Beta-gamma system, pure spinors and Hilbert series of arc spaces

Algorithms are presented for calculating the partition function of constrained beta-gamma systems in terms of the generating functions of the individual fields of the theory, the latter obtained as the Hilbert series of the arc space of the algebraic variety defined by the constraint. Examples of a beta-gamma system on a complex surface with an $A_1$ singularity and pure spinors are worked out and compared with existing results.


Introduction
A beta-gamma system is a two-dimensional conformal field theory modelled after the b-c ghost system with a set of possibly bosonic complex fields, denoted γ and their canonical conjugates, denoted β. It can be related to a certain large volume limit of the two-dimensional non-linear sigma-model with the fields γ identified as the complex coordinates of the target space [1][2][3]. A beta-gamma system is said to be free, field theoretically, or flat, geometrically, if the fields γ and β satisfy the commutation relations for free fields. In particular, a pair of γ's commute. The set of γ's then corresponds to the coordinates of the complex affine target space. A beta-gamma system is said to be curved, if the target space is curved. In this article we shall restrict to curved systems obtained as the coordinates of the target space satisfy one or more algebraic equations. This is obviously equivalent to imposing constraints on the γ's.
The action of a free beta-gamma system is linear in both the fields γ and β. The partition function of this field theory is obtained as the generating function of degeneracy of operators graded by quantum numbers associated to conserved charges of the classical action. The partition function of a constrained or curved system is then the generating function of degeneracy of operators satisfying the constraints.
The partition function can be computed by counting operators possessing equal conserved charges obtained through multiplication of β's, γ's and their derivatives with respect to the world-sheet coordinate. A direct construction of operators, however, becomes intractable in the presence of constraints, save for the simplest of instances, as the derivatives of the fields γ and β too are constrained by the derivatives of the constraints to all orders. Partition function of curved beta-gamma systems in several instances have been obtained by resorting to more indirect means [4][5][6][7]. A special case, which serves as the motivation for the majority of studies of the beta-gamma system in recent times, is the pure spinor constraint which is a quadratic one arising in an attempt to write a super-Poincarè invariant world-sheet string theory [8]. The partition function of pure spinors has been obtained as the character of representations of the SO(8) group [9][10][11][12]. In a variety of other examples the constraints are not quadratic. In the case where the target space can be realized as an orbifold, for example, C 2 /Z N or C 3 /Z M × Z N , with integral M and N, partition functions of beta-gamma systems have been obtained by lifting the geometric orbifold action to the partition function of the affine spaces C 2 and C 3 , respectively [7]. This, however, relies upon the affine parametrization of the orbifolds.
In the present article we consider two examples of constraints. The first is a quadratic one among three γ's, the other being pure spinors, which also obeys a set of quadratic constraints. We use the constraints directly without solving them, thereby avoiding any reference to the affine parametrization. Regarding the constraints in γ's as describing an algebraic variety embedded in the affine space of the unconstrained ones the contribution of the various modes of γ's to the partition function is given by the Hilbert series of the arc space of the variety. However, in both the instances considered here the varieties possess an isolated singular point. This renders the definition of the conjugate fields non-unique. The total partition function is then obtained by resorting to some prescription. One efficient prescription is to implement the so-called field-antifield symmetry of the partition function in a multiplicative fashion [4]. We show that it can also be obtained from the combination of various modes of the fields which are invariant under a certain gauge transformation that keeps the action unchanged modulo the constraints, provided the β's are subjected to the same constraints. We exhibit the computations explicitly for two cases. We obtain the partition function of a beta-gamma system on the rational double point surface singularity in both the ways and compare with the result obtained earlier [7] by realizing the target space as an orbifold. We find that the latter prescription fares slightly better when compared with the orbifold results. This computation uses the known description of resolution of surface singularities in terms of arc spaces. For the pure spinors this description is not known. We obtain the partition function by implementing the field-antifield symmetry on the contribution of the pure spinors obtained as the Hilbert series of the arc space of the pure spinor constraint. This is different from implementing the field-antifield symmetry at every order of mass separately. Obtaining the Hilbert series entails a computation of Gröbner basis of the ideal generated by the pure spinor constraint by considering 10m equations in 16m variables for every mass level m. The algorithm for this computation is rather simple and has been implemented in Macaulay2 [13]. The results match with the existing ones up to the first mass level.
In section 2 we begin by recalling some features of the beta-gamma system and its partition function and lay out the two prescriptions used to evaluate the partition function. In section 3 we recall the notion of arc spaces and the blow up of surface singularities in these terms. We use both prescriptions to compute the partition function of the beta-gamma system on the rational double point surface singularity in the following section, comparing the results. In section 5 we obtain the partition function of the pure spinor system up to the first mass level by implementing the fieldantifield symmetry. We conclude in section 6.
2 β-γ system on C d

Flat system
A beta-gamma system on the d-dimensional complex affine space C d is a two-dimensional conformal field theory of a set of complex fields {γ i } of vanishing conformal dimension and their canonical conjugates {β i }, i = 1, 2, · · · , d. On the two-dimensional space, henceforth referred to as the world-sheet, the conjugate fields are one forms, namely, β i = β iz dz + β iz dz, where z designates the coordinate of the world-sheet and a bar denotes its complex conjugate. For a flat betagamma system the fields γ are identified with the coordinates of the coordinate ring of the target The coordinates commute pairwise as do the conjugates thereby having trivial operator products. The operator product between a β and a γ, on the other hand, is taken to be the free one, namely The action for a beta-gamma system is written as in the conformal gauge, where ∂ = ∂ ∂z . The theory possesses two conserved currents, namely, the energy momentum tensor and a U(1) current corresponding to the scaling of the fields, The respective charges, namely, L 0 = dzzβ iz ∂γ i and J 0 = dzβ i γ i , characterize the field theory. Introducing the modular parameter q and another one, t, corresponding to the scaling the partition function of the beta-gamma system is written as the character where Tr signifies a trace with respect to the states of the Hilbert space of the theory.
Assuming that the fields possess mode expansions and the existence of a vacuum |0 to obey the highest weight conditions the character of the beta-gamma system on C d is obtained as [14,15] where the character of the affine complex plane is defined to be This can be interpreted as the generating function of degeneracy of monomials of a given degree for q and t, where each γ contributes a t, each β contributes q/t to the partition function while each derivative ∂ contributes a q [7].

Curved system
A class of curved beta-gamma systems is obtained from a flat one by imposing constraints on the fields γ. The constraints we consider are algebraic but non-linear, which are well-defined as the γ's commute between themselves. The constraint, however, renders the estimation of independent monomials for the computation of the partition function difficult. In the present article we deal with constraints which are algebraic, which can be thought of as algebraic varieties. We present a means to evaluate the partition function as the Hilbert series of the arc space of the variety. We consider the computation of partition function of curved systems wth quadratic constraints, written as with one or more constant d × d matrices Ω. Computation of the partition function then entails enumeration of combinations or monomials of the fields β and γ as well as their derivatives with respect to the world-sheet coordinate modulo the constraint and its derivatives to all orders. The multiplication of fields in forming monomials are to be normal ordered as usual, but this does not affect their number. As in the previous subsection we assume that β's are defined as objects of grade q/t. We consider two different ways to evaluate the partition function starting from the separate contributions of the γ's and β's. In the first method, implementing the so-called field-antifield symmetry [11], the contribution of the γ's, denoted Z γ , is split into two factors. The first is independent of q arising from the contribution of the zero modes. The other is a function of both q and t from the contribution of the massive modes. Thus, where the subscripts 0 and m refer to the zero and non-zero mass modes. The total partition function is obtained as An alternative is to use the gauge invariance of the action (2) under the gauge transformation modulo (9). This imposes a restriction on the possible combinations of suitably defined β's due to the constraints on γ's. Only β's appearing in gauge invariant combinations are counted in the partition function. Two such combinations at first and second mass levels, for example, are the U(1) current and energy momentum tensor, respectively, which are composite operators. Although the number of gauge invariant operators at each mass level is finite, new operators emerge at each higher mass level, rendering the counting of such states intractable. This is a hurdle in obtaining the partition function of generic β-γ systems in a closed form. We use a new method to implement gauge invariance in the β-γ directly at the level of the partition function. This also leads to a rationale for the omission of negative terms in 1/t. Assuming the existence of the conjugate fields we obtain their separate contribution to the partition function by subjecting them to the same constraint as the γ's, namely, Out of all monomials that can be constructed from the fields and their derivatives we need to keep the ones that vanish modulo the constraint. This is done by subtracting the ones that are nonvanishing after the gauge transformation from the set of all monomials. The generating function of degeneracy of all monomials involving both the fields and their derivatives satisfying the relations is then given by . The subtraction of unity in the first factor is for not counting monomials with only β's and their derivatives not paired with γ's. Such monomials can not be gauge invariant modulo (9) since the constraint is quadratic in γ's. Subtraction of unity in the second factor, on the other hand, is for not counting monomials involving only γ's since they are gauge invariant and shall be added separately to the partition function. The gauge transformation (12) on any monomial converts a β into a linear combination of γ's. We need to count the ones that vanish modulo the constraint after a gauge transformation. This is done by subtracting from Z ′ (q, t) the ones that do not vanish modulo the constraint, namely, Here C(Z γ , t) denotes the coefficient of t in the expansion of Z γ . The reason for subtraction of this and unity in the second factor is as before. Moreover, converting a β into a γ changes the grade by a factor of q/t 2 , which is to be corrected before subtraction. FInally, adding the monomials in only γ the resulting partition function is To evaluate the partition function in either way we need to find Z γ . For the second method we also need Z β . These are obtained as Hilbert series of the arc spaces of the varieties described by (9) and (13), respectively, to which we turn next.

Arc spaces and Hilbert Series
In this section we recall some features of the arc space of an algebraic variety [16] and define its Hilbert series. Relation between Hilbert series of arc spaces in a single variable and partitions has been noted earlier [17]. We restrict attention to complex numbers only but generalize the definition of Hilbert series to graded rings to incorporate the grades of q and t pertinent to beta-gamma systems. Let C[[ξ]] denote the formal power series (Puiseux series) ring of polynomials in a single variable ξ over the field of complex numbers C. In the simplest case the arc space of an algebraic variety defined by a polynomial equation f = 0 in the coordinate ring C[x 1 , x 2 , · · · , x n ] is the set of power series solutions to the equation f (x(ξ)) = 0, where x(ξ) = (x 1 (ξ), x 2 (ξ), · · · , x n (ξ)) ∈ C[[ξ]] n , with each component a power series in the formal variable ξ. This generalizes to more polynomials than one.
More formally, let M = Spec C[x 1 , x 2 , · · · , x n ]/(f 1 , f 2 , · · · , f m ) be an algebraic variety defined by m equations in the coordinate ring of C n . Let us write the coordinates x i as formal power series in a formal variable ξ as Substituting these series in the polynomials f k , k = 1, 2, · · · , m and truncating at the order ξ r we obtain the set of polynomials F (l) k as the coefficient of ξ l in the expansion of f k (x 1 , x 2 , · · · , x n ). The r-th jet scheme M r of M is then defined as where N denotes the set of natural numbers, N = {0, 1, 2, 3, · · · }. In more mundane terms, the arc space on M is defined by the infinite set of equations obtained at each order of ξ by substituting an infinite series of the form (16) into the defining equations f k = 0 of M for k = 1, 2, · · · m. A generating function for monomials in the variables x k , in general. However, for simple cases this complication may not exist. We shall associate a grade q j t to the variable x (j) i . The symbols are chosen to make the connection with the beta-gamma system conspicuous.

Arc space of the singular quadric in C 3
Let us illustrate the computation of the Hilbert space with two simple examples. Further examples with a singly graded variable exist in literature [17]. The arc space of the affine space C[x] consists of all the powers of x (j) for all j = 0, 1, · · · , ∞. The monomials are thereby obtained by arranging each of x (j) in a geometric series 1 + x (j) + (x (j) ) 2 + (x (j) ) 2 · · · = 1/(1 − x (j) ) and multiplying them as 1/ ∞ j=1 (1 − x (j) ). With the assignment of grades q j t to x (j) as above then yields the Hilbert series of the arc space of C[x] as Next let us work out the Hilbert series of the variety defined in C[x 1 , x 2 , x 3 ] by the quadratic polynomial f = x 1 x 2 − x 2 3 , corresponding to the rational double point singular variety This will provide part of the partition function Z γ of the beta-gamma system discussed in the previous section. Substituting the power series (16) in x 1 x 2 − x 2 3 we obtain the polynomials . . . at different orders of ξ. According to our assignment of grades to the variables every F (l) , being quadratic, has t-grade t 2 and q-grade q l . Now, considering only the three relations on the nine variables x (0) 3 , the Hilbert series is [7,18] ( Continuing ad infinitum for the countably infinite quadratic equations for the countable set of variables, we obtain the Hilbert series of the arc space of the variety f = 0 to be Similarly, assuming that the β's obey the same constraint and noting that they do not possess zero modes, the Hilbert series for them is obtained as where the grade of β (j) i is chosen to be q i /t for i = 1, 2, 3, · · · .

Contribution from blow up
Resolution of rational surface singularities can be treated using arcs. The surface (20) with an A 1 singularity at the origin is blown up with a P 1 . The single exceptional divisor corresponds to a truncation of the Puiseux series (16) to [16,19] x for i = 1, 2, 3. Putting this truncated series in the constraint (20) leads to the single equations 3 ) 2 = 0.
The Hilbert series is

Beta-Gamma system on surface with a rational double point
In this section we obtain the partition function of a beta-gamma system on the surface with an A 1 singularity using the Hilbert series obtained above in two different ways (11) and (15) as discussed before. Let us note that the coefficients x (j) i are in one-to-one correspondence with the derivatives ∂ j γ i of the fields as well as with the modes in (5). The former identification is more better suited for our purposes here. This allows the identification of the Hilbert series as the relevant part of the partition function through counting monomials in the fields and their derivatives. In the case of the affine space, there is no constraint. Identifying the coefficients x (j) i in (16) with ∂ j γ i , i = 1, 2, 3, the Hilbert series for each C is given by (19). Each component of the arc space will have a conjugate with an inverse t-charge corresponding to the unconstrained β's as well. Thus the total partition function of a flat beta-gamma system is obtained by augmenting H C in (19) with the contribution from the conjugates, resulting into (8). This can also be thought as an instance of implementing the field-antifield symmetry according to (11).
Let us now discuss the case of the quadratic constraint (9) with The constraint is (20) with the identification x i = γ i . This singular variety can also be looked upon as the orbifold C 2 /Z 2 , where the Z 2 acts on (u, v) ∈ C 2 by changing signs of both. The partition function has been computed earlier and written in a closed form [7], by directly implementing the orbifolding on the partition function of the affine space (7) with d = 2. Expanded in a series with respect to q and t the partition function is Z C 2 /Z 2 (q, t) = (1 + 3t 2 + 5t 4 + 7t 6 + 9t 8 + 11t 10 + · · · ) + q(4 + 12t 2 + 20t 4 + 28t 6 + 36t 8 + 44t 10 + · · · ) + q 2 ( 3 t 2 + 17 + 42t 2 + 70t 4 + 98t 6 + 126t 8 + 154t 10 + · · · ) + q 3 ( 12 t 2 + 52 + 120t 2 + 200t 4 + 280t 6 + 360t 8 + 440t 10 + · · · ) + q 4 ( 5 t 4 + 42 t 2 + 147 + 320t 2 + 525t 4 + 735t 6 + 945t 8 + 1155t 10 + · · · ) + q 5 ( 20 t 4 + 120 t 2 + 372 + 776t 2 + 1260t 4 + 1764t 6 + 2268t 8 + 2772t 10 + · · · ) + O q 6 . (31) The orbifold description and (9) are related by a quadratic identification of variables (u, v) ∈ C 2 with the x's as x 1 = u 2 , x 2 = v 2 , x 3 = uv. To compare results of x's to (u, v) variables, noting the t-charge assignment, a t is to be replaced with a t 2 in the formulas for partition function (15) as well as the Hilbert series. The comparison is, however, valid only in a local coordinate chart. Certain monomials which survive the orbifold action in terms of u, v variables are absent in the description in terms of x's. These correspond to missing states in the latter description. For example, there are four monomials u∂u, u∂v, v∂u, v∂v with grade qt 2 which survive the orbifold projection. Only three of them appear in terms of x's as ∂x 1 , ∂x 2 and ∂x 3 = u∂v + v∂u. The combination u∂v −v∂u is absent. Inclusion of this combination calls for extending the set of regular functions on the variety x 1 x 2 − x 2 3 = 0 by rational functions of γ's that is, x's and their derivatives [5]. This is achieved by including the blow up modes in the Hilbert series with (29). While this mends the partition function at this level, states at higher level still remain missing. This may be ascribed to the fact that a resolution of singularity by blowing up a point repairs the variety M up to its tangent space, M 1 in general. Thus we do not expect the partition function obtained without resorting to the parametric representation to completely match (31).

(37)
The partition function matches (31) up to the first mass level completely, which is expected since the first mass level corresponds to the tangent space of the variety. Terms with sufficiently high order of t 2 match as well for all powers of q, indicating that the number of missing states are finite at each mass level. The series has been verified at low orders in q and t 2 by explicitly constructing all possible combinations of β's and x's with arbitrary coefficients and then solving the consistency conditions arising from the Poisson bracket with the constraints.

Partition function of Pure spinors
In this section we present the results for the case of pure spinors. The action for the pure spinor system is where λ = x 1 , x 2 , · · · , x 16 T is a sixteen dimensional complex vector subject to the constraint and ω denotes its conjugate. Here T denotes matrix transpose and γ µ denotes the ten-dimensional gamma matrices with µ = 0, 1, · · · , 9. The action possesses the classical gauge symmetry We write down the partition function of the pure spinor system using the field-antifield symmetry. The arc space is obtained by substituting the expansion (16) for the sixteen complex coordinates in (39). Since there are more than one equations at every mass level, the computation of Hilbert series is more complicated than the previous case requiring the Gröbner basis of the ideal generated by the constraints at each level, that is for each power of q. We resort to Macaulay2 to compute the Hilbert series. For the first jet scheme of the variety described by (39) the Hilbert series is − 11q 3 t 5 + 65q 2 t 5 − 65q 3 t 6 + 11q 2 t 6 + 16q 4 t 7 − 45q 3 t 7 − q 2 t 7 − q 5 t 8 + 34q 4 t 8 − 5q 5 t 9 + 10q 4 t 9 − 5q 5 t 10 − q 5 t 11 /(1 − t) 11 (1 − qt) 16 .
Expanded in powers of q this gives where the higher order terms are omitted as we do not use the relations beyond the first power of q.
This matches with the expression obtained earlier [11].

Conclusion
We obtain the partition function of beta-gamma systems with algebraic constraints on the fields γ. We showed that the partition function of a beta-gamma system can be evaluated by identifying the contribution from the γ's as the Hilbert series of arc spaces of the algebraic variety given by the constraint. Two examples are worked out explicitly. In the first we consider the A 1 surface singularity given by a quadratic constraint in three γ's. We demonstrate two different ways of computing the partition function in this case. The first one implements the so-called field-antifield symmetry in a multiplicative fashion. This, however, gives rise to terms with negative coefficients in the partition function, which can not be accounted for as the partition is the generating function of degeneracy of operators. We show that the partition function can be obtained alternatively as the generating function of monomials invariant under the classical gauge symmetry of the action modulo the constraint. This is implemented by subtracting the number of monomials not vanishing under the gauge transformation from the totality of monomials. Hence the terms with negative coefficients signify an over-determined system and may thus be omitted. The positive terms of both the expressions, on the other hand, match. An advantage of the algorithm presented here lies in the fact that it can be straightforwardly extended to the case where the constraints are not reducible, such as pure spinor system. Moreover, this gives the partition function a geometric significance.
We also obtain the partition function of the pure spinor system using the Hilbert series of the arc space of the pure spinor constraint looked upon as a variety embedded in the sixteen-dimensional complex affine space. This requires the computation of Gröbner bases in the polynomial rings involved. We used Macualay2 to obtain the Hilbert series which, though straightforward as an algorithm, is extremely memory-intensive and we are restricted here to the first mass level. However, we show that the computation up to this level implementing the field-antifield symmetry in a product formula matches with previously known results [11]. The computation of gauge invariant monomials is more complicated since the resolution in terms of arc spaces is not known in addition to the variety of gauge invariants appearing.