This dissertation contains two parts of study on the Friedan states and on the conformal bootstrap with symmetries. The Friedan states are potential physical boundary states of the free boson. However, they exhibited a continuous spectrum in the open string sector, in contrast to more standard examples. The explicit expressions for the density of states of the Friedan states is obtained. Some pathologies and possible contradictions of these states are explored. The $g$ functions of the Friedan states are shown to be infinite, suggesting an infinite number of degrees of freedom in the theory. The modular conformal bootstrap using the moment curve approach and focused on two-dimensional conformal field theory with $c\geq 1$ is discussed. A refinement is made to incorporate conserved currents. Specializing to theories with symmetries, the anyon partition functions, which transform as a vector under modular transformations, are introduced. Applying the moment curve bootstrap to these leads to bounds on the minimum conformal weights in different sectors. In the analytical approach, the correlated bounds between different anyon sectors are obtained, with the abelian symmetries, $\mathbb{Z}_2$ and $\mathbb{Z}_3$ , and with the non-abelian symmetry, $S_3$ . The explicit bounds on the free boson theories with the presence of the symmetries are then calculated and compared with the bootstrap bounds. Finally, some numerical bootstrap results with $\mathbb{Z}_2$ and $\mathbb{Z}_3$ symmetries are presented.This dissertation contains two parts of study on the Friedan states and on the conformal bootstrap with symmetries. The Friedan states are potential physical boundary states of the free boson. However, they exhibited a continuous spectrum in the open string sector, in contrast to more standard examples. The explicit expressions for the density of states of the Friedan states is obtained. Some pathologies and possible contradictions of these states are explored. The $g$ functions of the Friedan states are shown to be infinite, suggesting an infinite number of degrees of freedom in the theory. The modular conformal bootstrap using the moment curve approach and focused on two-dimensional conformal field theory with $c\geq 1$ is discussed. A refinement is made to incorporate conserved currents. Specializing to theories with symmetries, the anyon partition functions, which transform as a vector under modular transformations, are introduced. Applying the moment curve bootstrap to these leads to bounds on the minimum conformal weights in different sectors. In the analytical approach, the correlated bounds between different anyon sectors are obtained, with the abelian symmetries, $\mathbb{Z}_2$ and $\mathbb{Z}_3$, and with the non-abelian symmetry, $S_3$. The explicit bounds on the free boson theories with the presence of the symmetries are then calculated and compared with the bootstrap bounds. Finally, some numerical bootstrap results with $\mathbb{Z}_2$ and $\mathbb{Z}_3$ symmetries are presented.

Jan 1, 20250 citations

Quantum Symmetry in Conformal Field Theory by Hamiltonian Methods

L. D. Faddeev

NATO ASI series. Series B : PhysicsJan 1, 199223 citations

Embedding Space Construction of Conformal Field Theory From First Principles

Sutirtha Mukherjee

This article presents a systematic development of conformal field theory from first principles in quantum field theory, emphasizing complete derivations of the conformal group in arbitrary dimensions from the conformal Killing equation, proving its isomorphism to $\mathrm{SO}(d+1,1)$ , and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra $[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}$ from infinitesimal generators $\ell_n = -z^{n+1}\partial_z$ , computing the central charge explicitly for free theories (boson $c=1$ , Majorana fermion $c=1/2$ , $bc$ -ghosts $c = 1-3(2\lambda-1)^2$ ) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states $|h,\bar{h}\rangle$ , deriving the transformation law $T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]$ that reveals the geometric origin of the conformal anomaly.This article presents a systematic development of conformal field theory from first principles in quantum field theory, emphasizing complete derivations of the conformal group in arbitrary dimensions from the conformal Killing equation, proving its isomorphism to $\mathrm{SO}(d+1,1)$, and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra $[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}$ from infinitesimal generators $\ell_n = -z^{n+1}\partial_z$, computing the central charge explicitly for free theories (boson $c=1$, Majorana fermion $c=1/2$, $bc$-ghosts $c = 1-3(2\lambda-1)^2$) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states $|h,\bar{h}\rangle$, deriving the transformation law $T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]$ that reveals the geometric origin of the conformal anomaly.

Feb 2, 20260 citations

QUANTUM SYMMETRY IN CONFORMAL FIELD THEORY BY HAMILTONIAN METHODS

L. D. Faddeev

WORLD SCIENTIFIC eBooksFeb 14, 20160 citations

g

functions of the Friedan states are shown to be infinite, suggesting an infinite number of degrees of freedom in the theory. The modular conformal bootstrap using the moment curve approach and focused on two-dimensional conformal field theory with

c\geq 1

is discussed. A refinement is made to incorporate conserved currents. Specializing to theories with symmetries, the anyon partition functions, which transform as a vector under modular transformations, are introduced. Applying the moment curve bootstrap to these leads to bounds on the minimum conformal weights in different sectors. In the analytical approach, the correlated bounds between different anyon sectors are obtained, with the abelian symmetries,

\mathbb{Z}_2

and

\mathbb{Z}_3

, and with the non-abelian symmetry,

S_3

. The explicit bounds on the free boson theories with the presence of the symmetries are then calculated and compared with the bootstrap bounds. Finally, some numerical bootstrap results with

\mathbb{Z}_2

and

\mathbb{Z}_3

symmetries are presented.

\mathrm{SO}(d+1,1)

, and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra

[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}

from infinitesimal generators

\ell_n = -z^{n+1}\partial_z

, computing the central charge explicitly for free theories (boson

c=1

, Majorana fermion

c=1/2

bc

-ghosts

c = 1-3(2\lambda-1)^2

) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states

|h,\bar{h}\rangle

, deriving the transformation law

T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]

that reveals the geometric origin of the conformal anomaly.