Open strings are topologically line intervals with two endpoints; closed strings are topologically circles. The fields defined on the strings may be periodic as a function of \(\sigma\), the spatial coordinate along the string, or they may obey various other boundary conditions. Let's look at them.
Strings are parameterized by the single spatial coordinate \(\sigma\), pronounce "sigma", and as they're moving in space, they're painting a two-dimensional generalization of particles' one-dimensional world lines, the world sheets. Among the two dimensions of the world sheet, one dimension is spatial, it's our \(\sigma\), and one is temporal and it's usually called \(\tau\), pronounce "tau". Note that "sigma" and "tau" are Latinized as "s" and "t" and those letters stand for "space" and "time", respectively.
As long as the laws on the string are uniform as a function of \(\sigma\), it's clear that \(\sigma\) can't be either infinite or semi-infinite because the total angular momentum, energy, and other quantities carried by the string would have infinite values and the string would have an infinite entropy. So real-world low-energy strings must definitely be encoded just by a line interval. It's a convention to rescale (or reparameterize) \(\sigma\) in such a way that it belongs to the interval\[
\sigma\in (0,\pi).
\] However, the end of the world – and end of the string – is a subtle place. The laws of physics may have trouble to determine what should happen there. I won't go into this technicality but if you will study string theory at a technical level, you may want to know it: When you derive the Euler-Lagrange equations by varying the world sheet action, you need to get rid of the boundary terms and it's only possible if the string obeys certain conditions at \(\sigma=0\) and \(\sigma=\pi\).
Closed strings
The simplest way to solve the problem with the "end of the string" is to have no ends of the string. We may just glue the endpoints. Any wave that is going to the left and reaching \(\sigma=0\) will simply reappear at the right side, near \(\sigma=\pi\). The string is "periodic". More precisely, the fields on the strings are periodic functions of \(\sigma\). What are the fields on the string? They're fields \(X^\mu(\sigma,\tau)\) encoding where a particular point of the string (or world sheet) is located in the target spacetime. There may be fermionic counterparts such as \(\psi^\mu(\sigma,\tau)\) or \(\theta^a(\sigma,\tau)\) and some friendly ghosts and superghosts denoted by letters such as \(b,c,\beta,\gamma\).
I will only talk about \(X^\mu(\sigma,\tau)\) in this blog entry.
So for closed strings, we simply have\[
X^\mu(\sigma=\pi,\tau) = X^\mu(\sigma=0,\tau).
\] The two end points are identified, the string becomes a circle (topologically), and the world sheet is a cylinder. All the boundary terms \(B|_0^\pi\) vanish simply because the values at \(\sigma=\pi\) cancel against those at \(\sigma=0\). When you decompose \(X^\mu(\sigma,\tau)\) into Fourier modes, the most natural modes go like \(\exp(2in\sigma)\) where \(n\) is an integer. You will find out that there are left-moving "complex" waves as well as right-moving "complex" waves on the world sheet.
The relevant Fourier modes of \(X^\mu(\sigma,\tau)\) and its canonical momentum (essentially velocity), \(P^\mu(\sigma,\tau)\), are known as \(\alpha^\mu_n\) and \(\tilde \alpha_n^\mu\) where \(\mu\) is a spacetime Lorentz index and \(n\) is an integer. Whenever \(n\) is negative, these \(\alpha\) operators are creation/raising operators of a harmonic oscillator; for positive values, they're the annihilation/lowering operators.
The single bosonic string has a ground state, \(\ket 0\), and it may be excited by those creation operators to obtain states such as\[
(g_{\mu\nu}+B_{\mu\nu}) \alpha_{-1}^\mu \tilde\alpha_{-1}^\nu \ket 0
\] where the fields \(g,B\) – the symmetric metric tensor and the antisymmetric B-field two-form that are somewhat naturally combined into a general 2-index tensor – may depend on the zero modes \(X_0^\mu\), the average of \(X^\mu(\sigma,\tau)\) over \(\sigma\).
While \(\ket 0\) corresponds to the tachyon with \(m^2\lt 0\), the first (doubly) excited states above are massless, as expected for a graviton (and the B-field as well). One may excite the harmonic oscillators many times to make the closed string behave as a massive particle with masses given by \(m^2=4N/ \alpha'\) where \(N\) is an integer (the sum of all the numbers \(n\) in the operators \(\alpha_{-n}^\kappa\); it must be equal to the same total excitation of \(\tilde\alpha\) operators).
We may also "twist" the boundary conditions by a symmetry. For example, the coordinate \(X^3(\sigma,\tau)\) may be antiperiodic rather than periodic,\[
X^3(\sigma=\pi,\tau) = -X^3(\sigma=0,\tau).
\] The minus sign is new. In this case, the function is an antiperiodic function of \(\sigma\) and the indices of the Fourier modes will be in \(n\in \ZZ+1/2\). This will still cancel the boundary terms because the action is bilinear in (derivatives of) \(X^\mu\) or, more conceptually, because the extra operation \(X^3\to -X^3\) we did before we glued the endpoints is a symmetry of the world sheet action.
Because \(X^3\) doesn't want to oscillate too much (it increases the energy i.e. the mass of the particle that the string behaves as) and because it's odd, it's clear that \(X^3\) will oscillate in the vicinity \(X^3=0\) in the spacetime. So these "twisted" closed strings will correspond to particles that are stuck near the \(X^3=0\) plane in the spacetime. Twisted boundary conditions for closed strings mean that the states of the closed string we may get in this way live in the so-called "twisted sectors", new kinds of states that appear in "string theory on orbifolds". In fact, the wound strings (strings wrapped around a cylinder in spacetime \(w\) times) may be viewed as special examples of a twisted sector.
But I don't want to focus on closed strings and orbifolds. Instead, my main target were the open strings and D-branes.
Open string boundary conditions
There exists another way to cancel the boundary terms at the end of the string: to force that the contribution to the boundary terms is zero from both \(\sigma=0\) and \(\sigma=\pi\) separately. The traditional condition for \(X^\mu\) that's been around since the early 1970s is the so-called Neumann boundary condition:\[
\pfrac{}{\sigma}X^\mu|_{\sigma=0} = 0
\] and similarly for \(\sigma=\pi\). The phrase "Neumann boundary condition" simply refers to the vanishing of the \(\sigma\)-derivative of the field \(X^\mu\), in this case. Note that the \(\sigma\)-derivative may also be called "normal" (perpendicular to the boundary at \(\sigma=0\)) so you may also say that the normal derivative vanishes.
If one omitted the differentiation with respect to \(\sigma\), he would get the so-called Dirichlet boundary condition such as \(X^\mu=0\) at \(\sigma=0\), named after a mathematician who was born 30 years before Carl Neumann. However, \(X^\mu=0\) and the endpoint of a string means that the plane \(X^\mu=0\) is special. Such laws for open strings break the translational, rotational, and Lorentz symmetries in the spacetime which is why people wouldn't allow such boundary conditions for open strings in the 1970s and 1980s.
On the other hand, open strings with Neumann boundary conditions were studied carefully for decades – it's really these open strings that were the first ones to appear in string theory (before closed strings). While closed strings have left-moving and right-moving waves, \(X^\mu(\sigma)\) at open strings may be decomposed into "standing waves" Fourier modes going like \(\cos n\sigma\). Note that the positive and negative values of \(n\) give you the same cosine-like function. However, you still get oscillators \(\alpha_{n}^\mu\) for both positive and negative values of \(n\) (which are independent) because there's another doubling of the degrees of freedom, the existence of the momenta \(P^\mu(\sigma,\tau)\).
As far as the mass of the particle that the string behaves as is concerned, the open string oscillators \(\alpha^\mu_n\) behave almost identically as their namesakes from the closed string case. However, for open strings, there is no \(\tilde\alpha_n^\nu\); we only have one set of oscillators. In this sense, the Hilbert space of a closed string is "close" to the tensor square of the open string Hilbert space; in other words, the open string is a "square root" of a closed string.
Note that in the case of the closed string, we needed at least one left-moving excitation \(\alpha\) and one right-moving excitation \(\tilde\alpha\) to get away from the tachyonic ground state \(\ket 0\). That's why the field associated with these particles – excited strings – had two Lorentz indices. That's why gravity inevitably appeared at the massless level of the closed strings. Here, for open strings, we only need one oscillator to get from the open string tachyon ground state \(\ket 0\) – the main hero of an article about Sen's tachyon condensation – to the massless level. The first excited state is\[
A_\mu \alpha_{-1}^\mu \ket 0
\] which is multiplied by a field with one Lorentz index. I called it \(A^\mu\) for a good reason: when you derive what laws are imposed on this field by the maths of string theory, you will find out that it really does behave exactly as a gauge field, with a gauge symmetry and the usual terms in the effective Lagrangian.
Just like the massless level of a closed string produces gravity (the metric tensor) and the B-field (a two-index potential under which the strings themselves are "charged"), aside from the (scalar) dilaton I neglected above (although it wouldn't have been too hard to add an \(\exp(\phi)\)-like factor in front of the \((g+B)\)), the open string's massless level produces gauge fields. If all the boundary conditions are Neumann, these open strings – and therefore the gauge field – is defined everywhere in the spacetime. Nevertheless, it's better to call their locus "a spacetime-filling D-brane" rather than the "spacetime" for reasons that could become clearer later.
T-duality
Let's return to closed strings for a little while. We have discussed the \(n\)-th Fourier modes (over the circle that is the closed string) for nonzero values of \(n\). However, we haven't spent much time with the \(n=0\) case, the "zero mode". Well, you may integrate \(X^\mu\) over \(\sigma\) (and divide the result by \(\pi\), the length of the \(\sigma\)-interval) to get the "center of mass" position of the string if you wish; and you may also integrate the momentum \(P^\mu\) over \(\sigma\) to find out the total momentum carried by the string.
A funny thing is that you may also integrate the \(\sigma\)-derivative of the \(X^\mu\) field\[
\Delta X^\mu = \int_0^\pi \dd\sigma \pfrac{}{\sigma} X^\mu.
\] I denoted the result as \(\Delta X^\mu\) because it's nothing else than the "jump" of the coordinate induced by the trip around the closed string. If the strings are closed and living in an infinite space, it must be zero. However, if you make the coordinate \(X^\mu\) for a particular value of \(\mu\) periodic with a period of \(2\pi R\), so that the dimension may be imagined as a circle of radius \(R\), you must also allow closed strings that obey\[
X^\mu(\sigma=\pi,0) = X^\mu(\sigma=0,\tau) + 2\pi R w, \quad w\in\ZZ.
\] I have already mentioned that these \(w\) times wound strings may be viewed as generalized "twisted sectors". At any rate, the "endpoint" of the closed string may be located at a point shifted relatively to the "beginning point" of the closed string by \(2\pi R w\) because such a shifted point corresponds to the same location of the spacetime: we made the spacetime periodic in \(X^\mu\).
These wound strings are completely new features of string theory that don't exist in point-like particle quantum field theories: points can't be "wound" around a circle.
The winding number \(w\) contributes to the mass of the string. The formula for the squared mass is actually:\[
m^2 = \frac{4(N-1)}{\alpha'} + \frac{n^2}{R^2} + w^2 (2\pi R T)^2.
\] The first term quantifies the total excitation of the nonzero modes along the string; there is \(N-1\) instead of \(N\) because the ground state is a tachyon. The second term adds the usual energy from the quantized momentum; note that even in point-like particle field theories, the momentum of a particle along a circle of radius \(R\) has to be of the form \(n/R\); the last term is the winding and the coefficient depends on the string tension \(T=1/2\pi \alpha'\).
For point-like particles, there would only be one "squared" term, the momentum term going like \(n^2\). For closed strings, we have two of them. You immediately see that the momentum and winding contributions are very analogous to each other. You may exchange \(w\) with \(n\) and if you also adjust the coefficients appropriately, which may be done by \(R\leftrightarrow \alpha' / R\), the second and third term will get exactly interchanged! The spectrum is symmetric under the operation; the interactions are actually also symmetric. The symmetry is known as T-duality.
The T-duality must also act on the detailed fields \(X^\mu(\sigma,\tau)\); we only saw its approximate action on the Fourier modes. If you think about it, the action of the T-duality is essentially to interchange \(\partial_\sigma X^\mu\) with \(\partial_\tau X^\mu\) (only for a particular value of \(\mu\): we are T-dualizing one spacetime dimension only which must be circular!). That's because the momentum is the integral of \(\partial_\tau X^\mu\) while the winding is the integral of \(\partial_\sigma X^\mu\) and we wanted to interchange these two integers.
You may also rephrase the previous sentences by saying that T-duality keeps \((\partial_\sigma + \partial_\tau) X^\mu\) constant but the difference of these two derivatives changes the sign under T-duality,\[
(\partial_\sigma - \partial_\tau) X^\mu \to - (\partial_\sigma - \partial_\tau) X^\mu.
\] So T-duality acts like the reflection \(X^\mu\to -X^\mu\) except that the reflection only applies to the "right moving" degrees of freedom of the string, not the left-moving ones. (Or vice versa: these two operations of T-duality only differ by the full reflection of the spacetime coordinate which is a simple symmetry of the bosonic string theory.)
T-duality for open strings
There's a lot of fun things to say about T-duality and closed strings but let's switch to open strings. What happens with an open string under T-duality? We said that the T-duality interchanged the \(\sigma\)-derivative and the \(\tau\)-derivative of \(X^\mu\). However, this affects a basic equation for open strings we have already mentioned, the boundary conditions, too!
We said that the "normal" open strings studied in the 1970s and 1980s had the Neumann boundary conditions\[
\pfrac{}{\sigma}X^\mu|_{\sigma=0} = 0.
\] Now, because the two derivatives were interchanged, the new boundary condition (only for one value of \(\mu\), the circular direction we are T-dualizing) should be\[
\pfrac{}{\tau}X^\mu|_{\sigma=0} = 0.
\] The derivative along the boundary vanishes. This really means that the whole boundary at \(\sigma=0\) must sit at the same value of \(X^\mu\). If you choose the value once, i.e. \(X^\mu=0\), it will be valid for all values of \(\tau\). In other words, we derived that the T-duality operation forces us to replace open strings with Neumann boundary conditions for everything by open strings with Neumann boundary conditions for "almost everything" except for \(X^\mu\), the T-dualized direction itself, which must now have the Dirichlet boundary condition:\[
X^\mu|_{\sigma=0} = 0.
\] This makes a difference! The endpoints of such an open string are stuck at \(X^\mu=0\), a co-dimension one surface in the spacetime. Again, because the string can't stretch too far without dramatically increasing the mass of the resulting particle, you may say that all the low-energy strings live at/near \(X^\mu=0\). It's the D24-brane. How did I get D24? Well, the D25-brane is the brane filling the whole 25+1-dimensional spacetime; the number 25, by conventions, only counts the number of spatial coordinates. And I subtracted 1 from 25 because the D-brane got localized in one direction.
(Open strings have two endpoints and both of them may have boundary conditions independent from one another; one endpoint may be stuck to one D-brane and the other endpoint may terminate at the same or a different D-brane. I won't go into these matters here.)
With the Dirichlet boundary conditions, the right functions to Fourier-expand into will be waves such as \(\sin n\sigma\); instead of cosines, we simply have the sines. It's almost the same thing (they're still dependent on the functions with the opposite value of \(n\), the sign doesn't influence much) except that we get zero for \(n=0\) while the cosines did give us a zero mode, too. This is a reflection of the fact that we no longer have the freedom to change the average \(X^\mu\) without changing the energy: the string is stuck at \(X^\mu=0\).
However, it's still possible to excite the open string tachyon ground state \(\ket 0\) by the oscillators \(\alpha_{-1}^\kappa\) both for \(\kappa\neq \mu\) which behave just like before and \(\kappa=\mu\) which are new because of the Dirichlet boundary conditions for \(X^\mu\). We get the same number of polarizations – massless bosonic fields in the spacetime – and most of them are components of a gauge field again.
Nevertheless, one subtlety is new: the field creating the particle identical to the string in the \(\alpha_{-1}^\mu\ket 0\) state, for the particular value of \(\mu\) that we T-dualized, will no longer be a component of the gauge field \(A_\mu\). Instead, it will be a scalar field \(X^\mu\) that is physically interpreted as the transverse location of the brane in the direction \(\mu\).
It's kind of cool. For the full-Neumann boundary conditions, we had a spacetime-filling D25-brane and a massless gauge field \(A_\mu\) that is defined everywhere in the spacetime (well, on the spacetime-filling D-brane). Now, the gauge field is only defined on a 24+1-dimensional hypersurface, the D24-brane. The gauge fields in 24+1 dimensions have one fewer polarization. But it's compensated by the fact that there's one scalar field that remembers the transverse location \(X^\mu\) of a point on the D24-brane as a function of the remaining 24+1 dimensions! You can actually make the D24-brane wiggle inside the spacetime if you "pour" a coherent state of open strings in the \(\alpha_{-1}^\mu\ket 0\) state on the D-brane.
Indeed, if you derive the effective action for all these open string modes from string theory, and it's a general fact that you may derive *everything* from string theory, you will find out that there are the usual Yang-Mills terms for the gauge field component; and there are new terms that give the D24-brane its tension. The D24-brane may wiggle, stretch, and try to return to a placid state just like the strings themselves. A difference is that for the fundamental strings, we inserted the equations doing these wiggly things as defining equations of the theory; for the D-branes, their wiggling (much like the dynamics of the gauge fields inside them) is derived from the open strings that can be attached to them!
At the beginning, I mentioned the closed string state that is responsible for the spacetime metric; "pouring" closed strings in this state into the spacetime is physically *identical* to deforming the shape of the spacetime. The closed strings in this graviton states are not just "similar" to the variations of the spacetime shape and curvature; the variations of the spacetime shape and curvature *are* the closed strings in this vibration mode. There's no other way to curve the spacetime differently than to pour closed strings. All the dynamics behaves in agreement with all the principles of general relativity.
Equivalently, the D24-brane doesn't have "completely new, non-stringy" degrees of freedom that would define its shape within the spacetime or the values of the gauge fields that only exist inside this D24-brane. Instead, all these degrees of freedom are among the "standard" stringy degrees of freedom that always existed, assuming that you also allow strings with the boundary conditions corresponding to a given D-brane. Again, the open strings are not just "analogous" to wiggles on the D24-brane; the wiggles on the D-brane are physically *identical* to some particular vibrating open strings. They're the same thing. Aside from the open strings, there are no other wiggles of the D-branes.
In perturbative string theory, everything – including all the spacetime fields and shapes and particles as well as D-branes' properties such as their shape and electromagnetic fields – is made out of strings. When you go to strong coupling, the compositeness becomes "less useful" for accurate calculations – you need to calculate lots of corrections etc. – and the fundamental strings become less fundamental. At a generic coupling, you can't unambiguously say which objects are fundamental and which objects are made out of them. There is actually often a symmetry, S-duality, that exchanges them.
However, it's still important to learn how string theory works at a very weak coupling and indeed, everything is made out of fundamental strings and it makes a perfect sense.
Needless to say, there are D-branes of all dimensionalities between 25 and –1. Yes, it's minus one. You may T-dualize several circular spacetime dimensions to get some intuition for what happens and how the degrees of freedom are related to the original spacetime-filling D-branes. The generalization is straightforward.
All these things may also be extended to the truly physically relevant theory, the 9+1-dimensional superstring. In that theory, we also need to deal with the world sheet fermionic fields and their boundary conditions and we find out that the even-dimensional and odd-dimensional branes are a bit different. Only one of the two groups may be supersymmetric and the others can't. (In type IIA and type IIB, the role of "even" and "odd" get interchanged.)
Moreover, superstring theory adds one more cool thing that isn't present in bosonic string theory: the supersymmetric D-branes (D-even-branes in type IIA and D-odd-branes in type IIB string theory) are charged under new antisymmetric tensor fields, the Ramond-Ramond \(p\)-form fields that arise at the massless level of the closed superstring (in the sector whose ground state looks like a spacetime spinor \(\otimes\) spacetime spinor). Prior to the discovery of D-branes, it looked like those fields were somewhat "useless" because nothing was charged under this particular generalization of electromagnetism. As Joe Polchinski showed, D-branes are charged under them: everything makes some sense in string theory, everything plays some role, and all these roles fit together. String theory never offers you an incomplete story, a Hollywood movie that wasn't finished because they found out there was no sensible way to explain what they already shot. String theory is a fully consistent theory that makes everything meaningful.
Everyone who wants to feel some familiarity with basic string theory should spend some time by thinking about the different boundary conditions and what they mean because the differences between things that were thought to be qualitatively different – like branes of different dimensions, with or without scalars, and so on – boil down to a very modest "technical difference" at the level of string theory. The diversity of physical phenomena one may derive from string theory by looking at objects with slightly different boundary conditions etc. is of course a part of the ability of string theory to unify everything.
And that's the memo.
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