## Final Report Summary - INTGAUGESTRING (Integrable Structures in Supersymmetric Gauge Theory and its String Dual)

Objective

~~~~~~~~~

Planar N=4 supersymmetric Yang-Mills theory (SYM) is a prototypical gauge theory: While it is not known to model real-world physics, it shares many salient features with more realistic theories. At the same time, it exhibits integrability, which admits unprecedented insight into the inner structure and dynamics of gauge theory. N=4 SYM thus provides a precision laboratory for studying general aspects of quantum field theory. Moreover, it is dual to string theory on anti-de-Sitter space, and thus probes a quantum theory of gravity. Over a decade of intensive study of the integrable structures has led to extraordinary success in our understanding of the energy spectrum of planar N=4 SYM and its dual string theory. The project IntGaugeString targets the dynamical content of the theory, which remains much less understood. The scientific objective has been to study the dynamical observables in planar gauge theory and its dual string theory, in particular the theory's correlation functions and scattering amplitudes.

Research Results

~~~~~~~~~~~~~~~~

During the course of the project, the following results have been obtained:

1) A new method for computing three-point correlation functions of "semi-heavy" local operators in N=4 super Yang-Mills theory in the regime of strong coupling has been developed. It employs a combination of semiclassical and flat space limits of the dual type IIB string theory. Using the method, a number of previously unknown structure constants has been computed.

2) An integrable deformation of the Graßmannian integral for scattering amplitudes in both N=4 super Yang-Mills and ABJM theory has been formulated and investigated. Furthermore, an integrable deformation of on-shell diagrams in ABJM theory has been constructed, including a formulation of deformed diagrams in terms of an R-matrix formalism.

3) The properties of the symbol for MHV amplitudes in N=4 super Yang-Mills theory in the multi-Regge limit have been investigated, in particular its behavior in the various different kinematic regions. The two-loop symbol in all kinematic regions and to all multiplicities could be reduced to only two basic building blocks. One of those was not known before, and its functional form could be significantly constrained. These findings have been understood from the complementary perspective of Regge pole and Regge cut contributions, which facilitated an extension of the decomposition into building blocks to higher loop orders. At three and four loops, the minimal set of building blocks was identified to all subleading orders. The leading-order building blocks could be extracted from existing data. The analysis also clarified how the next-to-leading order BFKL central emission block could be extracted from one of the two-loop building blocks.

The following results have been obtained during the course of the project, but their investigation remains ongoing:

4) The researcher started to study multi-Regge limit amplitudes using the integrability-based operator product expansion for Wilson loops (WLOPE) of Basso et al. Analytic continuation paths into all different kinematic regions have been identified, and the associated discontinuities are now being computed. Comparison with available BFKL data will then yield further constraints for the building block functions. These will in turn feed into an extension to higher loop orders, and perhaps even to finite coupling.

5) The researcher has begun a deeper study of the known two-loop symbol, with the aim of extracting information on terms of subleading transcendental order in the multi-Regge limit remainder function. This will probe Regge cut terms that have long been predicted but have never been tested or computed.

6) With an integrability-based description in mind, a study of higher-point correlation functions of local operators has been performed. It is based on an excitation picture around certain near-BPS operators in N=4 super Yang-Mills theory, which decompose into contributions of various multi-magnon exchange states, and which show much simplified systematics.

Impact and Potential Use

~~~~~~~~~~~~~~~~~~~~~~~~

Understanding the dynamics of interacting gauge theories at a fundamental level remains an important open problem in theoretical physics. The integrability of planar N=4 super Yang-Mills theory might take us closer to solving this problem by admitting a complete solution of the theory's dynamical observables. The results obtained in the project IntGaugeString contribute to this endeavor from several complementary directions.

Result (1) provides a new method for computing correlation functions at strong coupling that include stringy modes and thus go beyond the supergravity approximation. Such correlators are notoriously difficult to attain in curved spacetime, and the results of (1) yield a new handle on their computation. The resulting structure constants provide essential data points for an integrability-based description. Result (6) on the other hand attacks the same correlation functions at weak coupling, and promises to be a good starting point for an efficient treatment based on spin-chain excitations and their S-matrix.

The reformulation of scattering amplitudes in terms of Graßmannian spaces and on-shell diagrams promises to provide an entirely new perspective on quantum field theory. Result (2) provides a link to integrability by introducing a spectral parameter into the Graßmannian integral, and by recasting the on-shell building blocks for ABJM theory in an R-matrix formalism. The deformations may entail a consistent integration procedure for the on-shell integrand, which would solve one of the most important open problems of the on-shell formulation.

Results (3)-(5) regard scattering amplitudes in the multi-Regge limit. These are both structurally and phenomenologically interesting: They are relevant to actual collider experiments, but also admit a description in terms of factorized Regge pole and cut contributions. The results led to an improved understanding of Regge-limit amplitudes in planar N=4 SYM theory, both qualitatively and quantitatively. The decomposition into building blocks very non-trivially agrees with Regge physics predictions, and significantly constrains the functional form of the BFKL central emission block, which is an important element in all higher-order contributions. The current investigation of the WLOPE should shed further light on the analytic properties of multi-Regge limit amplitudes, and may eventually lead to a complete and exact solution of Regge-limit amplitudes.

Transfer of knowledge and international relations: During the project, collaborations with the following researchers and universities have been established or deepened: Joseph Minahan (Uppsala University), Raul Pereira (Uppsala University), Yu-tin Huang (now Taiwan National University), Masahito Yamazaki (now IPMU, Tokyo), Florian Loebbert (now Humboldt University Berlin), Georgios Papathanasiou (SLAC, Stanford), Volker Schomerus (DESY Hamburg).

~~~~~~~~~

Planar N=4 supersymmetric Yang-Mills theory (SYM) is a prototypical gauge theory: While it is not known to model real-world physics, it shares many salient features with more realistic theories. At the same time, it exhibits integrability, which admits unprecedented insight into the inner structure and dynamics of gauge theory. N=4 SYM thus provides a precision laboratory for studying general aspects of quantum field theory. Moreover, it is dual to string theory on anti-de-Sitter space, and thus probes a quantum theory of gravity. Over a decade of intensive study of the integrable structures has led to extraordinary success in our understanding of the energy spectrum of planar N=4 SYM and its dual string theory. The project IntGaugeString targets the dynamical content of the theory, which remains much less understood. The scientific objective has been to study the dynamical observables in planar gauge theory and its dual string theory, in particular the theory's correlation functions and scattering amplitudes.

Research Results

~~~~~~~~~~~~~~~~

During the course of the project, the following results have been obtained:

1) A new method for computing three-point correlation functions of "semi-heavy" local operators in N=4 super Yang-Mills theory in the regime of strong coupling has been developed. It employs a combination of semiclassical and flat space limits of the dual type IIB string theory. Using the method, a number of previously unknown structure constants has been computed.

2) An integrable deformation of the Graßmannian integral for scattering amplitudes in both N=4 super Yang-Mills and ABJM theory has been formulated and investigated. Furthermore, an integrable deformation of on-shell diagrams in ABJM theory has been constructed, including a formulation of deformed diagrams in terms of an R-matrix formalism.

3) The properties of the symbol for MHV amplitudes in N=4 super Yang-Mills theory in the multi-Regge limit have been investigated, in particular its behavior in the various different kinematic regions. The two-loop symbol in all kinematic regions and to all multiplicities could be reduced to only two basic building blocks. One of those was not known before, and its functional form could be significantly constrained. These findings have been understood from the complementary perspective of Regge pole and Regge cut contributions, which facilitated an extension of the decomposition into building blocks to higher loop orders. At three and four loops, the minimal set of building blocks was identified to all subleading orders. The leading-order building blocks could be extracted from existing data. The analysis also clarified how the next-to-leading order BFKL central emission block could be extracted from one of the two-loop building blocks.

The following results have been obtained during the course of the project, but their investigation remains ongoing:

4) The researcher started to study multi-Regge limit amplitudes using the integrability-based operator product expansion for Wilson loops (WLOPE) of Basso et al. Analytic continuation paths into all different kinematic regions have been identified, and the associated discontinuities are now being computed. Comparison with available BFKL data will then yield further constraints for the building block functions. These will in turn feed into an extension to higher loop orders, and perhaps even to finite coupling.

5) The researcher has begun a deeper study of the known two-loop symbol, with the aim of extracting information on terms of subleading transcendental order in the multi-Regge limit remainder function. This will probe Regge cut terms that have long been predicted but have never been tested or computed.

6) With an integrability-based description in mind, a study of higher-point correlation functions of local operators has been performed. It is based on an excitation picture around certain near-BPS operators in N=4 super Yang-Mills theory, which decompose into contributions of various multi-magnon exchange states, and which show much simplified systematics.

Impact and Potential Use

~~~~~~~~~~~~~~~~~~~~~~~~

Understanding the dynamics of interacting gauge theories at a fundamental level remains an important open problem in theoretical physics. The integrability of planar N=4 super Yang-Mills theory might take us closer to solving this problem by admitting a complete solution of the theory's dynamical observables. The results obtained in the project IntGaugeString contribute to this endeavor from several complementary directions.

Result (1) provides a new method for computing correlation functions at strong coupling that include stringy modes and thus go beyond the supergravity approximation. Such correlators are notoriously difficult to attain in curved spacetime, and the results of (1) yield a new handle on their computation. The resulting structure constants provide essential data points for an integrability-based description. Result (6) on the other hand attacks the same correlation functions at weak coupling, and promises to be a good starting point for an efficient treatment based on spin-chain excitations and their S-matrix.

The reformulation of scattering amplitudes in terms of Graßmannian spaces and on-shell diagrams promises to provide an entirely new perspective on quantum field theory. Result (2) provides a link to integrability by introducing a spectral parameter into the Graßmannian integral, and by recasting the on-shell building blocks for ABJM theory in an R-matrix formalism. The deformations may entail a consistent integration procedure for the on-shell integrand, which would solve one of the most important open problems of the on-shell formulation.

Results (3)-(5) regard scattering amplitudes in the multi-Regge limit. These are both structurally and phenomenologically interesting: They are relevant to actual collider experiments, but also admit a description in terms of factorized Regge pole and cut contributions. The results led to an improved understanding of Regge-limit amplitudes in planar N=4 SYM theory, both qualitatively and quantitatively. The decomposition into building blocks very non-trivially agrees with Regge physics predictions, and significantly constrains the functional form of the BFKL central emission block, which is an important element in all higher-order contributions. The current investigation of the WLOPE should shed further light on the analytic properties of multi-Regge limit amplitudes, and may eventually lead to a complete and exact solution of Regge-limit amplitudes.

Transfer of knowledge and international relations: During the project, collaborations with the following researchers and universities have been established or deepened: Joseph Minahan (Uppsala University), Raul Pereira (Uppsala University), Yu-tin Huang (now Taiwan National University), Masahito Yamazaki (now IPMU, Tokyo), Florian Loebbert (now Humboldt University Berlin), Georgios Papathanasiou (SLAC, Stanford), Volker Schomerus (DESY Hamburg).