We consider sphere partition functions of TT deformed large N conformal field theories in d = 2, 3, 4, 5 and 6 dimensions, computed using the flow equation. These are shown to non-perturbatively match with bulk computations of AdS$_{d+1}$ with a finite radial cut-off. We then demonstrate how the flow equation can be independently derived from a regularization procedure of defining TT operators through a local Callan-Symanzik equation. Finally, we show that the sphere partition functions, modulo bulk-counterterm contributions, can be reproduced from Wheeler-DeWitt wave functions.

/pdf/sphere-partition-functions-cut-off-ads-33501fgckq.pdf

Sphere partition functions & cut-off AdS

We investigate the “
 $$ T\overline{T} $$
 ” deformations of two-dimensional supersymmetric quantum field theories. More precisely, we show that, by using the conservation equations for the supercurrent multiplet, the $$ T\overline{T} $$
 deforming operator can be constructed as a super-symmetric descendant. Here we focus on $$ \mathcal{N} = \left(1,0\right) $$
 and $$ \mathcal{N} = \left(1,1\right) $$
 supersymmetry. As an example, we analyse in detail the $$ T\overline{T} $$
 deformation of a free $$ \mathcal{N} = \left(1,0\right) $$
 supersymmetric action. We also argue that the link between $$ T\overline{T} $$
 and string theory can be extended to su-perstrings: by analysing the light-cone gauge fixing for superstrings in flat space, we show the correspondence of the string action to the $$ T\overline{T} $$
 deformation of a free theory of eight $$ \mathcal{N} = \left(1,1\right) $$
 scalar multiplets on the nose. We comment on how these constructions relate to the geometrical interpretations of $$ T\overline{T} $$
 deformations that have recently been discussed in the literature.

/pdf/on-t-overline-t-deformations-and-supersymmetry-3zgubaf0ez.pdf

On T\overline{T} deformations and supersymmetry

Quantum field theory in zero spatial dimensions has a rich infrared landscape and a universal ultraviolet, inverting the usual Wilsonian paradigm. For holographic theories this implies a rich landscape of asymptotically AdS2 geometries. Isolating the interiors of these spacetimes suggests a study of the analog of the TT¯ deformation in quantum mechanics, which we pursue here. An equivalent description of this deformation in terms of coupling to worldline gravity is proposed, and applications to quantum-mechanical systems—including the Schwarzian theory—are studied.

$T\overline{T}$ in AdS$_2$ and Quantum Mechanics

We consider sphere partition functions of TT deformed large N conformal field theories in d=2,3,4,5 and 6 dimensions, computed using the flow equation. These are shown to non-perturbatively match with bulk computations of $AdS_{d+1}$ with a finite radial cut-off. We then demonstrate how the flow equation can be independently derived from a regularization procedure in defining TT operators through a local Callan-Symanzik equation. Finally, we show that the sphere partition functions, modulo bulk-counterterm contributions, can be reproduced from Wheeler-DeWitt wave functions.

/pdf/sphere-partition-functions-and-cut-off-ads-43vgrponv0.pdf

Sphere partition functions and cut-off AdS

This is a pedagogical review on $\mathrm{T}\overline{\mathrm{T}}$ deformation of two dimensional quantum field theories. It is based on three lectures which the author gave at ITP-CAS in December 2018. This review consists of four parts. The first part is a general introduction to $\mathrm{T}\overline{\mathrm{T}}$ deformation. Special emphasises are put on the deformed classical Lagrangian and the exact solvability of the spectrum. The second part focuses on the torus partition sum of the $\mathrm{T}\overline{{\mathrm{T}}}$/$\mathrm{J}\overline{\mathrm{T}}$ deformed conformal field theories and modular invariance/covariance. In the third part, different perspectives of $\mathrm{T}\overline{\mathrm{T}}$ deformation are presented, including its relation to random geometry, 2d topological gravity and holography. We summarize more recent developments until January 2021 in the last part.

/pdf/a-pedagogical-review-on-solvable-irrelevant-deformations-of-2403s4k24l.pdf

A pedagogical review on solvable irrelevant deformations of 2D quantum field theory

We construct a solvable deformation of two-dimensional theories with (2, 2) supersymmetry using an irrelevant operator which is a bilinear in the supercurrents. This supercurrent-squared operator is manifestly supersymmetric and equivalent to T (T) over bar after using conservation laws. As illustrative examples, we deform theories involving a single (2, 2) chiral superfield. We show that the deformed free theory is on-shell equivalent to the (2, 2) Nambu-Goto action. At the classical level, models with a superpotential exhibit more surprising behavior: the deformed theory exhibits poles in the physical potential which modify the vacuum structure. This suggests that irrelevant deformations of T (T) over bar type might also affect infrared physics.

T T flows and (2, 2) supersymmetry

We propose a manifestly supersymmetric generalization of the solvable $$ T\overline{T} $$
 deformation of two-dimensional field theories. For theories with (1, 1) and (0, 1) supersymmetry, the deformation is defined by adding a term to the superspace Lagrangian built from a superfield containing the supercurrent. We prove that the energy levels of the resulting deformed theory are determined exactly in terms of those of the undeformed theory. This supersymmetric deformation extends to higher dimensions, where we conjecture that it might provide a higher-dimensional analogue of $$ T\overline{T} $$
 , producing supersymmetric Dirac or Dirac-Born-Infeld actions in special cases.

/pdf/supersymmetry-and-t-overline-t-deformations-2e59zg6ho1.pdf

Supersymmetry and T\overline{T} deformations

The $$ T\overline{T} $$
 deformation of a supersymmetric two-dimensional theory preserves the original supersymmetry. Moreover, in several interesting cases the deformed theory possesses additional non-linearly realized supersymmetries. We show this for certain $$ \mathcal{N} $$
 = (2, 2) models in two dimensions, where we observe an intriguing similarity with known $$ \mathcal{N} $$
 = 1 models in four dimensions. This suggests that higher-dimensional models with non-linearly realized supersymmetries might also be obtained from $$ T\overline{T} $$
 -like flow equations. We show that in four dimensions this is indeed the case for $$ \mathcal{N} $$
 = 1 Born-Infeld theory, as well as for the Goldstino action for spontaneously broken $$ \mathcal{N} $$
 = 1 supersymmetry.

/pdf/non-linear-supersymmetry-and-t-t-like-flows-18h6tlv1fk.pdf

Non-linear supersymmetry and T T¯-like flows

We propose a manifestly supersymmetric generalization of the solvable $T \overline{T}$ deformation of two-dimensional field theories. For theories with $(1,1)$ and $(0,1)$ supersymmetry, the deformation is defined by adding a term to the superspace Lagrangian built from a superfield containing the supercurrent. We prove that the energy levels of the resulting deformed theory are determined exactly in terms of those of the undeformed theory. This supersymmetric deformation extends to higher dimensions, where we conjecture that it might provide a higher-dimensional analogue of $T \overline{T}$, producing supersymmetric Dirac or Dirac-Born-Infeld actions in special cases.

Supersymmetry and $T \overline{T}$ Deformations

The Dirac action describes the physics of the Nambu-Goldstone scalars found on branes. The Born-Infeld action defines a non-linear theory of electrodynamics. The combined Dirac-Born-Infeld (DBI) action describes the leading interactions supported on a single D-brane in string theory. We define a non-abelian analogue of DBI using the $T \overline{T}$ deformation in two dimensions. The resulting quantum theory is compatible with maximal supersymmetry and such theories are quite rare.

/pdf/a-non-abelian-analogue-of-dbi-from-t-overline-t-dwklcwr6py.pdf

Christian Ferko

Papers

T T flows and (2, 2) supersymmetry

Supersymmetry and T\overline{T} deformations

Non-linear supersymmetry and T T¯-like flows

Supersymmetry and $T \overline{T}$ Deformations

A non-abelian analogue of DBI from $T \overline{T}$