An Integrability Primer for the Gauge-Gravity Correspondence: an Introduction

We introduce a series of articles reviewing various aspects of integrable models relevant to the AdS/CFT correspondence. Topics covered in these reviews are: classical integrability, Yangian symmetry, factorized scattering, the Bethe ansatz, the thermodynamic Bethe ansatz, and integrable structures in (conformal) quantum field theory. In the present article we highlight how these concepts have found application in AdS/CFT, and provide a brief overview of the material contained in this series.Comment: v2, published versio

Reflecting magnons from D7 and D5 branes

We obtain the reflection matrices for the scattering of elementary magnons from certain open boundaries, corresponding to open strings ending on D7 and D5 branes in $AdS_5\times S^5$. In each case we consider two possible orientations for the vacuum state. We show that symmetry arguments are sufficient to determine the reflection matrices up to at most two unknown functions. The D7 reflection matrices obey the boundary Yang Baxter-Equation. This is automatic for one vacuum orientation, and requires a natural choice of ratio between two unknowns for the other. In contrast, the D5 reflection matrices do not obey the boundary Yang Baxter-Equation. In both cases we show consistency with the existent weak and strong coupling results.Comment: 32 pages, 1 figure; v2: added references and minor changes; v3: error in boundary Yang-Baxter equation for D5 reflection matrix note

Hidden Simplicity of Gauge Theory Amplitudes

These notes were given as lectures at the CERN Winter School on Supergravity, Strings and Gauge Theory 2010. We describe the structure of scattering amplitudes in gauge theories, focussing on the maximally supersymmetric theory to highlight the hidden symmetries which appear. Using the BCFW recursion relations we solve for the tree-level S-matrix in N=4 super Yang-Mills theory, and describe how it produces a sum of invariants of a large symmetry algebra. We review amplitudes in the planar theory beyond tree-level, describing the connection between amplitudes and Wilson loops, and discuss the implications of the hidden symmetries.Comment: 46 pages, 15 figures. v2 ref added, typos fixe

The SU(2 vertical bar 3) dynamic two-loop form factors

SCOAP

Long-Range Deformations for Integrable Spin Chains

We present a recursion relation for the explicit construction of integrable spin chain Hamiltonians with long-range interactions. Based on arbitrary short-range (e.g. nearest-neighbor) integrable spin chains, it allows to construct an infinite set of conserved long-range charges. We explain the moduli space of deformation parameters by different classes of generating operators. The rapidity map and dressing phase in the long-range Bethe equations are a result of these deformations. The closed chain asymptotic Bethe equations for long-range spin chains transforming under a generic symmetry algebra are derived. Notably, our construction applies to generalizations of standard nearest-neighbor chains such as alternating spin chains. We also discuss relevant properties for its application to planar D=4, N=4 and D=3, N=6 supersymmetric gauge theories. Finally, we present a map between long-range and inhomogeneous spin chains delivering more insight into the structures of these models as well as their limitations at wrapping order.Comment: 63 pages, v2: references added, v3: typos corrected in eqs (8.20) and (8.24

USP10 drives cancer stemness and enables super-competitor signalling in colorectal cancer

The contribution of deubiquitylating enzymes (DUBs) to β-Catenin stabilization in intestinal stem cells and colorectal cancer (CRC) is poorly understood. Here, and by using an unbiassed screen, we discovered that the DUB USP10 stabilizes β-Catenin specifically in APC-truncated CRC in vitro and in vivo. Mechanistic studies, including in vitro binding together with computational modelling, revealed that USP10 binding to β-Catenin is mediated via the unstructured N-terminus of USP10 and is outcompeted by intact APC, favouring β-catenin degradation. However, in APC-truncated cancer cells USP10 binds to β-catenin, increasing its stability which is critical for maintaining an undifferentiated tumour identity. Elimination of USP10 reduces the expression of WNT and stem cell signatures and induces the expression of differentiation genes. Remarkably, silencing of USP10 in murine and patient-derived CRC organoids established that it is essential for NOTUM signalling and the APC super competitor-phenotype, reducing tumorigenic properties of APC-truncated CRC. These findings are clinically relevant as patient-derived organoids are highly dependent on USP10, and abundance of USP10 correlates with poorer prognosis of CRC patients. Our findings reveal, therefore, a role for USP10 in CRC cell identity, stemness, and tumorigenic growth by stabilising β-Catenin, leading to aberrant WNT signalling and degradation resistant tumours. Thus, USP10 emerges as a unique therapeutic target in APC truncated CRC

All-mass n-gon integrals in n dimensions

We explore the correspondence between one-loop Feynman integrals and (hyperbolic) simplicial geometry to describe the "all-mass" case: integrals with generic external and internal masses. Specifically, we focus on $n$-particle integrals in exactly $n$ space-time dimensions, as these integrals have particularly nice geometric properties and respect a dual conformal symmetry. In four dimensions, we leverage this geometric connection to give a concise dilogarithmic expression for the all-mass box in terms of the Murakami-Yano formula. In five dimensions, we use a generalized Gauss-Bonnet theorem to derive a similar dilogarithmic expression for the all-mass pentagon. We also use the Schl\"afli formula to write down the symbol of these integrals for all $n$. Finally, we discuss how the geometry behind these formulas depends on space-time signature, and we gather together many results related to these integrals from the mathematics and physics literature.Comment: 49 pages, 8 figure

USP10 drives cancer stemness and enables super-competitor signalling in colorectal cancer

The contribution of deubiquitylating enzymes (DUBs) to β-Catenin stabilization in intestinal stem cells and colorectal cancer (CRC) is poorly understood. Here, and by using an unbiassed screen, we discovered that the DUB USP10 stabilizes β-Catenin specifically in APC-truncated CRC in vitro and in vivo. Mechanistic studies, including in vitro binding together with computational modelling, revealed that USP10 binding to β-Catenin is mediated via the unstructured N-terminus of USP10 and is outcompeted by intact APC, favouring β-catenin degradation. However, in APC-truncated cancer cells USP10 binds to β-catenin, increasing its stability which is critical for maintaining an undifferentiated tumour identity. Elimination of USP10 reduces the expression of WNT and stem cell signatures and induces the expression of differentiation genes. Remarkably, silencing of USP10 in murine and patient-derived CRC organoids established that it is essential for NOTUM signalling and the APC super competitor-phenotype, reducing tumorigenic properties of APC-truncated CRC. These findings are clinically relevant as patient-derived organoids are highly dependent on USP10, and abundance of USP10 correlates with poorer prognosis of CRC patients. Our findings reveal, therefore, a role for USP10 in CRC cell identity, stemness, and tumorigenic growth by stabilising β-Catenin, leading to aberrant WNT signalling and degradation resistant tumours. Thus, USP10 emerges as a unique therapeutic target in APC truncated CRC