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In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called . In conclusion, a line of symmetry splits the shape in half and those halves should be identical. In formal terms, a mathematical object is symmetric with respect to a given operation such as reflection, rotation or translation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa). The symmetric function of a two-dimensional figure is a line such that, for each perpendicular constructed, if the perpendicular intersects the figure at a distance 'd' from the axis along the perpendicular, then there exists another intersection of the shape and the perpendicular, at the same distance 'd' from the axis, in the opposite direction along the perpendicular. Another way to think about the symmetric function is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's s. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry. Triangles with reflection symmetry are isosceles. Quadrilaterals with reflection symmetry are kites, (concave) deltoids, rhombi, and isosceles trapezoids. All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, and one through edges. For an arbitrary shape, the axiality of the shape measures how close it is to being bilaterally symmetric.

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Mirror symmetry for moduli spaces of Higgs bundles via $p$-adic integration

Dimitri Stelio Wyss, Michael Gröchenig

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type

SL_n

and

PGL_n

. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises

p

-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for

d

coprime to

n

, that the number of rank

n

Higgs bundles of degree

d

over a fixed curve defined over a finite field, is independent of

d

. This proves a conjecture by Mozgovoy--Schiffman in the coprime case.

2017

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