Michael Kiermaier

E7(7) constraints on counterterms in N=8 supergravity

Abstract We prove by explicit computation that 6-point matrix elements of D 4 R 4 and D 6 R 4 in N = 8 supergravity have non-vanishing single-soft scalar limits, and therefore these operators violate the continuous E 7 ( 7 ) symmetry. The soft limits precisely match automorphism constraints. Together with previous results for R 4 , this provides a direct proof that no E 7 ( 7 ) -invariant candidate counterterm exists below 7-loop order. At 7-loops, we characterize the infinite tower of independent supersymmetric operators D 4 R 6 , R 8 , φ 2 R 8 , … with n > 4 fields and prove that they all violate E 7 ( 7 ) symmetry. This means that the 4-graviton amplitude determines whether or not the theory is finite at 7-loop order. We show that the corresponding candidate counterterm D 8 R 4 has a non-linear supersymmetrization such that its single- and double-soft scalar limits are compatible with E 7 ( 7 ) up to and including 6-points. At loop orders 7 , 8 , 9 we provide an exhaustive account of all independent candidate counterterms with up to 16 , 14 , 12 fields, respectively, together with their potential single-soft scalar limits.

On renormalization group flows and the a-theorem in 6d

A bstractWe study the extension of the approach to the a-theorem of Komargodski and Schwimmer to quantum field theories in d = 6 spacetime dimensions. The dilaton effective action is obtained up to 6th order in derivatives. The anomaly flow aUV − aIR is the coefficient of the 6-derivative Euler anomaly term in this action. It then appears at order p6 in the low energy limit of n-point scattering amplitudes of the dilaton for n ≥ 4. The detailed structure with the correct anomaly coefficient is confirmed by direct calculation in two examples: (i) the case of explicitly broken conformal symmetry is illustrated by the free massive scalar field, and (ii) the case of spontaneously broken conformal symmetry is demonstrated by the (2,0) theory on the Coulomb branch. In the latter example, the dilaton is a dynamical field so 4-derivative terms in the action also affect n-point amplitudes at order p6. The calculation in the (2,0) theory is done by analyzing an M5-brane probe in AdS7 × S4.

Recursion relations, generating functions, and unitarity sums in N = 4 SYM theory

We prove that the MHV vertex expansion is valid for any NMHV tree amplitude of = 4 SYM. The proof uses induction to show that there always exists a complex deformation of three external momenta such that the amplitude falls off at least as fast as 1/z for large z. This validates the generating function for n-point NMHV tree amplitudes. We also develop generating functions for anti-MHV and anti-NMHV amplitudes. As an application, we use these generating functions to evaluate several examples of intermediate state sums on unitarity cuts of 1-, 2-, 3- and 4-loop amplitudes. In a separate analysis, we extend the recent results of arXiv:0808.0504 to prove that there exists a valid 2-line shift for any n-point tree amplitude of = 4 SYM. This implies that there is a BCFW recursion relation for any tree amplitude of the theory.

Exact marginality in open string field theory: a general framework

We construct analytic solutions of open bosonic string field theory for any exactly marginal deformation in any boundary conformal field theory when properly renormalized operator products of the marginal operator are given. We explicitly provide such renormalized operator products for a class of marginal deformations which include the deformations of flat D-branes in flat backgrounds by constant massless modes of the gauge field and of the scalar fields on the D-branes, the cosine potential for a space-like coordinate, and the hyperbolic cosine potential for the time-like coordinate. In our construction we use integrated vertex operators, which are closely related to finite deformations in boundary conformal field theory, while previous analytic solutions were based on unintegrated vertex operators. We also introduce a modified star product to formulate string field theory around the deformed background.

Stringy KLT relations, global symmetries, and E 7(7)-violation

We study consequences of the Kawai-Lewellen-Tye (KLT) relations applied to tree amplitudes in toroidal compactifications of string theory to four dimensions. The closed string tree amplitudes with massless external states respect a global SU(4) × SU(4) symmetry, which is enhanced to the SU(8) R -symmetry of

A simple approach to counterterms in N=8 supergravity

\mathcal{N} = 8

General marginal deformations in open superstring field theory

supergravity in the field theory limit. Our analysis focuses on two aspects: (i) We provide a detailed account of the simplest SU(8)-violating amplitudes. We classify these processes and derive explicit superamplitudes for all local 5-and 6-point operators with SU(4) × SU(4) symmetry at order α′3 . Their origin is the dilatonic operator e−6φR4 in the closed-string effective action. (ii) We expand the 6-point closed string tree amplitudes to order α3 and use two different methods to isolate the SU(8)-singlet contribution from e−6φR4. This allows us to extract the matrix elements of the unique SU(8)-invariant supersymmetrization of R4. Their single-soft scalar limits are non-vanishing. This demonstrates that the

Proof of the MHV vertex expansion for all tree amplitudes in N = 4 SYM theory

\mathcal{N} = 8

Solutions from boundary condition changing operators in open string field theory

supergravity candidate counterterm R4 is incompatible with continuous E7(7) symmetry. From the soft scalar limits, we reconstruct to quadratic order the SU(8)-invariant function of scalars that multiplies R4, and show that it satisfies the Laplace eigenvalue equation derived recently from supersymmetry and duality constraints.

Solution to the Ward identities for superamplitudes

We present a simple systematic method to study candidate counterterms in