DBI Analysis of Open String Bound States on Non-compact D-branes

CHAPTER 3. SUPERSTRINGS 49F, when acting on a NS state, returns the number <strong>of</strong> times a fermionic operator hasbeen acted on the groundstate in order to obtain this state. So we see thatG|0〉 = −1, (3.49)and that acting on other states, G will be 1 for half-integer mass levels (levels, notmasses!) and -1 for integer mass levels. The GSO projection for the NS sector consistsin keeping only those states with positive G-parity. This way, we eliminate the tachyonfrom the spectrum, as well as all other states with half-integer mass, which was exactlywhat we wanted.For the R sector, G becomes a generalized chirality operator,∞∑G = Γ 11 (−1) F with F = d i −rd i r, (3.50)with Γ 11 the ten-dimensional chirality operator. The GSO projection in the R sectorconsists <strong>of</strong> keeping only one <strong>of</strong> both chiralities. Which one is purely a matter <strong>of</strong> choice.This effectively eliminates half <strong>of</strong> all states at every mass level.Since the result is too beautiful not to mention, we will take a quick look at themodified generating functions for the truncated (i.e. what is left after performing theGSO projection) spectra. To update f R (x) (Eq. 3.46), it suffices to change the factorin front <strong>of</strong> the product from 16 to 8:+∞ ∏f R (x) −→ fR GSO (x) = 8n=1n=1( 1 + xn1 − x n ) 8. (3.51)In the NS sector, we need to subtract all states obtained by acting with an even number<strong>of</strong> raising b operators from the full spectrum. This can be achieved by considering thegenerating function1+∞ ∏√ xn=1(1 − x n−1 21 − x n ) 8= 1x −1 2 − 8x 0 + 36x 1 2 − 128x 1 + . . . , (3.52)in which we see that states corresponding to an odd number <strong>of</strong> b operators carry a − sign,while states corresponding to an even number <strong>of</strong> b operators carry a + sign. Howeverwe see that when we subtract Eq. 3.52 from the unconstrained spectrum generated byEq. 3.41, we will double the number <strong>of</strong> states that survive, so we need to divide by anadditional factor <strong>of</strong> two to rectify this. Concretely, our new generating function becomesf NS (x) −→ f GSONS (x) = 12 √ x⎡∞∏⎣n=1(1 + x n−1 21 − x n ) 8−∞∏n=1( ) ⎤ 81 − x n−1 2⎦.1 − x n (3.53)At first sight, this looks like bad news, but incredibly enough, it was proven by CarlGustav Jacobi in 1829 (!) that indeed,⎡ ( ) 8 ( ) ⎤ 81∞∏2 √ ⎣1 + x n−1 2∞∏ 1 − x n−1 2∞∏( )x 1 − x n −⎦ 1 + xn 81 − x n = 81 − x n , (3.54)n=1n=1n=1