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I am reading Vafa's paper 'Topological Mirros and Quantum Strings'. In this paper, the author says the Hilbert Space of a fermionic string theory corresponds to the space of semi-infinite forms on the loop space.

I only know differential forms before so I would like to know is there some nice introduction material to this kind of 'semi-infinite form'?

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The space of semi-infinite forms is basically the name used by mathematicians for the fermionic Fock space please see for example: Friedrich Wagemann lecture, page 8.

Given an infinite dimensional vector space with spanned by: $\{ e_i, i\in \mathbb{Z}\}$, let its dual space be spanned by $\{ f_i, i\in \mathbb{Z}\} (\langle f_j, e_i \rangle = \delta_{ij}$) then an example of a semi-infinite dimensional form is:

$$ f = f_2 \wedge f_3 \wedge f_4 \wedge f_6\wedge f_7 \wedge f_8 …$$

(Please notice that after $f_6$, there are no gaps in the wedged vectors, this is the semi-infinite property)

The analogy of this representation to the usual fermionic Fock space representation is quite apparent. The mentioned vector represents the Fock space vector:

$$ f = a_1^{\dagger} a_5^{\dagger}|0\rangle$$,

($a_i^{\dagger}$ are creation operators and $a_i$ are annihilation operators). The Fock vacuum corresponds to the vector: $$ v = f_1 \wedge f_2\wedge f_3\wedge f_4\wedge f_5 \wedge f_6 …$$ (with no gaps- analogous with a full Dirac sea).

In the semi-infinite form notation, the creation operation is performed by contraction with the corresponding vector, for example, with our previous vector: $$ i(f_6) f = f_2 \wedge f_3 \wedge f_4 \wedge f_7 \wedge f_8 …$$ The annihilation is performed by wedging $$ f_5 \wedge f = -f_2 \wedge f_3 \wedge f_4 \wedge f_5\wedge f_6 \wedge f_7 \wedge f_8 …$$ One advantage of the semi-infinite form picture is that it can be given a geometrical interpretation (in contrast to the algebraic interpretation of the fermionic Fock space), for example the forms can be chosen to be differential forms on some infinite dimensional space, and (thus one can define, for example, infinite dimensional cohomology). However, I think that all these constructions can be interpreted via BRST without the actual need to use the semi-infinite form picture.

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  • $\begingroup$ Interesting. Thanks for the explanation, I was guessing what the terminology was about! However in my opinion it is a very bad denomination. In the context of (mathematical) quantum mechanics, when I see such term as semi-infinite forms, I think immediately of bounded from below quadratic forms (that define self-adjoint operators) :-P $\endgroup$ – yuggib Jan 27 '16 at 11:18
  • $\begingroup$ Thanks for the explanation. In your case, how to define a form of negative degree? $\endgroup$ – Li Xinghe Jan 27 '16 at 19:20

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