Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs At every genus if one considers the analytic terms, the integrand at a fixed order in the derivative expansion can be described diagrammatically by graphs, referred to as modular graph functions. Roughly, the vertices of the graphs are the positions of insertions of the vertex operators on the worldsheet, while the links are given by the scalar Green function connecting the vertices. These graphs depend on the moduli of the worldsheet and transform with fixed weights under Sp ( 2 g , Z ) transformations for the genus g Riemann surface, such that the integrand is Sp ( 2 g , Z ) invariant. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs At every genus if one considers the analytic terms, the integrand at a fixed order in the derivative expansion can be described diagrammatically by graphs, referred to as modular graph functions. Roughly, the vertices of the graphs are the positions of insertions of the vertex operators on the worldsheet, while the links are given by the scalar Green function connecting the vertices. These graphs depend on the moduli of the worldsheet and transform with fixed weights under Sp ( 2 g , Z ) transformations for the genus g Riemann surface, such that the integrand is Sp ( 2 g , Z ) invariant. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs At every genus if one considers the analytic terms, the integrand at a fixed order in the derivative expansion can be described diagrammatically by graphs, referred to as modular graph functions. Roughly, the vertices of the graphs are the positions of insertions of the vertex operators on the worldsheet, while the links are given by the scalar Green function connecting the vertices. These graphs depend on the moduli of the worldsheet and transform with fixed weights under Sp ( 2 g , Z ) transformations for the genus g Riemann surface, such that the integrand is Sp ( 2 g , Z ) invariant. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Our aim is to understand certain properties of some graphs at genus two. We shall consider the low momentum expansion of the genus two four graviton amplitude in type II superstring theory. The integrand is simpler than other string theories, thanks to the maximal supersymmetry the type II theory enjoys. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Our aim is to understand certain properties of some graphs at genus two. We shall consider the low momentum expansion of the genus two four graviton amplitude in type II superstring theory. The integrand is simpler than other string theories, thanks to the maximal supersymmetry the type II theory enjoys. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Do these graphs satisfy some eigenvalue equation(s) on moduli space? The answer to this question generalizes in several ways the structure of the eigenvalue equations obtained in other cases. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Do these graphs satisfy some eigenvalue equation(s) on moduli space? The answer to this question generalizes in several ways the structure of the eigenvalue equations obtained in other cases. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The genus two four graviton amplitude is the same in type IIA and IIB string theory (Green,Kwon,Vanhove). It is given by (D’Hoker,Phong;Berkovits;Berkovits,Mafra) | d 3 Ω | 2 A = π � ( det Y ) 3 B ( s , t , u ; Ω , ¯ 64 κ 2 10 e 2 φ R 4 Ω) , M 2 where I now define the various quantities. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The genus two four graviton amplitude is the same in type IIA and IIB string theory (Green,Kwon,Vanhove). It is given by (D’Hoker,Phong;Berkovits;Berkovits,Mafra) | d 3 Ω | 2 A = π � ( det Y ) 3 B ( s , t , u ; Ω , ¯ 64 κ 2 10 e 2 φ R 4 Ω) , M 2 where I now define the various quantities. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs 2 κ 2 10 = ( 2 π ) 7 α ′ 4 . The period matrix is given by Ω = X + iY , where X , Y are matrices with real entries. The measure is | d 3 Ω | 2 = � id Ω IJ ∧ d ¯ Ω IJ . I ≤ J The integral is over M 2 , the fundamental domain of Sp ( 4 , Z ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs 2 κ 2 10 = ( 2 π ) 7 α ′ 4 . The period matrix is given by Ω = X + iY , where X , Y are matrices with real entries. The measure is | d 3 Ω | 2 = � id Ω IJ ∧ d ¯ Ω IJ . I ≤ J The integral is over M 2 , the fundamental domain of Sp ( 4 , Z ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs 2 κ 2 10 = ( 2 π ) 7 α ′ 4 . The period matrix is given by Ω = X + iY , where X , Y are matrices with real entries. The measure is | d 3 Ω | 2 = � id Ω IJ ∧ d ¯ Ω IJ . I ≤ J The integral is over M 2 , the fundamental domain of Sp ( 4 , Z ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs 2 κ 2 10 = ( 2 π ) 7 α ′ 4 . The period matrix is given by Ω = X + iY , where X , Y are matrices with real entries. The measure is | d 3 Ω | 2 = � id Ω IJ ∧ d ¯ Ω IJ . I ≤ J The integral is over M 2 , the fundamental domain of Sp ( 4 , Z ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The dynamics is contained in |Y| 2 � ( det Y ) 2 e − α ′ � B ( s , t , u ; Ω , ¯ i < j k i · k j G ( z i , z j ) / 2 , Ω) = Σ 4 where each factor of Σ represents an integral over the genus two worldsheet. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The string Green function is given by � w � w G ( z , w ) = − ln | E ( z , w ) | 2 + 2 π Y − 1 � �� � Im ω I Im ω J , IJ z z where Y − 1 = ( Y − 1 ) IJ , E ( z , w ) is the prime form and ω I IJ ( I = 1 , 2 ) are the abelian differential one forms. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Finally, 3 Y = ( t − u )∆( 1 , 2 ) ∧ ∆( 3 , 4 ) + ( s − t )∆( 1 , 3 ) ∧ ∆( 4 , 2 ) +( u − s )∆( 1 , 4 ) ∧ ∆( 2 , 3 ) , where the bi–holomorphic form is given by ∆( i , j ) ≡ ∆( z i , z j ) dz i ∧ dz j = ǫ IJ ω I ( z i ) ∧ ω J ( z j ) . The Mandelstam variables are given by s = − α ′ ( k 1 + k 2 ) 2 / 4 , t = − α ′ ( k 1 + k 4 ) 2 / 4 , u = − α ′ ( k 1 + k 3 ) 2 / 4, i k i = 0 and k 2 where � i = 0. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Finally, 3 Y = ( t − u )∆( 1 , 2 ) ∧ ∆( 3 , 4 ) + ( s − t )∆( 1 , 3 ) ∧ ∆( 4 , 2 ) +( u − s )∆( 1 , 4 ) ∧ ∆( 2 , 3 ) , where the bi–holomorphic form is given by ∆( i , j ) ≡ ∆( z i , z j ) dz i ∧ dz j = ǫ IJ ω I ( z i ) ∧ ω J ( z j ) . The Mandelstam variables are given by s = − α ′ ( k 1 + k 2 ) 2 / 4 , t = − α ′ ( k 1 + k 4 ) 2 / 4 , u = − α ′ ( k 1 + k 3 ) 2 / 4, i k i = 0 and k 2 where � i = 0. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The amplitude is conformally invariant, as it is invariant under G ( z , w ) → G ( z , w ) + c ( z ) + c ( w ) even though the string Green function G ( z , w ) is not. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs To consider the analytic terms in the low momentum expansion, define ∞ Ω) σ p 2 σ q B ( s , t , u ; Ω , ¯ � B ( p , q ) (Ω , ¯ 3 Ω) = p ! q ! p , q = 0 where σ n = s n + t n + u n . Thus B ( p , q ) (Ω , ¯ Ω) is a sum of various graphs with distinct topologies. Each of them involves factors of G ( z , w ) in the integrand and hence is not generically conformally invariant, even though it is modular invariant. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs To consider the analytic terms in the low momentum expansion, define ∞ Ω) σ p 2 σ q B ( s , t , u ; Ω , ¯ � B ( p , q ) (Ω , ¯ 3 Ω) = p ! q ! p , q = 0 where σ n = s n + t n + u n . Thus B ( p , q ) (Ω , ¯ Ω) is a sum of various graphs with distinct topologies. Each of them involves factors of G ( z , w ) in the integrand and hence is not generically conformally invariant, even though it is modular invariant. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Of course, the total contribution from all the graphs is conformally invariant. It is natural to consider contributions coming from graphs each of which is conformally as well as modular invariant. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Of course, the total contribution from all the graphs is conformally invariant. It is natural to consider contributions coming from graphs each of which is conformally as well as modular invariant. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs This is obtained by considering |Y| 2 � ( det Y ) 2 e − α ′ � B ( s , t , u ; Ω , ¯ i < j k i · k j G ( z i , z j ) / 2 Ω) = Σ 4 and performing the low energy expansion, where G ( z , w ) is the conformally invariant Arakelov Green function. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs To define the Arakelov Green function, consider the Kahler form κ = 1 4 Y − 1 IJ ω I ∧ ω J , which satisfies � κ = 1 , Σ on using the Riemann bilinear relation � ω I ∧ ω J = 2 Y IJ . Σ Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The Arakelov Green function is defined by G ( z , w ) = G ( z , w ) − γ ( z ) − γ ( w ) + γ 1 , where � γ ( z ) = κ ( w ) G ( z , w ) , Σ w and � γ 1 = κ ( z ) γ ( z ) . Σ Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Defining the dressing factor ( z 1 , z 2 ) = Y − 1 IJ ω I ( z 1 ) ω J ( z 2 ) , we obtain the useful relation � µ ( z ) G ( z , w ) = 0 , Σ z where µ ( z ) = ( z , z ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Let us consider the modular graphs that arise at low orders in the momentum expansion. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The D 4 R 4 term is given by (D’Hoker,Gutperle,Phong) | ∆( 1 , 2 ) ∧ ∆( 3 , 4 ) | 2 Ω) = 1 � B ( 1 , 0 ) (Ω , ¯ = 32 . ( det Y ) 2 2 Σ 4 Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The D 6 R 4 term is given by | ∆( 1 , 2 ) ∧ ∆( 3 , 4 ) − ∆( 1 , 4 ) ∧ ∆( 2 , 3 ) | 2 − 1 � B ( 0 , 1 ) (Ω , ¯ Ω) = 3 ( det Y ) 2 Σ 4 � � × G ( z 1 , z 2 ) + G ( z 3 , z 4 ) − G ( z 1 , z 3 ) − G ( z 2 , z 4 ) 2 � � d 2 z i G ( z 1 , z 2 ) P ( z 1 , z 2 ) , = 16 Σ 2 i = 1 where P ( z 1 , z 2 ) = ( z 1 , z 2 )( z 2 , z 1 ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs This graph is given by the Kawazumi–Zhang invariant and satisfies an eigenvalue equation. (D’Hoker,Green,Pioline,Russo) All modular graphs are given by skeleton graphs with links given by Arakelov Green function, along with dressing factors involving the integrated vertices. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs This graph is given by the Kawazumi–Zhang invariant and satisfies an eigenvalue equation. (D’Hoker,Green,Pioline,Russo) All modular graphs are given by skeleton graphs with links given by Arakelov Green function, along with dressing factors involving the integrated vertices. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The D 8 R 4 term is given by | ∆( 1 , 2 ) ∧ ∆( 3 , 4 ) | 2 Ω) = 1 � B ( 2 , 0 ) (Ω , ¯ ( det Y ) 2 4 Σ 4 � 2 � × G ( z 1 , z 4 ) + G ( z 2 , z 3 ) − G ( z 1 , z 3 ) − G ( z 2 , z 4 ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Thus there are modular graph functions of three distinct topologies involving two factors of the Arakelov Green function, with skeleton graphs depicted by (i) (iii) (ii) Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs We denote 3 B ( 2 , 0 ) B ( 2 , 0 ) (Ω , ¯ � (Ω , ¯ Ω) = Ω) , i i = 1 where we define B ( 2 , 0 ) (Ω , ¯ Ω) next. i Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs We have that | ∆( 1 , 2 ) ∧ ∆( 3 , 4 ) | 2 � B ( 2 , 0 ) (Ω , ¯ G ( z 1 , z 4 ) 2 Ω) = 1 ( det Y ) 2 Σ 4 2 � � d 2 z i G ( z 1 , z 2 ) 2 Q 1 ( z 1 , z 2 ) , = 4 Σ 2 i = 1 where Q 1 ( z 1 , z 2 ) = µ ( z 1 ) µ ( z 2 ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs We have that | ∆( 1 , 2 ) ∧ ∆( 3 , 4 ) | 2 � B ( 2 , 0 ) (Ω , ¯ Ω) = − 2 G ( z 1 , z 4 ) G ( z 1 , z 3 ) 2 ( det Y ) 2 Σ 4 3 � � d 2 z i G ( z 1 , z 2 ) G ( z 1 , z 3 ) µ ( z 1 ) P ( z 2 , z 3 ) . = 4 Σ 3 i = 1 Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs We have that | ∆( 1 , 2 ) ∧ ∆( 3 , 4 ) | 2 � B ( 2 , 0 ) (Ω , ¯ Ω) = G ( z 1 , z 4 ) G ( z 2 , z 3 ) 3 ( det Y ) 2 Σ 4 4 � � d 2 z i G ( z 1 , z 4 ) G ( z 2 , z 3 ) P ( z 1 , z 2 ) P ( z 3 , z 4 ) . = Σ 4 i = 1 Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Our aim is to obtain eigenvalue equation(s) satisfied by genus two modular graphs on moduli space. Variations of the moduli are captured by variations of the Beltrami differentials. The holomorphic deformation with respect to the Beltrami differential µ is given by δ µ φ = 1 � d 2 w µ w w δ ww φ. ¯ 2 π Σ Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Our aim is to obtain eigenvalue equation(s) satisfied by genus two modular graphs on moduli space. Variations of the moduli are captured by variations of the Beltrami differentials. The holomorphic deformation with respect to the Beltrami differential µ is given by δ µ φ = 1 � d 2 w µ w w δ ww φ. ¯ 2 π Σ Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Our aim is to obtain eigenvalue equation(s) satisfied by genus two modular graphs on moduli space. Variations of the moduli are captured by variations of the Beltrami differentials. The holomorphic deformation with respect to the Beltrami differential µ is given by δ µ φ = 1 � d 2 w µ w w δ ww φ. ¯ 2 π Σ Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs We shall obtain the eigenvalue equation by first performing holomorphic and then anti–holomorphic variations with respect to the the Beltrami differentials of each modular graph. The relevant formulae can be derived using the known relations for the variations of the abelian differentials, period matrix and the prime form. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs We shall obtain the eigenvalue equation by first performing holomorphic and then anti–holomorphic variations with respect to the the Beltrami differentials of each modular graph. The relevant formulae can be derived using the known relations for the variations of the abelian differentials, period matrix and the prime form. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs A useful formula for both variations is � � Y − 1 = − Y − 1 δ ww IJ ω J ( z ) IJ ω J ( w ) ∂ z ∂ w G ( w , z ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs We also use the formulae 2 πδ 2 ( z − w ) − π ( z , w ) , ∂ w ∂ z G ( z , w ) = − 2 πδ 2 ( z − w ) + π ∂ z ∂ z G ( z , w ) = 2 µ ( z ) very often. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs For the holomorphic variations, we use δ ww G ( z 1 , z 2 ) = − ∂ w G ( w , z 1 ) ∂ w G ( w , z 2 ) − 1 � � � d 2 u ( w , u ) ∂ w G ( w , u ) ∂ u G ( u , z 1 ) + G ( u , z 2 ) . 4 Σ Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs For the anti–holomorphic variations, we also use ∂ u G ( u , z ) − 1 � � δ uu ∂ w G ( w , z ) = π ( w , u ) 2 ∂ u G ( u , w ) + π � d 2 x ( x , u )( w , x ) ∂ u G ( u , x ) . 4 Σ This leads to manifestly conformally covariant expressions. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs For the anti–holomorphic variations, we also use ∂ u G ( u , z ) − 1 � � δ uu ∂ w G ( w , z ) = π ( w , u ) 2 ∂ u G ( u , w ) + π � d 2 x ( x , u )( w , x ) ∂ u G ( u , x ) . 4 Σ This leads to manifestly conformally covariant expressions. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs By varying the Beltrami differentials, we perform the mixed variation for each of the three modular graphs. For each graph B ( 2 , 0 ) , we obtain contributions involving i four, two and zero derivatives ( B ( 2 , 0 ) has no contribution 1 involving zero derivatives). They act on the Arakelov Green functions. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs By varying the Beltrami differentials, we perform the mixed variation for each of the three modular graphs. For each graph B ( 2 , 0 ) , we obtain contributions involving i four, two and zero derivatives ( B ( 2 , 0 ) has no contribution 1 involving zero derivatives). They act on the Arakelov Green functions. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs By varying the Beltrami differentials, we perform the mixed variation for each of the three modular graphs. For each graph B ( 2 , 0 ) , we obtain contributions involving i four, two and zero derivatives ( B ( 2 , 0 ) has no contribution 1 involving zero derivatives). They act on the Arakelov Green functions. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Schematically, the contributions with four derivatives are of 2 the form ∂ 2 u , ∂ 2 w ∂ w ∂ u ∂ z + h . c ., and ∂ w ∂ u ∂ z i ∂ z j + h . c . . The contributions with two derivatives are of the form ∂ w ∂ u . Hermitian conjugation means w ↔ u in the various expressions as well. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Schematically, the contributions with four derivatives are of 2 the form ∂ 2 u , ∂ 2 w ∂ w ∂ u ∂ z + h . c ., and ∂ w ∂ u ∂ z i ∂ z j + h . c . . The contributions with two derivatives are of the form ∂ w ∂ u . Hermitian conjugation means w ↔ u in the various expressions as well. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Schematically, the contributions with four derivatives are of 2 the form ∂ 2 u , ∂ 2 w ∂ w ∂ u ∂ z + h . c ., and ∂ w ∂ u ∂ z i ∂ z j + h . c . . The contributions with two derivatives are of the form ∂ w ∂ u . Hermitian conjugation means w ↔ u in the various expressions as well. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs First let us consider the terms involving four derivatives that arise from the mixed variations of the graphs B ( 2 , 0 ) . i We shall consider the terms involving two and no derivatives later. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs First let us consider the terms involving four derivatives that arise from the mixed variations of the graphs B ( 2 , 0 ) . i We shall consider the terms involving two and no derivatives later. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Varying B ( 2 , 0 ) , we get that 1 E 1 8 δ uu δ ww B ( 2 , 0 ) � = Φ 1 ,α + . . . . 1 α = A Varying B ( 2 , 0 ) , we get that 2 E − 1 8 δ uu δ ww B ( 2 , 0 ) � = Φ 2 ,α + . . . . 2 α = A Varying B ( 2 , 0 ) , we get that 3 E 1 2 δ uu δ ww B ( 2 , 0 ) � = Φ 3 ,α + . . . . 3 α = A Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Varying B ( 2 , 0 ) , we get that 1 E 1 8 δ uu δ ww B ( 2 , 0 ) � = Φ 1 ,α + . . . . 1 α = A Varying B ( 2 , 0 ) , we get that 2 E − 1 8 δ uu δ ww B ( 2 , 0 ) � = Φ 2 ,α + . . . . 2 α = A Varying B ( 2 , 0 ) , we get that 3 E 1 2 δ uu δ ww B ( 2 , 0 ) � = Φ 3 ,α + . . . . 3 α = A Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Varying B ( 2 , 0 ) , we get that 1 E 1 8 δ uu δ ww B ( 2 , 0 ) � = Φ 1 ,α + . . . . 1 α = A Varying B ( 2 , 0 ) , we get that 2 E − 1 8 δ uu δ ww B ( 2 , 0 ) � = Φ 2 ,α + . . . . 2 α = A Varying B ( 2 , 0 ) , we get that 3 E 1 2 δ uu δ ww B ( 2 , 0 ) � = Φ 3 ,α + . . . . 3 α = A Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Here Φ i ,α ( α = A , B , C , D , E ) involves the various contributions with four derivatives, and we have ignored other contributions. It is very useful to denote the various contributions by skeleton graphs. We do not include the dressing factors for the sake of brevity. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Here Φ i ,α ( α = A , B , C , D , E ) involves the various contributions with four derivatives, and we have ignored other contributions. It is very useful to denote the various contributions by skeleton graphs. We do not include the dressing factors for the sake of brevity. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The skeleton graphs for ( i )Φ 1 , A , ( ii )Φ 2 , A and ( iii )Φ 3 , A are given by δ δ δ δ δ δ w u w u w u δ δ δ δ δ δ (iii) (ii) (i) Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs For example, � � d 2 z i Q 1 ( z 1 , z 2 ) ∂ w G ( w , z 1 ) ∂ w G ( w , z 2 ) Φ 1 , A = Σ 2 i = 1 , 2 × ∂ u G ( u , z 1 ) ∂ u G ( u , z 2 ) on including the dressing factor. 2 These graphs are of the form ∂ 2 w ∂ u . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs For example, � � d 2 z i Q 1 ( z 1 , z 2 ) ∂ w G ( w , z 1 ) ∂ w G ( w , z 2 ) Φ 1 , A = Σ 2 i = 1 , 2 × ∂ u G ( u , z 1 ) ∂ u G ( u , z 2 ) on including the dressing factor. 2 These graphs are of the form ∂ 2 w ∂ u . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The skeleton graphs for ( i )Φ 1 , B and ( ii )Φ 3 , B are given by δ δ δ δ u w u δ w δ δ δ (ii) (i) along with their hermitian conjugates. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The skeleton graphs for Φ 2 , B are given by δ δ δ u δ δ w δ w δ u δ Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs These graphs are of the form ∂ 2 w ∂ u ∂ z + h . c . . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The skeleton graphs for ( i )Φ 1 , C , ( ii )Φ 1 , D and ( iii )Φ 1 , E are given by δ δ δ δ δ δ δ δ δ δ δ δ u w w u w u (i) (ii) (iii) Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The skeleton graphs for Φ 2 , C are given by δ δ δ δ δ δ δ δ w u w u Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The skeleton graphs for Φ 2 , D are given by δ δ δ δ δ δ δ δ w w u u Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The skeleton graphs for Φ 2 , E are given by δ δ δ δ δ δ δ δ w w u u Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs The skeleton graphs for ( i )Φ 3 , C , ( ii )Φ 3 , D and ( iii )Φ 3 , E are given by δ δ δ δ u w Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs These graphs are of the form ∂ w ∂ u ∂ z i ∂ z j + h . c . . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Thus these terms that result from the mixed variations of B ( 2 , 0 ) do not simplify by themselves. i However it is expected that certain linear combinations of these terms involving different B ( 2 , 0 ) can potentially i simplify, much like the analysis for genus one graphs. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Thus these terms that result from the mixed variations of B ( 2 , 0 ) do not simplify by themselves. i However it is expected that certain linear combinations of these terms involving different B ( 2 , 0 ) can potentially i simplify, much like the analysis for genus one graphs. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Let us first consider the contributions that arise from varying B ( 2 , 0 ) and B ( 2 , 0 ) . 1 2 These are the contributions that involve Φ 1 ,α and Φ 2 ,α . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Let us first consider the contributions that arise from varying B ( 2 , 0 ) and B ( 2 , 0 ) . 1 2 These are the contributions that involve Φ 1 ,α and Φ 2 ,α . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Consider the auxiliary graph given by � � d 2 z i ∂ w G ( w , z 1 ) ∂ w G ( w , z 2 ) ∂ u G ( u , z 1 ) Φ 12 , A = Σ 3 i = 1 , 2 , 3 × ∂ u G ( u , z 3 ) µ ( z 1 )( z 2 , z 3 ) ∂ z 2 ∂ z 3 G ( z 2 , z 3 ) . We denote it by the skeleton graph δ δ u w δ δ δ δ Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Consider the auxiliary graph given by � � d 2 z i ∂ w G ( w , z 1 ) ∂ w G ( w , z 2 ) ∂ u G ( u , z 1 ) Φ 12 , A = Σ 3 i = 1 , 2 , 3 × ∂ u G ( u , z 3 ) µ ( z 1 )( z 2 , z 3 ) ∂ z 2 ∂ z 3 G ( z 2 , z 3 ) . We denote it by the skeleton graph δ δ u w δ δ δ δ Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs We get that Φ 12 , A = π ( 2 Φ 1 , A + Φ 2 , A ) . For the other auxiliary graphs, we simply give the skeleton graphs and ignore the dressing factors for brevity. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs We get that Φ 12 , A = π ( 2 Φ 1 , A + Φ 2 , A ) . For the other auxiliary graphs, we simply give the skeleton graphs and ignore the dressing factors for brevity. Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs From the auxiliary skeleton graph Φ 12 , B given by (along with its hermitian conjugate) δ δ δ δ δ δ u δ w δ w u δ δ δ δ we get that Φ 12 , B = − π ( 2 Φ 1 , B + Φ 2 , B ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs From the auxiliary skeleton graph Φ 12 , C given by δ δ δ δ δ δ δ δ δ δ δ δ w w u u we get that Φ 12 , C = π ( 2 Φ 1 , C + Φ 2 , C ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs From the auxiliary skeleton graph Φ 12 , D given by δ δ δ δ δ δ δ δ δ δ δ δ w w u u we get that Φ 12 , D = π ( 2 Φ 1 , D + Φ 2 , D ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs From the auxiliary skeleton graph Φ 12 , E given by δ δ δ δ u w δ δ we get that Φ 12 , E = − 4 π ( 2 Φ 1 , E + Φ 2 , E ) . Anirban Basu
Brief introduction The genus two four graviton amplitude in type II string theory Modular graph functions for the D 8 R 4 term Varying the Beltrami differentials The eigenvalue equation for some modular graphs Crucially, we always end up with the expression proportional to 2 Φ 1 ,α + Φ 2 ,α . Thus the mixed variation − 1 � B ( 2 , 0 ) 2 B ( 2 , 0 ) � δ uu δ ww 1 2 can be expressed in terms of these auxiliary graphs, as well as other contributions involving two or no derivatives. Anirban Basu
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