Holographic duals of the twisted supersymmetric theories

The winter breaks are essentially the “slingshots” which provide exponential growth to my knowledge-base. There is nothing like sipping coffee while staring at my digital paper and thinking about how universe works at various length-scales (especially with no semester pressure and coursework!). My research in String Theory has exposed me to the several elegant “candidate ways” which describe the working of nature, and I aim to explain one of them in this blog-post. Please note that I will use the jargon frequently enough to bore a sane layperson (and most of the physics majors!) but non-rigorously enough to annoy a decent mathematician. Clearly, the aim of my graduate career is to rectify these drawbacks and explain physics in a way which is fun without losing the mathematical rigor.

Now, there are certain quantum field theories with some extra (symmetry) constraints which provide a lucid way to discover and test the framework of String Theory. These symmetries are

Conformal symmetry
Supersymmetry (SUSY)

I will be focusing on $\mathcal{N}=4$ Super Yang-Mill (SYM) theory in $d=4$ spacetime manifold with the topology given by $\mathbb{R}^{1,1}\times\Sigma_2$ . Here $\Sigma_2$ is a 2 manifold with generic structure and curvature (for instance it could be a Riemann surface with constant negative curvature). For this theory the spin connection is in a $U(1)$ subgroup of the R-Symmetry group $SU(4)$ . Now since $\Sigma_2$ can be a curved manifold, it can (and will) break the supersymmetries. In order to preserve at least some of them, we need to, what is known as, twist the theory in a specific fashion. Essentially, we couple the external $SO(N)$ gauge fields with the R-Symmetry current and identify the spin connection with the gauge connection such that we get the covariantly constant spinors on the manifold (there is a more visually appealing picture in the language of branes which I will explain later in the post). In other words, the twist corresponds to the nature of the embedding of the $U(1)$ subgroup in the $SU(4)$ . The aim, then, is to find the holographic gravity duals of these twisted field theories.

Twists preserving (4,4) susy

Here we will consider the twist which corresponds to picking a $U(1)$ subgroup such that we break the R-Symmetry in the following way $SO(6)\rightarrow SO(2)\times SO(4)$ . To see what exactly is happening, consider the spinor field $\phi$ of the SYM with spin $s$ under the $SO(2)$ spin connection on $\Sigma_2$ and $U(1)$ charge $q$ . Now, the covariant derivative on the manifold is, obviously, $\mathcal{D}_\mu\phi=(\partial_\mu+is\omega_\mu+iqA_\mu)\phi$ . Here $\omega_\mu=\epsilon_{ab}\omega^{ab}_\mu/2$ . Now if the metric on $\Sigma_2$ is $ds^2=e^{2h}(dx^2+dy^2)$ , the spin connection can be computed and once we identify the $U(1)$ gauge connection with the spin connection, the constraint $s=-q$ will give us the “covariantly constant” spinors which, now, are essentially the scalars. We have twisted the field theory by fixing the spin of the fields!

Essentially, the symmetry group (associated with the $\mathbb{R}^{1,1}\times\Sigma_2$ ), $SO(1,3)\times SO(6)$ (corresponding to the tangent bundle and the normal bundle) is decomposed as $SO(1,1)\times SO(2)_{\Sigma_2}\times U(1)\times SU(2)_L \times SU(2)_R$ . This corresponds to having $(4,4)$ susy in the theory.

Brane realization through an example:

Consider a manifold $\mathbb{R}^6\times K3$ with D3 branes wrapping some holomorphic curve (Riemann surface) in K3. In the field theory limit, we obtain the gauge theory mentioned above. The transverse $\mathbb{R}^6$ direction, after the twist, will have the $SO(4)=SU(2)_L \times SU(2)_R$ rotational symmetry. Now I will make a statement without showing any mathematics, because it is tedious, but it is important for my research. When we consider the low energy limit, compared to the size of Riemann surface, then we get a 2 dimensional effective theory in IR which now becomes (4,4) SCFT!

Lagrangian description:

Let us write down the Lagrangian for the partially twisted theory which will enable us to find the gravity dual. Since we coupled the theory by introducing the spin connection with the $\Sigma_2$ , we expect the extra terms in the Lagrangian coming from the covariant derivatives along that direction or fields which are charged under the $U(1)$ part of the normal bundle with non-zero gauge connections.

Consider the Lagrangian with two twisted scalar fields given by $\mathcal{Z}=X^1+iX^2$ . The action looks like $S=\int Tr(|D_z\mathcal{Z}|^2+|D_{\bar{z}}\mathcal{Z}|^2+\frac{1}{4}R|\mathcal{Z}|^2)$ .

Supergravity (SUGRA) duals of the field theories

Maldacena’s conjecture of SYM theories being dual to supergravity theories on $AdS_5\times S^5$ give us a good starting point. Since we are dealing with the deformed SYM (defined on $\mathbb{R}^{1,1}\times\Sigma_2$ with coupling to $SO(6)$ gauge fields), the information gets translated into the boundary conditions in the dual gravity theory.

We start with a reasonable ansatz $AdS_5$ which at the boundary behaves like $ds^2\sim \frac{-dt^2+dz^2+dr^2+e^{2h}(dx^2+dy^2)}{r^2}$ . Also we impose that the $SO(6)$ gauge fields in $AdS_5$ asymptotes to the field theory gauge fields. This means that the metric of the geometry $AdS_5\times S^5$ with one index in $AdS_5$ and other in $S^5$ , that is $g_{\mu\phi}\sim A_\mu$ near the boundary.

Now a non trivial condition is that we should turn on an operator in the 20 of $SO(6)$ . Since we can easily see the coupling to the curvature in the action above, we get the ansatz for the operator as $\mathcal{O}=Tr(\frac{2}{3}|Z|^2-\frac{1}{3}(\phi_1^2+\ldots+\phi_4^2))$ .

Fortunately operators that are turned on correspond to the fields in the 5d gauged supergravity multiplet. Now people have already worked out such theory with $\mathcal{N}=8$ SUGRA and we will start with that!

Since the connection we started of is $U(1)$ , we can start with the $U(1)$ truncations of the supergravity. Hence the data with which we start off is

a 5d metric
a scalar field
a $U(1)$ gauge field

To find the supersymmetric solutions, we start with the fermionic supersymmetric variations which give the constraint equations. Once these equations are solved, we get the complete dual supergravity theory to the twisted SYM theory. To see the explicit calculations head over to page 8 of https://arxiv.org/pdf/hep-th/0007018v2.pdf.

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Ravi Mohan

If you haven't found something strange during a day, it hasn't been much of a day. – J. A. Wheeler

Holographic duals of the twisted supersymmetric theories