Speaker
Description
In experimental science, many phenomena remain beyond a self-consistent theoretical description. A prominent example is provided by the strong interaction: Yang–Mills theory, which models it, requires solving highly nonlinear equations that are still intractable in general. Several decades ago, however, advances in string theory led to the holographic correspondence (AdS/CFT). By analyzing the equations of motion of supergravity---the low-energy limit of string theory—in the classical supergravity regime, one can extract correlation functions of the dual conformal field theory at strong coupling. For instance, linearized perturbations around the $AdS_5\times S^5$ background allow one to compute correlators of $\mathcal N=4 \ SYM$ in the strong-coupling limit. Although the underlying theories remain strongly nonlinear, this framework reduces many otherwise formidable correlation-function calculations to solving differential equations for small supergravity perturbations, which are often tractable analytically or, at least, numerically. Consequently, further developing both holographic methods and techniques for solving supergravity equations is crucial for a more refined description of strong-interaction and nuclear phenomena, with the potential to yield new discoveries.
One of ways for generating new supergravity solutions is to study the solution space as a geometric object endowed with hidden symmetries, the best-studied of which is T-duality. In this context, a useful operation is the so-called $\beta$ -- deformation, encapsulated by
\begin{equation}
\bigl( (g+B)^{-1}+\beta \bigr)^{-1} = G+B ,
\end{equation}
where $g$ and $B$ are the metric and Kalb–Ramond two-form of a given bosonic supergravity background, $\beta$ is a bivector chosen along isometries of the background, and $G$ and $B$ denote the fields of the deformed solution. This construction exploits the fact that ten-dimensional supergravity arises as the low-energy effective theory of strings, which possess duality symmetries; at low energies these appear as families of solutions generated from highly symmetric seeds. The deformation is constrained by
\begin{equation}
\beta = \rho^{ab}\, k_a \wedge k_b,\qquad \rho^{a[b}\,\rho^{|c|d}\, f_{ac}{}^{e]}=0,
\end{equation}
where $k_a$ are Killing vectors, $\rho^{ab}$ encodes the bivector deformation along these isometries, and $f$are the structure constants defined by $[k_a,k_c]=f_{ac}{}^{e} k_e$.
For deformations produced in this way, a holographic interpretation can be formulated. Particularly interesting results arise when the Killing vectors generate a compact abelian Lie algebra: the image on the gauge-theory side is a manifold of conformal fixed points. By contrast, the (homogeneous) classical Yang–Baxter equation (CYBE) severely constrains—indeed forbids compact non-abelian bivector deformations, which limits strong this solution-generating method.
However, T-duality is not the only hidden symmetry inherited from string theory. U-duality, a richer symmetry of string theory, also has a supergravity representation and admits an analogue of the bivector construction in terms of polyvector deformations. It has been shown that certain tri- and quadrivector deformations generate new solutions of eleven-dimensional and type IIB supergravity. Moreover, there is no theorems or geometrical reasons to expect impossibility of using of compact non-abelian bivector deformations in the polyvector case. A natural objective, therefore, is to formulate an appropriate generalization of the CYBE that fully characterizes admissible three- and four-vector deformations, in parallel with the bivector theory, which is current task of my work.
