Math 2.0

Quantum theory Definition

String theory

Builds on Conformal field theoryRiemann surfaceModuli of algebraic curvesNambu–Goto action

Definition

Perturbative bosonic string theory is defined as follows.

  1. A string background is a two-dimensional conformal field theory, the matter theory, with central charge c=26c = 26. The standard example is the theory of 2626 free bosons XμX^\mu, describing maps from a surface to Minkowski space ℝ1,25\R^{1,25}. Its classical action is the Polyakov action.
  2. Tensor the matter theory with the bcbc ghost system of central charge −26-26. The physical states of the string are the classes in the cohomology of the BRST operator on this tensor product.
  3. The scattering amplitude of nn physical states is a sum over the genus g≥0g \ge 0, weighted by gs2g−2g_s^{2g - 2}. The term of genus gg is an integral over the moduli space ℳg,n\mathcal{M}_{g,n} of Riemann surfaces of genus gg with nn marked points. The integrand is a density built from correlation functions of the conformal field theory on these surfaces.

Here gsg_s is the string coupling constant. String theory in general refers to this construction and its variants. In superstring theories the worldsheet theory is superconformal with c=15c = 15 in a suitable normalization. Other variants are open strings, on surfaces with boundary, and non-perturbative extensions.

Remarks

The condition c=26c = 26 makes the total central charge 00. Then the Weyl anomaly cancels, the BRST operator QQ satisfies Q2=0Q^2 = 0, and the integrand in clause 3 is a density on the moduli space of curves. For 2626 free bosons, the lowest state of the closed string is a tachyon. The first massless level contains states of spin 22 (the graviton), an antisymmetric tensor and a scalar, the dilaton.

Example: the free bosons XμX^\mu on a torus target ℝ1,25−k×Tk\R^{1, 25 - k} \times T^k give another background with c=26c = 26, a toroidal compactification; it shows T-duality, under which radii RR and α′/R\alpha'/R give equivalent theories.

Non-example: DD free bosons with D≠26D \ne 26 give c=D≠26c = D \ne 26, and the construction fails. The Weyl anomaly does not cancel, and in the old covariant quantization the spectrum contains states of negative norm when D>26D > 26. The classical Nambu–Goto action makes sense in every dimension; the critical dimension appears only in the quantum theory.

A non-perturbative definition of string theory is not known in general.

Reading path

To understand this page from the ground up, read these 38 pages in order. The foundations come first.

Show the path as a graph
Action, concept, no evidence levelActionconceptAssociative algebra, concept, no evidence levelAssociative algebraconceptCommutator of a ring, concept, no evidence levelCommutator of a ringconceptConformal map, concept, no evidence levelConformal mapconceptEndomorphism, concept, no evidence levelEndomorphismconceptFirst fundamental form, concept, no evidence levelFirst fundamental formconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectIdentity matrix, object, no evidence levelIdentity matrixobjectLaurent polynomial ring, concept, no evidence levelLaurent polynomial ri…conceptLie algebra, concept, no evidence levelLie algebraconceptLie algebra extension, concept, no evidence levelLie algebra extensionconceptLie bracket of vector fields, concept, no evidence levelLie bracket of vector…conceptLie ideal, concept, no evidence levelLie idealconceptLorentzian manifold, concept, no evidence levelLorentzian manifoldconceptMatrix, definition, no evidence levelMatrixdefinitionMatrix, concept, no evidence levelMatrixconceptModuli of algebraic curves, concept, no evidence levelModuli of algebraic c…conceptQuantum field theory, concept, no evidence levelQuantum field theoryconceptRiemann surface, concept, no evidence levelRiemann surfaceconceptRing, concept, no evidence levelRingconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTrace of a matrix, concept, no evidence levelTrace of a matrixconceptVector space, definition, no evidence levelVector spacedefinitionNambu–Goto action, definition, no evidence levelNambu–Goto actiondefinitionJacobi identity, statement, no evidence levelJacobi identitystatementSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLie algebra of an associative algebra, example, no evidence levelLie algebra of an ass…exampleMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionGeneral linear Lie algebra, example, no evidence levelGeneral linear Lie al…exampleDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectSpecial linear Lie algebra, example, no evidence levelSpecial linear Lie al…exampleWitt algebra, object, no evidence levelWitt algebraobjectVirasoro algebra, object, no evidence levelVirasoro algebraobjectConformal field theory, definition, no evidence levelConformal field theorydefinitionString theory, definition, no evidence levelString theorydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Action concept
  2. Associative algebra concept
  3. Commutator of a ring concept
  4. Conformal map concept
  5. Endomorphism concept
  6. First fundamental form concept
  7. General linear group object
  8. Identity matrix object
  9. Laurent polynomial ring concept
  10. Lie algebra concept
  11. Lie algebra extension concept
  12. Lie bracket of vector fields concept
  13. Lie ideal concept
  14. Lorentzian manifold concept
  15. Matrix definition
  16. Matrix concept
  17. Moduli of algebraic curves concept
  18. Quantum field theory concept
  19. Riemann surface concept
  20. Ring concept
  21. Smooth manifold concept
  22. Trace of a matrix concept
  23. Vector space definition
  24. Nambu–Goto action definition
  25. Jacobi identity statement
  26. Square matrix definition
  27. Linear map definition
  28. Lie algebra of an associative algebra example
  29. Multilinear map definition
  30. Alternating multilinear map definition
  31. General linear Lie algebra example
  32. Determinant definition
  33. Special linear group object
  34. Special linear Lie algebra example
  35. Witt algebra object
  36. Virasoro algebra object
  37. Conformal field theory definition
  38. String theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/String_theory
About this page
TagPYBXE2C28T
URIm2://PYBXE2C28T
Kinddefinition
WritingW0 stub
MSC81T30, 83E30
DefinesString theory
UsesConformal field theory, Riemann surface, Moduli of algebraic curves, Nambu–Goto action
Changed3 November 2026

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