# Space of CFTs

## Main Question or Discussion Point

Take a cylindrical worldsheet parametrized by σ1 and σ2. The first goes along the length of the cylinder, and the other along its circumference.

The Hilbert space of states in string theory is defined by functions on the circular boundaries φ(σ1) and the evolution of this state along the cylinder is given by

φ(σ1) → e-βHφ(σ1)

where e-βH is the density operator of thermodynamics. These circular boundaries can be mapped to a puncture using conformal transformations, and states in the Hilbert space correspond to operator insertion at the location of the puncture. If we define

τ = β + iθ

where θ is the range of σ1, then the partition function is

tr qp+qp-

where q = e2πiτ. The transformations of τ of the form

τ → (aτ + b)/(cτ + d)

are important. The lagrangian for the CFT is of the form

L = √g gij dxi ∧ *dxj

where dxi → dxi + iσaθai

Here account must be taken of kappa symmetry. The various CFTs obtained in this way form a web, which only in certain limiting regions of this web take the form of recognizable CFTs, like the type IIA, type IIB, etc. Open string theory in a sense has super Yang-Mills theory as its low energy limit. What are some current proposals about how to describe this web of CFTs?

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Traditionally one regards the allowed superstring worldsheet theories as the 2D superconformal field theories with central charge 15. I think one can motivate this by starting with the Polyakov action for a string, which actually has worldsheet gravity coupled to the worldsheet fields, but in the critical dimension (no conformal anomaly) the worldsheet gravity decouples, leaving only the CFT.

I don't know what the latest perspectives on the space of all string CFTs are, but you could try recent papers that cite Witten 1992.

As for the larger landscape of string vacua in general, from Lubos Motl I picked up the idea that F-theory compactifications might be the most general description, connected by dualities to all the other islands in the string landscape.

Take a cylindrical world sheet, which corresponds to a propagating closed string. A state like α+α-|0⟩ corresponds to a wavefunction φ(σ1).

This state evolves along the cylinder according to φ(σ1) → e-βHφ(σ1), where the Hamiltonian for open strings is

H = L0 - 1

and the Hamiltonian for closed strings is

H = L0 + L'0 - 2

This cylinder can be mapped to a plane by the transformation z → exp (σ1 + iσ2). The state at the end of the cylinder corresponds to the insertion of a local operator O at the point shown in the figure.

In the theory of point particles, local operators can be inserted at the endpoints of the world line, but not in its interior.

There is a sense in which local operators on the boundary like O1 of the world line correspond to photons, and local operators inserted in the interior of the world-line like O would corresponds to gravitons. In the theory of point particles, it is not possible to insert local operators in the interior like O

In the theory of strings, we have a world sheet instead of a worldline. In string theory, it is possible to insert local operators in the interior. The closed strings correspond to local operator insertions in the interior of the world sheet, and the open strings correspond to local operator insertions on the boundary.

O' is an operator corresponding to the state of a photon, and the insertion of O corresponds to the state of a graviton.

My question is, what does the open-closed duality say about this kind of picture? And what is its relevance to the space of CFTs?

to elaborate on the previous, consider a world sheet of the following type.

A state on the edge parametrized by σ1 is built out of the modes $\alpha^i_{\ n}$ and $\tilde{\alpha}^j_{\ m}$. The state of a photon can be created as $\alpha^i_{\ -1}|0\rangle$. Normally states are defined on edges, but as explained in the previous post, they are equivalent to a picture of the following type

The state αi-1|0> at the point marked x corresponds to the insertion of a local operator at that point. A graviton state is of the form $(\alpha^i_{\ -1}\alpha^j_{\ -1} - \alpha^j_{\ -1}\alpha^i_{\ -1}) |0\rangle$ and these would correspond to operator insertions in the interior. The virasoro generators are defined in terms of the alphas as

Lm = 1/2 :αim-nαn, i:

These are the fourier modes of the energy momentum tensor. As explained in the previous post, the Hamiltonian for the open string is H = L0 - 1. This generates evolution of the state along the strip. The mass squared operator for open strings is

M2 = ∑ αi-nαn, i

what is the significance of the mass squared formula? and what is the connection with M2 = pipi?

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Consider a strip as shown below

A state $\alpha_{ \ \ -1}^{ j} |0\rangle$ evolves along the strip as αj-1|0〉 → e-βHαj-1|0〉. As a function of σ1, the state is

〈σ1j-1|0〉~ αj-1 cos ( -σ1)

Another description of the evolution of the state is in terms of the propagation function G. The propagation function G(1, 2) between the points 1 and 2 in the following picture is the correlator tr O1O2ρ of the product O1O2 of two operators.

In terms of G, the evolution of the state is given by

$\langle \tilde{\sigma}^1 | \mathrm{e}^{-\beta \mathrm{H}} \alpha_{-1}^{\ \ j} |0\rangle = \int d\sigma^1 G(\tilde{\sigma}^1, \sigma^1) \langle \sigma^1|\alpha_{-1}^{\ \ j}|0\rangle$

where the tilde sigma is on the other edge of the strip.The function G can be expressed in terms of the thermodynamic density operator as

$G(\tilde{\sigma}^1, \sigma^1) = \langle \tilde{\sigma}^1|\mathrm{e}^{-\beta \mathrm{H}} | \sigma^1 \rangle$

What is the connection between these two types of descriptions and how is it related to the previous posts?

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Consider a sheet as shown below

The propagation function $G(\tilde{\sigma}^1, \sigma^1)$ is

$\mathrm{tr} \ |\sigma^1\rangle \langle\tilde{\sigma}^1| \ \mathrm{e}^{-\beta H}$

The state $|\tilde{\sigma}^1\rangle$ as a function of $\sigma^1$ is $\langle \sigma^1 | \tilde{\sigma}^1\rangle = \delta(\tilde{\sigma}^1 - \sigma^1 )$. the path integral is

$\int_{\sigma^1}^{\tilde{\sigma}^1} e^{iS} = \langle \tilde{\sigma}^1|\mathrm{e}^{-\beta H}| \sigma^1 \rangle$

The trace of the density operator gives the partition function, so

$\int d\sigma^1 \langle \sigma^1|\mathrm{e}^{-\beta H}| \sigma^1 \rangle = \int d\sigma^1 \int_{\sigma^1}^{\sigma^1} e^{iS}$

involves a path integral with periodic boundary conditions with period β = 1/T where T is the temperature. since the upper and lower limits are identified by the trace. Consider a closed string as shown below.

Because $\sigma^1$ is around a circle, the path integral always involves fields periodic in $\sigma^1$. Is this periodicity connected to the periodic boundary conditions mentioned before?

how can one picture the web of CFTs? Consider the following picture

where there is a kind of "proliferation" of CFTs from the left to right, where the point x is a description by transformations of the kind g → 1/g. Is this approximately how it works?

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In what way are the states on the edges mapped to points with operators?|σ'1〉= ∫ dσ1 δ(σ1 - σ'1) |σ1〉 We have the evolution of the state

〈σ1j-1|0〉 → e-βH〈σ1j-1|0〉

As a function of σ1, the state is〈σ1j-1|0〉~ αj-1 cos ( -σ1). The propagator can be decomposed as

G(σ1, σ'1) = ∑ 〈σ'1|n〉〈n|σ1〉e-βEn

We have G(σ1, σ'1) = 〈σ'1|e-βH1〉. In terms of G, the evolution is given by

〈σ1|e-βHαi-1|0〉= ∫ dσ1 〈σ1i-1|0〉G(σ1, σ'1)

The states are defined on the edges parametrized by sigma. What is the precise manner in which these states are mapped to points with operators?

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The Lagrangian for the conformal field is the following. The solutions are self dual 1-forms dαi-1 which satisfy dαi-1 = -*dαi-1

L = 1/2 giji-1 ∧ *dαj-1

The local operator corresponding to the emission of a graviton is dαi-1 ∧ *dαj-1. The local operator corresponding to the emission of a photon is dαi-1 and OCFT is the local operator conformal field. The correlators for these operators are

〈dαi-1 〉= ∫ dOCFTi-1 e-S[OCFT]

〈dαi-1 ∧ *dαj-1〉= ∫ dOCFTi-1 ∧ *dαj-1 e-S[OCFT]

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Take a d-brane of charge 1/m and a string of charge m. The world sheets are shown below.

The action for the d-brane is ∫e√(g + B) and the action for the string is ∫ √g = S. The fields on the sheet satisfy

d*dαi-1 = 0

The weyl transformation is gij → eΦgij. In kaluza-klein theory, we think of the field Φ as equal to g55. The fields Φ, g and B are the massless modes of the closed string. B is the kalb-ramond field. Is it possible to think of various string theories as stretched between these branes? How are the massless modes related?

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i-1 ∧ αi-1 + αi-1 ∧ αj-1 ∧ αk-1 is the action for chern-simons theory. To elaborate on the previous post, imagine a cylindrical spacetime

g55 is the dilaton Φ. x5 is the circular dimension. The fifth dimensional momentum p5 is what corresponds to the electric charge Q. The electromagnetic field corresponds to gi5. Imagine a string wrapped around the circular dimension:

The number of windings is like a magnetic charge. The electric charge is quantized due to the circular x5.

Q = n/R

This Q is equal to R ∫ ∂x5/∂τ, and the magnetic charge is 1/R ∫ ∂x5/∂σ. The magnetic and electric charges are interchanged under

R → 1/R

The photon which corresponds to gi5 is interchanged with Bi5, where B is the kalb-ramond field. d-branes are in a sense magnetic charges and strings are electric charges of the kalb-ramond field.

The photon can be thought of as an open string with its end points stuck on a d-brane, as shown in the following figure.

A graviton is a closed string in the bulk away from the d-brane. d-branes are heavy objects with a cloud of strings around it and attached to it. The coupling constant g is the probability to emit a string. When g is small strings are light objects without any cloud because g is small. As g goes to 1/g and becomes large, the light strings develop a cloud and become d-branes, and the d-branes become light because it is harder to develop a cloud of strings because the strings have become heavy. d-branes are magnetic charges. The picture below shows the evaporation of a d-brane.

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berkeman
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