# Localising Path Integrals, Delocalising Wilson Network Vertices, and Moving the Boundary

In his third lecture Victor (ICTS) introduced the philosophy behind localization techniques and few examples to demonstrate it. We came to know that by localization we can exactly compute the partition function and expectation values of operators which are in a certain multiplet of the theory.

He started with a prototype action which is supersymmetric under $Q$.The partition function was that on $S^2$,

$Z_{S^2} = \int \mathcal{D}\varphi\, e^{-S[\varphi]}$

In order to localize the Partition function someone can add a $Q$ deformed functional to the action in the way,

$Z_{S^2} = \int\mathcal{D} \varphi\, e^{-S[\varphi]-tQV[\varphi]}$

where $t$ is some arbitrary parameter. By taking a derivative w.r.t. t, we get an expression which is $Q$ exact and hence reduces to zero. So, there is no harm in taking t to infinity limit which will allow us to analyze Saddle points of the action. Depending on the theory, the supersymmetry involved and the background etc. the classical solution can reduce just to a point, in that case we can say that the theory is localized well. we can also look for the expectation value of certain operators in that background. In the large $t$ limit we can write the $\varphi$ in terms of classical value and a subleading term in $t$.In a general action with more fermionic and bosonic fields involved, the ratio of bosonic fluctuations to fermionic fluctuations gives what is called one loop determinant,which he promised to calculate in his last lecture with gauge theory examples. For that he moved onto describing the $N=(2,2)$ theories on curved background with a R-symmetric multiplet and gravity multiplet,

$\widetilde{R}\equiv [T_{\mu\nu},S_{\alpha\mu},\widetilde{S}_{\alpha\mu},j_{\mu}^R,\widetilde{j_{\mu}^R}]$
$\widetilde{G}\equiv [g_{\mu\nu},\Psi_{\alpha\mu},\widetilde{\Psi_{\alpha\mu}}, V_{\mu},C_{\mu},\widetilde C_{\mu}]$

Here the $C_{\mu}$latex and $\widetilde{C_\mu}$ are graviphotons related with central charges.The dual field strengths were introduced, $H=-i{\epsilon^{\mu\nu} \partial_{\mu}C_{\nu}}$ and similarly $\tilde{H}$. Then he discussed the rigid limit of the gravity multiplet. The gravitino variations are given in terms of the spinors and the field strengths $H$ and $\tilde{H}$.For a particular choice of $H$ ( $H=\tilde{H}=0$)we can write $V_{\mu}$ in terms of the spin connection as $iS_{z}\Omega_\mu$.If the choice is $H=\tilde{H}=i/r$ then Killing spinor equations reduce to,

$\nabla_{\mu} \epsilon=\frac{i}{2r}\gamma_\mu \epsilon$
$\nabla_{\mu} \tilde{\epsilon}=\frac{i}{2r}\gamma_\mu \tilde{\epsilon}$

Then in somewhat more technical way he discussed the twisted and anti-twisted superpotentials in this theory.In the twisted case he calculated the magnetic flux.

Then the matter and gauge field variations were written down along with the Lagrangian.

$\mathcal{L}_{YM}^{bos} =\text{Tr } {{\Big( F_{12} - \frac{\sigma_2}{r} \Big)^2 +\Big(D + \frac{\sigma_1}{r} \Big)^2 + D_\mu\sigma_1 D^\mu \sigma_1 + D_\mu\sigma_2 D^\mu \sigma_2 -[\sigma_1,\sigma_2]^2}}$
$\mathcal{L}_{YM}^{fer}=\frac{1}{2}\text{Tr } {{\frac{i}{2}\tilde{\lambda} \gamma^\mu D_\mu \lambda + \frac{i}{2} \tilde{\lambda} [\sigma_1,\lambda] + \frac{1}{2} \tilde{\lambda} \gamma_3 [\sigma_2,\lambda]}}$
$\mathcal{L}_{m}^{bos}=D_\mu \tilde\phi D^\mu \phi + \tilde\phi \sigma_1^2 \phi + \tilde\phi \sigma_2^2 \phi - i \tilde\phi D \phi + \tilde F F + \frac{iq}r \tilde\phi \sigma_1 \phi + \frac{q(2-q)}{4r^2} \tilde\phi \phi$
$\mathcal{L}_{m}^{fer}=- i \tilde\psi \gamma^\mu D_\mu \psi + i \tilde\psi \sigma_1 \psi - \tilde\psi \gamma_3 \sigma_2 \psi + i \tilde\psi \lambda \phi - i \tilde\phi \tilde\lambda \psi + \frac q{2r} \tilde\psi \psi$

In the $r \to \infty$ the YM lagrangian reduces to flat space YM lagrangian. He discussed schematically the Q-exact form of the above lagrangians, and then we dispersed for lunch.

After lunch, we came back for a dyptich of evening talks that began in the afternoon. The first was by Atanu Bhatta (IMSc), on a proposal by himself and collaborators for calculating conformal blocks (more precisely, conformal partial waves) in a CFT using “open Wilson networks” in the bulk.

There’s already such a proposal, by Perlmutter and collaborators, where they showed that conformal blocks could be calculated by geodesic Witten diagrams with the exchange of only the field dual to the primary the block corresponds to. However, this formulation isn’t good enough for spinors, because it uses the metric formulation of gravity in the bulk.

The idea behind this work was to write down something that worked for spinors as well, by using the Hilbert-Palatini formalism, in which the gravitational dynamics can be written down in terms of an auxiliary gauge field made out of the vielbeins. Specialising to the case of $AdS_3$, we have an SL(2,C) Chern-Simons theory in 3 dimensions. As is well-known, the solution to the equations of motion here is that the gauge field configuration be locally pure gauge.

In this background, suppose there are three Wilson lines coming from three different points and fusing at the same point with a Clebsch-Gordan coefficient. Because the background is pure gauge, the Wilson line $P e^{\int_{x_1}^{y} A_{R_1}}$, where $R_1$ is the representation, is just $g_{R_1}(x_1) g_{R_1}(y)^{-1}$ we can use a fundamental identity of the Clebsch-Gordan coefficients to remove all dependence on y — so that an arbitrarily complicated network is completely specified by its endpoints, associated representations, and the Clebsch-Gordans at the vertices.

While these Wilson networks are well and good, they can’t be used for conformal block calculations in their present form, for the reason that they are in representations of a non-compact group which are generically infinite-dimensional; in particular, there are an infinite number of representations of the rotation group of the boundary in these representations. Therefore, they defined a class of “cap states” that project the end-points down to a definite irrep of the rotation group.

Then, the prescription for calculating conformal partial waves is: take a Wilson network for the four-point function with the Clebsch-Gordans chosen to have a particular representation in the internal leg, take the end-points on the boundary at the locations of the insertions, and sandwich it in the appropriate cap states. He ended his talk by showing some examples of this prescription in action.

The second evening talk of the day (and the second one to take place in the afternoon) was an overview of holographic renormalization based on lecture notes of Skenderis by Subramanya Hegde (IISER-Thiruvananthapuram). We can compactify a $d+1$ dimensional spacetime such that we have a smooth non degenerate metric on the compactified manifold. Such a conformal compactification induces a conformal class of metrics on the boundary. In particular for AdS spacetimes, the conformal class is that of conformally flat spacetimes. In this set up, one can do an isometry tranformation on the bulk $d+1$ dimensional $AdS$, which corresponds to a scaling transformation in the boundary theory. This connection allows us to associate the radial direction of the bulk with differnt energy scales in the boundary theory.

The talk started with discussions on the UV/IR connection in holography, conformally compact manifolds, asymptotically locally AdS spacetimes, and the Fefferman-Graham expansion. With the stage set, one wants to calculate renormalized boundary correlators using bulk asymptotics via AdS/CFT. If one naively tries to make the identification

$\langle O(x) \rangle =\left. \frac{\delta S_{on-shell}}{\delta\phi_{(0)}(x)}\right\vert_{\phi_{(0)}=0}$

one sees that the correlation functions diverge, essentially because the on-shell action is divergent. We also noted that the variational problem in the bulk is often ill defined.

We need to introduce a cut-off along the radial direction, say at $\rho_0=\epsilon$ and add counterterms carefully to extract meaningful correlation functions. We use the Fefferman-Graham expansion to write a generic field ${\cal F}(x,\rho)$ as

${\cal F}(x,\rho )= \rho ^m \left (f_{(0)}(x)+ \rho f_{(2)}(x) + \cdots + \rho ^n (f_{(2n)}(x)+\log \rho ~{\tilde f} _{(2n)}(x)\right)$

and then regularize and renormalize the action, $S$, and the correlators, order by order. For example, for a scalar, $\phi$, whose boundary dual has a scaling dimension $\Delta$, one has relations of the form

$\bar{\phi} \to \langle O(x) \rangle _s = \frac{1}{\sqrt{g_{(0)}(x)}}\frac{\delta S_{ren}}{\delta \phi_{(0)}(x)} \sim \phi_{(2\Delta - d)}(x)$

$\langle O(x_1) \ldots O(x_n) \rangle \sim \left. \frac{\delta \phi_{(2\Delta-d)}(x_1)}{\delta\phi_{(0)}(x_2)\ldots \phi_{(0)}(x_n)} \right \vert_{\phi_{(0)}=0}$

There are similar relations for the other fields in the theory.

Subbu then proceeded to illustrate these general comments with a concrete example of the renormalization of a scalar field in AdS${}_{d+1}$. After the example, an interesting discussion arose as to whether the CFT does indeed live at the boundary of the AdS, and interpretation of the radial direction of the bulk as the indicator of the energy scales on the boundary. After these intriguing but inconclusive discussions, Pranjal Nayak (TIFR) proceeded to talk about their proposal of holographic renormalization which is more in the spirit of Wilsonian RG. The evening session then concluded after almost four hours!