. This set of differential equations and the corresponding W-module was considered by Gelfand, Kapranov and Zelevinsky in [GKZ90], [GZK89]. For a complete discussion of results in this direction see [GKZ08], [SST00] and the references there. We prove the relevant results and state the theorems that we need. Their main results can be summarized in the following theorem from [Cat06]. Theorem 3.3. (GKZ) Let HA(β) be a GKZ hypergeometric system. (1) H A(β) is always holonomic. (2) The singular locus of HA(β) is independent of β ∈ Cn and agrees with the zero locus of the principal A-determinant E A(x) defined in chapter 10 of [GKZ08]. (3) For arbitrary A and generic β, the holonomic rank of HA(β) equals the normalized volume of the convex hull of A, vol(conv(A)). (4) For arbitrary A and β, rank(H A(β)) ≥ vol(conv(A)). (5) Given A, rank(HA(β)) = vol(conv(A)) for all β ∈ Cn if and only if the toric ideal I A is Cohen-Macaulay. φ (a) φ (a) Proposition 3.4. Let t a = t1 1 t2 2 φ (a) · · · tn n . Consider the ring R = k[pa, t a : a ∈ A] with an action of W given by (pa, P) 7→ pa P ∂ ∂ , P) 7→ P + t a P. ( ∂pa ∂pa 28 Moreover, consider the map Ψ:W →R Ö  ∂  ma Í Ψ(p I ) = p I t ma a . ∂pa a∈A Let f = Õ Pa t a a∈A ∂ ∂f + ti − βi Yi = ti ∂ti ∂ti Then we have: (1) Ψ is surjective (2) ker(Ψ) = W Í a a , with a as in (3.1) (3) image(W(Zi − βi )) = Yi R , with Zi as in (3.2) Í (4) Ψ gives an isomorphism between HA(β) and R/ i Yi R. Proof. The first two statements follow from the definition. Note that Ψ is k[pa : a ∈ A] linear and Yi acts k[pa : a ∈ A] linearly. Thus, to check (3) it is enough to compute the image of Ö  ∂  ma (Zi − βi ). ∂pa a∈A With φi as Definition 3.1, we have Ö  ∂  ma Ö  ∂  ma Õ Ö  ∂  ma ∂ Ψ( (Zi − βi )) = Ψ( φi (b)pb − βi ) ∂pa ∂pa ∂pb ∂pa a∈A a∈A a∈A b∈A   ma Õ ∂ Ö ∂ = φi (b)pb Ψ( ) ∂p ∂p b a a∈A b∈A Õ Ö  ∂  ma Ö  ∂  ma + φi (b)mb Ψ( ) − βi Ψ( ) ∂p ∂p a a a∈A a∈A b∈A Õ Õ Í Í Í b ma a = φi (b)pbt t + φi (b)mbt ma a − βi t ma a b∈A Õ Õb∈A Í a =( φi (a)pa t + φi ( ama ) − βi )t ma a a∈A Í = Yi t ma a . a∈A 29 To check (4) we need to show that, for P ∈ GK Z, we have ∂ ∂ P) = Ψ(P) + t b Ψ(P) . ∂pb ∂pb   ma Î If P has the form p I a∈A ∂p∂ a , we have Ψ(     ma Ö  ∂  ma ∂ ∂ I Ö ∂ ∂ I P) = Ψ( p ) +p Ψ( ∂pb ∂pb ∂pa ∂pa ∂pb a∈A a∈A = ∂ Ψ(P) + t b Ψ(P) . ∂pb  This construction makes GKZ a quotient of R. Note that it is not a R-module since the action of Yi does not commute with multiplication by t a . However, it is a k[pa : a ∈ A] module. We want to understand the structure as a k[pa : a ∈ A]module. Proposition 3.5. Assume −β1 + n is nonzero in k for all n ∈ Z≥0 . We have an isomorphism of k[pa : a ∈ A] modules ! Õ R/Y1 R  k[pa : a ∈ A][t a / f ]  k[pa : a ∈ A][t a ]/ pa t a = 1 =: R̂ . (3.3) a∈A Let φ1 be the first coordinate as Definition 3.1. The isomorphism is given by F:R→R F(t J ) = (−1)φ1 (J) γ(φ1 ( j)) t J, where γ(n) = −β1 (−β1 + 1) · · · (−β1 + n − 1) and γ(0) = γ(1) = −β1 . Proof. Note that we have γ(n + 1)/γ(n) = −β1 + n and since all points of A have 30 ∂f φ1 = 1, we have t1 ∂t = f 1 ∂f J ∂ J F(Y1 t J ) = F(t1 t + t1 t − β1 t J ) ∂t1 ∂t1 Õ = F( pa t J+a + φ1 (J)t J − β1 t J ) Õa∈A = pa t a (−1)φ1 (J)+1 γ(φ1 (J) + 1) t J a∈A + φ1 (J)(−1)φ1 (J) γ(φ1 (J)) t J − β1 (−1)φ1 (J) γ(φ1 (J)) t J   γ(φ1 (J) + 1) + φ1 (J) − β1 (−1)φ1 (J) γ(φ1 (J))t J = −f γ(φ1 (J)) = (−β1 + φ1 (J))(1 − f )(−1)φ1 (J) γ(φ1 (J))t J = (1 − f )(−β1 + φ1 (J))F(t J ). By definition F is surjective. The equations above shows that the image of Y1 R is the ideal generated by 1 − f and we have the isomorphism.  Proposition 3.6. Assume −β1 + n is nonzero in k, for all n ∈ Z≥0 . Define Ỹi by Ỹi : R̂ → R̂  ∂f ∂ ∂ − βi − ti (−β1 + t1 ) . Ỹi = ti ∂ti ∂ti ∂t1  We have n Õ H A(β) = R̂/ Ỹi R̂ . i=2 Proof. We need to find the image of Yi R under F. We have F(Yi t J ) = F(ti ∂ ∂f J t + ti t J − βi t J ) ∂ti ∂ti ∂f ∂ = (−1)φ1 (J)+1 γ(φ1 (J) + 1)ti t J + (−1)φ1 (J) γ(φ1 (J))ti t J − βi (−1)φ1 (J) γ(φ1 (J))t J ∂ti ∂ti   ∂f ∂ = −(−β1 + φ1 (J))ti + ti − βi (−1)φ1 (J) γ(φ1 (J))t J ∂ti ∂ti   ∂ ∂f ∂ = −ti (−β1 + t1 ) + ti − βi (−1)φ1 (J) γ(φ1 (J))t J . ∂ti ∂t1 ∂ti Thus, we have that the image is   ∂ ∂f ∂ ti − βi − ti (−β1 + t1 ) R. ∂ti ∂ti ∂t1  31 To show that the equations come from geometry we observe that the ring R is the coordinate ring of an affine toric variety. Let NA be the semigroup generated by the set A as a sub-semigroup of Zn . By definition, R is the semigroup algebra k[pa : a ∈ A][NA]. It is well known that Abelian semigroup algebras are local models for toric varieties. Definition 3.7. We denote the semigroup above by Σ. It is an Abelian semigroup, which is generated by the set A and 0. We denote the semigroup algebra by SΣ = k[t a : a ∈ A] and the corresponding toric variety by XΣ = spec[SΣ ]. φ1 induces a grading on Σ. We denote the projective toric variety Proj(SΣ, φ1 ) by PΣ and the corresponding line bundle by O(1). Remark 3.8. Toric varieties defined by semigroup algebras are not necessarily normal, while all toric varieties defined by a rational polyhedral fan are normal. It turns out that a spectrum of an Abelian semigroup ring is normal iff the semigroup is saturated, i.e. iff all Zn points in the real cone generated by the semigroup are in the semigroup. (see [Hoc72]) Note that the k-vector space of global sections of O(1) is canonically isomorphic to V, since degree one elements in the semigroup are a basis for V. We consider f as the universal section of O(1). Let Y = V( f ) be the codimension one subvariety of the zeros of f in V × PΣ and let U be the complement. Lemma 3.9. We have isomorphisms spec(R) =V × XΣ spec(R̂) =U = V × PΣ \ Y Proof. The first part follows from the definition of XΣ as the spectrum of k[t a : a ∈ A]. For the second part, note that, since V × PΣ \ Y is an affine chart in V × PΣ , its coordinate ring can be described by degree zero elements in R[1/ f ]. This agrees with the definition of the R̂ as in (3.3).  We state some well known facts from the theory of toric varieties. Definition 3.10. Let PA (respectively, P̄A) be the convex hull of the points in A (respectively, A ∪ {0}), as a subset of Rn . PA has the structure of n − 1 dimensional polytope and P̄A has the structure of n dimensional polytope. 32 By a k-dimensional face of a m-dimensional polytope P we mean points in P that lie on a hyperplane, with k dimensional span (as an affine subspace) such that all points of P are on the same side of the hyperplane. There is exactly one m-dimensional face. Codimension one faces are called facets. Lemma 3.11. Faces of the PA (respectively, P̄A) are in one to one correspondence with torus orbits in PΣ (respectively, XΣ ). PA is a subset of Rn , where the the first coordinate is 1. Thus, we can find hyperplanes defining faces that pass through the origin. Such a hyperplane can be defined as the set of x ∈ Rn with hw, xi = 0, for a vector w in Rˇn . In the case where points are all integer points, we can choose w to have integer coefficients as well. The following definition gives an algebraic description of the torus orbits. Definition 3.12. Let w ∈ Zn be a lattice point with the property hw, ai ≥ 0 for all a ∈ Σ and denote by Σwc the set of elements of Σ that have nonzero inner product with w, i.e. hw, ai > 0, for a ∈ Σwc . Moreover, denote by Σw the set of elements with zero inner product , i.e. hw, ai = 0 for a ∈ Σw . Note that the sub k-vector space of SΣ generated by Σwc is a graded ideal in SΣ . We denote this ideal by Iw . The quotient ring, which we denote by Sw , is isomorphic to k[Σw ]. We denote by Pw = Proj(Sw, Φ1 ) the corresponding projective toric variety. The projective toric varieties defined above are not necessarily smooth. Assuming that the toric variety X is smooth, we have the following exact sequence of coherent sheaves: Ê 0 → Ω1X → Ω1 (logD) → OPw → 0, {Facet} where D is the union of codimension one orbits and the second map is the residue map. For a non-smooth toric variety Ω1 (logD) is well defined. See [Ish87]. Note that the complement of D is a n − 1 torus. By definition its coordinate ring is the ring of degree zero elements in the graded ring k[Σ − Σ] ≃ k[Zn ]. Here by Σ − Σ ,we mean differences of the elements of Σ. It is equal to Zn since Σ generates Zn as a Z-module. Thus, it is isomorphic to k[Zn /Z]. We denote Zn /Z by M̃. An element m̃ ∈ M̃ gives us a rational function on PΣ , then dlog(m̃) is an element of Ω1X (logD), 33 which we denote by same notation m̃. It turns out that Ω1 (logD) is always free and we have an isomorphism of coherent sheaves Ω1 (logD) ≃ M̃ ⊗Z OPΣ . We define the free sheaf of algebraic differential forms with logarithmic poles along D to be n−1 n−1 Û Ê Ê i i M̃ ⊗Z OPΣ . Ω (logD) ≃ i=0 i=0 We have a diagram of varieties j V × T \ Y0 U V × PΣ p p p V id V id (3.4) V, where Y0 = Y ∩ V × T and T = PΣ \ D. We consider the sheaf of relative differential forms with logarithmic poles along D on V × PΣ . Since p is the projection on the first factor, we have Û• • ΩV×P (logD) = M̃ ⊗Z OV×PΣ . Σ /V Sections of this sheaf on U are • ΩV×P (logD)(U) = Σ /V Û• M̃ ⊗Z R̂ . This graded sheaf comes with a differential • •+1 d : ΩV×P (logD) → ΩV×P (logD), Σ /V Σ /V which makes it a complex of sheaves. For each element a ∈ Σ ⊂ M, there is a corresponding ã ∈ M̃. Note that the restriction of d to U acts as  Õ d(m ⊗ (t I / f φ1 (I) )) = ( I˜ ∧ m) ⊗ (t I / f φ1 (I) ) − φ1 (I) (ã ∧ m) ⊗ (pa t I+a / f φ1 (I)+1 ) . a∈A 34 Note that this is well defined since we have d(m ⊗ ( f t I / f φ1 (I)+1 )) = Õ d(m ⊗ ( pa t a+I / f φ1 (I)+1 )) Õ a∈A ˜ ∧ m) ⊗ (pa t a+I / f φ1 (I)+1 ) = ((ã + I) a∈A − Õ pa (1 + φ1 (I)) a∈A = ( I˜ ∧ m) ⊗ ( Õ Õ (b̃ ∧ m) ⊗ (pbt I+a+b / f φ1 (I)+2 ) b∈A a I pa t t / f φ1 (I)+1 ) + a∈A − (1 + φ1 (I)) (ã ∧ m) ⊗ (pa t a+I / f φ1 (I)+1 ) a∈A Õ (b̃ ∧ m) ⊗ ( Õ pa t a pbt I+b / f φ1 (I)+2 ) ! a∈A b∈A = ( I˜ ∧ m) ⊗ (t / f I Õ  φ1 (I) ) − φ1 (I) Õ (b̃ ∧ m) ⊗ (pbt I+b / f φ1 (I)+1 )  b∈A = d(m ⊗ (t / f I φ1 (I) )) . Moreover, it is not hard to check that d 2 is zero. Note that this differential is the standard definition of d coming from derivation below on rational functions  ∂ ti ∂ti tJ  f φ1 (J) = ti ∂t∂i t J f φ1 (J) − φ1 (J)ti ∂∂tfi t J f φ1 (J)+1 . We can change the differential to ! n Õ Õ dti dti ã ⊗ pa t a / f ∧ − βi ∧, ∇ β = d + β1 d f / f ∧ − βi ∧ = d + β1 ti ti a∈A i=2 i=2 n Õ and we still have a complex, since we have ∇β = β f β1 β β t2 2 · · · t n n ◦d◦ β t2 2 · · · tn n f β1 . (3.5) • Proposition 3.13. Consider the complex (ΩV×P (logD)(U), ∇ β ), whose terms are Σ /V free R̂-modules and with OV linear differential ∇ β . The n − 1 hyper-cohomology of this complex is isomorphic to H A(β), i.e. • Rn−1 p∗ (ΩV×P (logD)|U, ∇ β ) = H A(β) . Σ /V By Rn−1 p∗ we mean the (n − 1)-th derived functor of p∗ . Everything is in Zariski topology. 35 Proof. Since p is affine, hyper-cohomology agrees with cohomology of the complex. Choose the standard basis e2, e3, · · · , en for M̃, which gives us coordinates t2, · · · , tn on T. We have Ûn−2 n M̃ ⊗Z R̂ = ⊕i=2 dt2 dtn dtˆ i ··· ··· ⊗ R̂, t2 ti tn and with φi as Definition 3.1, we have ! n Õ dti dt2 dtn dtˆ i d + β1 d f / f ∧ − βi ∧ ··· ··· ⊗ t I / f φ1 (I) t t t t i 2 i n i=2 dt2 dtn dtˆ i = ( I˜ ∧ ··· ··· ) ⊗ (t I / f φ1 (I) ) t2 ti tn  Õ dt2 dtˆ i dtn I+a φ1 (I)+1 − φ1 (I) (ã ∧ ··· ··· ) ⊗ (pa t / f ) t t t 2 i n a∈A ! Õ dtˆ i dtn dt2 ··· ··· ⊗ t I / f φ1 (I) + β1 ã ⊗ pa t a / f ∧ t t t 2 i n a∈A − dt2 dtn ··· βi ⊗ t I / f φ1 (I) t2 tn ! Õ dtn dt2 ··· ⊗ −βi + φi (I) − (φ1 (I) − β1 ) φi (ã)pa t a / f t I / f φ1 (I) = t2 tn a∈A   dt2 dtn ∂ ∂f ∂ = ··· ⊗ ti − βi − ti (−β1 + t1 ) t I / f φ1 (I) . t2 tn ∂ti ∂ti ∂t1 Identifying top forms with R̂, we see that the top cohomology is exactly HA(β) by Proposition 3.6.  Definition 3.14. Given an element w of the dual of M̃ we have the contraction map Û• Û•−1 ιw : M̃ → M̃ k Õ ιw (a1 ∧ · · · ∧ a k ) = (−1)i hw, ai ia1 ∧ · · · ∧ âi ∧ · · · ∧ a k . i=1 We extend it R̂-linearly to R̂ ⊗ Ó• M̃. Definition 3.15. Let w1, . . . , wm be elements of Zn defining the facets of PA. An element β ∈ Cn is called non-resonant (respectively, semi non-resonant) if hwi, −β + Zn i , 0 (respectively, if hwi, −β + Σi , 0), for all i. 36 Proposition 3.16. Let w correspond to a face of PA, the polytope defined in Definition 3.10, and let Iw , Pw , and Sw be as in Definition 3.12. Let i w : Uw → U be the fiber product iw Uw U p V × P Σ, V × Pw which is the inclusion into U of the intersection of the boundary components corresponding to w and U. This inclusion is given by the ideal sheaf Iw . Furthermore, assume β is semi non-resonant. Then the inclusion q.i.s • • (Iwn ΩV×P (logD)|U, ∇ β ) −−−→ (ΩV×P (logD)|U, ∇ β ) Σ /V Σ /V is a quasi-isomorphism. Proof. First note that Uw is affine. By induction it is enough to show that • • (Iwn ΩV×P (logD)|U, ∇ β ) ,→ (Iwn−1 ΩV×P (logD)|U, ∇ β ) Σ /V Σ /V is a quasi-isomorphism, or equivalently that the cokernel ( • Iwn−1 ΩV×P (logD)|U Σ /V • Iwn ΩV×P (logD)|U /V , ∇β) Σ is quasi-isomorphic to zero. • Let t J ⊗ α be a section on Iwn−1 ΩV×P (logD)|U . We show that a multiple of ιw̃ t J ⊗ α Σ /V is ∇ β -primitive of t J ⊗ α, where w̃ = (w2, · · · , wn ). We have ∇ β ιw̃ t J ⊗ α = t J ⊗ J ∧ ιw̃ α + (−φ1 (J) + β1 ) Õ pa t a+J ⊗ a ∧ ιw̃ α − Õ βi t J ⊗ ei ∧ ιw̃ α = a∈A − t ⊗ ιw̃ J ∧ α − (−φ1 (J) + β1 ) J Õ pa t a+J ⊗ ιw̃ a ∧ α + Õ βi t J ⊗ ιw̃ ei ∧ α+ a∈A t ⊗ (ιw̃ J) ∧ α + (−φ1 (J) + β1 ) J Õ pa t a+J ⊗ (ιw̃ a) ∧ α − Õ βi t J ⊗ (ιw̃ ei ) ∧ α = a∈A ! − ιw̃ ∇β t J ⊗ α + t J h(0, w̃), Ji + (−φ1 (J) + β1 ) Õ Pa t a+J h(0, w̃), ai − t J a∈A Note that h(0, w̃), Ji =hw, Ji − w1 φ1 (J) h(0, w̃), ai =hw, ai − w1 Õ Pa t a+J =t J . a∈A Õ βi φi (w) ⊗ α . 37 Using these equalities we can rewrite the equation as ! ∇ β ιw̃ t J ⊗α = −ιw̃ ∇ β t J ⊗α+ t J hw, Ji + (−φ1 (J) + β1 ) Õ Pa t a+J hw, ai − t J hw, βi ⊗α . a∈A Note that hw, Ji ∈ Z≥0 , and terms that appear in the first sum are all zero, since if • a ∈ Sw we have hw, ai = 0 and if a ∈ Swc we have t a+J ∈ Iwn ΩV×P (logD)|U . If Σ /V t J ⊗ α is closed, we see that Õ hw, Jit J ⊗ α − t J βi hw, ei i ⊗ α = (hw, Ji − hw, βi)t J ⊗ α = c t J ⊗ α is exact, where c is a nonzero constant, by the semi non-resonance assumption.  In [AB01], authors construct a category of complexes of sheaves. De Rham complexes live in this category. Given an open embedding j : X → Y with X smooth and a D-module M on X, they define a R j! functor on this category. If Y is smooth they show DR( j! M) = R j! DR(M), where j! is the left adjoint of j ! operator on holonomic D-modules as in [Bor+87]. See appendix D of [AB01]. Corollary 3.17. Let j : V × T \ Y0 → U be the inclusion of the complement of the torus boundary as in (3.4). Assuming that β is semi non-resonant, we have q.i.s • • R j! j ∗ (ΩV×P (logD)|U, ∇ β ) −−−→ (ΩV×P (logD)|U, ∇ β ) . Σ /V Σ /V Here we think of complexes as objects of the derived category of sheaves of Abelian groups. The functor j ∗ is the pull back functor and R j! is as in Definition D.2.14 of [AB01]. • Proof. To compute R j! we need to take an extension of j ∗ (ΩV×P (logD)|U, ∇ β ) Σ /V to U and take the limit by the ideal defining the complement. We already have an extension and we have shown that the powers of the ideal defining the boundary do not change the cohomology. Since both varieties are affine, the limit is • (ΩV×P (logD)|U, ∇ β ).  Σ /V Lemma 3.18. Consider OV×T as a D-module and let DRV×T/V (OV×T ) be its relative de Rham complex. Assume β has integer coefficients. Then we have a quasiisomorphism q.i.s • • j ∗ (ΩV×P (logD)|U, ∇ β ) −−−→ (ΩV×T/V |V×T\Y0, d) = DRV×T\Y0 /V (OV×T\Y0 ) . Σ /V 38 Proof. This simply follows from the fact that the ti ’s are invertible on T. The isomorphism is given by twisting the differential by f β1 β β t2 2 . . . t n n and its inverse as in (3.5). Note that, since A generates Zn as a Z-module, multiplication by the rational function above is an isomorphism.  Theorem 3.19. Assume β has integer coefficients, is semi non-resonant and β1 is negative. Then we have Rn−1 p∗ (R j! DRV×T\Y0 /V (OV×T\Y0 )) = HA(β) . Proof. Proposition 3.13 and Lemma 3.18 together with Corollary 3.17 imply this.  For the following theorem, we assume that the toric variety is normal, which is equivalent to Σ being saturated. Theorem 3.20. Assume β has integer coefficients, is semi non-resonant and β1 is negative. Assume the semigroup Σ is saturated. Let Uv be the fiber of p over v ∈ V and let D be the boundary divisor, i.e. the complement of V × T. We have an H 0 (Sol(H A(β)))v := HomDV (HA(β), OV,v ) = Hn−1 (Uv, Uv ∩ D) . Proof. If PΣ was smooth, we could consider D-modules on U. Note that taking the relative de Rham complex commutes with j! by , i.e. we have R j! DRV×T\Y0 /V (OV×T\Y0 ) = DRU/V ( j! OV×T\Y0 ). We have p+ functor of [Bor+87]. By definition of p+ for projections, Rn−1 p∗ ◦ DRU/V = H 0 p+ . This implies HA(β) = H 0 p+ ( j! OV×T\Y0 ) as quasi-coherent sheaves. The fact that connections agree follows from direct computation. The rest of the proof is the same as in [Hua+15]. The idea is to use the Riemann-Hilbert correspondence and the fact that Sol ◦ j! = j∗ ◦ Sol from section 15 of [Bor+87] and the sheaf theoretic definition of relative homology from lemma 3.4 of [Hua+15]. Furthermore, the isomorphism is given by the cycle to period map 39 defined in [HLZ13]. X = PΣ is normal since Σ is saturated. By [Ish87], there exists an equivariant resolution of singularities g : X 0 → X, such that X 0 is smooth and Rg∗ O X 0 = O X . We have a diagram of fiber products of the form V × T \ Y0 j0 id V × T \ Y0 j U0 V × X0 g0 id×g U V × X. By the same computations in Proposition 3.16, one can show that the de Rham complex for j!0 O can be computed using the de Rham complex with logarithmic poles with twisted differential. The twisted de Rham complex is again a complex of free O X 0 modules. Since Rg∗0 OU 0 = OU we have • • Rg∗0 DRU 0/V ( j!0 OV×T\Y0 ) = Rg∗0 (ΩV×X 0 /V log(D)|U 0, ∇ β ) = (ΩV×X/V log(D)|U , ∇ β ) . Thus, we have • HA(β) = Rn−1 p∗ (ΩV×X/V log(D)|U, ∇ β ) = Rn−1 p∗ Rg∗0 DRU 0/V ( j!0 OV×T\Y0 ) = Rn−1 (p ◦ g0)∗ DRU 0/V ( j!0 OV×T\Y0 ) = H 0 (p ◦ g0)+ ( j!0 OV×T\Y0 ) . Applying the Sol functor we get Sol( j!0 OV×T\Y0 ) = R j∗0 CV×T\Y0 . Moreover, using Sol(p ◦ g0)+ = R(p ◦ g0)! Sol[n − 1] from section 14 of [Bor+87] , we have Sol(HA(β)) = p R0 R(p ◦ g0)! R j∗0 CV×T\Y0 [n − 1] = p R0 Rp! ◦ Rg!0 ◦ R j∗0 CV×T\Y0 [n − 1] . However, g0 is proper, therefore Rg!0 = Rg∗0 and we have Rg∗0 ◦ R j∗0 = R(g0 ◦ j 0)∗ = R j∗ . Moreover, we have Sol(HA(β)) = p R0 Rp! ◦ R j∗ CV×T\Y0 [n − 1] . 40 Thus, the sheaf of classical solution is H 0 (Sol(HA(β))) = Rn−1 p! ◦ R j∗ CV×T\Y0 . Note that p! commutes with taking fiber, by the compact support base change theorem. We want to find the restriction of R j∗ CV×T\Y0 to the fiber Uv of p over a point v ∈ V. Denote the inclusion of V × T into V × PΣ by j¯ and denote the map from U to V × PΣ by i. We have a diagram of varieties. j V × T \ Y0 jv Uv \ D U Uv i iv V ×T T j¯v j¯ V × PΣ v×Id PΣ Since R j∗ is local on the target, we have R j∗ CV×T\Y0 = i ∗ R j¯∗ CV×T . Therefore R j∗ CV×T\Y0 |Uv = i ∗ R j¯∗ CV×T |Uv = i ∗v (v × Id)∗ R j¯∗ CV×T . Since j¯ does not depend on the V, we have (v × Id)∗ R j¯∗ CV×T = R j¯v∗ CT . From the square in the front, we see that i ∗v R j¯v∗ CT = R jv∗ CUv \D . Thus, j∗ commutes with restriction to a fiber. R j∗ CV×T\Y0 |Uv = R jv∗ CUv \D By sheaf theoretic definition of relative homology we deduce an HomDV (HA(β), OV,v ) = (Rn−1 p! ◦ R j∗ CV×T\Y0 )v = Hn−1 (Uv, Uv ∩ D) .  We showed that, for semi-nonresonant integer β, the sheaf of classical solutions to HA(β) is isomorphic to a relative homology. To find the isomorphism one needs to follow the proof of the Riemann-Hilbert correspondence which can be found in [HLZ13]. Assume that v ∈ V is point and consider a relative chain δv ∈ Hn−1 (Uv, Uv ∩ D). We can extend this cycle to an analytic neighborhood of v in V. Let φ(v) be the function defined in the neighborhood of v by ∫ dtn f β1 dt2 ... . φ(v) = tn δv t β̃ t2 41 Some computations similar to what we had in Proposition 2.22 show that this function satisfies HA(β). In fact the isomorphism in the Riemann-Hilbert correspondence comes from this morphism. We do not use this fact in the general form, since in our case the cycle δ is always the positive real numbers. For more precise formulation see [HLZ13]. 3.2 Regularization To have a better notation, we change the dimension from n to n + 1. Assume A ⊂ Zn+1 such that all points have first coordinate equal to 1. We denote the semigroup generated by A by Σ and we assume Σ − Σ = Zn+1 . As before, let PA be the convex hull of points in A and let C A be the cone consisting of rays originating from zero and passing through PA. For (β0, . . . , βn ) = β ∈ Cn+1 , we can consider the corresponding differential equations and integral forms of the solutions. Assume all Pa ’s are positive i.e. v is a positive real point of V. We claim that the closure of R+n is a relative chain for the pair (Uv, Uv ∩ D). To check this, it is enough to show that f does not vanish on the closure of R+n . One can check that closure of R+n is homeomorphic to the polytope PA by moment map. The restriction of f to each torus orbit corresponding to a face defined by w is Õ Pa t ã . hw,ai=0 Since Pa are positive and t ã are positive, we see that f does not vanish. Assuming β is semi non-resonant and integer, it follows that φ is well defined and the integral ∫ dtn t α̃ dt1 ... α 0 t tn 1 R+n f is convergent, where α = −β and α = (α0, α̃). We prove this fact for more general values of β. Lemma 3.21. Assume the Pa ’s are positive real numbers indexed by a ∈ A. The integral ∫ t α̃ dt1 dtn I := ... (3.6) Í α 0 ã t1 tn R+n ( a∈A Pa t ) converges for α = (α0, α̃) iff <(α) is in the interior of C A. 42 Proof. To show this, we reparametrize R+n by Rn , with the map ti = exp (xi ), which gives ∫ exphα̃, xi (3.7) Í α0 dx1 . . . dxn . Rn ( a∈A Pa exphã, xi) For any a ∈ A, we have exphα̃, xi exph<(α̃), xi | Í ≤ C exph<(α̃ − α0 ã), xi, α0 | ≤ ( a∈A Pa exphã, xi) (Pa exphã, xi)<(α0 ) where C is a constant that only depends on the Pa ’s. Taking the minimum on A we can rewrite the inequality as ∫ ∫   exphα̃, xi 0 dx . . . dx ≤ C exp min h<( α̃ − α ã), xi dx1 . . . dxn . Í 1 n 0 α0 a Rn ( a∈A Pa exphã, xi) Rn Note that <(α) is in the interior of the cone iff <(α̃) is in the interior of the convex hull of {<(α0 )ã}a∈A. For a fixed x, there exists at least one a such that h<(α̃ − α0 ã), xi = h<(α̃), xi − h<(α0 )ã, xi < 0, otherwise all point of <(α0 )ã would be in the half-space h · , xi ≥ hα̃, xi and <(α̃) would be on the boundary, which would contradict the fact that α̃ is in the interior of the convex hull. To show that the integral converges, we use radial coordinates and we write   ∫ x 0 I≤C exp r min h<(α̃ − α0 ã), i r n−1 dr dΩ . a n−1 |x| S ×R+ We showed that, for any x, mina h<(α̃−α0 ã), xi is negative. Let −ε be the supremum of this function on the sphere of radius one, which is negative by compactness of the sphere. Substituting, we get ∫ ∫ Γ(n) 0 n−1 −εr 0 I≤C r e dr dΩ ≤ C dΩ < ∞ . n S n−1 ×R+ S n−1 ε  We want to find relations among integrals with different α. Let K(α, Pa ) be the integral ∫ t α̃ dt1 dtn K(α, Pa ) = ... . (3.8) Í α 0 ã t1 tn R+n ( a∈A Pa t ) Lemma 3.22. Assume <(α) is in the interior of the cone. Let w ∈ Zn+1 . We have the equality Õ hw, αiK(α, Pa ) = α0 hw, aiPa K(α + a, Pa ) . a∈A 43 Proof. Both sides are linear in w, hence it is enough to check this for w = ei . Note that, for w = e0 , this equality is trivial. For i , 0 and w = ei , we use exponential change of variable as in equation (3.7). Consider the differential form on Rn given by exphα̃, xi ˆ i . . . dxn . θ= Í dx1 . . . dx ( a∈A Pa exphã, xi)α0 A basic computation shows that we have ! Í P a exph α̃ + a, xi exphα̃, xi a i − α0 Ía∈A dθ = αi Í dx1 . . . dxn . ( a∈A Pa exphã, xi)α0 ( a∈A Pa exphã, xi)α0 +1 Assuming that K(α, Pa ) converges implies that <(α) is the interior of the cone generated by A. Since <(α) is in the interior of the cone, <(α) + a is also in the interior of the cone and all terms in dθ have absolutely convergent integrals. Therefore, we can integrate dθ in the interior of ball of radius r and take the limit as r goes to infinity. By Stokes’ theorem the integral of dθ over a ball of radius r is equal to integral of θ on a sphere of radius r. By the same computations as in the previous lemma, we can see that θ has exponential decay, while the volume of the sphere grows polynomially. This implies that the limit is zero. Note that the integral of dθ is equal to Õ hei, αiK(α, Pa ) − α0 hei, aiPa K(α + a, Pa ), a∈A which implies the statement.  Theorem 3.23. Let K(α, Pa ) be as in (3.8), defined for <(α) in the interior of the cone generated by A. Then K(α, Pa ) has meromorphic continuation to Cn+1 , with poles along the −Σ translates of the hyperplanes defining the facets of the convex hull of A, or equivalently the semi-resonant −α’s. Furthermore, we have the identity in Lemma 3.22. Proof. First note that the integral is absolutely convergent and that the integrand is analytic in α, which implies that the integral is holomorphic for <(α) in the interior of the cone, i.e. for hwi, <(α)i > 0. We use induction to show the statement for the sets hwi, <(α)i > mi , which cover Cn+1 for mi ∈ Z. The statement is true for ® = (0, . . . , 0). Assume it is true for m ® = (m1, . . . , mn+1 ). By Lemma 3.22 we have m Õ hwi, αiK(α, Pa ) = α0 hwi, aiPa K(α + a, Pa ) . a∈A 44 Note that hw, ai = 0 for a in the facet defined by wi . Thus, hwi, αiK(α, Pa ) can be expressed as a linear combination of K(b, Pa ), where, for all b, we have hwi, <(b)i ≥ mi + 1 and hw j , <(b)i ≥ m j for j , i. We can define the integral for hwi, <(α)i > mi − 1 and hw j , <(α)i > m j by dividing both sides by hwi, αi. This means that there is pole when hwi, αi is zero. By repeating this operation, we get poles on the −Σ translates of hwi, αi = 0. Note that the equation above is analytic in α and is valid for <(α) in the interior of the cone, hence it is valid everywhere.  Remark 3.24. Note that in the identity of Lemma 3.22, the sum on the right is multiplied by α0 . If we start from an α with α0 ∈ Z≤0 , we eventually multiply be zero, since all points in the interior of the cone have α0 > 0. Thus, K(α, Pa ) has degree one zero along hyperplanes α0 ∈ Z≤0 . We know that, for positive real Pa ’s, if K(α, Pa ) converges, then it is a solution to HA(β). We claim that this is true for the analytic continuation of K(α, Pa ) as well. From the identity of Lemma 3.22, we know that for α with real part in the interior of the positive cone, the left hand side is a solution to HA(−α). Note that both differential equations and relations are analytic in α and Pa , therefore they are valid for the analytic continuation of K(α, Pa ). As a result we have that K(α, Pa ) is a solution to H A(−α) when −α is non semi-resonant. For a resonant −α we can find a vector u® ∈ Cn+1 such that −(α +  u®) is semi non-resonant for small enough . We can take the Laurent expansion in the u® direction and we obtain K(α +  u®, Pa ) = ∞ Õ  i Kiu®(α, Pa ) . (3.9) i=−k As in the previous chapter, we denote the set of integer relations among points of A by R. For r ∈ R, we have a corresponding box differential operator r as in (3.1) and Õ ∂ . (3.10) Zw = hw, aiPa ∂P a a∈A Note that Zi in (3.2) is Zei in 3.10. In this notation, a solution φ to HA(β) is equivalent to a function satisfying r φ = 0 Zw φ = hw, βiφ . By the same computation of Theorem 2.24, we arrive to the following proposition. Proposition 3.25. For w ∈ Cn+1 , we have r Kiu®(α, Pa ) = 0 u® Zw Kiu®(α, Pa ) = hw, −αiKiu®(α, Pa ) + hw, −® uiKi−1 (α, Pa ) . 45 Proof. This follows from expanding both sides of (3.9).  In particular we see that the lowest coefficient gives us a solution to HA(−α). Remark 3.26. All the calculations we have done here can be done for any chain δ replacing the positive real points. In fact, Lemma 3.22 is valid for the integral over any chain, and the rest of the calculation is exactly the same. In this way we can construct a set of solutions for resonant β. 46 Chapter 4 AMPLITUDES AND REGULARIZATION As we showed in chapter 2.2, amplitudes satisfy certain differential equations. From a Feynman diagram, we constructed a subset of Zn+1 in the following way. For each monomial t S in the first Symanzik polynomial ΨΓ of the graph, we consider the ® ∈ Zn+1 and, for each monomial t T in Q Γ , we consider the lattice point point (1, S) ® We denote this set of lattice points by A. In general A does not generate (0, T). the lattice Zn+1 but it generates a sublattice of dimension n. To see this, note that (1, 0, 0, ...0, −1, 0, ..., 0), where −1 is in the i-th place, is in the sublattice generated ® − (0, T), ® where the monomial T is the product of m2 ti and by A. This term is (1, S) i Ín S t . On the other hand, all points of A lie on the hyperplane i=0 ai = ` + 1, where ` is the number of loops in Γ. Thus, the sublattice generated by A in Zn+1 is the Í set of lattice points x, such that ` + 1| xi . Denote this sublattice by L and let r : L → Zn+1 be the function defined by r0 (x) = (x0 + x1 + ... + xn )/(` + 1) and ri (x) = xi , for 1 ≤ i ≤ n. It is easy to see that r is invertible and that the determinant is 1/(` + 1), hence the image of the standard cube has volume 1 and r(A) generates the lattice. We replace A by r(A). Note that all points of r(A) have first coordinate equal to 1. By Proposition 2.21 we know that I(c1, c2, Pa, v®) satisfies HA(β), where Õ β = ((c1 − c2 )(` + 1) + vi + n, −1 − v1, . . . , −1 − vn ) . i Thus, I satisfies Hr(A) (r(β)) and we have r(β) = (c1 − c2, −1 − v1, . . . , −1 − v2 ) . For the rest of this chapter we replace A by r(A) and β by r(β). Note that, for the ® and thus the original amplitude A(Γ, Pa ), we have c1 = 0, c2 = D/2 and v = 0, corresponding vector β is β = (−D/2, −1, . . . , −1) . Assuming the normality condition and the semi non-resonant condition in Theorem 3.20, all solutions of HA(β) come from integrals. Thus, A(Γ, Pa ) is equal to ∫ dt1 dtn t1 . . . t n ... , Í ã D/2 t1 tn δ ( a∈A Pa t ) 47 for a relative chain δ. In fact, using the projective version of the integral, we can show that it is equal to K(−β, Pa ), up to multiplication by a rational number. Lemma 4.1. Feynman’s parametric integral formula gives ∫ 1 Γ(a + b) ∞ λ a−1 dλ = . Aa B b Γ(a)Γ(b) 0 [λA + B]a+b  Proof. See Chapter 8 of [KS01]. Corollary 4.2. Assume K(α, Pa ) is convergent, i.e. α is in the interior of the cone generated by A. Then we have the equality K(α, Pa ) = Γ(−| α̃| + α0 (` + 1)) I(0, α0, Pa, α̃ − (1, . . . , 1)) . Γ(α0 ) Proof. By Lemma 4.1 we have t α̃ α0 R+n (Q Γ + ΨΓ ) ∫ r | α̃ |+n−1 t α̃ =
drΩ `+1 Q + r ` Ψ α0 R+ ×∆ n−1 r Γ Γ ∫ r | α̃ |+n−1−α0 ` t α̃ = α0 drΩ R+ ×∆ n−1 (rQ Γ + ΨΓ ) ∫ Γ(−n − | α̃| + α0 (` + 1))Γ(| α̃| + n − α0 `) = Γ(α0 ) = ∫ | α̃ |+α0 ` Q−n− Γ −n− | α̃ |+α0 (`+1) ∆ n−1 Ψ Γ t α̃ Ω Γ(−n − | α̃| + α0 (` + 1))Γ(| α̃| + n − α0 `) J(−n − | α̃| + α0 `, −n − | α̃| + α0 (` + 1), Pa, α̃) . Γ(α0 ) By the definition of K(α, Pa ) and the equality (2.11), we have K(α, Pa ) = Γ(−| α̃| + α0 (` + 1))Γ(| α̃| − α0 `) J(−| α̃| + α0 `, −| α̃| + α0 (` + 1), Pa, α̃ − (1, . . . , 1)) Γ(α0 ) Γ(−| α̃| + α0 (` + 1)) = I(0, α0, Pa, α̃ − (1, . . . , 1)) . Γ(α0 )  ® we can By corollary above and the fact that A(Γ, Pa ) = π D`/2 I(0, D/2, Pa, 0), compute the amplitude from K as A(Γ, Pa ) = π D`/2 Γ(D/2) K((D/2, 1, . . . , 1), Pa ) . Γ(−n + D/2(` + 1)) (4.1) Note that the Gamma function never vanishes. As a result, when K(α, Pa ) converges, we can define the amplitude. In fact, we can define the amplitude by this equation. 48 For the rest of this chapter we study K(α, Pa ), and the structure of its poles. To do that, we first find the defining inequalities for PA. We need a few definitions and notation. Definition 4.3. For a graph Γ with n edges, we denote by SPΓ ⊂ Rn the polytope constructed by the incidence vectors of the complements of the spanning trees of ® where S corresponds to a monomial t S in the Γ, i.e. it is the convex hull of { S}, first Symanzik polynomial of Γ. We denote by PΓ the polytope constructed from the terms in the first and second Symanzik polynomials (including mass terms). Note that we have PΓ = SPΓ + En , where En is the convex hull of {0, e1, . . . , en } and plus is the Minkowski sum. For a subgraph γ ⊂ Γ, we have natural inclusions Pγ ⊂ PΓ and SPγ ⊂ SPγ . Definition 4.4. For a subgraph γ ⊂ Γ, let 1γ be the incidence vector of γ in Zn , i.e. the vector where the coefficient corresponding to an edge e ∈ Γ is 1 if e ∈ γ and is zero otherwise. We denote by `γ the number of loops in γ, i.e. the dimension of the first homology of γ. Lemma 4.5. Given two subgraph γ1 and γ2 , we have `γ1 + `γ2 ≤ `γ1 ∩γ2 + `γ1 ∪γ2 . Proof. By Mayer-Vietoris for the pair γ1 and γ2 , we have the exact sequence 0 = H2 (γ1 ∪γ2 ) → H1 (γ1 ∩γ2 ) → H1 (γ1 )⊕H1 (γ2 ) → ker (H1 (γ1 ∪ γ2 ) → H0 (γ1 ∩ γ2 )) → 0 . Counting dimensions implies the inequality.  We define another polytope using inequalities and we will show it is the same polytope we considered before. Spanning tree polytopes (SPΓ ) and in general matroid polytopes have been studied in combinatorial optimization theory, see [Cho89], [Edm71] and [Ful71]. The polytope PΓ is similar to these polytopes and we can translate some of the results in combinatorial optimization to our setting. Definition 4.6. For a graph Γ, let PΓ0 be the subset of Rn defined by the inequalities h1γ, x̃i ≥ `γ h1Γ, x̃i ≤ `Γ + 1, where the first inequality is valid for all subgraphs. 49 Assume that the graph γ in the first inequality above is a single edge. In this case, the inequality implies that all coefficients are positive. The second inequality implies the sum of the coefficients is bounded. Thus, these equations define a bounded set and PΓ0 is a polytope. Any face of the polytope P = PΓ0 is defined by setting some of the inequalities to equalities. Assume we have a set of equalities h1γ, x̃i = `γ for γ ∈ F, where F is a family of subgraphs. Lemma 4.7. Given a point x̃ in P, let F be the set F = {γ ⊂ Γ : h1γ, x̃i = `γ } . Then F is closed under intersection and union of its elements. Proof. We have h1γ1 ∩γ2, x̃i ≥ `γ1 ∩γ2 and h1γ1 ∪γ2, x̃i ≥ `γ1 ∪γ2, and we have 1γ1 + 1γ2 = 1γ1 ∩γ2 + 1γ1 ∪γ2 . Combining this with the inequality of Lemma 4.5, we have `γ1 + `γ2 = h1γ1, x̃i + h1γ1, x̃i ≥ `γ1 ∩γ2 + `γ1 ∪γ2 ≥ `γ1 + `γ2, and hence both inequalities are equalities. This shows that F is closed under intersection and union of its elements.  By the previous lemma, the defining equations of faces (coming from subgraphs) can be chosen to be closed under intersection and union. By a chain of subgraphs we mean a family C of subgraphs such that, for γ1, γ2 ∈ C, we have either γ1 ⊂ γ2 or γ2 ⊂ γ1 . Lemma 4.8. Let F be the set of equalities corresponding to subgraphs and let PF be the corresponding face. Let C ⊂ F be a maximal chain in F. The family of linear equations {h1γ, x̃i = `γ : γ ∈ F} is equivalent to {h1γ, x̃i = `γ : γ ∈ C}, i.e. we have PF = PC . Proof. Given a subgraph γ, by a chain violation we mean a subgraph c ∈ C such that neither c ⊂ γ nor γ ⊂ c. Since C is maximal, for any subgraph γ in F \ C there exists c ∈ C such that it is a chain violation for γ and PC∪{c} , PC , otherwise 50 the statement follows. Among all of the subgraphs choose the one with minimal number of chain violations. By Lemma 4.7, we have c ∩ γ ∈ F and c ∪ γ ∈ F. For all x̃ ∈ PF we have h1γ, x̃i = `γ h1c, x̃i = `c h1γ∩c, x̃i = `γ∩c h1γ∪c, x̃i = `γ∪c . These four linear equations are dependent, i.e. the sum of the first two is equal to the sum of the last two. The first equation is not satisfied by all points of PC , so we either have PC∪{γ∩c} , PC or PC∪{γ∪c} , PC . Replacing c by γ ∩ c or γ ∪ c decreases the number of chain violations, which contradicts the maximality condition.  Proposition 4.9. Two polytope are equal, i.e. PΓ = PΓ0 . Furthermore let CΓ be the cone over PΓ in Rn+1 and let γ® = (−`γ, 1γ ). Then CΓ is given by the equalities hγ®, xi ≥ 0 for all γ ⊂ Γ, and hΓ® − e0, xi ≤ 0 . Proof. Note that the inequalities define a cone since 0 satisfies all the equations. The intersection of this cone with the hyperplane hx, e0 i = 1 is PΓ0 , which is bounded. Since the intersection is bounded, the cone defined by the inequalities is the cone over the intersection. As a result it is enough to show that the intersection is equal to PΓ . The first inequality for a subgraph is satisfied by all points of A, since an intersection of a spanning tree with a subgraph is a spanning forest of the subgraph, hence number of edges in its complement is greater than the number of loops. The second inequality is trivial. PΓ0 contains all the extreme points of PΓ , and thus it contains PΓ . First we find the integer points of PΓ0 . Any edge e ∈ E is a subgraph and the corresponding first inequality implies xi ≥ 0. Thus, integer points of PΓ0 have the form (a1, . . . , an ), where ai ∈ Z≥0 . For any loop γ ∈ Γ we have Õ ae ≥ 1. e∈γ Thus, for at least one e ∈ γ, we have ae , 0. If we remove this edge from the graph, the remaining graph has `Γ − 1 loops and we can find another loop that does 51 not have e in it. This in turn implies that another ae 0 is non-zero. Iterating this procedure, we get at least `Γ many nonzero ai ’s, which are chosen from different loops. The complement of the edges corresponding to these ai ’s is a spanning tree. Í The second inequality implies i ai ≤ `Γ + 1, hence (a1, . . . , an ) corresponds to the incidence vector of a complement of a spanning tree, or ei plus the incidence vector of a complement of a spanning tree. The integer points of PΓ0 are exactly the integer points of PΓ . To finish the proof we need to show that the extreme points of PΓ0 are integers. Let x̃ be an extreme point of PΓ0 . Then x̃ is the unique solution to a set of linear equations corresponding to defining equations of PΓ0 . There are two cases. The first case occurs when the second equation is not used and x̃ is defined by h1γ, x̃i = `γ for γ ∈ F. By Lemma 4.8, we can replace F by a chain of subgraphs C. Since the solution is unique, we need n many equalities and we have γ1 ⊂ γ2 ⊂ · · · ⊂ γn = Γ C = {γ1, . . . , γn }. Thus, γi \ γi−1 has only one edge. We denote this edge by ei . By the equalities we see that xi = `γi − `γi−1 ∈ {0, 1}, which implies we have an integer point. The set of extreme points we obtain in this way corresponds to a spanning trees. The second case is when we have the equality h1Γ, x̃i = `Γ + 1. By the same argument, we see that xi ∈ {0, 1}, for all i except one of them. Since these add up to an integer, the other one has to be an integer too. The set of extreme points we obtain in this way corresponds to the set of monomials in the second Symanzik polynomial.  A subgraph is called 2-connected, if it is connected and remains connected after removing any vertex. Note that single edges are 2-connected. We have found a set of inequalities that define PΓ , but this set is not minimal. We just need the equations defining the facets of PΓ . Any facet is defined by a single equation, so we need to find subgraphs for which the equality defines a facet. Lemma 4.10. For a 2-connected graph γ with no self-loops, SPγ is |E(γ)| − 1 dimensional. Proof. Note that all integer points have the form 1S , where S is complement of a spanning tree. Thus, all points of SPγ lie on the hyperplane where the sum of 52 coefficients is `γ . This implies that SPγ is at most |E(γ)| − 1 dimensional. We show that ei − e j can be constructed from differences of points of SPγ . Let e be an edge of γ and let C be a loop in γ which contain e. This exists, since γ is 2-connected. Let T be a spanning tree that contains all edges of C except e. Note that such a spanning tree exists, since any tree can be extended to a spanning tree. Let e0 be another edge of C. We claim that T ∪ e \ e0 is a spanning tree. The reason is that T ∪ e has only one loop C, and removing any edge from it makes it a tree. Let S = T c and S0 = (T ∪ e \ e0)c . Then 1S − 1S 0 is 1e − 10e , hence, for any two edges e, e0 in the same loop, 1e − 1e 0 can be computed as a difference of points in SPγ . Since γ is 2-connected and does not have self loops, we can compute 1e − 1e 0 for any pair of edges, and we find that SPγ is of codimension one.  For a self loop e in Γ, SPe is just a point, which indeed is |E(e)| − 1 = 0 dimensional. Note that PΓ is always n dimensional, since it is equal to SPΓ + En , where En is n dimensional. Lemma 4.11. For a subgraph γ ⊂ Γ, PΓ ∩ { x̃ : h1γ, x̃i = `γ } is SPγ × PΓ//γ , where Γ//γ is constructed from contracting connected components of γ to points. Proof. It is enough to find extreme points of PΓ ∩ { x̃ : h1γ, x̃i = `γ }, since the equation defines a face of PΓ . The equation h x̃, 1γ i = `γ for an extreme point (ae : e ∈ Γ), implies the vector (ae : e ∈ γ) is equal to 1S , where S is the complement of a spanning tree in γ. A monomial t T can be written as t S t T−S , where S has coefficients in γ and T − S corresponds to edges in Γ − γ that are the edges in the contracted graph. Note that t T−S always corresponds to a monomial for the first or second Symanzik polynomial of Γ//γ. On the other hand S + S0, where S comes from a spanning tree in γ and S0 comes from a monomial in Γ//γ, is a monomial in the first or second Symanzik polynomial of Γ. Thus, we can identify the corresponding face PΓ with SPγ × PΓ//γ  Theorem 4.12. For a graph Γ, the polytope PΓ is given by the inequality h1Γ, x̃i ≤ `Γ + 1 and inequalities hγ, x̃i ≥ `γ 53 indexed by 2-connected subgraphs without self-loops γ ⊂ Γ, as well as inequalities h1e, x̃i = xe ≥ 1 indexed by self loops e ∈ Γ. Replacing any inequality corresponding to a subgraph or self-loop γ with equality, defines a facet of PΓ that is equal to SPγ × PΓ//γ . Proof. By Proposition 4.9, we know that these equation define the polytope. We just need to find the ones that define facets, i.e. are codimension one. By Lemma 4.11 the codimension is equal to the codimension of SPγ . The latter is equal to one for self loops and for 2-connected subgraphs without self-loops, by Lemma 4.10. On the other hand, if we have a subgraph which is not 2-connected, then the complement of a spanning tree has to have `γi many edges in γi , where γi are 2connected components of γ. Thus, SPγ is at least of codimension 2. For a subgraph that is not a self loop but that contains a self loop e, we have two linear equalities for points of SPγ , i.e. h1γ, x̃i = `γ and xe = 1. This makes SPγ at least of codimension 2.  For a Feynman diagram Γ, the vector α is equal to (D/2, 1, . . . , 1) . To find the inequalities defining the interior of the cone C A, we need to replace inequalities with strict inequalities. Applying these to a vector α, we find the necessary and sufficient condition for convergence of K(α, Pa ), namely D/2(` + 1) > |E(Γ)| . If we have self-loops, then the inequality D/2 = D`/2 < |E | = 1 is not satisfied and the integral diverges. For all 2-connected subgraphs without self-loops γ we have D`/2 < |E(γ)| . Note that for single edges this equality is satisfied, hence it is enough to check this for 2-connected subgraphs that are not single edges, i.e. for the so called 1PI subgraphs. Lemma 4.13. Let Σ be the semigroup generated by A in Zn+1 . Then Σ is saturated. 54 Proof. Assume a = (a0, ã) ∈ Zn+1 is an integer point in C A. We want to find Í (α1, α2, . . . , αk ) ∈ Ak such that a = i αi . Assume ® ai > 0 . hΓ, We claim that there exists i ∈ {1, . . . , n}, such that a − ei is in C A. Assume this is not the case. Then, for each i, one of the inequalities is not satisfied by a − ei . Note that we have hΓ® − e0, a − ei i = hΓ® − e0, ai − 1 ≤ 0 . Thus, for each i, there exists a subgraph γi such that hγ®i, a − ei i < 0 . Since a has integer coefficients and a is in C A, we must have hγ®i, ai = 0, which is equivalent to h1γi , ã/a0 i = `γi . By Lemma 4.5 we have the same equality for a union of γi ’s, i.e. h1Γ, ã/a0 i = `Γ . This contradicts our assumption. As a result, we can write a as b + c, where b is ® bi = 0 and c has positive integer coordinates an integer point in C A that satisfies hΓ, with c0 = 0. Note that Õ 0 ≥ hΓ® − e0, b + ci = −b0 + ci, i Í which implies b0 ≥ i ci . Assume we can write b = i αi0 with αi0 in Σ satisfying ® α0i = 0. Since b0 ≥ Íi ci , we can distribute Íi ci many ei ’s among the α0 hΓ, i i and define αi = αi0 + e ji . To finish the proof, we need to show that the semigroup ® xi = 0 is saturated. This has been generated by the points of A on the facet hΓ, shown in [Whi77].  Í Corollary 4.14. PΣ is projectively normal, i.e. k[Σ] is integrally closed. By [Hoc72] the toric ideal has the Cohen-Macaulay property. Theorem 4.15. For a graph Γ, with the properties that D/2(`Γ + 1) > |E(Γ)| and D`γ /2 < |E(γ)|, 55 and for all 1PI subgraphs γ, the amplitude is equal to K((D/2, 1, . . . , 1), Pa ), up to multiplication by rational numbers and powers of π. This is a period of the motive H n (Uv, Uv ∩ D). Furthermore, for any graph Γ, the lowest coefficient of the -expansion of the amplitude agrees with the lowest coefficient of the -expansion of K((D/2 + , 1, . . . , 1)), up to multiplication by rational numbers and powers of π. This can be computed as a linear combination Õ pi (Pa )K(αi, Pa ), where the pi ’s are polynomials in the Pa ’s with rational coefficients and all the K(αi, Pa )’s are convergent and are periods of the motive H n (Uv, Uv ∩ D). Proof. Note that the value of the gamma function at positive integers is an integer. The first part then follows from the Relation (4.1) between K(α, Pa ) and A(Γ, Pa ). For the second part, note that, using the relations in Lemma 3.22, we can replace a divergent K with a linear combination of convergent integrals, with polynomial coefficients in Pa . The poles arise by diving by terms hw, αi, which have a rational residue in the variable . Moreover, we have division by the gamma function, which has rational residue at negative integers. Note that all the resulting convergent integrals are periods of H n (Uv, Uv ∩ D) by Theorem 3.20.  Proof of Theorem 1.1. We have ® = c0 A(D/2) := I(0, D/2, Pa, 0) Γ(α0 ) K(α, Pa ), Γ(−hΓ® − e0, αi) where α = (D/2, 1, · · · , 1). By Remark 3.24, zeros of K(α, Pa ) cancel poles of the gamma function in the numerator. By Theorem 3.23, poles of K(α, Pa ) appear on semi non-resonant −α’s. To find the semi non-resonant locus, we need to find the −Σ translates of the facets. By description of the facets in Theorem 4.12, −α is semi non-resonant iff hΓ® − e0, αi ∈ Z≥0 or hγ®, αi ∈ Z≤0 for a 2-connected subgraph γ. However, the poles of the gamma function at negative integers cancel the poles coming from the first equation. The second equation for α = (D/2, 1, · · · , 1) is equivalent to 56 −D/2`γ + |E(γ)| ∈ Z≤0 . Thus, the first part of the theorem follows. The second part is a special case of Theorem 4.15.  Remark 4.16. To find the  expansion of the integral, using the relations in Lemma 3.22, we can replace the differential form dt1 dtn t α̃ Í ... a α + ( a Pa t ) 0 t1 tn with differential forms that have logarithmic poles along the boundary, since division by zero appears when we want to replace α on a facet with a linear combination of points that are not on the facet. The boundary components correspond to the products of subgraphs and quotient graphs. Based on this observation, we think there is a relation between the Connes-Kreimer renormalization and limiting mixed Hodge structures of [BK08]. 57 BIBLIOGRAPHY [AB01] Yves André and Francesco Baldassarri. De Rham cohomology of differential modules on algebraic varieties. Vol. 189. Progress in Mathematics. Birkhäuser Verlag, Basel, 2001, pp. viii+214. isbn: 3-7643-6348-7. doi: 10.1007/978-3-0348-8336-8. url: http://dx.doi.org/ 10.1007/978-3-0348-8336-8. [AM09] Paolo Aluffi and Matilde Marcolli. “Feynman motives of banana graphs”. In: Commun. Number Theory Phys. 3.1 (2009), pp. 1–57. issn: 19314523. doi: 10.4310/CNTP.2009.v3.n1.a1. url: http://dx.doi. org/10.4310/CNTP.2009.v3.n1.a1. [BB03] Prakash Belkale and Patrick Brosnan. “Matroids, motives, and a conjecture of Kontsevich”. In: Duke Math. J. 116.1 (2003), pp. 147–188. issn: 0012-7094. doi: 10.1215/S0012- 7094- 03- 11615- 4. url: http://dx.doi.org/10.1215/S0012-7094-03-11615-4. [BEK06] Spencer Bloch, Hélène Esnault, and Dirk Kreimer. “On motives associated to graph polynomials”. In: Comm. Math. Phys. 267.1 (2006), pp. 181–225. issn: 0010-3616. doi: 10.1007/s00220-006-0040-2. url: http://dx.doi.org/10.1007/s00220-006-0040-2. [BK08] Spencer Bloch and Dirk Kreimer. “Mixed Hodge structures and renormalization in physics”. In: Commun. Number Theory Phys. 2.4 (2008), pp. 637–718. issn: 1931-4523. doi: 10.4310/CNTP.2008.v2.n4.a1. url: http://dx.doi.org/10.4310/CNTP.2008.v2.n4.a1. [BK10] Spencer Bloch and Dirk Kreimer. “Feynman amplitudes and Landau singularities for one-loop graphs”. In: Commun. Number Theory Phys. 4.4 (2010), pp. 709–753. issn: 1931-4523. doi: 10.4310/CNTP.2010. v4.n4.a4. url: http://dx.doi.org/10.4310/CNTP.2010.v4. n4.a4. [BK95] D. J. Broadhurst and D. Kreimer. “Knots and numbers in φ4 theory to 7 loops and beyond”. In: Internat. J. Modern Phys. C 6.4 (1995), pp. 519–524. issn: 0129-1831. doi: 10.1142/S012918319500037X. url: http://dx.doi.org/10.1142/S012918319500037X. [Blo10] Spencer Bloch. “Motives associated to sums of graphs”. In: The geometry of algebraic cycles. Vol. 9. Clay Math. Proc. Amer. Math. Soc., Providence, RI, 2010, pp. 137–143. [Bor+87] A. Borel et al. Algebraic D-modules. Vol. 2. Perspectives in Mathematics. Academic Press, Inc., Boston, MA, 1987, pp. xii+355. isbn: 0-12-117740-8. 58 [Bro12] Francis Brown. “Mixed Tate motives over Z”. In: Ann. of Math. (2) 175.2 (2012), pp. 949–976. issn: 0003-486X. doi: 10.4007/annals. 2012.175.2.10. url: http://dx.doi.org/10.4007/annals. 2012.175.2.10. [BS12] Francis Brown and Oliver Schnetz. “A K3 in φ4 ”. In: Duke Math. J. 161.10 (2012), pp. 1817–1862. issn: 0012-7094. doi: 10 . 1215 / 00127094 - 1644201. url: http : / / dx . doi . org / 10 . 1215 / 00127094-1644201. [BW09] Christian Bogner and Stefan Weinzierl. “Periods and Feynman integrals”. In: J. Math. Phys. 50.4 (2009), pp. 042302, 16. issn: 0022-2488. doi: 10.1063/1.3106041. url: http://dx.doi.org/10.1063/1. 3106041. [Cat06] E. Cattani. “Three Lectures on Hypergeometric Functions”. In: Lecture notes (2006). [Cho89] Sunil Chopra. “On the spanning tree polyhedron”. In: Oper. Res. Lett. 8.1 (1989), pp. 25–29. issn: 0167-6377. doi: 10.1016/0167-6377(89) 90029- 1. url: http://dx.doi.org/10.1016/0167- 6377(89) 90029-1. [CM08] Alain Connes and Matilde Marcolli. Noncommutative geometry, quantum fields and motives. Vol. 55. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2008, pp. xxii+785. isbn: 978-08218-4210-2. [DD98] A. I. Davydychev and R. Delbourgo. “A geometrical angle on Feynman integrals”. In: J. Math. Phys. 39.9 (1998), pp. 4299–4334. issn: 00222488. doi: 10.1063/1.532513. url: http://dx.doi.org/10. 1063/1.532513. [Edm71] Jack Edmonds. “Matroids and the greedy algorithm”. In: Math. Programming 1 (1971), pp. 127–136. issn: 0025-5610. [Ful71] D. R. Fulkerson. “Blocking and anti-blocking pairs of polyhedra”. In: Math. Programming 1 (1971), pp. 168–194. issn: 0025-5610. [GKZ08] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Discriminants, resultants and multidimensional determinants. Modern Birkhäuser Classics. Reprint of the 1994 edition. Birkhäuser Boston, Inc., Boston, MA, 2008, pp. x+523. isbn: 978-0-8176-4770-4. [GKZ90] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. “Generalized Euler integrals and A-hypergeometric functions”. In: Adv. Math. 84.2 (1990), pp. 255–271. issn: 0001-8708. doi: 10.1016/0001-8708(90) 90048- R. url: http://dx.doi.org/10.1016/0001- 8708(90) 90048-R. 59 [GM04] A. B. Goncharov and Yu. I. Manin. “Multiple ζ-motives and moduli spaces M 0,n ”. In: Compos. Math. 140.1 (2004), pp. 1–14. issn: 0010437X. doi: 10.1112/S0010437X03000125. url: http://dx.doi. org/10.1112/S0010437X03000125. [GZK89] I. M. Gelfand, A. V. Zelevinsky, and M. M. Kapranov. “Hypergeometric functions and toric varieties”. In: Funktsional. Anal. i Prilozhen. 23.2 (1989), pp. 12–26. issn: 0374-1990. doi: 10.1007/BF01078777. url: http://dx.doi.org/10.1007/BF01078777. [HLZ13] A. Huang, B. H. Lian, and X. Zhu. “Period Integrals and the RiemannHilbert Correspondence”. In: ArXiv e-prints (Mar. 2013). arXiv: 1303. 2560 [math.AG]. [Hoc72] M. Hochster. “Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes”. In: Ann. of Math. (2) 96 (1972), pp. 318–337. issn: 0003-486X. [Hua+15] A. Huang et al. “Chain Integral Solutions to Tautological Systems”. In: ArXiv e-prints (Aug. 2015). arXiv: 1508.00406 [math.AG]. [Ish87] Masa-Nori Ishida. “Torus embeddings and de Rham complexes”. In: Commutative algebra and combinatorics (Kyoto, 1985). Vol. 11. Adv. Stud. Pure Math. North-Holland, Amsterdam, 1987, pp. 111–145. [KS01] Hagen Kleinert and Verena Schulte-Frohlinde. Critical properties of φ4 theories. World Scientific Publishing Co., Inc., River Edge, NJ, 2001, pp. xx+489. isbn: 981-02-4658-7. doi: 10 . 1142 / 9789812799944. url: http://dx.doi.org/10.1142/9789812799944. [SST00] Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama. Gröbner deformations of hypergeometric differential equations. Vol. 6. Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, 2000, pp. viii+254. isbn: 3-540-66065-8. doi: 10 . 1007 / 978 - 3 - 662 - 04112 - 3. url: http://dx.doi.org/10.1007/978-3-662-04112-3. [Sta98] Richard P. Stanley. “Spanning trees and a conjecture of Kontsevich”. In: Ann. Comb. 2.4 (1998), pp. 351–363. issn: 0218-0006. doi: 10.1007/ BF01608530. url: http://dx.doi.org/10.1007/BF01608530. [Ste98] John R. Stembridge. “Counting points on varieties over finite fields related to a conjecture of Kontsevich”. In: Ann. Comb. 2.4 (1998), pp. 365–385. issn: 0218-0006. doi: 10 . 1007 / BF01608531. url: http://dx.doi.org/10.1007/BF01608531. [Sym58] K. Symanzik. “Dispersion relations and vertex properties in perturbation theory”. In: Progr. Theoret. Phys. 20 (1958), pp. 690–702. issn: 0033-068X. doi: 10.1143/PTP.20.690. url: http://dx.doi.org/ 10.1143/PTP.20.690. 60 [Whi77] Neil L. White. “The basis monomial ring of a matroid”. In: Advances in Math. 24.3 (1977), pp. 292–297. issn: 0001-8708.