content_image/416883.jpg content_image/416884.jpg content_image/416885.jpg content_image/416881.jpg content_image/416886.jpg
are dynamically suppressed or “quenched”, so that the time ( t$_{1}$ ) evolution does not violate the Craig-Weinstein constraint (10). In order to understand what this re- quirement implies, we consider the dynamics of the field within the ( n − 1)-dimensional subspace of the hyperplane ( x , t $^{′}$) ∈ R p,q − $^{1}$, on which the initial-value data are given. For specificity, let us consider the case p = 3 , q = 2, i. e. the extension of Minkowski space by one additional time coordinate, which we denote by t$_{2}$ ≡ θ . We also sim- ply write t$_{1}$ ≡ t . We are looking for a field theory, in which all momentum modes with | χ | > | k | are dynami- cally quenched and only modes with | χ | ≤ | k | survive. If we could find such a field theory, we would not need to arbitrarily impose the constraint (10) on the initial data, rather, the constraint would arise dynamically from the field theory. 

The thermal N = 4 supersymmetric Yang-Mills the- ory in (3+1) dimensions at large number of colors N$_{c}$ and large ’tHooft coupling constitutes precisely such a theory. The spectral function of current and stress-energy corre- lation functions of this theory have been studied in detail [32, 33], and it was shown that the theory does not con- tain propagating modes, except the hydrodynamic mode which is only weakly damped. The hydrodynamic mode with the space-like dispersion relation χ 2 = k $^{2}$/ 3 exhibits minimal damping from a shear viscosity η that saturates the so-called KSS bound η = s/ 4 π [34, 35]. 

Time-like excitations in the thermal super-Yang-Mills theory at strong coupling are rapidly damped [36, 37]. (Note that we here refer to the θ -direction as “time” and to T$_{5}$ ≡ T$_{θ}$ as “temperature”.) In the gauge theory, this process corresponds to the “democratic” splitting of the original elementary excitation into lower-energy modes, which become part of the thermal bath within a finite time [38, 39]. One may visualize this as “shooting a wa- ter gun under water”, where the initially well identified stream of water emerging from the gun quickly fragments into smaller and smaller eddies and quickly loses its iden- tity and dissipates into the surrounding water. 

In order to see how the exponential growth in physical time of modes with χ 2 > k 2 can get suppressed by such rapid damping, we consider a schematic example. Let us assume that all dynamical excitations in θ -time decay like Gaussians with a time constant τ , and we focus on the “most time-like” modes by setting k = 0: 

<img src='content_image/416883.jpg'>

The Fourier transform is 

<img src='content_image/416884.jpg'>

The full time-dependence of a massless mode in t and θ is then obtained by setting ω = ± iχ : 

<img src='content_image/416885.jpg'>

Averaging this expression over θ yields a constant, which is independent of the physical time t . In other word, the θ -time dependence decouples from the physical time dependence, which will be determined by the dynamics in Minkowski space, i. e. four-dimensional ( x , t ) space-time. 

## C. Holographic theory with two time dimensions 

This picture of a strongly coupled, thermal, super- symmetric gauge theory in higher dimensions is sugges- tive of Witten’s holographic construction [40] of the 4- dimensional large- N$_{c}$ Yang-Mills theory as the boundary field theory of a supergravity theory on AdS$_{7}$ space in which two spatial dimensions have been compactified on a circle. The first compactification preserves supersym- metry, while the second one is assumed to correspond to a “thermal” compactification, which breaks supersymme- try by requiring periodic boundary conditions for bosons and antiperiodic ones for fermions. Although this addi- tional benefit is important if one wants to imbed the sce- nario of two time dimensions into string theory, it is not essential in a more general context. Witten’s construc- tion has been used to obtain estimates for the glueball spectrum and other properties of the pure Yang-Mills theory in the large ’tHooft coupling limit [41, 42]. 

<img src='content_image/416881.jpg'>

One possible realization of a holographic theory with two time dimensions, one of which is thermal, is six- dimensional Anti-deSitter space space AdS$_{6}$ containing a black hole [43–45]. The imbedding of the black hole iden- tifies which direction in the two-dimensional plane of time dimensions is thermal. As elsewhere in this manuscript, we label this direction by the coordinate θ ; the orthogonal time direction is labeled t . The metric of this space-time is given by 

<img src='content_image/416886.jpg'>

