Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Modular Forms of type IIB superstring theory, and U (1)-violating amplitudes Congkao Wen Queen Mary University of London To appear with Michael Green String Theory from a Worldsheet Perspective The Galileo Galilei Institute for Theoretical Physics 1 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Introduction We are interested in low-energy effective action of type IIB superstring theory. 0 ( τ ) R 4 + α ′ 5 F (2) L EFT ∼ R + α ′ 3 F (0) 0 ( τ ) d 4 R 4 0 ( τ ) d 6 R 4 + · · · · · · + α ′ 6 F (3) Good understanding on the modular functions of the coefficient of R 4 , d 4 R 4 and d 6 R 4 . see Michael’s talk. We want to extend our understanding for general BPS-terms. 2 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Outline Brief review on the known results. Derive these results from a different point of view: Using constraints from the consistency of superamplitudes. This allows us to extend the results to more general BPS interactions L ( p ) n i ∼ F ( p ) ( i ) P ( w ) w i ( τ ) d 2 p ( { Φ } ) , n where i denotes a possible degeneracy. 3 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Review on non-holomorphic modular forms Non-holomorphic modular forms are functions of τ : w , w ′ ( τ ) → ( c τ + d ) w ( c ¯ τ + d ) w ′ F ( p ) F ( p ) w , w ′ ( τ ) under SL (2 , Z ) transformation τ → a τ + b c τ + d . Covariant derivatives i τ 2 ∂ τ + w � � D w F ( p ) F ( p ) w , w ′ ( τ ) := F ( p ) w , w ′ ( τ ) := w +1 , w ′ − 1 ( τ ) , 2 � τ + w ′ � D w ′ F ( p ) ¯ F ( p ) w , w ′ ( τ ) := F ( p ) w , w ′ ( τ ) := − i τ 2 ∂ ¯ w − 1 , w ′ +1 ( τ ) . 2 We will only consider the cases with w ′ = − w 4 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Examples: Eisenstein series Well-known examples: non-holomorphic Eisenstein series � w τ s � m + n ¯ τ � 2 E w ( s , τ ) = | m + n τ | 2 s m + n τ ( m , n ) � =(0 , 0) It has weight ( w , − w ). Satisfies Laplace equation, ∆ ( w ) − E w ( s , τ ) := 4 D w − 1 ¯ D − w E w ( s , τ ) = ( s − w )( s + w − 1) E w ( s , τ ) or ∆ ( w ) + E w ( s , τ ) := 4 ¯ D − w − 1 D w E w ( s , τ ) = ( s + w )( s − w − 1) E w ( s , τ ) 5 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks d 2 p R 4 terms R 4 and d 4 R 4 [Green, Gutperle + Vanhove][Green, Sethi][Basu][Pioline][Berkovits, Vafa]...... 2 , τ ) R 4 , 2 , τ ) d 4 R 4 E 0 ( 3 E 0 ( 5 Perturbative expansions in large τ 2 agree with explicit computations 3 − 1 E 0 ( 3 2 , τ ) = 2 ζ (3) τ 2 + 4 ζ (2) τ 2 2 + instantons 2 The coefficient of d 6 R 4 satisfies an inhomogeneous Laplace equation [Green, Vanhove][Yin, Wang] � � ∆ (0) F (3) 2 , τ ) 2 − − 12 0 ( τ ) = − E 0 ( 3 as a consequence of first-order differential equations we will derive. 6 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Fluctuations On-shell amplitudes and fluctuations To compute scattering amplitudes, we expand the effective action around a fixed background τ 0 . F ( τ 0 + δτ ) = F ( τ 0 ) − 2 i τ 2 ∂ τ F ( τ 0 )ˆ τ F ( τ 0 )¯ τ + 2 i τ 2 ∂ ¯ τ ˆ τ 2 + · · · τ 2 − 2¯ τ F ( τ 0 )¯ − 2 τ 2 2 ∂ 2 τ F ( τ 0 )ˆ τ 2 2 ∂ 2 ˆ ¯ τ := i δτ/ (2 τ 0 here ˆ 2 ). Example: if F (0) 0 ( τ 0 + δτ ) is the coefficient of R 4 , the expansion generates five and higher-point interactions: τ ˆ ˆ τ 7 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Fluctuations SL (2 , Z ) covariant derivatives Each vertex from expansion is in the form of simple derivatives instead of SL (2 , Z ) covariant derivatives: The fluctuation ˆ τ does not transform properly under U (1). Two-derivative classical action, when expanded around τ 0 contains infinity set of U (1)-violating vertices. e.g. ∂ µ τ∂ µ ¯ τ τ 2 + 2ˆ τ∂ µ ¯ τ + ¯ τ ¯ τ + ¯ � τ 2 ) + · · · � = ∂ µ ˆ ˆ τ 1 + 2(ˆ τ ) + 3(ˆ ˆ ˆ ˆ 4 τ 2 they are all vanishing on-shell, no U (1)-violating amplitudes in type IIB supergravity. 8 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Fluctuations SL (2 , Z ) covariant derivatives The 2-derivative U (1)-violating vertices do contribute to higher-derivative amplitudes, by attaching them to higher-derivative vertices: R 4 ˆ τ etc. τ ˆ ˆ τ These additional contributions precisely make the simple derivatives to be covariant derivatives. 9 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Field redefinition Field redefinitions A systematics: A field redefinition that removes all these on-shell vanishing vertices: τ = τ − τ 0 τ ˆ B = − 1 − ˆ τ − ¯ τ 0 the normal coordinate of the sigma model G / H . Used in [Schwarz, 83’] for SU (1 , 1) formulation of the classical theory. The field B kills two birds with one stone: The B field transforms linearly τ 0 + d � c ¯ � B → B . c τ 0 + d Removes all U (1)-violating (on-shell vanishing) vertices in the classical action. 10 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Field redefinition Field redefinition and covariant derivatives Expanded in terms of B fields: F ( τ 0 + δτ ) = F ( τ 0 ) − 2 i τ 2 ∂ τ F ( τ 0 )ˆ τ 2 + O (ˆ τ − 2 τ 2 2 ∂ 2 τ F ( τ 0 )ˆ τ 3 ) = F ( τ 0 ) − 2 i τ 2 ∂ τ F ( τ 0 ) B + B 2 + O ( B 3 ) − τ 2 2 ∂ 2 τ F ( τ 0 ) + i τ 2 ∂ τ F ( τ 0 ) � � + 2 Now each term is covariant derivatives i τ 2 ∂ τ F ( τ 0 ) = D 0 F ( τ 0 ) − τ 2 2 ∂ 2 τ F ( τ 0 ) + i τ 2 ∂ τ F ( τ 0 ) = D 1 D 0 F ( τ 0 ) 11 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Field redefinition Field redefinition: other fields Similar field redefinition for other fields in theory. Example: a fermionic term Q µ = ∂ µ τ 1 Λ a γ µ ( ∂ µ + iq Λ Q µ )¯ Λ a , . with 2 τ 2 The redefined field � q Λ / 2 � 1 − B Λ ′ a = Λ a . 1 − ¯ B 12 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Superamplitudes Summary: All interactions are manifestly SL (2 , Z ) invariant, with appropriate choices of the fluctuation fields. Ready to study scattering amplitudes: 10D spinor helicity and type IIB SUSY: [Boels and O’Connell] massless momentum p BA := ( γ µ ) BA p µ = λ Ba λ A a . A = 1 , . . . , 16 is the spinor of SO (9 , 1) and a = 1 , . . . , 8 the SO (8) little group index. Supercharges n n ∂ � ¯ � λ A , a Q A λ A i , a η a Q A n = i , n = . i ∂η a i i =1 i =1 13 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Superamplitudes The on-shell massless states: a + 1 2! η a η b φ ab + · · · + 1 8!( η ) 8 ¯ Φ( η ) = B + η a Λ ′ B . a = − 3 B = 2, and q η = − 1 q B = − 2 , q Λ ′ 2 , · · · , q h = 0 , · · · , q ¯ 2 . The super amplitudes � � n A n = δ 10 � δ 16 ( Q n ) ˆ Q A ¯ n ˆ A n ( η, λ ) , A n ( η, λ ) = 0 , p r with r =1 U (1)-conserved amplitudes ˆ A n ∼ η 4( n − 4) . 14 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Superamplitudes: maximal U (1)-violating Maximal U (1)-violating amplitudes [Boels] A ( p ) n , i = F ( p ) A ( p ) n − 4 , i ( τ ) δ 16 ( Q n ) ˆ n , i ( s ij ) , where i denotes a possible degeneracy. The maximal U (1)-violating amplitudes have no poles. A ( p ) Therefore ˆ n , i ( s ij ) is a degree- p symmetric polynomial of s ij . They are super vertices. In 4D, they are KLT of MHV ⊗ MHV . 15 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Superamplitudes: maximal U (1)-violating Higher-point amplitudes are related to the lower-point ones by soft limits. The coefficients are related by covariant derivatives, F ( p ) n − 4 , i ( τ ) ∼ D n − 5 F ( p ) n − 5 , i ( τ ) The kinematics are related by soft limits (soft B field) A ( p ) A ( p ) ˆ p n → 0 → ˆ � n , i ( s ij ) n − 1 , i ( s ij ) � Covariant derivative is a result of combination of soft dilaton ( τ 2 ∂ τ 2 A n ) [Di Vecchia][Di Vecchia, Marotta, Mojaza, Nohle] and soft axion limit ( w � i R i A n ). 16 / 28
Introduction Review known results Expansions of fluctuations Superamplitudes Conclusion and remarks Superamplitudes: maximal U (1)-violating A (0) ˆ n , i ( s ij ) = 1 is for dimension-8 interactions, related to R 4 , (no degeneracy) F (0) n − 4 ( τ ) ∼ D n − 5 · · · D 0 E ( 3 2 , τ ) ∼ E n − 4 ( 3 2 , τ ) A (2) ˆ i < j s 2 n , i ( s ij ) = � ij is for dimension-12 interactions, related to d 4 R 4 , (no degeneracy) F (2) n − 4 ( τ ) ∼ D n − 5 · · · D 0 E ( 5 2 , τ ) ∼ E n − 4 ( 5 2 , τ ) 17 / 28
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