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arivero
arivero
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The descent from 24 to 8 seems to happen when the straight sums are substituted by alternating sums. This is known from the theory of the Riemann zeta funtion, whose only pole in z=1$s=1$ is cancelled via a multiplication that produces the Dirichlet eta function, $$\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$$.

This function has better analiticity than Zeta. But it is alternating, $$ \eta(s) = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots$$ And thus it could be related with the simple expresion I have mentioned above in the comments. But, more important,

$$\eta(-1)=\left(1-2^{2}\right) \zeta(-1) = -3 \times {-1\over 12} = {1\over 4}$$

And you can suspect that the Eta function does for the superstring the same role that the Zeta does for the bosonic string. And indeed it appears in very similar situations. For instance, Michael B. Green, in his 1986 Trieste lectures "String and Supertring Theory", section 5.11, calculates the NS sector spectrum and then the normal ordering constant, that appears formally as a difference between the bosonic term and the fermionic term. Such difference can be manipulated to obtain the Eta function as above, times $(D-2)/2$

So if anyone can tell how the Zeta regulator is related to the integer sum up to 24, then we could probably guess how the Eta regulator would be related to the alternating sum up to 8.

The descent from 24 to 8 seems to happen when the straight sums are substituted by alternating sums. This is known from the theory of the Riemann zeta funtion, whose only pole in z=1 is cancelled via a multiplication that produces the Dirichlet eta function, $$\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$$.

This function has better analiticity than Zeta. But it is alternating, $$ \eta(s) = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots$$ And thus it could be related with the simple expresion I have mentioned above in the comments. But, more important,

$$\eta(-1)=\left(1-2^{2}\right) \zeta(-1) = -3 \times {-1\over 12} = {1\over 4}$$

And you can suspect that the Eta function does for the superstring the same role that the Zeta does for the bosonic string. And indeed it appears in very similar situations. For instance, Michael B. Green, in his 1986 Trieste lectures "String and Supertring Theory", section 5.11, calculates the NS sector spectrum and then the normal ordering constant, that appears formally as a difference between the bosonic term and the fermionic term. Such difference can be manipulated to obtain the Eta function as above, times $(D-2)/2$

So if anyone can tell how the Zeta regulator is related to the integer sum up to 24, then we could probably guess how the Eta regulator would be related to the alternating sum up to 8.

The descent from 24 to 8 seems to happen when the straight sums are substituted by alternating sums. This is known from the theory of the Riemann zeta funtion, whose only pole in $s=1$ is cancelled via a multiplication that produces the Dirichlet eta function, $$\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$$.

This function has better analiticity than Zeta. But it is alternating, $$ \eta(s) = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots$$ And thus it could be related with the simple expresion I have mentioned above in the comments. But, more important,

$$\eta(-1)=\left(1-2^{2}\right) \zeta(-1) = -3 \times {-1\over 12} = {1\over 4}$$

And you can suspect that the Eta function does for the superstring the same role that the Zeta does for the bosonic string. And indeed it appears in very similar situations. For instance, Michael B. Green, in his 1986 Trieste lectures "String and Supertring Theory", section 5.11, calculates the NS sector spectrum and then the normal ordering constant, that appears formally as a difference between the bosonic term and the fermionic term. Such difference can be manipulated to obtain the Eta function as above, times $(D-2)/2$

So if anyone can tell how the Zeta regulator is related to the integer sum up to 24, then we could probably guess how the Eta regulator would be related to the alternating sum up to 8.

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arivero
arivero
  • 1.9k
  • 18
  • 39

The descent from 24 to 8 seems to happen when the straight sums are substituted by alternating sums. This is known from the theory of the Riemann zeta funtion, whose only pole in z=1 is cancelled via a multiplication that produces the Dirichlet eta function, $$\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$$.

This function has better analiticity than Zeta. But it is alternating, $$ \eta(s) = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots$$ And thus it could be related with the simple expresion I have mentioned above in the comments. But, more important,

$$\eta(-1)=\left(1-2^{2}\right) \zeta(-1) = -3 \times {-1\over 12} = {1\over 4}$$

And you can suspect that the Eta function does for the superstring the same role that the Zeta does for the bosonic string. And indeed it appears in very similar situations. For instance, Michael B. Green, in his 1986 Trieste lectures "String and Supertring Theory", section 5.11, calculates the NS sector spectrum and then the normal ordering constant, that appears formally as a difference between the bosonic term and the fermionic term. Such difference can be manipulated to obtain the Eta function as above, times $(D-2)/2$

So if anyone can tell how the Zeta regulator is related to the integer sum up to 24, then we could probably guess how the Eta regulator would be related to the alternating sum up to 8.