Witten Index, 4-D N = 1 Gauge Theories, and Riemann Surfaces

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




and by Jeffrey-Kirwan path-integration, we obtain



with {Z_{cl,1l}} the classical and 1-loop contribution derived from the index of a topological twist on {\Sigma _g} \times {T^2} via the partition function




and crucially noting that the effective action

    \[\tilde Z = \hat A\exp \int_{{\Sigma _g}} {\left[ {\hat BF + c\wp {\eta ^\dagger } \wedge \eta } \right]} \]

is topological on {\Sigma _g}. Take 4-D N = 1 gauge theories with a non-anomalous U{(1)_R} R-symmetry, located on {\Sigma _g} \times {T^2} with a {\Sigma _g} twist. Given that {T^2} has modulus \tau, the Lagrangian is given by

    \[L = L{\,_{YM}} + {L_{mat}} + {L_{{W^p}}}\]

with {L_{YM}} the supersymmetric Yang-Mills Lagrangian{L_{mat}} the matter kinetic Lagrangian, and {L_{{W^p}}} superpotential interactions. For Abelian gauge-group factors reasons, we must also include a Fayet-Iliopoulos term

    \[{L_{FI}} = i\frac{\zeta }{{2\pi }}D\]

where the parameters of the background are the flux

    \[\tilde n = \frac{1}{{2\pi }}\int_\Sigma {{F^{flav}}} \]

and flavor flat connection are

    \[v = 2\pi \oint_{A{\rm{ - cycle}}} A - 2\pi \tau \oint_{B{\rm{ - cycle}}} A \]

and v lives on a copy of the spacetime {T^2}. Now, define

    \[\left\{ {\begin{array}{*{20}{c}}{q = {e^{2\pi i\tau }}}\\{x = {e^{iu}}}\\{y = {e^{iv}}}\end{array}} \right.\]

Then the semi-classical and one-loop contribution {Z_{{\rm{cl}},1l}} consists of the following pieces:

the semi-classical action contribution is from the FI term

    \[{Z_{FI}} = {e^{ - {\rm{vol}}\left( {{T^2}} \right)\,\zeta \hat m}}\]

the one-loop determinant for chiral multiplets is




with elliptic functions

    \[\eta (q) = {q^{1/24}}\prod\nolimits_{n = 1}^\infty {\left( {1 - {q^n}} \right)} \]





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



whereas the vector multiplets contribution along the Cartan generators is

    \[Z_{1{\rm{ - loop}}}^{{\rm{Gauge,Cartan}}} = \eta {\left( q \right)^{2r\left( {1 - g} \right)}}{\left( {idu} \right)^r}\]

and the fermionic zero modes on {\Sigma _g} is

    \[{\left( {{{\det }_{\left[ {ab} \right]}}\frac{{{{\not \partial }^2}\log {Z_{1{\rm{loop}}}}}}{{\not \partial i{u_a}\not \partial {{\hat m}_b}}}} \right)^g}\]

Thus, combining, the final Abelian formula is




and there are no boundary contributions since the integration domain \hat {\rm M} is compact.

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




Now take U(1) a supersymmetric Yang-Mills-Chern-Simons theory at level k which is equivalent to bosonic Chern-Simons at level k at low energies. The semi-classical and one-loop contribution is

    \[{Z_{{\rm{cl,}}1l}} = {x^{\hat t}}{\xi ^{\hat m}}{x^{k\,\hat m}}\]

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

    \[Z = \sum\limits_{\hat m\, \in \mathbb{Z}} {\int_{{\rm{JK}}} {\frac{{dx}}{{2\pi ix}}\,} } {k^g}{x^{k\,\hat m\, + \hat t}}{\xi ^{\hat m}}\]

with the charges of the points at infinity being {Q_0} = - k and {Q_\infty } = k

Given the natural assumptions: k > 0, \eta < 0 and x = 0

we get

    \[Z = \left\{ {\begin{array}{*{20}{c}}{ - {k^g}{\xi ^{ - \hat t}}/k\quad {\rm{, if }}\hat t = 0\,\quad \bmod k}\\{0\,\,\,\,\,\,\,{\rm{,}}\,\,\,{\rm{otherwise}}}\end{array}} \right.\]

where the known and central {k^g} is the number of ground states of U{(1)_k} Chern-Simons on {\Sigma _g}.

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

    \[{Z_{{\rm{cl,}}1l}} = {\xi ^{\hat m}}{x^{k\,\hat m\, + \hat t}}{\prod\limits_{i = 1}^N {\left( {\frac{{{x^{{Q_i}/2}}{y_i}^{1/2}}}{{1 - {x^{{Q_i}}}{y_i}}}} \right)} ^{{Q_i}\,\hat m\, + \,{{\hat n}_i}}}\]

with \left( {{y_i},{{\hat n}_i}} \right) controlling the flavor-symmetries-background for up to one combination that could be reabsorbed into \left( {x,\hat m} \right), and \left( {\xi ,\hat t} \right) controls the background for the topological symmetry, and, in order not to face a gauge-parity anomaly, we insist that the following equivalence holds:

    \[k + \frac{1}{2}\sum\nolimits_i {Q_i^2} \in \mathbb{Z} \cong k + \frac{1}{2}\sum\nolimits_i {{Q_i}} \in \mathbb{Z}\]

To compute the Witten index, set {\hat n_i} = \hat t = 0, and then the poles are at

    \[\left\{ {\begin{array}{*{20}{c}}{{{\hat n}_i} = \hat t = 0}\\{x = 0}\\\infty \end{array}} \right.\]

hence the sum over \hat m generates the following




with {x_{(\alpha )}} being the roots of the BaezAEs:

    \[{e^{iB}} = \xi {x^k}{\prod\limits_{i = 1}^N {\left( {\frac{{{x^{{Q_i}/2}}y_i^{1/2}}}{{1 - {x^{{Q_i}}}{y_i}}}} \right)} ^{{Q_i}}} = 1\]

hence the index {Z_{{T^3}}} 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:

    \[\left\{ {\begin{array}{*{20}{c}}{{I_ + }\quad :\quad {Q_i} > 0}\\{{I_ - }\quad :\quad {Q_i} < 0}\end{array}} \right.\]

Thus our WI- equation is





Defining the positive numbers

    \[\left\{ {\begin{array}{*{20}{c}}{{n_ + } = \frac{1}{2}\sum\limits_{i \in \,{I_ + }} {Q_i^2} }\\{{n_ - } = \frac{1}{2}\sum\limits_{i \in \,{I_ - }} {Q_i^2} }\end{array}} \right.\]

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

    \[{I_{{W_{itt}}}} = \left\{ {\begin{array}{*{20}{c}}{\left| k \right| + {n_ + } + {n_ - }\quad {\rm{, }}\;{\rm{if }}\left| k \right| \ge \left| {{n_ + } - {n_ - }} \right|}\\{\max \left( {2{n_ + },2{n_ - }} \right)\quad {\rm{,}}\;\;{\rm{if}}\;\;\left| k \right| \le \left| {{n_ + } - {n_ - }} \right|}\end{array}} \right.\]

yielding the total Witten index





    \[\left\{ {\begin{array}{*{20}{c}}{{s_i} \equiv \Theta \left( {{n_i}F\left( {{\sigma _{{Q_i}}}} \right)} \right)}\\{\Theta (x) = \left\{ {\begin{array}{*{20}{c}}{1{\rm{ : }}x > 0}\\{0{\rm{ : }}x < 0}\end{array}} \right.}\end{array}} \right.\]

and {\sigma _I} implicitly defined by

    \[{\sigma _I}{ = _{{\rm{def}}}} - \frac{{{\zeta _{eff}}\left( {{\sigma _{{I_ + }}}} \right)}}{{{k_{eff}}\left( {{\sigma _{{I_ - }}}} \right)}}\]

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