The latest work from IIT Madras researchers Suresh Govindarajan and Jagannath Santara digs into the math‑heavy heart of two‑dimensional quantum theories. It sits at the crossroads of abstract number theory and the physics of tiny, symmetrical universes. Their arena is the holomorphic modular bootstrap, a method born to classify rational conformal field theories by reading their symmetry fingerprints—the modular S and T matrices—like a cosmic map of possible realities. It’s a story about how deep mathematics can illuminate the very space of physical possibilities, and how a clever reframing can turn a stubborn classification problem into a playground for new ideas.
In practical terms, the authors offer a new way to generate and test candidate theories starting from a known RCFT, rather than grinding through equations from scratch. They blend two strands of the field: the traditional modular‑differential equation (MLDE) approach, which anchors everything to a fixed count of characters and a Wronskian index, and a modern vector‑valued modular form (VVMF) perspective, which treats the whole family of RCFT characters as a single, multi‑component object that transforms together under modular symmetries. The result is a recipe for producing fresh admissible solutions—new potential theories—by reusing the symmetry backbone of an existing RCFT. The IIT Madras team demonstrates the idea across examples with up to six characters, showing that these techniques can discover, repackage, and sometimes even extend the known zoo of RCFTs.
What matters beyond the math is the bigger picture: this is about widening the “grammar” we use to describe the possible two‑dimensional quantum worlds. If the old bootstrap was a way to prune a forest of possibilities by symmetry and positivity, this new approach gives us a way to grow that forest from familiar trees, using the same roots to sprout new branches. And because the work builds on well‑established mathematics—vector‑valued modular forms, S and T matrices, and the mysterious Wronskian index—it sits at a uniquely productive meeting point of physics, math, and even potential applications in quantum information and string theory.
A new way to search the space of theories
Rational conformal field theories, or RCFTs, are the tidy, highly symmetric cousins of the wild, general conformal field theories. They come with a small set of primary fields, each with a definite “weight” or dimension, and a handful of characters that encode the theory’s spectrum. In the classical holomorphic bootstrap, one fixes the number of characters n and a Wronskian index ℓ, then writes down an n‑th order modular differential equation whose solutions are constrained by modularity. The real test is whether these solutions have non‑negative integer coefficients in their q‑series, which would signal a physical, unitary RCFT rather than a purely mathematical curiosity. This approach has yielded clean classifications for some small, neat cases, but quickly runs into a brick wall as n grows or ℓ shifts around.
Govindarajan and Santara pull a different lever. They start with the notion that each RCFT’s characters can be packaged into a vector X(τ) of length n, a rank‑n vector valued modular form. This vector transforms under the modular group with a fixed multiplier, determined by the RCFT’s S and T matrices. If you understand one RCFT well enough to know its S and T, you can use the theory of VVMFs to generate other vectors that share that same multiplier. In other words, you get a family of potential solutions, all dancing to the same symmetry music, but with different individual characters. The trick is to pair the original X with new vectors Y1, Y2, …, Yn−1 and assemble a matrix Ξ(τ) whose columns are X and the Yi’s. Because each column transforms under the same multiplier, Ξ obeys the same modular rules as the original RCFT. The net effect is a principled way to mine for new admissible solutions by exploring the linear algebra of these shared symmetries.
Two concrete ideas drive the method. First, the new Yi’s often start as quasi‑characters: their q‑series have integral coefficients but can include negative terms, so they aren’t themselves admissible RCFT characters. The second idea is to form clever linear combinations of the original X and the Yi’s, sometimes weighted by functions of the modular j‑invariant (the J(τ) function). The aim is to cancel the problematic pieces and land on new, admissible solutions that could correspond to actual RCFTs. The Ising model, the prototype two‑character RCFT, serves as a transparent demonstration: starting from the known X, the procedure produces a family of Yi’s and a small set of linear combinations that can yield genuine, consistent theories with the same S and T data as the original. It’s like discovering that a melody can be harmonized with different chords while preserving the same underlying tune.
From quasi-characters to real theories
The heart of the second act is precisely the tension between mathematical possibility and physical viability. The Yi’s are constructed to share the same modular multiplier as X, so they inherit the same S and T matrices, the algebraic data that encode how the theory behaves under the torus’s modular transformations. But admissibility demands more: the q‑series must have non‑negative integer coefficients, mirroring that all physical states appear with nonnegative multiplicities. The authors show how, in many cases, you can combine the original characters with the Yi’s to push the central charge upward by discrete chunks of 24, and simultaneously raise the Wronskian index by multiples of 6. In formula‑light terms: you can ride a staircase from (n, ℓ) to (n, ℓ+6r) while preserving the same modular “fingerprint.”
Concretely, one productive maneuver is to form combinations Ur = X + sum bi Yi with positive integers bi. For some (n, ℓ) RCFTs these Ur turn out to be admissible as soon as you choose the bi in the right range. When successful, you get a family of theories with the same underlying S and T but higher central charge c+24m and higher ℓ′ = ℓ+6r. A vivid example occurs in the two‑character case: starting from certain (2,0) theories, you can reach a sequence of admissible theories with higher indices, matching results that had previously only appeared via the more brute‑force MLDE route. The same logic extends to more characters, including the more intricate four‑character landscapes and beyond, where the authors explicitly demonstrate new admissible theories with central charges and Wronskian indices that push beyond the most familiar models.
It’s worth pausing on a subtle but important caveat. Not every linear combination will yield an admissible theory. In some instances the naïve add‑and‑combine strategy yields quasi‑characters that cannot be promoted to true RCFTs. The paper documents several such exercises, especially in higher ranks, where some attempted combinations fail to produce physically meaningful spectra. Still, the upshot is encouraging: there are explicit, constructive routes to extend the catalog of admissible RCFTs, at least in regimes up to six characters, guided by the shared symmetry data rather than trial‑and‑error guesswork alone.
What this could mean for physics and math
None of this makes RCFTs trivial to classify, but it reframes the problem in a way that leans on a mature piece of mathematics—the theory of vector‑valued modular forms. By showing how to generate new admissible vectors from a known RCFT’s character data, Govindarajan and Santara provide a concrete bridge between a traditional MLDE‑based search and a more flexible, linear‑algebraic exploration of modular structures. The practical payoff is a broader, more navigable map of the RCFT landscape, including higher‑rank theories that previously felt out of reach for explicit construction.
Beyond the physics payoff, there are meaningful mathematical ripples. The method keeps the S and T data fixed while varying the spectrum, so it touches the structure of modular tensor categories—the algebraic backbone of topological phases and many string‑theory constructions. It also interacts with classical questions about vector‑valued modular forms, hypergeometric representations of modular objects, and the monodromy around elliptic points on the torus. In short, the work translates a physics problem into a concrete algebraic playground where ideas from number theory and representation theory can be tested against the constraints that physics imposes, and vice versa.
And there’s a human, institutional thread worth noting. The study is grounded in the Indian Institute of Technology Madras, with the lead researchers Govindarajan and Santara advancing a collaboration that sits at the nexus of pure math and theoretical physics. It’s a reminder of how modern breakthroughs often arrive not from isolated laboratories but from cross‑pollination among communities that speak different dialects of the same language: symmetry, structure, and the taste for what can or cannot be realized in the quantum world.
There’s also a sense in which this work returns to ancient mathematics with renewed energy. Vector‑valued modular forms, the J‑function, hypergeometric representations, and the modular group’s representations—these are not new toys, but the authors show how to deploy them in a contemporary physics agenda: to search, assemble, and test a hierarchy of candidate theories. The payoff isn’t just a longer list of possible RCFTs; it’s a more coherent story about how deeply symmetry constrains what is physically allowable, and how we can use that constraint as a generator rather than a gatekeeper.
What to watch for next
If the approach scales gracefully, we could see a more expansive catalog of admissible RCFTs with higher character counts, shedding light on the boundaries between viable two‑dimensional theories and those that live only in the realm of mathematical possibility. The authors themselves point to promising directions: relaxing some of the Wronskian constraints, exploring polynomial deformations of J, and weaving their VVMF toolkit more tightly into the broader program of classifying Modular Tensor Categories. There’s also the tantalizing possibility that this method could intersect with physical systems where two‑dimensional conformal symmetry plays a role, from critical phenomena in condensed matter to worldsheet descriptions in certain string theory backgrounds.
What’s clear is that this paper from IIT Madras turns a stubborn, highly abstract problem into a tangible workflow. It invites a new way of thinking about admissible theories: not as a fixed closet of known models, but as a living structure you can explore, extend, and, with luck, discover anew. It’s a reminder that even in a rigorous corner of physics and mathematics, creativity still thrives when you let symmetry do the heavy lifting and give yourself permission to experiment within the rules it imposes.
Institutional note: The project is led by researchers at the Indian Institute of Technology Madras, with Suresh Govindarajan and Jagannath Santara as lead authors.
Bottom line: This holomorphic modular bootstrap work doesn’t just classify a niche corner of RCFTs; it provides a practical, scalable way to imagine a richer landscape of theories, guided by deep symmetry and clean mathematics. It’s a promising toolkit for anyone who loves the idea that the universe’s most profound ideas might be written in the elegant language of modular forms and their vector‑valued cousins.
Note for readers: The full article contains many technical details and examples spanning two, three, four, and even six characters, including Ising, Potts, and tricritical Ising cases. The spirit, however, is accessible: start with a known theory, keep its symmetry story intact, and see what new, admissible versions you can author by reassembling the same alphabet in clever ways.
