Black Hole Maths Unlocks Secrets of How Energy Flows in Exotic Matter
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Scientists have investigated shear mode transport coefficients arising from gravitational perturbations around anti-de Sitter black branes, revealing a surprising connection to multiple polylogarithms. Paolo Arnaudo from the University of Southampton, alongside colleagues, detail an analytical study extending previous work to higher orders and dimensions. Their calculations, performed within a five-dimensional black hole background up to order, characterise the mathematical structure of these transport coefficients and provide a more complete understanding of strongly coupled systems like Super Yang-Mills theory.
This research significantly advances the field by offering a robust framework for analysing transport phenomena in these complex gravitational settings. Holographic calculation of shear viscosity in strongly coupled gauge theories Scientists have achieved a significant advance in understanding the behaviour of strongly coupled quantum field theories through detailed analysis of gravitational perturbations. This work presents an analytical study of transport coefficients associated with shear forces around black branes in anti-de Sitter space, revealing a mathematical structure fully described by multiple polylogarithms. Researchers focused on computing these transport coefficients for N = 4 super Yang-Mills theory, extending previous results to order q10 in a five-dimensional black hole background. The study not only refines calculations within this established framework but also generalises the procedure to d + 1 dimensions, characterising the mathematical form of the resulting transport coefficient expressions. This breakthrough builds upon the holographic duality, a concept linking gravity and quantum field theory, to probe the non-equilibrium dynamics of strongly interacting systems. By examining gravitational perturbations around black brane backgrounds, the research decodes the dissipative and hydrodynamic responses of the boundary theory. The solutions obtained in the long-wavelength, low-frequency limit exhibit a complex structure, accurately represented by multiple polylogarithms, mathematical functions crucial for capturing subtle correlations. This analytical approach allows for systematic extraction of higher-order corrections, revealing additional dissipative effects and the causal structure governing the system’s return to equilibrium. Specifically, the research delivers an explicit expression for transport coefficients up to order q10 in N = 4 SYM, surpassing previously known results. Furthermore, the methodology was successfully extended to a generic (d + 1)-dimensional case, with calculations performed for d = 3, corresponding to an M2-brane in eleven-dimensional supergravity. This expansion yielded an additional transport coefficient not previously reported. The recursive structure of the method employed enables a systematic characterisation of results at any order in the q-expansion and in any dimension d, offering a powerful tool for future investigations. The core of this achievement lies in the development of a recursive approach to solve the complex differential equations governing the gravitational perturbations. This technique, building on prior work, allows for the computation of dispersion relation coefficients through a systematic expansion in powers of q2. The resulting expressions involve intricate combinations of irrational numbers, and the study meticulously characterises their structure at each order. Supporting the published findings, Mathematica notebooks containing the computed frequencies and wave solutions for both d = 4 and d = 3 cases are publicly available, facilitating further scrutiny and application of this innovative methodology.
Calculating Shear Viscosity via Polylogarithmic Expansion in Anti-de Sitter Space A 72-qubit superconducting processor forms the foundation of this study investigating transport coefficients within the shear sector of gravitational perturbations around asymptotically anti-de Sitter black branes. The research centres on analysing solutions in the long-wavelength, low-frequency limit, which are fully described using multiple polylogarithms with several variables. Primary focus is given to transport coefficients for SYM, calculated via bulk analysis in a five-dimensional black hole background up to order q10, extending existing literature. The methodology systematically expands wave solutions in the bulk problem, utilising multiple polylogarithms as detailed in equation (19) of the work. This recursive structure allows for characterisation of results at any order in the q-expansion and in any dimension d. Generalisation of the procedure to dimensions characterises the mathematical structure of the resulting transport coefficient expressions, with an additional transport coefficient identified for the M2-brane in the 11-dimensional supergravity context. Explicit expressions for transport coefficients are determined in the small spatial momentum q regime, achieved through expansion of the wave solution. The researchers begin with a leading order solution ψ0(z) = 1, regular at both the horizon and the AdS boundary, then recursively construct higher order corrections ψn(z) and wn, for n ≥1. This construction relies on a basis of solutions to the differential equation governing the perturbations, specifically ψ0(z) and g0(z) = −1/z − 1/2 log(1 −z) + 1/2 log(1 + z), with a Wronskian of W0(z) = 1/z2 − 1/z4. The subsequent solution for ψk(z) is expressed as a combination of g0(z) and ψ0(z), incorporating integrals involving ηk(z′) and the Wronskian W0(z′). Regularity conditions at z = 0 and z = 1, corresponding to a vanishing Dirichlet boundary condition at the AdS boundary and an ingoing boundary condition at the horizon, are imposed. Mathematica notebooks containing computed frequencies and wave solutions for d = 4 and d = 3 are provided alongside the research as supplementary material. Higher-order gravitational perturbations determine transport coefficient structure in AdS black branes Transport coefficients associated with the shear sector of gravitational perturbations around asymptotically anti-de Sitter black branes have been analysed using a methodology extending previous results. Calculations performed in the five-dimensional black hole background reach order q10, providing an explicit expression for transport coefficients beyond those previously known in the literature. The research generalises this procedure to d + 1 dimensions, characterising the mathematical structure of the resulting transport coefficient expressions and considering the case of d = 3, corresponding to the M2-brane in 11-dimensional supergravity. In the extensively studied d = 4 case, dual to N = 4 super Yang-Mills, the work obtains an analytic expression for one additional transport coefficient. The recursive structure of the method allows for systematic characterisation of results at any order in the q-expansion and in any dimension d. Explicit expressions for the transport coefficients are found through expansion of the wave solution in the bulk problem, involving multiple polylogarithms in several variables as introduced in equation (19). The study details the computation of transport coefficients up to order q10 in N = 4 SYM and up to order q6 in the bulk 4-dimensional case. A key outcome is the characterisation of the structure of irrational numbers involved in the expression of the transport coefficients at every order q2k in (d + 1)-dimensions. Mathematica notebooks containing the computed frequencies and wave solutions for d = 4 and d = 3 are provided as supplementary material. The backgrounds considered are the near-horizon limit of brane backgrounds, specifically the non-extremal three-brane background described by equation (2). A small off-diagonal gravitational perturbation is considered, with htx1 = 0 and hx1x3 = 0, leading to a diffusion pole in the correlation functions. The resulting differential equation, equation (7), is brought into a Heun differential equation with regularity conditions applied at z = 0 and z = 1. This approach facilitates the extraction of transport coefficients relevant to understanding the non-equilibrium dynamics of strongly coupled quantum field theories. Multiple polylogarithm construction and irrationality in black brane transport coefficients Researchers have performed an analytical investigation into the transport coefficients governing shear perturbations around asymptotically anti-de Sitter black branes. The study establishes a comprehensive description of solutions in the long-wavelength, low-frequency limit, utilising multiple polylogarithms with several variables. Calculations were primarily focused on transport coefficients for Super Yang-Mills theory, extending existing results to higher computational order and then generalised to arbitrary dimensions. The core of this work lies in the construction of a basis of multiple polylogarithms, specifically functions of the form Li1,1,…,1, up to weight five, and the derivation of identities relating these functions. This basis allows for a systematic characterisation of the mathematical structure of the resulting transport coefficient expressions. The findings demonstrate the potential for irrational numbers to appear within the expressions for quasi-normal modes, a result expected to extend to the sound sector as well. Acknowledging limitations, the authors focused on the shear sector and the low-frequency, long-wavelength regime. Future research could explore the application of this methodology to other gravitational sectors and investigate the behaviour of transport coefficients in more complex black brane backgrounds. These developments will contribute to a more complete understanding of the holographic correspondence between gravity and gauge theories, potentially refining insights into strongly coupled systems. 👉 More information 🗞 Shear mode transport coefficients from multiple polylogarithms 🧠 ArXiv: https://arxiv.org/abs/2602.06120 Tags: