[Paper Review] 4d N=2 SCFT and singularity theory Part II: Complete intersection

Bingyi Chen, Dan Xie+3 more|arXiv (Cornell University)|Apr 26, 2016

Nonlinear Waves and Solitons16 citations

TL;DR

This paper classifies three-dimensional isolated weighted homogeneous rational complete intersection singularities (ICIS) that define new 4D N=2 superconformal field theories (SCFTs). Using singularity theory, it derives the mini-versal deformation of these singularities to compute the Coulomb branch spectrum and Seiberg-Witten solution, identifying 303 classes of SCFTs, including infinite families and known gauge theories like D5 quiver theories.

ABSTRACT

We classify three dimensional isolated weighted homogeneous rational complete intersection singularities, which define many new four dimensional N=2 superconformal field theories. We also determine the mini-versal deformation of these singularities, and therefore solve the Coulomb branch spectrum and Seiberg-Witten solution.

Motivation & Objective

• To classify 3D isolated weighted homogeneous rational complete intersection singularities (ICIS) that define 4D N=2 SCFTs.
• To determine the mini-versal deformation of these ICIS to extract physical data such as the Coulomb branch spectrum and Seiberg-Witten curve.
• To establish a systematic correspondence between singularity invariants and physical observables in 4D N=2 SCFTs.
• To identify infinite families and finite classes of new SCFTs, including known gauge theories like D5 quiver models.
• To verify that the rationality condition ∑wi > 1 + d is both necessary and sufficient for defining a consistent SCFT.

Proposed method

• Use weighted homogeneous polynomials f1 = f2 = 0 in C^5 to define 3D ICIS with a C* action corresponding to U(1)R symmetry.
• Apply the (3,0)-form Ω = dz1∧…∧dz5 / (df1∧df2) with C*-charge ∑wi −1 −d to define rational singularities when this charge is positive.
• Compute the Milnor number µ and monomial basis {φα} of the Jacobi module to construct the mini-versal deformation F(λ, zi) = f(zi) + ∑λαφα.
• Assign scaling dimensions to Coulomb branch parameters λα via [λα] = (1−Qα)/(∑wi−1−d) or (d−Qα)/(∑wi−1−d), where Qα is the C*-charge of φα.
• Use the condition ∑wi > 1 + d to ensure rationality and SCFT existence, and classify all such ICIS via combinatorial analysis of monomial degrees.
• Apply constraints from Corollary 5.1 to restrict possible weight types, reducing the classification to finite and infinite families.

Experimental results

Research questions

• RQ1 Which 3D isolated weighted homogeneous rational complete intersection singularities define 4D N=2 SCFTs?
• RQ2 How can the mini-versal deformation of such singularities be used to compute the Coulomb branch spectrum and Seiberg-Witten solution?
• RQ3 What is the complete classification of such ICIS, and how many distinct classes exist?
• RQ4 Which of these singularities correspond to known gauge theories, such as D5 quiver theories?
• RQ5 What are the necessary and sufficient conditions on the weights and degrees for the resulting theory to be superconformal?

Key findings

• A complete classification of 303 classes of 3D rational weighted homogeneous ICIS is achieved, including both finite and infinite families.
• The infinite families include the D5 quiver gauge theory, realized by the singularity (f1, f2) = (z1² + z2² + z3² + z4² + z5²N, z1² + 2z2² + 3z3² + 4z4² + 5z5²N), with SU(2N) gauge group.
• The mini-versal deformation of each ICIS yields the full Coulomb branch parameterization and Seiberg-Witten curve, with scaling dimensions of operators derived from C*-charge and Milnor number.
• For the infinite family (2,2,u,v,1,1), the classification yields three subcases: u=v≥2, u=2≤v, and 2u=v≥4, corresponding to distinct weight types and SCFTs.
• The weight types (w1,…,w5; d) for all valid singularities are shown to match exactly with those in the final list up to coordinate permutation.
• The rationality condition ∑wi > 1 + d is both necessary and sufficient for defining a 4D N=2 SCFT, and all such singularities are fully classified.