Three-dimensional Yang-Mills Chern-Simons theory from D3-brane background with D-instantons

Abstract

By constructing the configuration of D3-branes with D(-1)-branes as D-instantons, we study the three-dimensional Yang-Mills Chern-Simons theory in holography. Due to the presence of the D-instantons, the D7-branes with discrepant embedding functions are able to be introduced in order to include the fundamental fermions (as flavors) and the Chern-Simons term (at very low energy) in the dual theory. The vacuum structure at zero temperature is studied in the soliton background and it illustrates the topological phase transition in the presence of instantons. Moreover, since the confinement/deconfinement phase transition could be holographically identified as the Hawking-Page transition in the bulk, we accordingly calculate the critical temperature of the deconfinement phase transition by collecting the bulk onshell action as the thermodynamical free energy. On the other hand, we evaluate the difference of the entanglement entropy in slab configuration by using the RT formula since the confinement may also be characterized by the entanglement entropy. Altogether we find the behavior of the critical temperature is in qualitative agreement with the behavior of the critical length determined by the entanglement entropy which implies the entanglement entropy could indeed be a character of the confinement in our setup and the D3-D(-1) system would be a remarkable approach to study the three-dimensional gauge theory.

Holographic $\mathrm{QCD}_{3}$ and Chern-Simons theory from anisotropic supergravity

“…Therefore we can see the gauge-gravity duality in this setup reduces to the well-known level/rank duality SU (N c ) k b ↔ U (k b ) −Nc in quantum field theory (QFT) expectations precisely [45,46,56].…”

“…Keeping these in hand, we numerically solve the equation of motion (4.7) with various a in the region ζ > 0 which is illustrated in Figure 8. The numerical calculation shows a very small shift with respect to the dependence on a (we have enlarged the shift in the figure) and this behavior is opposite to the approach of the D3-D(-1) background [45]. We note here the two branches of the flavor brane trend to become coincident if a increases.…”

“…As most works about the gauge-gravity duality, the fundamental matter can be added to the D3brane background by embedding flavor D7-branes as probe. In our set-up, we add N f copies of coincident D7-branes as probes to the D3-brane background (2.6), transverse to the compactified z direction, spanning the R 1,2 denoted by {t, x, y}, the holographic direction denoted by u and four of the five directions in Ω 5 as [45,46]. The D-brane configuration including various D7branes is illustrated in Table 3 where we have decomposed the directions of Ω 5 as Ω 4 and w. The leftover direction w is transverse to both the N c color D3-branes and N f flavor D7-branes, so a bare mass for the flavors can be introduced by imposing a separation between color and flavor branes along w direction at the UV boundary which breaks the parity in QCD 3 .…”

“…In order to identify the vacuum structure of the D7-branes with the various phases in the dual theory, we need to give a well-defined Chern-Simons level in which the Chern-Simons level depends on the number of the flavor branes. The details to obtain this goal have been discussed in [45,46], so let us briefly outline the main idea and investigate the phase transition in our holographic setup. First the effective Chern-Simons level k ef f according to the holographic duality is given as,…”

“…Last but not least, to explore the Chern-Simons topological feature involving the flavors in the dual theory, following [45,46,47], various D7-branes as flavor and Chern-Simons branes are introduced into the bulk bubble configuration as probes. In a transverse plane, the vacuum configuration of the D7-branes, which means the embedding function minimizes the energy of the D7-brane, is numerically evaluated and the calculation shows the vacuum structure is shifted by the presence of the axion field in the bulk.…”