Mathematical structures motivated by String theorystring theory, known as a B-field. There are field equations involving a generalized notion of the curvature of such a string connection.

Mathematical structures motivated by String theory

Ralph L. Cohen ∗

Stanford University

November 30, 2004

String theory has been an exciting, active area of research within theoretical physics for morethan twenty years. It can be viewed as a quantum theory of gravitation. Aside from its fundamentalimplications within physics, it has also had a dramatic and important impact upon mathematics.String theory has predicted mathematical theorems that were eventually proved in algebraic ge-ometry, it has pointed to previously unseen connections between seemingly very different areas ofmathematics such as number theory and algebraic topology, and it has motivated new directions ofresearch in such areas as symplectic geometry and differential topology. In this note I will describevery basic and elementary examples of the kind of mathematical structure motivated either directlyor indirectly by string theory.

One object of study common to all geometric areas of mathematics and physics, are manifolds.A compact, n-dimensional smooth manifold M can be thought of as a smooth region in a largedimensional Euclidean space, M ⊂ RN . Being smooth implies that at every point x ∈ M , thereis an n-dimensional vector space consisting of vectors in RN that tangent to M . This is called thetangent space, TxM . See figure 1.

In these geometric disciplines, one not only wants to study the manifold M itself, but alsoindividual points in M , and paths or curves connecting points. For example, in classical physics onemight think of M as a model of the universe, and a path or a curve in M as a description of how apoint moves or evolves over time.

In algebraic topology, the goal is to study manifolds and other topological spaces using algebra.One assigns an algebraic structure to the object of study, and then one studies the algebra in orderto deduce topological information. In the case of points moving in an n-manifold M , our algebraicinvariant might be the continuous assignment to every point x ∈ M , a vector space Vx. Such anassignment is called a vector bundle. For example one may assign to a point x, its tangent space,TxM . This is the tangent bundle, TM . Tangent bundles are very important, but they are not theonly interesting vector bundles. For example, let M be the middle circle in a Moebius band. Onecan assign to a point x ∈ M , the line (one dimensional vector space) containing the interval in the

∗The author was partially supported by a grant from the NSF

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x

Figure 1: A two dimensional manifold with its tangent space, TxM

Moebius strip that is perpendicular to the the middle circle M . This is called the normal bundle ofthe circle in the Moebeus band.

In order to understand how points evolve during an interval of time in our space M , we want toassign to every path γ between points, say x0, x1 ∈M, a nonsingular transformation

Φγ : Vx0 → Vx1 .

An important axiom that these transformations are required to satisfy is the “Gluing Axiom”which can be described as follows. Suppose γ1 is a path in M from x0 to x1, and γ2 is a path fromx1 to x2. Then we can define the “glued” path γ1#γ2 to be the concantenation of γ1 and γ2, viewedas a path from x0 to x2. See figure 2.

The gluing axiom states that the linear operator associated to the glued path, Φγ1#γ2 is equalto the composition of the linear operators Φγ1 and Φγ2 :

Φγ1#γ2 = Φγ2 ◦ Φγ1 : Vx0 → Vx1 → Vx2 . (1)

It is usually also assumed that the linear operator Φγ does not depend on the parameterization ofthe path γ. That is, suppose h : [0, 1] → [0, 1] is a diffeomorphism, which is to say a smooth, bijectivemap of the unit interval to itself. Clearly the paths γ : [0, 1] → M and γ ◦ h : [0, 1] → [0, 1] → M

have the same image curve. The gluing axiom states that in this case,

Φγ = Φγ◦h. (2)

This linear representation of the dynamics of points in a manifold by vector spaces and linearoperators between them, is known as a vector bundle with a “parallel transport” operator, or a

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x10 x

x2

2

1

Figure 2: The glued path γ1#γ2

“connection”. In classical physics, fields are represented by connections. In particular an electro-magnetic field is represented by a connection on a vector bundle whose curvature satisfies Maxwell’sequations, or more generally, the Yang-Mills equations.

In string theory the basic configuration space does not consist of points in a manifold, but ratheropen and closed “strings”, which are mathematically modeled by paths and loops in the manifold.So in the case of closed strings, we are studying the loop space,

LM = {α : S1 → M}

where S1 is the unit circle in the plane. This space is clearly infinite dimensional, but it does havethe structure of a manifold. The infinite dimensionality of LM has lead to many mathematicalchallenges, and has motivated the study of infinite dimensional geometry and topology.

If we want the analogue of an algebraic invariant given by a vector bundle with parallel transportoperator in this setting, we need to associate to each loop α : S1 → M a vector space, Vα. Becauseof the infinite dimensionality of LM , we might want our vector space to be infinite dimensional,with perhaps more structure, like that of being a Hilbert space. A more interesting question is howdo we represent how loops evolve over time?

A path of loops can be viewed as a map from a cylinder, S1 × I → M . See figure 3.However loops in M can evolve in more complicated ways. They can split apart to form more

than one loop, or two loops can interact with each other to form a single loop. Such a continuoussequence of loops in M may be viewed as a map of a more general surface to M . Physically, thissurface is known as a “worldsheet”. See figure 4.

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M

Figure 3: A path in of loops is a cylinder in M

We view these surfaces as directed, in the sense that some of its boundary circles are viewed as“incoming”, and the remaining are viewed as “outgoing”. So aside from assigning to each loop α inM a vector space Vα, we assign to each map of a directed surface, γ : Σ → M , a linear transformationin the following way. Suppose Σ has p incoming boundary circles and q outgoing boundary circles.Assume the restriction of the map γ to the p incoming circles are the p loops, α1, · · · , αp, and therestriction of γ to the outgoing circles are the q loops, β1, · · · , βq. Then we assign to γ : Σ → M alinear transformation between the tensor products of the vector spaces,

Φγ : Vα1 ⊗ · · · ⊗ Vαp−→ Vβ1 ⊗ · · · ⊗ Vβq

.

The mathematical name for a map γ : Σ → M that restricts on the boundary components in theway described, is a cobordism between the loops, α1, · · · , αp and the loops β1, · · · , βq. So in thistheory we assign vector spaces to loops, and linear transformations to cobordisms between loops. Aswas described above, this assignment must respect gluing of cobordisms. That is, if Σ1 is a surfacewith p incoming and q outgoing boundary circles, Σ2 is a directed surface with q incoming and r

outgoing boundary components, then we can form the “glued” directed surface Σ1#Σ2 as in figure5.

If γ : Σ1 → M is a cobordism between the loops α1, · · · , αp and β1, · · · , βq, and θ : Σ2 → M is acobordism between the loops β1, · · · , βq, and δ1, · · · , δr, then we can glue the maps γ and θ togetheralong their common boundaries, to obtain a map from the glued surface

γ#θ : Σ1#Σ2 → M

which is a cobordism between the loops α1, · · · , αp and the loops δ1, · · · , δr. The gluing axiom

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M

M

Figure 4: A sequence of loops in M may be viewed as a map of a surface to M

for our theory is that the linear operator for the glued cobordism is the composition of the linearoperators of the individual cobordism,

Φγ#θ = Φθ ◦ Φγ : Vα1 ⊗ · · · ⊗ Vαp→ Vβ1 ⊗ · · · ⊗ Vβq

→ Vδ1 ⊗ · · · ⊗ Vδr. (3)

Compare this with gluing axiom for classical parallel transport operators given in equation 1. Theanalogue of the invariance of parameterization axiom for parallel transport operators given in equa-tion 2 for this theory is the following. If h : Σ → Σ is a diffeomorphism (i.e a smooth bijective map),and if γ : Σ → M is a map from the worldsheet, then

Φγ = Φγ◦h. (4)

A theory that assigns to a loop α a vector space Vα, and to a cobordism γ : Σ → M an operatorΦγ with the above properties is called a “string connection” on the associated vector bundle over theloop space. Physically, string connections on vector bundles, represent a kind of field in (bosonic)string theory, known as a B-field. There are field equations involving a generalized notion of thecurvature of such a string connection.

It turns out that B-fields give rise to “conformal field theories” (CFT). This is a mathematicaltheory that involves both geometric and algebraic concepts. The notion was given mathematical

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p 1q q

r2

p

r

1# 2

Figure 5: Gluing of cobordisms

axioms by Segal in [4]. A simplified, but related notion is that of a “topological quantum field theory”(TQFT) whose axioms were described by Atiyah in [1]. Roughly, an n+1-dimensional TQFT assignsto every closed n-manifold M , (i.e a compact manifold with no boundary) a vector space VM . Thevector spaces associated to diffeomorphic n-manifolds are assumed to be isomorphic. To every n+1dimensional manifold W whose boundary has two components: an “incoming component” M1, andand “outgoing” component M2, a TQFT associates a homomorphism

ΦW : VM1 → VM2

that satisfy a variety of axioms. Perhaps the most notable among these is a gluing axiom likedescribed above, and the sum axiom that says that VM1tM2 = VM1 ⊗ VM2 , where M1 t M2 is thedisjoint union of the two manifolds M1 and M2. Also, diffeomorphic manifolds W1 and W2 yieldisomorphic linear operators, in the appropriate sense. A conformal field theory is the same sort ofstructure, in the case of d = 1, except in such a theory, the surfaces are equipped with metrics,( a metric is a way of determining the distance between points), and the vector spaces and linearoperators depend on what is called the “conformal class” of the metric. See [4] for more details.

These field theories can be viewed as a combination of algebra and geometry. For example, whend = 0 the story of a TQFT is mostly algebraic. The only closed zero dimensional manifolds are

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finite collections of points. By the sum axiom, the vector space associated to a collection of n pointsis of the form V ⊗ · · · ⊗ V (the n-fold tensor product of V with itself), where V is the vector spaceassociated to a single point. The only one dimensional manifolds with boundary are collections ofline segments. If there are n line segments they would induce nonsingular linear transformationsfrom V ⊗n to itself.

If d = 1 the geometry of a TQFT induces a deeper algebraic structure. The only closed onedimensional manifolds are a collection of circles, and the two manifolds with boundary are thesurfaces described above. Given a TQFT, let V be the vector space associated to a circle. LetP be the surface given by a sphere with three holes, (see figure 6). It is a surface with threeboundary circles. If two of the boundary circles are viewed as incoming, and the remaining oneas outgoing, then then there is a linear operator ΦP : V ⊗ V → V , which turns out to be anassociative, commutative multiplication. The linear operators associated to other surfaces give amore sophisticated algebraic structure. Indeed the underlying vector space V has the structure ofwhat is called a “Frobenius algebra”. Field theories have played an important role in geometry andtopology over the last fifteen years. They have been important in knot theory, algebraic geometry,three and four dimensional topology, as well as more recently, more general algebraic topology.

Figure 6: The surface P: A sphere with three holes

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This essay was meant to give the reader just a small taste of only one facet of the exciting andfar reaching influence string theory and quantum field theory have had on mathematics. For a moretechnical discussion of the topics introduced here (and others), the reader is encouraged to read thearticles of Atiyah [1], and Segal [4], which served as important references for the preparation of thisarticle. For a more sophisticated introduction to string theory geared toward mathematicians, werecommend the article by d’Hoker [2].

References

[1] M.F. Atiyah, Topological quantum field theories, Publ. Math. IHES 68, (1988), 175-186.

[2] E. d’Hoker, String theory, Quantum fields and strings: a course for mathematicians,ed.P. Deligne et al., Princeton, N.J: American Mathematical Society, vol 2, (1999), 807-1011.

[3] G. Segal, Topological structures in string theory, Phil. Trans. R. Soc. Lond. A 359, (2001),1389-1398.

[4] G. Segal, The definition of conformal field theory, Differential geometrical methods in theoreticalphysics, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci, Kluwer Acad. Publ. 250, (1988), 165-171.

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