Vertex operator algebras, Higgs branches, and modular differential equations (1707.07679v4)

Abstract: Every four-dimensional ${\cal N}=2$ superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any ${\cal N}=2$ SCFT should obey a finite order modular differential equation. By way of the "high temperature" limit of the superconformal index, this allows the Weyl anomaly coefficient $a$ to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the "Deligne-Cvitanovi\'c exceptional series" of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class ${\cal S}$ theories, and ${\cal N}=4$ super Yang-Mills with $\mathfrak{su}(n)$ gauge group for small-to-moderate values of $n$.

• The paper establishes that the associated VOA variety, defined by its reduced C₂ algebra, aligns with the Higgs branch as a Poisson variety.
• It demonstrates that Schur indices of 4D SCFTs satisfy finite-order modular differential equations, linking modular forms to VOA null vectors.
• Key examples include Deligne–Cvitanovic and Argyres–Douglas theories, highlighting novel correspondences between algebraic structures and physical moduli.

Vertex Operator Algebras, Higgs Branches, and Modular Differential Equations

The paper explores intricate relationships between vertex operator algebras (VOAs), Higgs branches in the context of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs), and modular differential equations. VOAs serve as a robust algebraic invariant tied to each four-dimensional SCFT, preserving special properties of the system but still requiring a grasp of their connection with established physical quantities.

Proposed Relationship Between VOAs and Higgs Branches

The authors propose a correspondence between Higgs branches, which are algebraic geometric entities, and VOAs related to four-dimensional SCFTs. The hypothesis is that the associated variety of VOA, defined by its reduced $C_2$ algebra, matches the Higgs branch of the parent SCFT as a Poisson variety.

• Higgs Chiral Ring: The paper identifies the Higgs chiral ring, a commutative, associative algebra, with a reduced version of the $C_2$ algebra. This extends the identification of such a complex structure with the algebraic variety of the SCFT's moduli space, suggesting the variety of the associated VOA matches the Higgs branch moduli space.

Constraints and Null Vectors

Through modularity concepts, the paper ventures into modules over VOAs, exploring null vectors which indicate the presence of modular differential equations acting on the Schur index. The Schur index of $\mathcal{N}=2$ SCFTs aligns with modular forms, supported by conjectural pathways and established results.

• Modular Differential Equations: The authors also assert that Schur indices should satisfy finite-order modular differential equations, a statement pivotal for understanding the index’s transformation properties and high-temperature behavior.

Deligne-Cvitanovic Series

The paper discusses significant examples known as the Deligne-Cvitanovic exceptional series, linked with rank-one SCFTs via affine current algebras. Here, key dynamics of unitarity bounds are illustrated, connecting Weyl anomaly coefficients and flavor central charges to VOAs.

• Null Vectors and Modular Constraints: Saturated unitarity bounds imply the presence of null vectors in the vacuum module of VOAs, driving the modular equation statistics bound with affine algebras. The resulting differential equations solidify pathway connections between anomalous dimensions of the $4d$ theory and $2d$ characters.

Argyres-Douglas Examples

Various Argyres-Douglas theories are analyzed, conjectured to map into Virasoro and affine current algebras, with consideration given to their central charges and Higgs branch variabilities. The intricacy of ensuring such mappings hinges on modular solutions. The authors present intriguing computations regarding the Schur index and its modular differential equations for specific classes of AD theories.

Implications and Future Directions

Finally, the paper speculates on broader impacts on AI developments and theoretical frameworks in physics, hinting at potential classes of VOAs tethered to novel subclasses of SCFTs for future exploration.

To summarize, the authors adeptly intertwine foundational constructs of vertex algebras with physical representations in SCFTs, providing deep insights into modular structures that bridge multi-dimensional field theories and algebraic data. The blend of rigorous mathematics with speculative physics seen here offers a compelling narrative likely to persist in scholarly and theoretical discourse.