In [1], we argued that the (0, 2) heterotic string gives rise in spacetime to left and right-moving symmetric product CFT’s. In this paper we confirm this claim by showing that it computes correlation functions in these CFT’s.

Israel

Amit Giveon

United States

Akikazu Hashimoto

David Kutasov

CFT correlators from (0, 2) heterotic string

We show that 1 + 1 dimensional vacua of the (0$\textit{,}$ 2) heterotic string have many properties in common with more recently studied physical systems. The free theory exhibits a symmetric product CFT structure. The single-string Hilbert space splits into sectors labeled by the string winding, $\textit{w}$, that contain purely left-moving (for $\textit{w >}$ 0) and right-moving (for $\textit{w <}$ 0) excitations. These sectors exhibit infinite-dimensional left and right-moving symmetry algebras, respectively, with central extensions that depend on $\textit{w}$. String interactions between left and right-movers appear to lead to a $$ T\overline{T} $$ deformation of the free theory.

$$ \mathcal{N} $$ = 2 heterotic strings revisited

We revisit the problem of computing the determinant of Klein-Gordon operator ∆ = −∇2 + M 2 on Euclideanized AdS 3 with the Euclideanized time coordinate compactified with period β, H 3/Z, by explicitly computing its eigenvalues and computing their product. Upon assuming that eigenfunctions are normalizable on H 3/Z, we found that there are no such eigenfunctions. Upon closer examination, we discover that the intuition that H 3/Z is like a box with normalizable eigenfunctions was false, and that there is, instead, a set of eigenfunctions which forms a continuum. Somewhat to our surprise, we find that there is a different operator   <math> <mover> <mtext>∆</mtext> <mo>~</mo> </mover> </math>  $$ \overset{\sim }{\Delta } $$  = r 2∆, which has the property that (1) the determinant of ∆ and the determinant of r 2∆ have the same dependence on β, and that (2) the Green’s function of ∆ can be spectrally decomposed into eigenfunctions of   <math> <mover> <mtext>∆</mtext> <mo>~</mo> </mover> </math>  $$ \overset{\sim }{\Delta } $$ . We identify the   <math> <mover> <mtext>∆</mtext> <mo>~</mo> </mover> </math>  $$ \overset{\sim }{\Delta } $$  operator as the “weighted Laplacian” in the context of warped compactifications, and comment on possible applications.

France

Soumangsu Chakraborty

Brazil

Horatiu Nastase

Path integral of free fields and the determinant of Laplacian in warped space-time

We derive a compact formula for the one-loop, bosonic string partition function of Euclideanized   <math> <msub> <mi>J</mi> <mn>3</mn> </msub> <msub> <mover> <mi>J</mi> <mo>¯</mo> </mover> <mn>3</mn> </msub> </math>  $$ {J}_3{\overline{J}}_3 $$  deformed AdS 3 with periodic Euclidean time as an integral transform of the partition function of the undeformed Euclideanized AdS 3. Such a deformation is interpretable as an irrelevant “single-trace   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ T\overline{T} $$  deformation” of the boundary. We will do this by first establishing a formal procedure to compute a worldsheet torus zero point function for an exactly marginal   <math> <mi>J</mi> <mover> <mi>J</mi> <mo>¯</mo> </mover> </math>  $$ J\overline{J} $$  deformation of a sigma model with U(1) L  × U(1) R  global symmetry. We then describe how this procedure is implemented on SL(2, R) sigma model and its Euclidean continuation. Finally, we describe the embedding of the deformed SL(2, R) torus amplitude into critical string theory and interpret the result as the leading perturbative contribution to the thermal partition function of the deformed theory.

Thermal partition function of   <math> <msub> <mi>J</mi> <mn>3</mn> </msub> <msub> <mover> <mi>J</mi> <mo>¯</mo> </mover> <mn>3</mn> </msub> </math>  $$ {J}_3{\overline{J}}_3 $$  deformed AdS 3

Aspects of superstring theory on deformed BTZ black holes, formed near k NS5 branes by p fundamental strings, and single-trace   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ T\overline{T} $$  holography, are presented. The excitation energy of a singly wound long string plus its contribution to the energy of the black hole, 1/p fraction of the black-hole energy, is the same as that in   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ T\overline{T} $$  deformed CFT2 with c = 6k. This supports the claim that the black hole can be thought of as a state in a symmetric product of p CFT2’s of central charge 6k, where the black-hole energy is split equally among all p factors. A comment on superstring theory on deformed global AdS3 is also presented.

On string theory on deformed BTZ and   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \textrm{T}\overline{\textrm{T}} $$

Time-like linear dilaton backgrounds admit a classical solution that describes a closed folded string that is created at an instant. We refer to such strings as Instant Folded Strings (IFS). We study an exact worldsheet CFT description of an IFS that involves two vertex operators which describe two open string modes that propagate on a time-like FZZT-brane, which plays the role of a regulator to the IFS. We take advantage of this description to calculate the most basic quantity associated with IFSs — their production rate. Some implications of this calculation to stringy cosmology and black hole interior are briefly discussed.

Nissan Itzhaki

Uri Peleg

A worldsheet description of instant folded strings

We consider the weighted average of a two dimensional conformal field theory (CFT), whose target space is <math><msup><mi>T</mi><mn>2</mn></msup></math>, over its Narain moduli space. We take as the weighing function the integral kernel which gives rise to <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> deformation when applied to the world sheet moduli data of the partition function viewed as vacuum amplitude when the world sheet is a torus. We compute the smeared partition function where this kernel is applied to the target space moduli. Smearing the partition function over the parameter space of a field theory generally leads to the breakdown in the ability to write the partition function as a sum over Boltzmann factor with unit coefficients. The weight function inspired by the <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> deformation appears to be an exception to this general expectation. We show that this smearing leads to a marginal deformation corresponding to the overall rescaling of the target space <math><msup><mi>T</mi><mn>2</mn></msup></math>.

India

Weighted average over the Narain moduli space as a <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> deformation of the CFT target space

We show that the   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ T\overline{T} $$  deformation of conformal field theories whose entropy grows as S(E) ∼ E  γ  for γ &gt; 1/2 exhibits negative specific heat in its microcanonical thermodynamic function S(ℇ). We analyze the large N symmetric product CFT as a concrete example of a CFT with this property and compute the thermodynamic functions such as S(ℇ) and ℇ(T). The negative specific heat in the microcanonical data is interpreted as signaling the first order phase transition when the system is coupled to a heat bath.

Comments on the negative specific heat of the   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ T\overline{T} $$  deformed symmetric product CFT

We derive the geodesic equation for determining the Ryu-Takayanagi surface in AdS 3 deformed by single trace   <math> <mi>μT</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mu T\overline{T} $$  +   <math> <msub> <mi>ε</mi> <mo>+</mo> </msub> <mi>J</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ {\varepsilon}_{+}J\overline{T} $$  +   <math> <msub> <mi>ε</mi> <mo>−</mo> </msub> <mi>T</mi> <mover> <mi>J</mi> <mo>¯</mo> </mover> </math>  $$ {\varepsilon}_{-}T\overline{J} $$  deformation for generic values of (μ, ε + , ε  − ) for which the background is free of singularities. For generic values of ε  ± , Lorentz invariance is broken, and the Ryu-Takayanagi surface embeds non-trivially in time as well as spatial coordinates. We solve the geodesic equation and characterize the UV and IR behavior of the entanglement entropy and the Casini-Huerta c-function. We comment on various features of these observables in the (μ, ε + , ε  − ) parameter space. We discuss the matching at leading order in small (μ, ε + , ε  − ) expansion of the entanglement entropy between the single trace deformed holographic system and a class of double trace deformed theories where a strictly field theoretic analysis is possible. We also comment on expectation value of a large rectangular Wilson loop-like observable.

Entanglement entropy for   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$ ,   <math> <mi>J</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{J}\overline{\mathrm{T}} $$ ,   <math> <mi>T</mi> <mover> <mi>J</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{J}} $$  deformed holographic CFT

We derive a formula for the torus partition sum of the symmetric product of <math><mi>T</mi><mover><mrow><mi>T</mi></mrow><mrow><mo>¯</mo></mrow></mover></math> deformed CFT's, using previous work on long strings in (deformed) <math><mi>A</mi><mi>d</mi><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math>, and universality. The result is given by an integral transform of the partition function for the block of the symmetric product, summed over its Hecke transforms, and is manifestly modular invariant. The spectrum is interpretable as a gas of multiply wound long strings with a particular orientation.