The Cascade is a MMS Instanton

TL;DR

<p>The paper argues that RR fluxes and D-brane charges in type II string theory are naturally classified by a covariant, S-duality–invariant extension of twisted K-theory, realized via MMS instantons that mediate brane decay and flux rearrangement. It shows that a KS-like cascade emerges from successive MMS-type decays in the conifold/deformed conifold setup, with a precise accounting of residual flux and half-integer quantization that preserves consistency under dualities. Through MQCD brane cartoons and a T-dual IIB description, Evslin demonstrates that the cascade can proceed for arbitrary integers $m$ and $n$, and that nonbaryonic roots give new cascades; these results provide the first concrete evidence for an S-dual K-theory classification that does not rely on strong/weak duality. The analysis unifies WZW-model cascades, MQCD brane dynamics, and conifold physics, and clarifies how the MMS instanton flux balance ensures charge conservation while reducing gauge-group rank in the IR, with implications for F-theory lifts and S-duality-covariant brane classifications.

Abstract

Wrap m D5-branes around the 2-cycle of a conifold, place n D3-branes at a point and watch the system relax. The D5-branes source m units of RR 3-form flux on the 3-cycle, which cause dielectric NS5-branes to nucleate and repeatedly sweep out the 3-cycle, each time gaining m units of D3-charge while the stack of D5-branes loses m units of D3-charge. A similar description of the Klebanov-Strassler cascade has been proposed by Kachru, et al. when m>>m-n. Using the T-dual MQCD we argue that the above process occurs for any m and n and in particular may continue for more than one step. The nonbaryonic roots of the SQCD vacua lead to new cascades because, for example, the 3-cycle swept does not link all of the D5's. This decay is the S-dual of a MMS instanton, which is the decay into flux of a brane that is trivial in twisted K-theory. This provides the first evidence for the S-dual of the K-theory classification that does not itself rely upon any strong/weak duality.

Figures (5)

• Figure 1: The baryon and MMS Instanton of type II string theory on $\mathbb R^7\times S^3$ with $\int_{S^3}H=2$. The 2 units of $H$-flux imply that 2 D$p$-branes must end on every D$(p+2)$-brane wrapping the $S^3$. The MMS instanton violates D$p$-brane charge conservation by 2 units. Charges are time-independent in the baryon configuration, and in fact the baryon number itself is conserved as it represents a nontrivial homology class $[1]\in \textup{H}_3(S^3)$, albeit a class that does not lift to K-homology.
• Figure 2: The type IIA and M-theory realizations of a $SU(2)\times SU(3)$$N=2$ supersymmetric gauge theory, with 2 bifundamental hypermultiplets. In IIA the theory and vacuum are described by D4 and NS5-branes, while the degrees of freedom are F-strings and D2-branes. In the M-theory lift the theory and vacuum are described by an M5-brane while the degrees of freedom are M2-branes. Here all D4-branes end on NS5-branes, but in general a D4-brane may also wrap the circle $x^6$.
• Figure 3: The type IIA and M-theory realizations of the $SU(7)\times SU(10)\longrightarrow SU(7)$ cascade. This is not a Klebanov-Strassler type cascade, which would end with only the decoupled $U(1)$ as seen in Fig. \ref{['ks']}. On the upper-left $v$ and $x^6$ are represented, while the upper-right is a projection of the same configuration onto Re($v$) and $x^6$. The reduction to IIA is performed at two energy scales, $v_1$ and $v_2$. The intersection of the M5 with $|v|=v_1$ consists of the loops $A$ and $B$, which have winding numbers of 3 and $-3$ about the M-theory circle. Thus the reduction at that energy scale yields 2 NS5-branes, one connected to 3 D4's on the right and the other to 3 D4's on the left. The M5-brane does not intersect $|v|=v_2$, and so a reduction at that energy scale does not result in any NS5-branes. In every cartoon we see the 7 flat branes that support the $SU(7)$$N=4$ theory at the bottom of the cascade.
• Figure 4: Here we see the M-theory configuration corresponding to the Klebanov-Strassler cascade $SU(7)\times SU(10)\longrightarrow U(1)$, where the $U(1)$ is present at every stage but generally omitted. The continuation to the higher energy scale $v_1$ is not unique, and we have chosen the continuation such that the UV $SU(10)\times SU(13)$ is in a baryonic root vacuum. The effective gauge group at each energy scale $v_k$ is written next to the energy scale. When the group is $SU(a)\times SU(a+b)$ then the IIA reduction consists of two NS5-branes with $a$ D4-branes connecting them on one side and $a+b$ on the other. Notice that this M5-brane configuration is different (only 1 flat brane) from the M5-brane configuration in Fig. \ref{['r0']}, which describes a different vacuum and so a different cascade of the same initial $SU(7)\times SU(10)$ theory.
• Figure 5: The cascade occurs when an NS5-brane bubble spontaneously forms in the vacuum and sweeps out a 3-surface linking the D5-branes. The NS5-brane wraps a trivial cycle and so carries no net NS5-brane charge, but the $G_3$ flux of the D5-branes produces D3-brane charge on the NS5-branes. Similarly the $H$ flux of the NS5-branes destroys an equal amount of D3-charge on the D5-branes, reducing the gauge group as seen by a probe not big enough to extend to the NS5-branes from the D5-branes, where the gauge theory lives (the horizon at large $m$). The fact that the total D3-charge is conserved, as is clear to a UV observer who notices no reduction in gauge group, is apparent from the Hanany-Witten transition to a stack of D3-branes escaping.