Cascading gauge theory on dS4 and String Theory landscape

Placing anti-D3 branes at the tip of the conifold in Klebanov-Strassler geometry provides a generic way of constructing meta-stable de Sitter (dS) vacua in String Theory. A local geometry of such vacua exhibit gravitational solutions with a D3 charge measured at the tip opposite to the asymptotic charge. We discuss a restrictive set of such geometries, where anti-D3 branes are smeared at the tip. Such geometries represent holographic dual of cascading gauge theory in dS₄ with or without chiral symmetry breaking. We find that in the phase with unbroken chiral symmetry the D3 charge at the tip is always positive. Furthermore, this charge is zero in the phase with spontaneously broken chiral symmetry. We show that the effective potential of the chirally symmetric phase is lower than that in the symmetry broken phase, i.e., there is no spontaneous chiral symmetry breaking for cascading gauge theory in dS₄. The positivity of the D3 brane charge in smooth de-Sitter deformed conifold geometries with fluxes presents difficulties in uplifting AdS vacua to dS ones in String Theory via smeared anti-D3 branes.

First, turning on fluxes on Calabi-Yau compactifications of type IIB string theory produces highly warped geometry with stabilized complex structure (but not Kähler) moduli of the compactification [3];

Next, including non-perturbative effects (which are under control given the unbroken supersymmetry), one obtains anti-de Sitter (AdS₄) vacua with all moduli fixed;

Finally, one uses anti-D3 branes of type IIB string theory to uplift AdS₄ to de Sitter (dS₄) vacua.

As the last step of the construction completely breaks supersymmetry, it is much less controlled. In fact, in [4-7] it was argued that putting anti-D3 branes at the tip of the Klebanov-Strassler (KS) [8] geometry (as done in KKLT construction) leads to a naked singularity. Whether or not the resulting singularity is physical is subject to debates.

When M₄=dS₄ and the chiral symmetry is unbroken, the D3 brane charge at the tip of the conifold is always positive, as long as ln H²Λ²/P²g₀ ⩾-0.4.

When M₄=dS₄ and the chiral symmetry is broken, the D3 brane charge at the tip of the conifold is always zero; we managed to construct geometries of this type for ln H²Λ²/P²g₀⩾-0.03.

Comparing effective potential of the gauge theory in broken Veffb and unbroken Veffs phases we establish that in all cases, when we can construct the phase with spontaneously broken chiral symmetry, Veffb>Veffs, when ln H²Λ²/P²g₀⩾-0.03, i.e., spontaneous symmetry breaking does not happen for given values of the gauge theory parameters. To put these parameters in perspective, note that the (first-order) confinement/deconfinement and chiral symmetry breaking phase transition in cascading gauge theory plasma occurs at temperature T such that [16] ln Tdeconfinement,χSB2Λ²/P²g₀=0.2571(2), and the (first-order) chiral symmetry breaking in cascading gauge theory on S³ occurs for compactification scale μ₃≡ℓ3-1 such that [21] ln μ3,χSB2Λ²/P²g₀=0.4309(8).

When M₄=R×S³ and the chiral symmetry is unbroken, the D3 brane charge at the tip of the conifold is negative when ln μ32Λ²/P²g₀ <ln μ3,negative2Λ²/P²g₀ =0.0318(3).

However, since cascading gauge theory undergoes a first order phase transition with spontaneous breaking of the chiral symmetry at μ>μ, and the D3 brane charge at the tip of the conifold in broken phase is zero, the charge in the ground state is in fact zero whenever μ₃⩽μ. Furthermore, chirally symmetric states of cascading gauge theory on S³ develop symmetry breaking tachyonic instabilities at μ (below the first order chiral symmetry breaking scale μ) ln μ3,tachyon2Λ²/P²g₀=0.3297(3) which is again above μ.Our results represented here, together with those reported in [10], point that the singularity of smeared anti-D3 branes at the tip of the conifold is unphysical: had it been otherwise, we should have been able to implement an infrared cutoff in the geometry with a D3 brane charge measured at the cutoff being negative. The role of the cutoff is played by the temperature (as discussed in [10]), by the compactification scale (when M₄=R×S³), or by the Hubble scale (when M₄=dS₄). Interesting, we find that the D3 brane charge can become negative when the KT throat geometry is S³ deformed; however this occurs in the regime where this phase is unstable both via the first order phase transition and the tachyon condensation to S³ deformed KS throat geometry - the latter geometry has zero D3 brane charge at the tip. All this raises questions about construction of generic de Sitter vacua in String Theory [2].We stress, however, that our analysis does not definitely exclude local non-singular supergravity description of de Sitter vacua in String Theory. The issue stems from the anti-D3 brane “smearing approximation” used. Early discussion of the relevant smearing approximation appeared in [6,9]. There, the authors carefully analyzed non-supersymmetric deformations of KS geometry, invariant under the SU(2)×SU(2) global symmetry of the latter. They further identified a class of perturbations that is being sources by anti-D3 branes, placed at the tip of the conifold, and then computed the leading-order backreaction of those perturbations on KS geometry. Insistence on preserving the SU(2)×SU(2) global symmetry is a smearing approximation - from the brane perspective it implies that anti-D3 branes are uniformly distributed (uniformly smeared) over the transverse compact five-dimensional manifold. Our discussion here shares the same smearing approximation as in [6,9], but extends the analysis to the full (rather than leading-order) backreaction. Smearing approximation is a practical tool enabling the analysis of the complicated cascading geometries involved. However, it must be questioned: it is not clear that non-supersymmetric uniform distribution along T directions of anti-D3 branes is stable against ‘clumping’. While it is highly desirable to lift this approximation, it is very difficult to do this in practice: one is forced to analyze a coupled nonlinear system of partial differential equations, rather than ordinary differential equations. We feel that until fully localized anti-D3 brane analysis in cascading geometries are performed, the singularity question of local supergravity description of de Sitter vacua in String Theory will remain open.