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I am reading "Mirror Symmetry" by Hori et al, and have a question on Chap.17 (Chiral rings and geometry of the vacuum bundle). On p.425 the authors say

Consider the path-integral on the hemisphere. The boundary of the hemisphere is a circle on which our Hilbert space is based. The path-integral will give us a number, and so defines a functional from boundary filed configurations to numbers, equivalently, a state in the Hilbert space...

Then they say

To obtain a ground state at the boundary we consider the "neck" of the hemisphere to be infinitely stretched. In other words, we imagine connecting the hemisphere to a semi-infinite flat tube. Noe that on the flat tube the twisted and untwisted theories are equivalent.

They continue

Similarly, if we consider the topological path-integral together with the insertion of the corresponding chiral fields, we obtain a correspondence between chiral fields and the ground state....

Question 1 Why are the twisted and untwisted theories equivalent on the flat tube?

Question 2 What does it mean by inserting chiral field? I don't think this is explained anywhere in the book. Does the insertion mean that the operator acts on the field after some time corresponding to the position of the insertion?

Should I think of this hemisphere as a Riemann surface? I am confused because this also looks like the operator formalism and manifolds with boundary. I am kind of lost because they suddenly introduce the hemisphere and identify the states with the Hilbert space on the boundary.

I think I lack of firm understanding of the subject, so I would appreciate it if someone could kindly explain things from the very basic.

I am reading "Mirror Symmetry" by Hori et al, and have a question on Chap.17 (Chiral rings and geometry of the vacuum bundle). On p.425 the authors say

Consider the path-integral on the hemisphere. The boundary of the hemisphere is a circle on which our Hilbert space is based. The path-integral will give us a number, and so defines a functional from boundary filed configurations to numbers, equivalently, a state in the Hilbert space...

Then they say

To obtain a ground state at the boundary we consider the "neck" of the hemisphere to be infinitely stretched. In other words, we imagine connecting the hemisphere to a semi-infinite flat tube. Noe that on the flat tube the twisted and untwisted theories are equivalent.

They continue

... Similarly, if we consider the topological path-integral together with the insertion of the corresponding chiral fields, we obtain a correspondence between chiral fields and the ground state....

I am lost because they suddenly introduce the hemisphere and identify the states with the Hilbert space on the boundary.

Question 1 Should I think of this hemisphere as a Riemann surface? Or is this the operator formalism and manifolds with boundary?

Question 2 Why are the twisted and untwisted theories equivalent on the flat tube?

Question 3 What does it mean by inserting chiral field? I don't think this is explained anywhere in the book. Does the insertion mean that the operator acts on the field after some time corresponding to the position of the insertion?

I think I lack of firm understanding of the subject, so I would appreciate it if someone could kindly explain things from the very basic.

I am reading "Mirror Symmetry" by Hori et al, and have a question on Chap.17 (Chiral rings and geometry of the vacuum bundle). On p.425 the authors say

Consider the path-integral on the hemisphere. The boundary of the hemisphere is a circle on which our Hilbert space is based. The path-integral will give us a number, and so defines a functional from boundary filed configurations to numbers, equivalently, a state in the Hilbert space...

Then they say

To obtain a ground state at the boundary we consider the "neck" of the hemisphere to be infinitely stretched. In other words, we imagine connecting the hemisphere to a semi-infinite flat tube. Noe that on the flat tube the twisted and untwisted theories are equivalent.

They continue

Similarly, if we consider the topological path-integral together with the insertion of the corresponding chiral fields, we obtain a correspondence between chiral fields and the ground state....

Question 1 Why are the twisted and untwisted theories equivalent on the flat tube?

Question 2 What does it mean by inserting chiral field? I don't think this is explained anywhere in the book. Does the insertion mean that the operator acts on the field after some time corresponding to the position of the insertion?

Should I think of this hemisphere as a Riemann surface? I am confused because this also looks like the operator formalism and manifolds with boundary. I am kind of lost because they suddenly introduce the hemisphere and identify the states with the Hilbert space on the boundary.

I think I lack of firm understanding of the subject, so I would appreciate it if someone could kindly explain things from the very basic.

I am reading "Mirror Symmetry" by Hori et al, and have a question on Chap.17 (Chiral rings and geometry of the vacuum bundle). On p.425 the authors say

Consider the path-integral on the hemisphere. The boundary of the hemisphere is a circle on which our Hilbert space is based. The path-integral will give us a number, and so defines a functional from boundary filed configurations to numbers, equivalently, a state in the Hilbert space...

Then they say

To obtain a ground state at the boundary we consider the "neck" of the hemisphere to be infinitely stretched. In other words, we imagine connecting the hemisphere to a semi-infinite flat tube. Noe that on the flat tube the twisted and untwisted theories are equivalent.

They continue

Similarly, if we consider the topological path-integral together with the insertion of the corresponding chiral fields, we obtain a correspondence between chiral fields and the ground state....

Question 1 Why are the twisted and untwisted theories equivalent on the flat tube?

Question 2 What does it mean by inserting chiral field? I don't think this is explained anywhere in the book. Does the insertion mean that the operator acts on the field after some time corresponding to the position of the insertion?

Should I think of this hemisphere as a Riemann surface? I am confused because this also looks like the operator formalism and manifolds with boundary. I am kind of lost because they suddenly introduce the hemisphere and identify the states with the Hilbert space on the boundary.

I think I lack of firm understanding of the subject, so I would appreciate it if someone could kindly explain things from the very basic.

I am reading "Mirror Symmetry" by Hori et al, and have a question on Chap.17 (Chiral rings and geometry of the vacuum bundle). On p.425 the authors say

Consider the path-integral on the hemisphere. The boundary of the hemisphere is a circle on which our Hilbert space is based. The path-integral will give us a number, and so defines a functional from boundary filed configurations to numbers, equivalently, a state in the Hilbert space...

Then they say

To obtain a ground state at the boundary we consider the "neck" of the hemisphere to be infinitely stretched. In other words, we imagine connecting the hemisphere to a semi-infinite flat tube. Noe that on the flat tube the twisted and untwisted theories are equivalent.

They continue

Similarly, if we consider the topological path-integral together with the insertion of the corresponding chiral fields, we obtain a correspondence between chiral fields and the ground state....

Question 1 Why are the twisted and untwisted theories equivalent on the flat tube?

Question 2 What does it mean by inserting chiral field? I don't think this is explained anywhere in the book.

I think, the authors are talking about the operator formalism and manifolds with boundary. Does the insertion mean that the operator acts on the field after some time corresponding to the position of the insertion?

I think I lack of firm understanding of the subject, so I would appreciate it if someone could kindly explain things from the very basic.

I am reading "Mirror Symmetry" by Hori et al, and have a question on Chap.17 (Chiral rings and geometry of the vacuum bundle). On p.425 the authors say

Consider the path-integral on the hemisphere. The boundary of the hemisphere is a circle on which our Hilbert space is based. The path-integral will give us a number, and so defines a functional from boundary filed configurations to numbers, equivalently, a state in the Hilbert space...

Then they say

To obtain a ground state at the boundary we consider the "neck" of the hemisphere to be infinitely stretched. In other words, we imagine connecting the hemisphere to a semi-infinite flat tube. Noe that on the flat tube the twisted and untwisted theories are equivalent.

They continue

Similarly, if we consider the topological path-integral together with the insertion of the corresponding chiral fields, we obtain a correspondence between chiral fields and the ground state....

Question 1 Why are the twisted and untwisted theories equivalent on the flat tube?

Question 2 What does it mean by inserting chiral field? I don't think this is explained anywhere in the book. Does the insertion mean that the operator acts on the field after some time corresponding to the position of the insertion?

Should I think of this hemisphere as a Riemann surface? I am confused because this also looks like the operator formalism and manifolds with boundary. I am kind of lost because they suddenly introduce the hemisphere and identify the states with the Hilbert space on the boundary.

I think I lack of firm understanding of the subject, so I would appreciate it if someone could kindly explain things from the very basic.