In 26 flat dimensions, the effect of the operator Ω in the trace is an extra −1 at the even mass levels plus an appropriate accounting of the Chan–Paton factors.
That is, the operator exp(2πiRp) which translates strings once around the periodic dimension must leave states invariant, so the center-of-mass momentum is quantized k= n , R n∈Z. (3.7.7) separates into a d-dimensional antisymmetric tensor transformation ζµ and an ordinary gauge invariance ζd .
Let us now discuss the OPE and vertex operators, first in a slightly heuristic way and then filling in the details.
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However, as in the Polyakov Action, there is no square root over the $ X $ ' s, it is much easier to quantise compared to the Nambu-Goto Action.
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From the mode expansion one may derive the equal time (|z1 | = |z2 |) commutator [ XL (z1 ), XL (z2 ) ] = πiα sign(σ11 − σ21 ) . To describe the states of the closed string CFT, consider the Laurent 236 8 Toroidal compactification and T -duality X _1 0 +1 Y (a) X 0 Y _1 +1 (b) Fig.
7.3. 7.5 (a) Obtain the leading behavior of the theta functions (7.2.37) as Im τ → ∞. It is known as a Chern–Simons term, this signifying the antisymmetrized combination of one gauge potential and any number of field strengths. %PDF-1.5 The amplitude is then ZM2 = ±inV26 ∞ dt 0 4t (8π 2 α t)−13 ϑ00 (0, 2it)−12 η(2it)−12 . Here is given a little introduction to the most basic string theory: th bosonic string. Where $ \tau $ is the trace of the Stress-Energy-Momentum Tensor and $ \gamma_{ab} $ is the induced metric on the worldsheet. Parameterize the metric as M N µ ν d µ 2 ds2 = GD MN dx dx = Gµν dx dx + Gdd (dx + Aµ dx ) .
For the same reason there is an extra 2 in both the torus and cylinder amplitudes in the unoriented theory.
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The general X µ path integral (6.2.18) factorizes in an obvious way into holomorphic times antiholomorphic, so that one can simply replace n |zij |α ki kj → i,j=1 i.
˜ µ , the covariant derivative is Defining Aµ = R A ˜µ , ∂µ + ipd Aµ = ∂µ + inA (8.1.10) so that the charges are integers. The Bosonic String Theory was developed in the late 60s and a couple of years later, it was established that it will make sense by keeping it in 26 dimensions. (a) Closed oriented strings of winding number w = +1, 0, −1. w∼ (7.4.18) = w + 4πit , w + π ∼ = −(w It follows that the upper and lower edges of the region (7.4.16b) are periodically identified. 7.2. At the time, finding a mathematical structure for this S-matrix was considered to be a significant step toward creating a coherent model of particle physics. stream The reason for these extra dimensions can be seen by analogy. Even though bosonic string theory was flawed and incomplete, string theorists occasionally do mathematical work with this model to test new methods and theories before moving on to the more modern superstring models. /Length 259
We'll assume you're ok with this, but you can opt-out if you wish. For illustration, one compact dimension X and one noncompact dimension Y are shown. The initial and final state of particle interactions can be recorded in an array of numbers called an S-matrix. (Dimensions are generally thought of in terms of up/down, left/right, forward/backward.). It is easy to see that winding number is always conserved as in this example. /Type /ObjStm
The key thing is that these waves, or vibrations, move only back and forth along the length of the spring. We live in a three-dimensional world. masters level) students. /Length 888 7.14 Show that the amplitude (7.4.11) with the boundary state (7.4.13) reproduces the cylinder amplitude (7.4.3). (8.2.14) The field X then splits into holomorphic and antiholomorphic parts, X(z, ¯z ) = XL (z) + XR (¯z ) , with α α XL (z) = xL − i pL ln z + i 2 2 α α XR (¯z ) = xR − i pR ln ¯z + i 2 2 1/2 ∞ m=−∞ m=0 1/2 ∞ m=−∞ m=0 (8.2.15) αm , mz m (8.2.16a) ˜αm . The graviton-dilaton action (3.7.20) becomes 1 dD x (−GD )1/2 e−2Φ (R + 4∇µ Φ∇µ Φ) S1 = 2κ20 πR = 2 dd x (−Gd )1/2 e−2Φ+σ κ0 1 2σ µ µ µν × Rd − 4∂µ Φ∂ σ + 4∂µ Φ∂ Φ − e Fµν F 4 πR = 2 dd x (−Gd )1/2 e−2Φd κ0 1 × Rd − ∂µ σ∂µ σ + 4∂µ Φd ∂µ Φd − e2σ Fµν F µν , 4 (8.1.9) giving kinetic terms for all the massless fields.
There is again a t → 0 divergence; it corresponds to the process of figure 7.3(a) with one tadpole from the disk and one from the projective plane. The Nambu-Goto Action is the action principle generally associated with Classical Bosonic String Theory. It can be shown that this is classically equivalent to the Nambu-Goto Action as follows (Please click here to view properly): $ 0 = \frac{\delta S}{\delta h^{ab}} = T_{ab} $, $ 0 = \frac{\delta S}{\delta h^{ab}} = \frac{\tau}{2} \sqrt{-h} \left( \gamma_{ab} - \frac12 h_{ab} h^{cd} \gamma_{cd} \right) $.
. (7.2.4)). The full set of lectures notes can be downloaded here and weigh in at around 200 pages. Let us see this in detail, taking the more general case of D = d + 1 spacetime dimensions with xd periodic. (8.2.4) m+1 ¯ z m=−∞ The total change in the coordinate X in going around the string is 2πRw = ¯ = 2π(α /2)1/2 (α0 − ˜α0 ) . . For closed bosonic strings, it becomes: $ m= \sqrt{N+\tilde N-a-\tilde a} =\sqrt{N+\tilde N-2 } $ p. Where we took $ a = \tilde a = 1 $ because both the left- (Without the tilde) and right- (with the tilde) moving sectors are from the Bosonic String Theory. a� The Ricci scalar for the metric (8.1.2) is − p µ pµ = 1 R = Rd − 2e−σ ∇2 eσ − e2σ Fµν F µν , 4 (8.1.8) 8.1 Toroidal compactification in field theory 233 where R is constructed from GD MN and Rd from Gµν . When we do any calculations, we just assume some things to be constant. We could set R to some convenient value, say α1/2 , or to unity if we use dimensionless variables, but it is convenient to leave it general; often it will be more convenient to set Gdd to unity instead.
We still need to understand a lot of physics and mathematics to understand the concept of the string theory or the Bosonic string theory in its full essence, we need to understand a lot of mathematics.
endstream The antisymmetric tensor also gives rise to a gauge symmetry by a generalization of the Kaluza–Klein mechanism.
What about the other 22 spatial dimensions? [Compare exercise A.3.] For now the fields Gµν , Gdd , and Aµ are allowed to depend only on the noncompact coordinates xµ .
Use this to find the numerical relation between gc and go2 .
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<< It is important, however, that the full vertex operator OPE is single-valued: the net phase when z1 circles z2 is exp[πiα (kL kL − kR kR )] = exp[2πi(nw + wn )] = 1 .
We first consider the same compactification in field theory, encountering in particular the Kaluza–Klein unification of gauge interactions and gravity. The different values of Φ and σ label degenerate configurations (or states in the quantum theory), and a state in which these fields are slowly varying has energy only from the gradient.
The simple, short, and obvious answer is because the math only fits in that way.
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