Worldsheet References

In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime.^[1] The term was coined by Leonard Susskind^[2] as a direct generalization of the world line concept for a point particle in special and general relativity.

The type of string, the geometry of the spacetime in which it propagates, and the presence of long-range background fields (such as gauge fields) are encoded in a two-dimensional conformal field theory defined on the worldsheet. For example, the bosonic string in 26 dimensions has a worldsheet conformal field theory consisting of 26 free scalar bosons. Meanwhile, a superstring worldsheet theory in 10 dimensions consists of 10 free scalar fields and their fermionic superpartners.

Mathematical formulation

Bosonic string

We begin with the classical formulation of the bosonic string.

First fix a $d$ -dimensional flat spacetime ( $d$ -dimensional Minkowski space), $M$ , which serves as the ambient space for the string.

A world-sheet $\Sigma$ is then an embedded surface, that is, an embedded 2-manifold $\Sigma \hookrightarrow M$ , such that the induced metric has signature $(-,+)$ everywhere. Consequently it is possible to locally define coordinates $(\tau ,\sigma )$ where $\tau$ is time-like while $\sigma$ is space-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

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

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

References

^ 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. Retrieved August 14, 2022.
^ Polchinski, Joseph (1998). String Theory, Volume 1: Introduction to the Bosonic string.

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[Di_FrancescoMathieu1997-1] 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-2] Susskind, Leonard (1970). "Dual-symmetric theory of hadrons, I.". Nuovo Cimento A. 69 (1): 457–496.

[tong-3] Tong, David. "Lectures on String Theory". Lectures on Theoretical Physics. Retrieved August 14, 2022.

[Polchinski-4] Polchinski, Joseph (1998). String Theory, Volume 1: Introduction to the Bosonic string.

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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 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