In this post, I will discuss the Witten Index in the context of 4-D N = 1 gauge theories on , a Riemann surface of genus g times a torus where in 2-D, the WI is

and by Jeffrey-Kirwan path-integration, we obtain

with the classical and 1-loop contribution derived from the index of a topological twist on via the partition function

and crucially noting that the effective action

is topological on . Take 4-D N = 1 gauge theories with a non-anomalous -symmetry, located on with a twist. Given that has modulus , the Lagrangian is given by

with the supersymmetric Yang-Mills Lagrangian, the matter kinetic Lagrangian, and superpotential interactions. For Abelian gauge-group factors reasons, we must also include a Fayet-Iliopoulos term

where the parameters of the background are the flux

and flavor flat connection are

and lives on a copy of the spacetime . Now, define

Then the semi-classical and one-loop contribution consists of the following pieces:

the semi-classical action contribution is from the FI term

the one-loop determinant for chiral multiplets is

with elliptic functions

and

So, the one-loop determinant for the off-diagonal vector multiplets is

whereas the vector multiplets contribution along the Cartan generators is

and the fermionic zero modes on is

Thus, combining, the final Abelian formula is

and there are no boundary contributions since the integration domain is compact.

In the non-Abelian contexts, W-bosons contributions are factored by re-summing over and excluding the roots of the associated Baez AEs for which the Vandermonde determinant is zero

Now take a supersymmetric Yang-Mills-Chern-Simons theory at level which is equivalent to bosonic Chern-Simons at level at low energies. The semi-classical and one-loop contribution is

turning on a background for the topological symmetry. Hence, we can derive

with the charges of the points at infinity being and

Given the natural assumptions: , and

we get

where the known and central is the number of ground states of Chern-Simons on .

To Witten all roads lead: take with chiral multiplets of charges and -charge 1 so one does not face the parity anomalies in the -symmetry. In the Witten-index context, the semi-classical and one-loop contribution is

with controlling the flavor-symmetries-background for up to one combination that could be reabsorbed into , and controls the background for the topological symmetry, and, in order not to face a gauge-parity anomaly, we insist that the following equivalence holds:

To compute the Witten index, set , and then the poles are at

hence the sum over generates the following

with being the roots of the BaezAEs:

hence the index *is the Witten index of the theory*

*is the Witten index of the theory*

On the flip-side,

becomes, up to sign-rescaling, the number of solutions to the BaezAEs.

Now, to compute the number of solutions, divide the chiral multiplets into two groups:

Thus our WI- equation is

Defining the positive numbers

we can conclude that the number of solutions to **WI-E **is

yielding the total Witten index

with

and implicitly defined by

In subsequent posts, the total Witten index will be used in rather deep ways crossing many fields of mathematics intersecting theoretical physics.…

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