Q2. What is the origin of the Hurwitz numbers?
In particular, they arise naturally in combinatorics, as they count factorizations of permutations into transpositions, and the original idea of Hurwitz expresses them in terms of the representation theory of the symmetric group.
Q3. What is the generating function for Riemann surfaces?
by relating the simple Hurwitz numbers to descendant Gromov-Witten invariants, the authors have obtained the explicit generating functions (3.10) for the number of inequivalent reducible covers for arbitrary source and target Riemann surfaces.
Q4. Why do the authors need to distinguish when the degree n is odd or even?
(2.8)Because v determines the range of the first sum in the master formula, the authors need to distinguish when the degree n is odd or even.
Q5. What is the generating function for the Hodge integrals?
An interesting application of this development is to use the generating function for Hurwitz numbers µg,d0,1(d) to derive a generating function for Hodge integrals over the moduli spaceMg,1.
Q6. What is the symmetry factor for the genus g?
More precisely, when the summation variable s in their formula equals (g + 1)/2, for an odd genus g, there is a symmetry factor of 1/2 in labeling the edges because the two disconnected graphs are identical except for the labels.
Q7. What is the main reason for the renewed interest in enumerative geometry?
Many classical problems in enumerative geometry have been receiving renewed interests in recent years, the main reason being that they can be translated into the modern language of Gromov-Witten theory and, moreover, that they can be consequently solved.
Q8. How do the authors associate a Young diagram with j rows?
To each irreducible representation labeled by ρ = (ρ1, . . . , ρj) ` 2k, the authors can associate a Young diagram with j rows, the ith row having length ρi.
Q9. What is the equivalence relation of covers?
the equivalence relation of covers gets translated into conjugation by a permutation in Sn, i.e. two elements of Tn,h,σ are now equivalent iff they are conjugates.
Q10. What is the generating function for the number of inequivalent simple covers of an ?
for a given lattice L associated with the target elliptic curve, the authors need to find the number of inequivalent sublattices L′ ⊂ L of index [L : L′] = n.
Q11. What is the generalized “partition function” for h?
The generalized “partition function” Z(h) = exp(Φ(h)) for all h is given byZ(h) = 1 + q + ∑ n≥2 ∑ γ∈Rn ( n! fγ )2h−2 cosh [( n 2 ) χγ(2) fγ λ ] qn. (3.10)Proof:
Q12. What is the recursion relations for a elliptic curve?
these recursion relations require as initial data the knowledge of simple Hurwitz numbers, and their work would be useful for applying the relations as well.
Q13. How do the authors find the degree of the irreduciblecovers?
Upon fixing the degree of the irreduciblecovers, the authors have obtained closed form answers for simple Hurwitz numbers for arbitrary source and target Riemann surfaces, up to degree 7.
Q14. How do you obtain a set of recursion relations for the numbers g?
In particular, Li et al. have obtained a set of recursion relations for the numbers µg,nh,w(α) by applying the gluing formula to the relevant relative GW invariants [LZZ].
Q15. how can the authors get the generating function G(t, k)?
The generating function G(t, k) can be evaluated at k = −1 to beG(t,−1) = 1 2 − 1 t2( cos t+ t22 − 1) = 12( sin(t/2)t/2)2 , (4.8)and similarly at k = 0 to beG(t, 0) = 12( tsin t) = 12t/2sin(t/2)1cos(t/2) . (4.9)Proof:
Q16. what is the asymptotic limit of G(t, k)?
The asymptotic limit of G(t, k) isG(t k 1 2 , k−1) k→0−→ exp( t2/3 ) 2 t √ πErf [ t 2 ] , (4.15)and thus, the integrals can be evaluated to be∫ Mg,2 1 (1− ψ1)(1− ψ2) = 1 2 g∑ m=01 m! 12m (g −m)! (2g − 2m+ 1)! . (4.16)Proof: