Relations for quadratic Hodge integrals via stable maps

Following Faber-Pandharipande, we use the virtual localization formula for the moduli space of stable maps to $ \mathbb{P}^1 $ to compute relations between Hodge integrals. We prove that certain generating series of these integrals are polynomials.


G. Politopoulos
In the present note, we consider the following power series in C[] [[]] defined using double Hodge integrals where  = ( 1 , . . .,   ) is a vector of non-negative integers.If  = 1, we use the convention: Here we provide the first values of   (− − 1, ).In the list below we omit the variables − − 1 and  in the notation: Considering these first values, we conjecture that   is a polynomial of total degree || in both variables  and .

Preliminaries
We denote by M , (P 1 , 1) the moduli space of stable maps of degree 1 to P 1 .It is a proper DM stack of virtual dimension 2 + .Here we can define in an analogous way the Hodge bundle E, the cotangent line bundles L  and we denote again   and   the respective Chern classes.We also have the forgetful and evaluation maps  : M ,+1 (P 1 , 1) → M , (P 1 , 1), and   : M ,+1 (P 1 , 1) → P 1 .
Throughout this note the enumeration of markings starts from 0. Furthermore,  is the morphism that forgets the marking  0 and   is the evaluation of a stable map to the -th marked point.The vector bundle  :=  1  * ( * 0 O P 1 (−1))) is of rank  and we denote by  its top Chern class.We will denote: where  denotes the class of a point in P 1 .
Besides, we have the String and Dilaton equation for Hodge integrals.

The calculation
Note that the GW-invariant ⟨  =1    ()|⟩ P 1 ,1 is 0 unless || =  for dimensional reasons.Indeed, dim C [M , (P 1 , 1)] vir = 2 +  and the cycle we are integrating is in codimension  + || +.Using the above localization formula, and Lemma 2.1 of [TZ03] the intersection number ⟨  =1    ()|⟩ P 1 ,1 is expressed as: In the last equation we used Proposition 1.2 in order to replace ∫ Then, we have ) is a polynomial in  of degree , which actually determines the degree of   ().
We now present a proof for the main result.Relations for quadratic Hodge integrals via stable maps 5 proven in section 3.1 of [FP00a].Now, we consider the product of exp  24 and to obtain a new power series whose coefficients in degree  are given by This is exactly  , () •  =1 (2  + 1)!!(−4)   .Hence, we can rewrite the power series   (, ) in the form As it is computed in the start of Section 2 we have that the numbers  , () vanish when  > ||.Hence, we get that all coefficients of the power series   (, ) vanish when  > ||, i.e.   (, ) is a polynomial of degree ||.Furthermore, the top coefficient of   (, ), i.e. the coefficient of  | | is given by This value is computed in [KL11] and is actually equal to 1.In particular, the number  =1 (−4)   (2  + 1)!! is here to make the polynomial monic.■ We now prove several other properties of the polynomials   .
Proposition 2.1 The constant term  0 of   (, ) is non zero if and only if  = 1 where then  0 = (−1)   =1 (−4)   (2  + 1)!! or if  > 1 and  =1   ≤  − 2 where then Proof We only compute the integrals appearing in the constant term of this polynomial since then we only have to multiply with  =1 (2  + 1)!!(−4)   .The integral in the constant term of   (, ) is given by ∫ When  > 1, if  =1   >  − 2, then  0 is zero for dimensional reasons.Otherwise, we have Proof We define the power series Note that the following equation holds.