[Paper Review] Holomorphic primary fields in free CFT4 and Calabi-Yau orbifolds

Robert de Mello Koch, Phumudzo Rabambi+2 more|arXiv (Cornell University)|May 18, 2017

Black Holes and Theoretical Physics45 references3 citations

TL;DR

This paper derives counting formulae for primary fields in free four-dimensional conformal field theories (CFT4) of scalars, vectors, and matrices using $\mathfrak{so}(4,2)$ characters. It constructs extremal primary fields as holomorphic polynomial functions on permutation orbifolds $\mathbb{C}^{2n}/(\mathbb{C}^2 \times S_n)$, which are shown to be Calabi-Yau manifolds, revealing a ring structure on the space of extremal primaries with palindromic Hilbert series and non-trivial relations encoded in the numerator of the generating function.

ABSTRACT

Counting formulae for general primary fields in free four dimensional conformal field theories of scalars, vectors and matrices are derived. These are specialised to count primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. in terms of relations between the charges under the Cartan subgroup of $SO(4,2)$). The construction of primary fields for scalar field theory is mapped to a problem of determining multi-variable polynomials subject to a system of symmetry and differential constraints. For the extremal primaries, we give a construction in terms of holomorphic polynomial functions on permutation orbifolds, which are shown to be Calabi-Yau spaces.

Motivation & Objective

• To derive general counting formulae for primary fields in free CFT4 using $\mathfrak{so}(4,2)$ representation theory.
• To construct extremal primary fields as holomorphic polynomial functions on symmetric orbifolds $\mathbb{C}^{2n}/(\mathbb{C}^2 \times S_n)$, which are Calabi-Yau spaces.
• To establish a ring structure on the space of extremal primary operators via holomorphic functions on these orbifolds.
• To analyze the Hilbert series of the ring of holomorphic functions, proving its palindromic property and identifying relations among generators and syzygies.
• To connect the algebraic structure of the operator spectrum to symmetric group representations and Gorenstein properties of the orbifold ring.

Proposed method

• Map the construction of composite primary fields in CFT4 to multi-variable polynomial functions $\Psi(x_\mu^I)$ on $\mathbb{R}^{4n}$, subject to Laplace's equation, translation invariance, and $S_n$ permutation symmetry.
• Use a complex structure on $\mathbb{R}^4 \cong \mathbb{C}^2$ to define holomorphic functions in $z,w$ variables, yielding primary fields on the orbifold $\mathbb{C}^{2n}/(\mathbb{C}^2 \times S_n)$.
• Construct extremal primaries as holomorphic functions on $\mathbb{C}^n/S_n$ by restricting to $z$-dependence only, corresponding to a special class of operators.
• Compute the Hilbert series of the ring of holomorphic functions on the orbifold, revealing a palindromic structure linked to symmetric group representations.
• Identify generators and relations (syzygies) of the ring from the numerator of the Hilbert series, with explicit expressions for 7 generators and 6 relations at degree 3.
• Use the group algebra $\mathbb{C}(S_n) \otimes \mathbb{C}(S_n) \otimes \mathbb{C}(S_k) \otimes \mathbb{C}(S_l)$ to describe the representation content of the operator spectrum.

Experimental results

Research questions

• RQ1 How can extremal primary fields in free CFT4 be systematically constructed using holomorphic functions on symmetric orbifolds?
• RQ2 What is the algebraic structure of the ring of extremal primary operators, and how does it relate to the geometry of the underlying orbifold?
• RQ3 Why does the Hilbert series of the ring of holomorphic functions on $\mathbb{C}^{2n}/(\mathbb{C}^2 \times S_n)$ exhibit a palindromic property?
• RQ4 How do the relations among generators of the ring (syzygies) manifest in the numerator of the Hilbert series, and what is their physical and algebraic significance?
• RQ5 What is the role of symmetric group representations in organizing the spectrum of extremal primaries in free CFT4?

Key findings

• Extremal primary fields in free CFT4 are constructed as holomorphic polynomial functions on the orbifold $\mathbb{C}^{2n}/(\mathbb{C}^2 \times S_n)$, which is a Calabi-Yau space.
• The ring of holomorphic functions on this orbifold forms a Gorenstein, Calabi-Yau ring with a palindromic Hilbert series, proven via symmetric group representation theory.
• The Hilbert series for three scalar fields, $Z_3(s,x,y)$, has a numerator with 6 terms encoding 6 relations among 7 generators, and 5 terms encoding relations among the relations (syzygies).
• The generators $G_1, \dots, G_7$ are explicitly identified as symmetric polynomials in $z_{ij}, w_{ij}$, with degrees corresponding to monomials in the Hilbert series.
• The relations include $\chi_1 = 3G_3G_4 - 2G_2G_5 + G_1G_6 = 0$ and $\chi_6 = G_2^7 - G_1G_2^5G_3 + \frac{1}{9}G_2^4G_5G_6 - G_2^4G_4G_7 = 0$, with degrees $s^5x^{5/2}\sqrt{y}$ and $s^{14}x^7$ respectively.
• The syzygy relations, such as $4\chi_5G_2 + \chi_4G_3 = 0$, are explicitly derived and match the positive terms in the numerator of $Z_3(s,x,y)$.