mathematics

Virasoro conjecture

Virasoro conjecture is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Virasoro conjecture rather than just read about it. In short: In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra. The Virasoro conjecture is named after theoretical physicist Miguel Ángel Virasoro.

Key takeaways

Virasoro conjecture belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Virasoro conjecture to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Virasoro conjecture from memory before moving on to harder problems.

Reference excerpt

In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra. The Virasoro conjecture is named after theoretical physicist Miguel Ángel Virasoro. Tohru Eguchi, Kentaro Hori, and Chuan-Sheng Xiong (1997) proposed the Virasoro conjecture as a generalization of Witten's conjecture. Ezra Getzler (1999) gave a survey of the Virasoro conjecture. The proof of the genus 0 Virasoro conjecture for all smooth projective varieties (or more generally, compact symplectic manifolds) was first given by Xiaobo Liu and Gang Tian (1998).

References

Getzler, Ezra (1999), "The Virasoro conjecture for Gromov-Witten invariants", in Wiśniewski, Jarosław; Szurek, Michał; Pragacz, Piotr (eds.), Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), Contemporary Mathematics, vol. 241, Providence, R.I.: American Mathematical Society, pp. 147–176, arXiv:math/9812026, Bibcode:1998math.....12026G, doi:10.1090/conm/241/03634, ISBN 978-0-8218-1149-8, MR 1718143 Eguchi, Tohru; Hori, Kentaro; Xiong, Chuan-Sheng (1997), "Quantum cohomology and Virasoro algebra", Physics Letters B, 402 (1): 71–80, arXiv:hep-th/9703086, Bibcode:1997PhLB..402...71E, doi:10.1016/S0370-2693(97)00401-2, ISSN 0370-2693, MR 1454328 Liu, Xiaobo; Tian, Gang (1998-10-20), Virasoro Constraints For Quantum Cohomology, arXiv:math/9806028, Bibcode:1998math......6028L

Worked examples

Example 1 — a first encounter with Virasoro conjecture

Start with the simplest possible case. Write down what Virasoro conjecture claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Virasoro conjecture before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Virasoro conjecture ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Virasoro conjecture

In research: Virasoro conjecture appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Virasoro conjecture in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Virasoro conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, Algebraic geometry stubs, Conjectures, so understanding it makes those chapters shorter.
In everyday life: Look for Virasoro conjecture outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Virasoro conjecture” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Virasoro conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Virasoro conjecture means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Virasoro conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Virasoro conjecture in simple terms?

Why does Virasoro conjecture matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Virasoro conjecture?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Virasoro conjecture.