DT invariants of curves¶

DT invariants of curves

We compute invariants of moduli stacks of semistable vector bundles on a curve using the formula of Zagier in the form presented in arXiv:1310.4991.

Then we compute integer DT invariants (by taking plethystic logarithm) and invariants of stable moduli spaces following arXiv:0711.0634.

EXAMPLES:

sage: from msinvar.curve_DT import Curve
sage: C=Curve(g=0,prec=[2,2]); C
Curve of genus 0
sage: C.intDT().dict() # DT invariants for rank<=2 and 0<=deg<=2
{(1, 0): 1, (1, 1): 1, (1, 2): 1}

sage: C=Curve(g=1,prec=[2,2]); C
Curve of genus 1
sage: C.intDT().dict() # DT invariants for rank<=2 and 0<=deg<=2
{(1, 0): y^2 - 2*y + 1,
(1, 1): y^2 - 2*y + 1,
(1, 2): y^2 - 2*y + 1,
(2, 1): y^2 - 2*y + 1}

sage: C=Curve(g=2,prec=[2,2]); C
Curve of genus 2
sage: I=C.intDT()
sage: I([1,0])
y^4 - 4*y^3 + 6*y^2 - 4*y + 1
sage: I([2,0])
y^10 - 4*y^9 + 7*y^8 - 8*y^7 + 8*y^6 - 8*y^5 + 8*y^4 - 8*y^3 + 7*y^2 - 4*y + 1
sage: I([2,1])
y^10 - 4*y^9 + 7*y^8 - 12*y^7 + 24*y^6 - 32*y^5 + 24*y^4 - 12*y^3 + 7*y^2 - 4*y + 1

sage: C.stable_val([1,0]) # motivic classes of stable moduli spaces
y^4 - 4*y^3 + 6*y^2 - 4*y + 1
sage: C.stable_val([2,0]) # this value is different from int_DT
y^10 - 4*y^9 + 6*y^8 - 4*y^7 - 4*y^6 + 20*y^5 - 30*y^4 + 20*y^3 - 5*y^2
sage: C.stable_val([2,1])
y^10 - 4*y^9 + 7*y^8 - 12*y^7 + 24*y^6 - 32*y^5 + 24*y^4 - 12*y^3 + 7*y^2 - 4*y + 1

class msinvar.curve_DT.Curve(g=0, prec=None)[source]

Wall-crossing structure of a curve.

It is a polynomial algebra in 2 variables (actually a quantum affine plane) over the field Q(y,t), where our zeta function of the curve lives. The zeta function is $$(1-yt)^{2g}/(1-t)/(1-y^2t)$$.

eform(a, b)[source]

Euler form between sheaves having Chern characters a, b, where a[0] is a rank, a[1] is a degree and similarly for b.

sform(a, b)[source]

Skew-symmetric Euler form.

P(**kw)[source]

Numerator of the zeta function.

Z(**kw)[source]

Zeta function of a curve. It has the form $$(1-yt)^{2g}/(1-t)/(1-y^2t)$$.

Zhat(**kw)[source]

Twsited version of Z().

Bun(r)[source]

Motivic class of the moduli stack of rank r vector bundles and some fixed degree.

Bun_tw(r)[source]

Twisted version of Bun().

stacky()[source]

Motivic classes of stacks of semistable vector bundles.

stacky_val(r, d=None)[source]

Motivic class of the stack of semistable vector bundles having rank r and degree d. If d is None, then r should be a 2-vector.

intDT()[source]

Integer DT invariants.

intDT_val(r, d=None)[source]

Motivic class of the moduli space of stable vector bundles having rank r and degree d (not necessarily coprime).

stable(slope=0)[source]

Motivic classes of moduli spaces of stable vector bundles having slope d/r equal slope.

stable_val(r, d=None)[source]

Motivic class of the moduli space of stable vector bundles having rank r and degree d (not necessarily coprime).