Gauge/gravity duality and the interplay of various fractional branes

We consider different types of fractional branes on a ${\mathbb{Z}}_2$ orbifold of the conifold and analyze in detail the corresponding gauge/gravity duality. The gauge theory possesses a rich and varied dynamics, both in the UV and in the IR. We find the dual supergravity solution, which contains both untwisted and twisted 3-form fluxes, related to what are known as deformation and $\mathcal{N}=2$ fractional branes, respectively. We analyze the resulting renormalization group flow from the supergravity perspective, by developing an algorithm to easily extract it. We find hints of a generalization of the familiar cascade of Seiberg dualities due to a nontrivial interplay between the different types of fractional branes. We finally consider the IR behavior in several limits, where the dominant effective dynamics is either confining in a Coulomb phase or runaway, and discuss the resolution of singularities in the dual geometric background.

Figure 1

The quiver diagram of the gauge theory for the most generic choice of ranks. Circles represent unitary gauge groups, arrows represent bifundamental chiral superfields. For later purposes we have parametrized the four independent ranks in terms of a common $N$.

Figure 2

The singular conifold in real angular coordinates: it is a real cone in $r$ over $T^{1,1}$, which in turn is a $U(1)$ fibration in $\psi$ over the Kähler-Einstein space ${\mathbb{P}}^1\times {\mathbb{P}}^1$ parameterized by ${\theta }_i$ and ${\varphi }_i$. The fixed point locus of the orbifold action $\Theta$ is given by two lines $p$ and $q$, localized at antipodal points on the two $S^2$’s. At the tip the spheres shrink and $p$ and $q$ meet.

Figure 3

The $(p,q)$ web (right) associated to the specific triangulation (which corresponds to a specific resolution) of the toric diagram of the orbifolded conifold (left).

Figure 4

Example of the pattern of the cascade of Seiberg dualities for ranks ($N+P$, $N$, $N+P$, $N$) as derived from the field theory. Black numbers indicate Seiberg dualities, performed on gauge groups with diverging couplings. Inverse squared gauge couplings are plotted versus the logarithm of the energy scale.

Figure 5

Example of the pattern of the cascade of Seiberg dualities for ranks ($N+P$, $N$, $N$, $N$) as derived from the field theory.

Figure 6

Flow in the Klebanov-Strassler theory as computed with the algorithm. $x$ is in units of $2\pi /3g_{s}M$, while $\chi$ in units of $2\pi /g_s$. At integer values of $x$ a large gauge transformation is required. At each step the Page D3 charge and the field theory is indicated.

Figure 7

Seiberg dual quivers. The central quiver is the most extensively discussed one in the paper. The left quiver is obtained with a Seiberg duality on node 1 or 3, while the right one on node 2 or 4.

Figure 8

A set of six dictionaries for the orbifolded conifold theory. $F_{(3)}$, $F_{(4)}$ refer to the left quiver, with adjoints on nodes 2–4; $F_{(1)}$, $F_{(2)}$ refer to the central quiver, without adjoints; $F_{(5)}$, $F_{(6)}$ refer to the right quiver, with adjoints on nodes 1–3. The four columns represent the nodes activated by a D5 brane on ${\mathcal{C}}_2$, ${\mathcal{C}}_4$, ${\mathcal{C}}_{\alpha }$, and a D3 brane, respectively.

Figure 9

Figures: the following figures represent the RG flow as computed from supergravity with the algorithm, for typical values of the integration constants $a_2$, $a_4$, $a_{\alpha }$ and the initial radius $x=\log r/r_0$. The gauge couplings are in units of $2\pi /g_s$. On the right side we report, for each step, the dictionary used and the ranks in the quiver; the addition of $N$ is understood. Underlined ranks signal an adjoint chiral superfield at the corresponding node. The red line represents the first group, the orange the second one, the light green the third one, the dark green the fourth one. RG flow for the ($N+1$, $N$, $N+1$, $N$) theory from SUGRA.

Figure 10

RG flow for the ($N+1$, $N+1$, $N$, $N$) dtheory from supergravity.

Figure 11

RG flow for the ($N+2$, $N$, $N+1$, $N$) theory from supergravity.

Figure 12

RG flow for the ($N+1$, $N$, $N$, $N$) theory from supergravity.

Figure 13

The 3-node quiver that corresponds to the IR bottom of the cascade.

Figure 14

The toric diagram and the dual $(p,q)$ web. The specific toric diagram triangulation is the one related to having all ${\xi }_a>0$ in the associated GLSM.

Figure 15

The toric diagram and the dual $(p,q)$ web in the region of the FI parameter space, where ${\xi }_{\beta }<0$.

Figure 16

The six-dimensional manifold seen as a singular ${\mathbb{C}}^{*}$ fibration over the $(z_1,z_2)$ space. The surfaces $H_k(z_1,z_2)=z_1{z}_2-{ϵ}_k=0$, $k=1$, 3, are the loci where the ${\mathbb{C}}^{*}$ fiber degenerates to a cone $xy=0$ and a nontrivial $S^1$ shrinks.

Figure 17

The projection of the $A$ and $\mathcal{C}$ cycles in the $(x,y)$ space. The noncompact B-cycles are obtained as $\mathcal{C}$-cycle fibers over $r$.