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Worldsheet

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Mathematical concept

String theory

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v t e

Instring theory, aworldsheet is a two-dimensionalmanifold which describes the embedding of astring inspacetime.^[1] The term was coined byLeonard Susskind^[2] as a direct generalization of theworld line concept for a point particle inspecial andgeneral relativity.

The type of string, the geometry of the spacetime in which it propagates, and the presence of long-range background fields (such asgauge fields) are encoded in atwo-dimensional conformal field theory defined on the worldsheet. For example, thebosonic string in 26 dimensions has a worldsheet conformal field theory consisting of 26free scalar bosons. Meanwhile, asuperstring worldsheet theory in 10 dimensions consists of 10 free scalar fields and theirfermionic superpartners.

Mathematical formulation

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Bosonic string

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We begin with the classical formulation of the bosonic string.

First fix a $d {\displaystyle d}$ -dimensionalflat spacetime ( $d {\displaystyle d}$ -dimensionalMinkowski space), $M {\displaystyle M}$ , which serves as theambient space for the string.

Aworld-sheet $\Sigma$ is then anembedded surface, that is, an embedded 2-manifold $\Sigma \hookrightarrow M$ , such that theinduced metric has signature $(-,+)$ everywhere. Consequently it is possible to locally define coordinates $(\tau ,\sigma )$ where $\tau$ istime-like while $\sigma$ isspace-like.

Strings are further classified into open and closed. The topology of the worldsheet of an open string is $\mathbb {R} \times I$ , where $I:=[0,1]$ , a closed interval, and admits a global coordinate chart $(\tau ,\sigma )$ with $-\infty <\tau <\infty$ and $0\leq \sigma \leq 1$ .

Meanwhile the topology of the worldsheet of a closed string^[3] is $\mathbb {R} \times S^{1}$ , and admits 'coordinates' $(\tau ,\sigma )$ with $-\infty <\tau <\infty$ and $\sigma \in \mathbb {R} /2\pi \mathbb {Z}$ . That is, $\sigma$ is a periodic coordinate with the identification $\sigma \sim \sigma +2\pi$ . The redundant description (using quotients) can be removed by choosing a representative $0\leq \sigma <2\pi$ .

World-sheet metric

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In order to define thePolyakov action, the world-sheet is equipped with aworld-sheet metric^[4] $\mathbf {g}$ , which also has signature $(-,+)$ but is independent of the induced metric.

SinceWeyl transformations are considered a redundancy of the metric structure, the world-sheet is instead considered to be equipped with aconformal class of metrics $[\mathbf {g} ]$ . Then $(\Sigma ,[\mathbf {g} ])$ defines the data of aconformal manifold with signature $(-,+)$ .

References

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^Di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997).Conformal Field Theory. p. 8.doi:10.1007/978-1-4612-2256-9.ISBN 978-1-4612-2256-9.
^Susskind, Leonard (1970). "Dual-symmetric theory of hadrons, I.".Nuovo Cimento A.69 (1):457–496.
^Tong, David."Lectures on String Theory".Lectures on Theoretical Physics. RetrievedAugust 14, 2022.
^Polchinski, Joseph (1998).String Theory, Volume 1: Introduction to the Bosonic string.

v t e String theory
Background	Strings Cosmic strings History of string theory First superstring revolution Second superstring revolution String theory landscape
Theory	Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string Type IIA string Type IIB string Heterotic string N=2 superstring F-theory String field theory Matrix string theory Non-critical string theory Non-linear sigma model Tachyon condensation RNS formalism GS formalism
String duality	T-duality S-duality U-duality Montonen–Olive duality
Particles and fields	Graviton Dilaton Tachyon Ramond–Ramond field Kalb–Ramond field Magnetic monopole Dual graviton Dual photon
Branes	D-brane NS5-brane M2-brane M5-brane S-brane Black brane Black holes Black string Brane cosmology Quiver diagram Hanany–Witten transition
Conformal field theory	Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model
Gauge theory	Anomalies Instantons Chern–Simons form Bogomol'nyi–Prasad–Sommerfield bound Exceptional Lie groups (G₂,F₄,E₆,E₇,E₈) ADE classification Dirac string p-form electrodynamics
Geometry	Worldsheet Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G₂ manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics) Twisted K-theory
Supersymmetry	Supergravity Eleven-dimensional supergravity Type I supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup
Holography	Holographic principle AdS/CFT correspondence
M-theory	Matrix theory Introduction to M-theory
String theorists	Aganagić Arkani-Hamed Atiyah Banks Berenstein Bousso Curtright Dijkgraaf Distler Douglas Duff Dvali Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson Harvey 't Hooft Hořava Gibbons Kachru Kaku Kallosh Kaluza Kapustin Klebanov Knizhnik Kontsevich Klein Linde Maldacena Mandelstam Marolf Martinec Minwalla Moore Motl Mukhi Myers Nanopoulos Năstase Nekrasov Neveu Nielsen van Nieuwenhuizen Novikov Olive Ooguri Ovrut Polchinski Polyakov Rajaraman Ramond Randall Randjbar-Daemi Roček Rohm Sagnotti Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher Steinhardt Strominger Sundrum Susskind Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach

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