Monopoles and baryonic matter at strong coupling

Based in Ref.[ 1 ]. In collaboration with Antón Faedo and Carlos Hoyos

We studied the phase diagram of a strongly coupled confining theory in (2+1) dimensions, as a function of temperature and baryon chemical potential.

This is accomplished using a fully-fledged supergravity holographic dual, by means of which we uncover a rich phase diagram.

In particular, we use it to predict a line of first-order phase transitions separating a confining phase from a deconfined phase. Both phases exhibit a non-zero baryon density thus providing a first example of baryonic matter in a confining string dual that does not require the introduction of flavor branes.

A confining theory in three dimensions

Our starting point is the confining theory found in Ref.[ 2 ]. The geometry contains a collapsing cycle that introduces confinement in the same way as in the Klebanov-Strassler background, as we discuss next.

This solution is sourced by the following fluxes in type IIA supergravity.

\[ \begin{aligned} F_2 &= 0 \\ F_4 & \propto *\Omega_{\mathbb{C}\text{P}^3}+\text{additional terms}\\ H&\neq 0\end{aligned} \,. \]

Note the appearance of CP3 as the compact part of the geometry. We can think of CP3 as a two-sphere fibered over a four-sphere. Remarkably, the two-sphere collapses at the origin, while the four-sphere remains finite. This introduces confinement, as we explain next.

Figure 1: Representation of the behaviour of the compact part of the geometry.


The collapse of a cycle in the compact part of the geometry (while the rest keeps finite) makes the theory to be confining. This is because it forces the potential between two quarks to grow linearly at large separatins: the strings have nowhere to end at the bottom of the geometry and thus, they cannot split.

Figure 2: Holographically, the potential between two quarks is given by the length of a string attached to them in the boundary (UV). The fact that in this case the geometry finishes smoothly causes the linear growth of the potential between them.

The gauge theory dual

Now, we source a massless vector field in the geometry,

\[ \begin{aligned} F_2 &= \text{d} C_1 \\ C_1 &= a_t(r)\text{d}t + B ( x_1 \text{d}x _2 + x_2 \text{d}x _1 )\end{aligned} \,. \]

Depending on the boundary conditions that we choose in the gravity side, the gauge theory dual will have two different interpretations Ref.[ 3 ]:

  • With Dirichlet boundary conditions for the massless vector field, it was argued there that the dual was a quiver theory with U(N)×U(N+M) gauge group. Moreover, the global current was identified as the topological current counting the magnetic flux of the diagonal U(1) gauge group.
  • With Neumann boundary conditions one can perform an electromagnetic duality transformation in the four-dimensional gravity theory. In this case the field theory becomes a quiver with SU(N)×SU(N+M) gauge group. In this case, the global conserved current corresponds to a U(1)B baryonic symmetry acting on fields in the bifundamental representation.

This duality interchanges the role between electric and magnetic branes:

Table 1: The change in boundary conditions exchanges the role of electric and magnetic branes, and is implemeneted via Hodge duality.

This changes the type of branes that are allowed to end at the boundary. On the one hand, with Dirichlet boundary conditions D0 and D2 branes dual to monopoles are allowed to reach the boundary. On the other hand, D4 and D6 branes can, which are dual in this case to baryons. This is summarised in the following picture.

Figure 3: Representation of the duality between monopoles and baryons.

In conclusion, the gravity solutions have a different interpretation depending on the boundary conditions chosen for the vector field. Let us examine both of them next.

Dirichlet boundary conditions: monopoles

In Ref.[ 4 ], we argued that with Dirichlet boundary conditions confinement is caused by monopole condensation. To investigate this claim, we considered an external magnetic field for the monopoles. We obtained the following rich phase diagram, endowed with a line of second order phase transitions.

Figure 3: Phase diagram in the temperature, monopole magnetic field plane. Solid curves show the values at which a first order phase transition occurs. There is also a line of second order phase transitions. This provides the first example of phase transition in holographic duals of a deconfinement transition which is not first order.

Newmann boundary conditions: baryons

If now we turn to Neumann boundary conditions, the global symmetry dual to the gauge symmetry in the bulk becomes a baryonic symmetry. If we turn on a chemical potential for it, we get the following rich phase diagram.

Phase diagram: entropy density for the different phases.
Figure 1: Density plot for the (logarithm of the) entropy density as a function of the baryon chemical potential, and temperature. The confined and plasma phases are separated by a line of first-order phase transitions. The confining phase has an entropy density of strictly zero, as no black hole is present in the dual theory in this case. In contrast, the entropy density in the plasma phase grows with temperature.

Remarkably, both phases have finite baryonic charge.

Phase diagram: baryon charge density for the different phases.
Figure 2: Density plot of the baryon charge of the preferred phase at every chemical
potential and temperature. Remarkably, the charge density is finite in the confining phase.

Thus, this corresponds to the first fully-backreacted string theory example of a holographic dual to a strongly coupled confining theory with non-zero baryon density. Note that it does not rely on probe flavor branes and does not require considering multi-instanton solutions or phenomenological approximations to those, as other approaches do.


  1. Antón F. Faedo, Carlos Hoyos & Javier G. Subils, arxiv:2304.0725 [hep-th]
  2. M. Cvetic, G. W. Gibbons, J. T. Liu, H. Lu, and C. N. Pope, Class. Quant. Grav. 19, 5163 (2002), arXiv:hep-th/0106162
    C. P. Herzog, Phys. Rev. D 66, 065009 (2002), arXiv:hep-th/0205064
  3. O. Bergman, Y. Tachikawa, and G. Zafrir, JHEP 07, 077, arXiv:2004.05350 [hep-th]
  4. A. F. Faedo, C. Hoyos, and J. G. Subils, JHEP 03, 218, arXiv:2212.04996 [hep-th].