Strominger's equations

Key takeaways

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

Reference excerpt

In heterotic string theory, the Strominger's equations are the set of equations that are necessary and sufficient conditions for spacetime supersymmetry. It is derived by requiring the 4-dimensional spacetime to be maximally symmetric, and adding a warp factor on the internal 6-dimensional manifold. Consider a metric ω {\displaystyle \omega } on the real 6-dimensional internal manifold Y and a Hermitian metric h on a vector bundle V. The equations are:

The 4-dimensional spacetime is Minkowski, i.e., g = η {\displaystyle g=\eta } . The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish N = 0 {\displaystyle N=0} . The Hermitian form ω {\displaystyle \omega } on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy,

∂ ∂ ¯ ω = i Tr F ( h ) ∧ F ( h ) − i Tr R − ( ω ) ∧ R − ( ω ) , {\displaystyle \partial {\bar {\partial }}\omega =i{\text{Tr}}F(h)\wedge F(h)-i{\text{Tr}}R^{-}(\omega )\wedge R^{-}(\omega ),}

d † ω = i ( ∂ − ∂ ¯ ) ln | | Ω | | , {\displaystyle d^{\dagger }\omega =i(\partial -{\bar {\partial }}){\text{ln}}||\Omega ||,} where R − {\displaystyle R^{-}} is the Hull-curvature two-form of ω {\displaystyle \omega } , F is the curvature of h, and Ω {\displaystyle \Omega } is the holomorphic n-form; F is also known in the physics literature as the Yang-Mills field strength. Li and Yau showed that the second condition is equivalent to ω {\displaystyle \omega } being conformally balanced, i.e., d ( | | Ω | | ω ω 2 ) = 0 {\displaystyle d(||\Omega ||_{\omega }\omega ^{2})=0} . The Yang–Mills field strength must satisfy,

ω a b ¯ F a b ¯ = 0 , {\displaystyle \omega ^{a{\bar {b}}}F_{a{\bar {b}}}=0,}

F a b = F a ¯ b ¯ = 0. {\displaystyle F_{ab}=F_{{\bar {a}}{\bar {b}}}=0.}

These equations imply the usual field equations, and thus are the only equations to be solved. However, there are topological obstructions in obtaining the solutions to the equations;

The second Chern class of the manifold, and the second Chern class of the gauge field must be equal, i.e., c 2 ( M ) = c 2 ( F ) {\displaystyle c_{2}(M)=c_{2}(F)}

A holomorphic n-form Ω {\displaystyle \Omega } must exists, i.e., h n , 0 = 1 {\displaystyle h^{n,0}=1} and c 1 = 0 {\displaystyle c_{1}=0} . In case V is the tangent bundle T Y {\displaystyle T_{Y}} and ω {\displaystyle \omega } is Kähler, we can obtain a solution of these equations by taking the Calabi–Yau metric on Y {\displaystyle Y} and T Y {\displaystyle T_{Y}} . Once the solutions for the Strominger's equations are obtained, the warp factor Δ {\displaystyle \Delta } , dilaton ϕ {\displaystyle \phi } and the background flux H, are determined by

Δ ( y ) = ϕ ( y ) + constant {\displaystyle \Delta (y)=\phi (y)+{\text{constant}}} ,

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Strominger's equations

Start with the simplest possible case. Write down what Strominger's equations 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 Strominger's equations 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 Strominger's equations 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 Strominger's equations

In research: Strominger's equations 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 Strominger's equations 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: Strominger's equations is common in secondary-school and first-year university syllabi. It links to neighbouring topics String theory, so understanding it makes those chapters shorter.
In everyday life: Look for Strominger's equations 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.

How to study Strominger's equations in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Strominger's equations 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 Strominger's equations out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Strominger's equations in simple terms?

In heterotic string theory, the Strominger's equations are the set of equations that are necessary and sufficient conditions for spacetime supersymmetry. It is derived by requiring the 4-dimensional spacetime to be maximally symmetric, and adding a warp factor on the internal 6-dimensional manifold.

Why does Strominger's equations 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 Strominger's equations?

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 Strominger's equations.