The on-shell action of supergravity & the B-side of TsT and single-trace $T\bar T$

This paper proposes specific boundary terms for the supergravity action on $M_3 \times S^3 \times T^4$ spacetimes to render the on-shell action and stress tensor finite, demonstrating that TsT-transformed backgrounds reproduce the thermodynamic and trace flow properties of single-trace $T\bar T$-deformed CFTs.

The Gist

Here is an explanation of the paper "The on-shell action of supergravity & the B-side of TsT and single-trace T $\bar{T}$" using simple language, analogies, and metaphors.

The Big Picture: Fixing a Leaky Boat to Sail New Waters

Imagine you are a physicist trying to understand the universe. You have a very powerful map called Holography (specifically the AdS/CFT correspondence). This map says that a universe with gravity (like a black hole in 3D space) is mathematically equivalent to a universe without gravity (like a flat sheet of quantum particles) living on the edge of that space.

For decades, this map worked perfectly for "AdS" universes (universes with a specific, negative curvature). But recently, physicists discovered a new kind of universe called T $\bar{T}$-deformed. These are universes that are "squished" or "stretched" in a very specific way. They are interesting because they might explain how gravity emerges from quantum mechanics, but they are messy and hard to map.

The author of this paper, Luis Apolo, is trying to fix the map. He wants to show that the "squished" universe (T $\bar{T}$) is actually just a different way of looking at a specific type of gravitational background called a Linear Dilaton background.

The Problem: The "Leaky" Math

In physics, when you calculate the total energy or "cost" of a system (called the Action), you often get infinite numbers. It's like trying to measure the weight of a cloud; if you don't have a container, the numbers go to infinity.

To fix this, physicists add "boundary terms"—think of them as sealant or gaskets around the edge of the universe to stop the math from leaking.

• The Old Sealant: Previously, physicists had a standard gasket for normal AdS universes.
• The New Problem: When they tried to use this standard gasket on the "squished" (T $\bar{T}$) universes, it didn't work. The math still leaked, and the results were wrong.

The Paper's Solution: Apolo proposes a new, custom-made gasket. He adds two specific extra terms to the math equation. One term deals with the shape of space, and the other deals with a mysterious field called the B-field (which is like a magnetic field for strings).

The Journey: The "TsT" Transformation

How do you get from a normal AdS universe to a T $\bar{T}$ universe? You use a recipe called TsT.

• T (T-duality): Imagine folding a piece of paper. If you fold it, the inside becomes the outside.
• S (Shift): You slide one part of the paper relative to the other.
• T (T-duality): You unfold it back.

This recipe transforms a normal universe into a "squished" one. However, the author discovered that this recipe is very sensitive. If you don't adjust the B-field (the magnetic field) correctly before and after the folding and sliding, you don't get the right universe.

The Analogy: Imagine you are baking a cake (the T $\bar{T}$ universe). The TsT recipe is the mixing instructions. But if you don't add the right amount of salt (the B-field adjustment) at the right time, the cake tastes like soap. Apolo figured out exactly how much "salt" to add so the cake tastes right.

The "Chemical Potentials": The Hidden Cost

When Apolo applied his new sealant and the TsT recipe, he found something surprising. The resulting universes had "chemical potentials."

• What is a chemical potential? Think of it as a hidden fee or a tax you have to pay to enter the universe. In physics, it's a value that determines how much energy is needed to add a particle to the system.
• The Discovery: The "squished" universes generated by TsT naturally come with this hidden fee.
• The Twist: Apolo showed that you can waive this fee by performing a "Large Gauge Transformation."
  • Analogy: Imagine you have a toll booth on a bridge. The TsT recipe puts a toll booth there. But Apolo found a secret backdoor (a gauge transformation) that lets you drive through without paying, provided you set the B-field to zero at the center of the universe.

The Results: Why This Matters

Once Apolo fixed the math (added the sealant) and removed the hidden fees (set the B-field correctly), two magical things happened:

1. The Energy Matches: The energy of the gravitational universe (the black hole side) matched perfectly with the energy of the "squished" quantum universe (the T $\bar{T}$ side).
2. The "Trace Flow" Equation: The way the energy flows in these universes followed a specific rule known as the Trace Flow Equation. This is the "fingerprint" of a T $\bar{T}$ universe. It's like finding a specific fingerprint on a crime scene that proves the suspect was there.

The Conclusion: A Unified Map

The paper concludes that:

• The "squished" T $\bar{T}$ universes are not weird, isolated anomalies. They are actually just Linear Dilaton backgrounds (a specific type of gravitational space) that we can reach by folding and sliding (TsT) a normal AdS universe.
• However, to make the math work, we must be very careful with the B-field. If we ignore it, the math breaks. If we fix it, the "holographic dictionary" works perfectly.
• The "on-shell action" (the total cost of the universe) calculated by Apolo matches the "partition function" (the probability of all possible states) of the T $\bar{T}$ theory.

In Simple Terms:
Luis Apolo fixed the leaky math of a specific type of universe. He showed that if you fold and slide a normal universe correctly, and adjust a hidden magnetic field just right, you get a "squished" universe that behaves exactly like the mysterious T $\bar{T}$ theories physicists have been studying. This confirms that these two seemingly different worlds are actually the same thing, just viewed from different angles.

Technical Summary

Here is a detailed technical summary of the paper "The on-shell action of supergravity & the B-side of TsT and single-trace $T\bar{T}$" by Luis Apolo.

1. Problem Statement

The paper addresses the holographic description of the $T\bar{T}$ deformation (an irrelevant deformation of 2D CFTs built from the stress tensor) within the context of supergravity. Specifically, it seeks to resolve two main issues:

1. Finiteness and Variational Principle: The standard supergravity action on $M_3 \times S^3 \times T^4$ spacetimes (where $M_3$ is $AdS_3$ or a linear dilaton background) lacks a well-defined variational principle and yields divergent on-shell actions without specific boundary terms.
2. Chemical Potentials in TsT Transformations: The linear dilaton backgrounds generated by TsT (T-duality + shift + T-duality) transformations of $AdS_3$ are known to reproduce features of single-trace $T\bar{T}$-deformed CFTs. However, these backgrounds inherit non-vanishing chemical potentials (associated with the B-field) from the parent $AdS_3$ solutions. It was unclear how these potentials affect the on-shell action, the Brown-York stress tensor, and whether they can be consistently removed to match the universal partition function of $T\bar{T}$-deformed theories.

2. Methodology

The author employs the following theoretical tools and steps:

• Supergravity Action Construction: The paper proposes a complete set of boundary terms for the NS sector of supergravity on $M_3 \times S^3 \times T^4$. The total action $I$ includes:

  • The bulk action ($I_3$).
  • The Gibbons-Hawking term ($I_{GH}$).
  • Standard counterterms ($I_{ct1}$) proportional to the boundary area.
  • New Boundary Terms: A counterterm ($I_{ct2}$) depending on the field strength $H=dB$ to render the action finite, and a B-field dependent term ($I_B$) to fix the electric charge $Q_e$ and ensure a well-defined variational principle.
  • The proposed action is: $I = I_3 + I_{GH} + 2I_{ct1} + I_{ct2} + I_B$.

• Variational Principle: The variation of the action is analyzed to determine boundary conditions. The inclusion of $I_B$ ensures that the electric charge $Q_e$ (conjugate to the B-field) is fixed, rather than the B-field value itself. This corresponds to a canonical ensemble.

• TsT Transformations: The author applies TsT transformations to $AdS_3$ backgrounds. Crucially, a shift in the B-field ($B \to B - \Gamma du \wedge dv$) is performed after the TsT transformation to control the asymptotic behavior of the fields.

• Evaluation of On-Shell Action and Stress Tensor:

  • The on-shell action is evaluated for both global $AdS_3$ and BTZ black holes, and subsequently for the resulting linear dilaton backgrounds.
  • The Brown-York stress tensor is computed from the variation of the action with respect to the boundary metric.
  • The trace of the stress tensor is checked against the $T\bar{T}$ trace flow equation.

3. Key Contributions

A. Proposal of Boundary Terms

The paper identifies that standard counterterms are insufficient for supergravity on these backgrounds. The author proposes specific boundary terms ($I_{ct2}$ and $I_B$) involving the B-field field strength $H$.

• Significance: These terms render the on-shell action and the Brown-York stress tensor finite for both $AdS_3$ and linear dilaton backgrounds.
• Charge Fixing: The term $I_B$ ensures the electric charge $Q_e$ is fixed, which is physically required as $Q_e$ determines the number of fundamental strings and the central charge of the dual CFT.

B. Role of Chemical Potentials and B-field Shifts

The paper demonstrates that TsT transformations generate linear dilaton backgrounds with non-vanishing chemical potentials ($\nu$) conjugate to $Q_e$.

• Origin: These potentials arise from the value of the B-field at the origin (for global AdS) or the horizon (for black holes) of the parent $AdS_3$ solution.
• Removal: It is shown that these chemical potentials can be eliminated via large gauge transformations of the B-field (specifically by choosing a shift parameter $\Gamma$ such that the B-field vanishes at the origin/horizon).
• Integrability: The paper clarifies that for the charges to be integrable and the variational principle to be well-defined in the deformed space, the shift parameter $\gamma$ of the undeformed B-field must be set to zero. This specific choice ($\gamma=0$) is necessary to recover the correct entropy and energy spectrum of $T\bar{T}$-deformed CFTs.

C. Reproduction of $T\bar{T}$ Features

The author shows that with the correct boundary conditions ($\gamma=0$ and appropriate $\Gamma$):

1. Trace Flow Equation: The Brown-York stress tensor satisfies the $T\bar{T}$ trace flow equation:
   $$T^\mu_\mu = -\frac{3\lambda}{c} \left( T_{\mu\nu}T^{\mu\nu} - (T^\mu_\mu)^2 \right)$$
   where $\lambda$ is the TsT parameter and $c$ is the central charge. This holds for both single-trace and double-trace interpretations (though the supergravity approximation cannot distinguish between them).
2. Partition Function: The Euclidean on-shell action, when chemical potentials are turned off, reproduces the universal partition function of $T\bar{T}$-deformed CFTs in the semiclassical limit:
   $$\log Z \approx \max \left\{ i\pi(\tau - \bar{\tau})E_{vac}(\lambda), \quad -i\pi\left(\frac{1}{\tau} - \frac{1}{\bar{\tau}}\right)E_{vac}\left(\frac{\lambda}{\tau\bar{\tau}}\right) \right\}$$
   The contributions from the ground state and black hole geometries are related by the modular S-transformation characteristic of $T\bar{T}$ theories.

4. Results

• Finite Action: The proposed boundary terms successfully regularize the supergravity action, making it finite and dependent on the deformation parameter $\lambda$.
• Chemical Potential Dependence: The on-shell action initially contains terms proportional to the chemical potential $\nu Q_e$. These terms represent the cost of the B-field shift. By performing a large gauge transformation to set the B-field to zero at the horizon/origin, these terms vanish, leaving the pure $T\bar{T}$ partition function.
• Ground State Energy: The ground state energy derived from the supergravity action matches the known result for single-trace $T\bar{T}$-deformed CFTs:
  $$E_{vac}(\lambda) = -\frac{c}{6\lambda}\left(1 - \sqrt{1-\lambda}\right)$$
• Modular Invariance: The relationship between the thermal AdS (ground state) and BTZ (black hole) contributions in the deformed theory respects the generalized modular invariance $(\tau, \bar{\tau}, \lambda) \to (-1/\tau, -1/\bar{\tau}, \lambda/\tau\bar{\tau})$.

5. Significance

• Holographic Dictionary for $T\bar{T}$: This work provides a rigorous supergravity derivation of the $T\bar{T}$ deformation's thermodynamic properties, bridging the gap between string theory constructions (TsT) and the effective field theory description of $T\bar{T}$.
• Clarification of B-field Dynamics: It resolves the ambiguity regarding the B-field in TsT transformations, showing that specific gauge choices (large gauge transformations) are required to align the supergravity description with the standard $T\bar{T}$ CFT results.
• Universality: The results suggest that the essential features of $T\bar{T}$ holography (trace flow, partition function, modular invariance) are captured by the semiclassical limit of supergravity on linear dilaton backgrounds, independent of whether the deformation is single-trace or double-trace.
• Foundation for Future Work: By establishing the correct boundary terms and variational principle, the paper lays the groundwork for studying more complex deformations and the role of discrete states (which are present in full string theory but absent in the supergravity limit) in the holographic correspondence.

In summary, the paper successfully constructs a finite, well-defined supergravity action for $T\bar{T}$-related backgrounds, demonstrates how to remove spurious chemical potentials via gauge transformations, and proves that the resulting on-shell action and stress tensor precisely reproduce the universal features of $T\bar{T}$-deformed Conformal Field Theories.