Knot InvariantsFrom Maximally Supersymmetric

Yang-Mills Theory

Edward WittenStrings 2011, Uppsala, June 28, 2011

I decided that rather than any technical details, Iwould give an overview of the content of severalrecent papers.

I won’t try to give references to allthe basic results that I will mention along the way,but I should at least mention the paper by S.Gukov, A. Schwarz and C. Vafa, hep-th/0412243,which was part of the inspiration.

I decided that rather than any technical details, Iwould give an overview of the content of severalrecent papers. I won’t try to give references to allthe basic results that I will mention along the way,but I should at least mention the paper by S.Gukov, A. Schwarz and C. Vafa, hep-th/0412243,which was part of the inspiration.

In N = 4 super Yang-Mills theory, there are 1/16BPS Wilson loop operators that are supported onan arbitrary loop in spacetime.

We just combine thegauge field Aµ with four of the scalar fields onN = 4 super Yang-Mills theory, which we will callφµ, µ = 1, . . . , 4. Then for any loop K inspacetime, and any representation R of the gaugegroup G , we define

WR(K ) = TrRP exp

∮K

(Aµ + iφµ)dxµ.

This is an ordinary Wilson loop operator except forthe replacement Aµ → Aµ + iφµ.

In N = 4 super Yang-Mills theory, there are 1/16BPS Wilson loop operators that are supported onan arbitrary loop in spacetime. We just combine thegauge field Aµ with four of the scalar fields onN = 4 super Yang-Mills theory, which we will callφµ, µ = 1, . . . , 4.

Then for any loop K inspacetime, and any representation R of the gaugegroup G , we define

WR(K ) = TrRP exp

∮K

(Aµ + iφµ)dxµ.

This is an ordinary Wilson loop operator except forthe replacement Aµ → Aµ + iφµ.

In N = 4 super Yang-Mills theory, there are 1/16BPS Wilson loop operators that are supported onan arbitrary loop in spacetime. We just combine thegauge field Aµ with four of the scalar fields onN = 4 super Yang-Mills theory, which we will callφµ, µ = 1, . . . , 4. Then for any loop K inspacetime, and any representation R of the gaugegroup G , we define

WR(K ) = TrRP exp

∮K

(Aµ + iφµ)dxµ.

This is an ordinary Wilson loop operator except forthe replacement Aµ → Aµ + iφµ.

In N = 4 super Yang-Mills theory, there are 1/16BPS Wilson loop operators that are supported onan arbitrary loop in spacetime. We just combine thegauge field Aµ with four of the scalar fields onN = 4 super Yang-Mills theory, which we will callφµ, µ = 1, . . . , 4. Then for any loop K inspacetime, and any representation R of the gaugegroup G , we define

WR(K ) = TrRP exp

∮K

(Aµ + iφµ)dxµ.

This is an ordinary Wilson loop operator except forthe replacement Aµ → Aµ + iφµ.

Being 1/16-BPS, this operator preserves onesupersymmetry, which I will call Q. This operatorobeys Q2 = 0.

We can actually define a topologicalfield theory by only looking at Q-invariant operatorsO, modulo O → O + {Q, ·}. This gives atopological field theory because the stress tensor istrivial, i.e. Tµν = {Q,Λµν}, for some Λ. (It isactually the same topological field theory thatKapustin and I studied in relation to geometricLanglands.)

Being 1/16-BPS, this operator preserves onesupersymmetry, which I will call Q. This operatorobeys Q2 = 0. We can actually define a topologicalfield theory by only looking at Q-invariant operatorsO, modulo O → O + {Q, ·}.

This gives atopological field theory because the stress tensor istrivial, i.e. Tµν = {Q,Λµν}, for some Λ. (It isactually the same topological field theory thatKapustin and I studied in relation to geometricLanglands.)

Being 1/16-BPS, this operator preserves onesupersymmetry, which I will call Q. This operatorobeys Q2 = 0. We can actually define a topologicalfield theory by only looking at Q-invariant operatorsO, modulo O → O + {Q, ·}. This gives atopological field theory because the stress tensor istrivial, i.e. Tµν = {Q,Λµν}, for some Λ.

(It isactually the same topological field theory thatKapustin and I studied in relation to geometricLanglands.)

Being 1/16-BPS, this operator preserves onesupersymmetry, which I will call Q. This operatorobeys Q2 = 0. We can actually define a topologicalfield theory by only looking at Q-invariant operatorsO, modulo O → O + {Q, ·}. This gives atopological field theory because the stress tensor istrivial, i.e. Tµν = {Q,Λµν}, for some Λ. (It isactually the same topological field theory thatKapustin and I studied in relation to geometricLanglands.)

As a consequence, the expectation value of theWilson operator WR(K ) is a topological invariant.But this doesn’t give us anything very interesting.

Ifwe were in three dimensions, topological invariantsof loops would be very interesting.

There is nothinglike that in four dimensions.

As a consequence, the expectation value of theWilson operator WR(K ) is a topological invariant.But this doesn’t give us anything very interesting. Ifwe were in three dimensions, topological invariantsof loops would be very interesting.

There is nothinglike that in four dimensions.

As a consequence, the expectation value of theWilson operator WR(K ) is a topological invariant.But this doesn’t give us anything very interesting. Ifwe were in three dimensions, topological invariantsof loops would be very interesting.

There is nothinglike that in four dimensions.

As a consequence, the expectation value of theWilson operator WR(K ) is a topological invariant.But this doesn’t give us anything very interesting. Ifwe were in three dimensions, topological invariantsof loops would be very interesting.

There is nothinglike that in four dimensions.

To get something interesting, we are going to consider N = 4super Yang-Mills theory not on R4 but on a half-spaceR3 × R+, where R+ is a half-line y ≥ 0.

At the boundaryy = 0, we need a boundary condition, of course. Theboundary condition that we will use is the one that comesfrom the D3-NS5 system.

To get something interesting, we are going to consider N = 4super Yang-Mills theory not on R4 but on a half-spaceR3 × R+, where R+ is a half-line y ≥ 0. At the boundaryy = 0, we need a boundary condition, of course. Theboundary condition that we will use is the one that comesfrom the D3-NS5 system.

To get something interesting, we are going to consider N = 4super Yang-Mills theory not on R4 but on a half-spaceR3 × R+, where R+ is a half-line y ≥ 0. At the boundaryy = 0, we need a boundary condition, of course. Theboundary condition that we will use is the one that comesfrom the D3-NS5 system.

In other words, we will study the D3-brane gaugetheory with this boundary condition.

This is a halfBPS boundary condition, so it preserves 8supersymmetries. One linear combination of them isthe supersymmetry that is preserved by the 1/16BPS Wilson operator that we mentioned before.

In other words, we will study the D3-brane gaugetheory with this boundary condition. This is a halfBPS boundary condition, so it preserves 8supersymmetries.

One linear combination of them isthe supersymmetry that is preserved by the 1/16BPS Wilson operator that we mentioned before.

In other words, we will study the D3-brane gaugetheory with this boundary condition. This is a halfBPS boundary condition, so it preserves 8supersymmetries. One linear combination of them isthe supersymmetry that is preserved by the 1/16BPS Wilson operator that we mentioned before.

So we can have a topological field theory on ahalf-space R3 × R+ with Wilson loop operators foran arbitrary loop K .

The D3-NS5 boundarycondition is more easily described if the gaugetheory θ-angle is zero (it gives Neuman boundaryconditions for gauge fields, for instance). In thatcase, the supersymmetry Q of the 1/16-BPS Wilsonoperators is a linear combination of the eightsupersymmetries allowed by the boundary condition.

So we can have a topological field theory on ahalf-space R3 × R+ with Wilson loop operators foran arbitrary loop K . The D3-NS5 boundarycondition is more easily described if the gaugetheory θ-angle is zero (it gives Neuman boundaryconditions for gauge fields, for instance).

In thatcase, the supersymmetry Q of the 1/16-BPS Wilsonoperators is a linear combination of the eightsupersymmetries allowed by the boundary condition.

So we can have a topological field theory on ahalf-space R3 × R+ with Wilson loop operators foran arbitrary loop K . The D3-NS5 boundarycondition is more easily described if the gaugetheory θ-angle is zero (it gives Neuman boundaryconditions for gauge fields, for instance). In thatcase, the supersymmetry Q of the 1/16-BPS Wilsonoperators is a linear combination of the eightsupersymmetries allowed by the boundary condition.

So we can do topological field theory on R3 ×R+ inthis situation with Wilson operators for an arbitraryK .

Their expectation values are topologicalinvariants, but not interesting, for the same reasonas before.

So we can do topological field theory on R3 ×R+ inthis situation with Wilson operators for an arbitraryK .Their expectation values are topologicalinvariants, but not interesting, for the same reasonas before.

We actually do get something interesting if we takethe gauge theory θ-angle to be nonzero. TheD3-NS5 boundary condition (which was generalizedto this situation in D. Gaiotto and EW,arXiv:0804.2902) still preserves 8 supersymmetries,but a different 8.

It no longer preserves the samesupersymmetry that is preserved by the 1/16 BPSWilson operators, so they disappear.

We actually do get something interesting if we takethe gauge theory θ-angle to be nonzero. TheD3-NS5 boundary condition (which was generalizedto this situation in D. Gaiotto and EW,arXiv:0804.2902) still preserves 8 supersymmetries,but a different 8. It no longer preserves the samesupersymmetry that is preserved by the 1/16 BPSWilson operators, so they disappear.

Instead, we get something more subtle andinteresting.

Though a generic loop K doesn’tsupport a Wilson operator that preserves asupersymmetry that is also preserved by theboundary condition, a loop that is in the boundaryactually does.

Instead, we get something more subtle andinteresting. Though a generic loop K doesn’tsupport a Wilson operator that preserves asupersymmetry that is also preserved by theboundary condition, a loop that is in the boundaryactually does.

Instead, we get something more subtle andinteresting. Though a generic loop K doesn’tsupport a Wilson operator that preserves asupersymmetry that is also preserved by theboundary condition, a loop that is in the boundaryactually does.

Because our loops are now in three dimensions, theycan be knotted.

Finally we get somethinginteresting.

Because our loops are now in three dimensions, theycan be knotted. Finally we get somethinginteresting.

Now we would like to compute the expectationvalues of the Wilson operators that appear on theboundary in this situation.

Since these operators areinvariant (as is the whole construction) under asupersymmetry Q that obeys Q2 = 0, we can use aprocedure known as supersymmetric localization.The basic idea is that the path integral over allfields can be replaced by an integral over only thesupersymmetric configurations. This technique hasall sorts of applications, some of which will bediscussed at this meeting.

Now we would like to compute the expectationvalues of the Wilson operators that appear on theboundary in this situation. Since these operators areinvariant (as is the whole construction) under asupersymmetry Q that obeys Q2 = 0, we can use aprocedure known as supersymmetric localization.

The basic idea is that the path integral over allfields can be replaced by an integral over only thesupersymmetric configurations. This technique hasall sorts of applications, some of which will bediscussed at this meeting.

Now we would like to compute the expectationvalues of the Wilson operators that appear on theboundary in this situation. Since these operators areinvariant (as is the whole construction) under asupersymmetry Q that obeys Q2 = 0, we can use aprocedure known as supersymmetric localization.The basic idea is that the path integral over allfields can be replaced by an integral over only thesupersymmetric configurations.

This technique hasall sorts of applications, some of which will bediscussed at this meeting.

Now we would like to compute the expectationvalues of the Wilson operators that appear on theboundary in this situation. Since these operators areinvariant (as is the whole construction) under asupersymmetry Q that obeys Q2 = 0, we can use aprocedure known as supersymmetric localization.The basic idea is that the path integral over allfields can be replaced by an integral over only thesupersymmetric configurations. This technique hasall sorts of applications, some of which will bediscussed at this meeting.

If one applies supersymmetric localization in thissituation, one learns something interesting: theexpectation value of one of these Wilson operatorsin the boundary of a four-dimensional space can becomputed in a purely three-dimensional topologicalfield theory, namely (bosonic) Chern-Simons theory.

So in fact, what we get this way are the knotinvariants – such as the Jones polynomial – that canbe computed by Chern-Simons theory.

If one applies supersymmetric localization in thissituation, one learns something interesting: theexpectation value of one of these Wilson operatorsin the boundary of a four-dimensional space can becomputed in a purely three-dimensional topologicalfield theory, namely (bosonic) Chern-Simons theory.So in fact, what we get this way are the knotinvariants – such as the Jones polynomial – that canbe computed by Chern-Simons theory.

In particular, 3d Chern-Simons theory is completelysoluble via its relation to 2d conformal field theory,so all these invariants are explicitly calculable.

Iwish there were time to review this today, but therereally isn’t. Likewise, I won’t be able to explain whylocalization of the D3-NS5 system gives a purely 3ddescription via a Chern-Simons theory.

In particular, 3d Chern-Simons theory is completelysoluble via its relation to 2d conformal field theory,so all these invariants are explicitly calculable. Iwish there were time to review this today, but therereally isn’t.

Likewise, I won’t be able to explain whylocalization of the D3-NS5 system gives a purely 3ddescription via a Chern-Simons theory.

In particular, 3d Chern-Simons theory is completelysoluble via its relation to 2d conformal field theory,so all these invariants are explicitly calculable. Iwish there were time to review this today, but therereally isn’t. Likewise, I won’t be able to explain whylocalization of the D3-NS5 system gives a purely 3ddescription via a Chern-Simons theory.

There is something else we can do that is actuallyconceptually more straightforward. We just applyelectric-magnetic duality.

The D3-NS5 boundarycondition becomes a D3-D5 boundary condition.

There is something else we can do that is actuallyconceptually more straightforward. We just applyelectric-magnetic duality. The D3-NS5 boundarycondition becomes a D3-D5 boundary condition.

There is something else we can do that is actuallyconceptually more straightforward. We just applyelectric-magnetic duality. The D3-NS5 boundarycondition becomes a D3-D5 boundary condition.

The Wilson operators in the boundary become ’tHooft operators.

The Wilson operators in the boundary become ’tHooft operators.

We can apply supersymmetric localization in thissituation, and it gives a more straightforwardanswer.

The localization occurs on the solutions ofa certain set of equations that are the conditions forQ-invariance:

F − φ ∧ φ = ?Dφ, Dµφµ = 0.

(These equations were introduced by Kapustin andme in studying geometric Langlands. They havealso been used in K. Lee and H. Yee,hep-th/0606159 to discuss six-dimensional stringwebs.) Localization on the solutions of an equationis the simplest sort of answer that one sometimesgets from supersymmetric localization.

We can apply supersymmetric localization in thissituation, and it gives a more straightforwardanswer. The localization occurs on the solutions ofa certain set of equations that are the conditions forQ-invariance:

F − φ ∧ φ = ?Dφ, Dµφµ = 0.

(These equations were introduced by Kapustin andme in studying geometric Langlands. They havealso been used in K. Lee and H. Yee,hep-th/0606159 to discuss six-dimensional stringwebs.) Localization on the solutions of an equationis the simplest sort of answer that one sometimesgets from supersymmetric localization.

We can apply supersymmetric localization in thissituation, and it gives a more straightforwardanswer. The localization occurs on the solutions ofa certain set of equations that are the conditions forQ-invariance:

F − φ ∧ φ = ?Dφ, Dµφµ = 0.

(These equations were introduced by Kapustin andme in studying geometric Langlands. They havealso been used in K. Lee and H. Yee,hep-th/0606159 to discuss six-dimensional stringwebs.)

Localization on the solutions of an equationis the simplest sort of answer that one sometimesgets from supersymmetric localization.

We can apply supersymmetric localization in thissituation, and it gives a more straightforwardanswer. The localization occurs on the solutions ofa certain set of equations that are the conditions forQ-invariance:

F − φ ∧ φ = ?Dφ, Dµφµ = 0.

(These equations were introduced by Kapustin andme in studying geometric Langlands. They havealso been used in K. Lee and H. Yee,hep-th/0606159 to discuss six-dimensional stringwebs.) Localization on the solutions of an equationis the simplest sort of answer that one sometimesgets from supersymmetric localization.

Evaluating the path integral reduces to counting thesolutions of those equations.

Let an be the numberof solutions for which the instanton number

1

8π2

∫R3×R+

Tr F ∧ F

is equal to n. Then the path integral Z is

Z =∑

n

anqn,

where in the purely 3d description by Chern-Simonstheory,

q = exp(2πi/(k + 2)).

Evaluating the path integral reduces to counting thesolutions of those equations. Let an be the numberof solutions for which the instanton number

1

8π2

∫R3×R+

Tr F ∧ F

is equal to n. Then the path integral Z is

Z =∑

n

anqn,

where in the purely 3d description by Chern-Simonstheory,

q = exp(2πi/(k + 2)).

Since the path integral of Chern-Simons theory canbe explicitly computed by independent methods,one can view this as a prediction for the number ofsolutions of those four-dimensional equations.

Theprediction is based on electric-magnetic duality andsupersymmetric localization. This prediction hasbeen verified in D. Gaiotto and EW,arXiv:1106.4789, by directly analyzing the equationsand counting their solutions. One can view this asan unusual test of electric-magnetic duality. Thereisn’t time to explain what we did, but I can say thata key fact was that the equations are actuallytractable in the time-independent case. Also, ouranalysis showed that these equations have a novelrelation to two-dimensional conformal field theoryand integrable spin systems.

Since the path integral of Chern-Simons theory canbe explicitly computed by independent methods,one can view this as a prediction for the number ofsolutions of those four-dimensional equations. Theprediction is based on electric-magnetic duality andsupersymmetric localization.

This prediction hasbeen verified in D. Gaiotto and EW,arXiv:1106.4789, by directly analyzing the equationsand counting their solutions. One can view this asan unusual test of electric-magnetic duality. Thereisn’t time to explain what we did, but I can say thata key fact was that the equations are actuallytractable in the time-independent case. Also, ouranalysis showed that these equations have a novelrelation to two-dimensional conformal field theoryand integrable spin systems.

Since the path integral of Chern-Simons theory canbe explicitly computed by independent methods,one can view this as a prediction for the number ofsolutions of those four-dimensional equations. Theprediction is based on electric-magnetic duality andsupersymmetric localization. This prediction hasbeen verified in D. Gaiotto and EW,arXiv:1106.4789, by directly analyzing the equationsand counting their solutions.

One can view this asan unusual test of electric-magnetic duality. Thereisn’t time to explain what we did, but I can say thata key fact was that the equations are actuallytractable in the time-independent case. Also, ouranalysis showed that these equations have a novelrelation to two-dimensional conformal field theoryand integrable spin systems.

Since the path integral of Chern-Simons theory canbe explicitly computed by independent methods,one can view this as a prediction for the number ofsolutions of those four-dimensional equations. Theprediction is based on electric-magnetic duality andsupersymmetric localization. This prediction hasbeen verified in D. Gaiotto and EW,arXiv:1106.4789, by directly analyzing the equationsand counting their solutions. One can view this asan unusual test of electric-magnetic duality.

Thereisn’t time to explain what we did, but I can say thata key fact was that the equations are actuallytractable in the time-independent case. Also, ouranalysis showed that these equations have a novelrelation to two-dimensional conformal field theoryand integrable spin systems.

Since the path integral of Chern-Simons theory canbe explicitly computed by independent methods,one can view this as a prediction for the number ofsolutions of those four-dimensional equations. Theprediction is based on electric-magnetic duality andsupersymmetric localization. This prediction hasbeen verified in D. Gaiotto and EW,arXiv:1106.4789, by directly analyzing the equationsand counting their solutions. One can view this asan unusual test of electric-magnetic duality. Thereisn’t time to explain what we did, but I can say thata key fact was that the equations are actuallytractable in the time-independent case.

Also, ouranalysis showed that these equations have a novelrelation to two-dimensional conformal field theoryand integrable spin systems.

Since the path integral of Chern-Simons theory canbe explicitly computed by independent methods,one can view this as a prediction for the number ofsolutions of those four-dimensional equations. Theprediction is based on electric-magnetic duality andsupersymmetric localization. This prediction hasbeen verified in D. Gaiotto and EW,arXiv:1106.4789, by directly analyzing the equationsand counting their solutions. One can view this asan unusual test of electric-magnetic duality. Thereisn’t time to explain what we did, but I can say thata key fact was that the equations are actuallytractable in the time-independent case. Also, ouranalysis showed that these equations have a novelrelation to two-dimensional conformal field theoryand integrable spin systems.

But again, there is something interesting to say thatis much more straightforward. (This will lead to themain result of my recent paper on “Fivebranes AndKnots.”)

We just compactify one of the directionstransverse to the D3-D5 system on a circle. Thenwe apply T -duality. The D3-D5 system becomes aD4-D6 system. The D3-brane gauge theory isreplaced by a D4-brane gauge theory and now wecan calculate the path integral, if we are so inclined,by counting solutions of some supersymmetricequations in five dimensions instead of fourdimensions.

But again, there is something interesting to say thatis much more straightforward. (This will lead to themain result of my recent paper on “Fivebranes AndKnots.”) We just compactify one of the directionstransverse to the D3-D5 system on a circle.

Thenwe apply T -duality. The D3-D5 system becomes aD4-D6 system. The D3-brane gauge theory isreplaced by a D4-brane gauge theory and now wecan calculate the path integral, if we are so inclined,by counting solutions of some supersymmetricequations in five dimensions instead of fourdimensions.

But again, there is something interesting to say thatis much more straightforward. (This will lead to themain result of my recent paper on “Fivebranes AndKnots.”) We just compactify one of the directionstransverse to the D3-D5 system on a circle. Thenwe apply T -duality. The D3-D5 system becomes aD4-D6 system.

The D3-brane gauge theory isreplaced by a D4-brane gauge theory and now wecan calculate the path integral, if we are so inclined,by counting solutions of some supersymmetricequations in five dimensions instead of fourdimensions.

But again, there is something interesting to say thatis much more straightforward. (This will lead to themain result of my recent paper on “Fivebranes AndKnots.”) We just compactify one of the directionstransverse to the D3-D5 system on a circle. Thenwe apply T -duality. The D3-D5 system becomes aD4-D6 system. The D3-brane gauge theory isreplaced by a D4-brane gauge theory and now wecan calculate the path integral, if we are so inclined,by counting solutions of some supersymmetricequations in five dimensions instead of fourdimensions.

In the D4-brane description, the knot is stillrepresented by an ’t Hooft operator (which now issupported on K × S1, where S1 is the circle thatwas generated by the T -duality).

What do we gain by introducing a fifth dimension?

Since the D3-branes lived on R3 × R+. theD4-branes live on R3 × R+ × S1. Just focus on thefact that there is now a circle factor. A pathintegral on a circle gives a trace or in thesupersymmetric context a supertrace. So if we writeH for the space of physical states (the cohomologyof the supercharge Q) in quantization of theD4-brane system on R3 × R+, then the partitionfunction is a trace or more exactly an index:

Z = Tr (−1)FqP ,

where P is the instanton number.

What do we gain by introducing a fifth dimension?Since the D3-branes lived on R3 × R+. theD4-branes live on R3 × R+ × S1.

Just focus on thefact that there is now a circle factor. A pathintegral on a circle gives a trace or in thesupersymmetric context a supertrace. So if we writeH for the space of physical states (the cohomologyof the supercharge Q) in quantization of theD4-brane system on R3 × R+, then the partitionfunction is a trace or more exactly an index:

Z = Tr (−1)FqP ,

where P is the instanton number.

What do we gain by introducing a fifth dimension?Since the D3-branes lived on R3 × R+. theD4-branes live on R3 × R+ × S1. Just focus on thefact that there is now a circle factor.

A pathintegral on a circle gives a trace or in thesupersymmetric context a supertrace. So if we writeH for the space of physical states (the cohomologyof the supercharge Q) in quantization of theD4-brane system on R3 × R+, then the partitionfunction is a trace or more exactly an index:

Z = Tr (−1)FqP ,

where P is the instanton number.

What do we gain by introducing a fifth dimension?Since the D3-branes lived on R3 × R+. theD4-branes live on R3 × R+ × S1. Just focus on thefact that there is now a circle factor. A pathintegral on a circle gives a trace or in thesupersymmetric context a supertrace.

So if we writeH for the space of physical states (the cohomologyof the supercharge Q) in quantization of theD4-brane system on R3 × R+, then the partitionfunction is a trace or more exactly an index:

Z = Tr (−1)FqP ,

where P is the instanton number.

What do we gain by introducing a fifth dimension?Since the D3-branes lived on R3 × R+. theD4-branes live on R3 × R+ × S1. Just focus on thefact that there is now a circle factor. A pathintegral on a circle gives a trace or in thesupersymmetric context a supertrace. So if we writeH for the space of physical states (the cohomologyof the supercharge Q) in quantization of theD4-brane system on R3 × R+, then the partitionfunction is a trace or more exactly an index:

Z = Tr (−1)FqP ,

where P is the instanton number.

As always, we lose information if we take an index,since some states cancel out of the index.

(Also, the states carry an integer-valued fermionnumber F , and the index only depends on (−1)F ,i.e. on the values of F mod 2. And introducing afifth dimension lets us consider more general ’tHooft operators.)

So there is a more powerful theory: we just studythe space H of physical states, instead of the index.

As always, we lose information if we take an index,since some states cancel out of the index.

(Also, the states carry an integer-valued fermionnumber F , and the index only depends on (−1)F ,i.e. on the values of F mod 2. And introducing afifth dimension lets us consider more general ’tHooft operators.)

So there is a more powerful theory: we just studythe space H of physical states, instead of the index.

As always, we lose information if we take an index,since some states cancel out of the index.

(Also, the states carry an integer-valued fermionnumber F , and the index only depends on (−1)F ,i.e. on the values of F mod 2. And introducing afifth dimension lets us consider more general ’tHooft operators.)

So there is a more powerful theory: we just studythe space H of physical states, instead of the index.

Thus, Chern-Simons theory can be derived from amore powerful theory by taking an index.

The morepowerful theory is known as Khovanov homology(developed in 2000 following ideas of I. Frenkel, andfirst interpreted physically by Gukov, Schwarz, andVafa). It is known that this more powerful theorycontains significantly more information about knots.That would be interesting to discuss, but asphysicists, there is something else we shouldconsider.

Thus, Chern-Simons theory can be derived from amore powerful theory by taking an index. The morepowerful theory is known as Khovanov homology(developed in 2000 following ideas of I. Frenkel, andfirst interpreted physically by Gukov, Schwarz, andVafa).

It is known that this more powerful theorycontains significantly more information about knots.That would be interesting to discuss, but asphysicists, there is something else we shouldconsider.

Thus, Chern-Simons theory can be derived from amore powerful theory by taking an index. The morepowerful theory is known as Khovanov homology(developed in 2000 following ideas of I. Frenkel, andfirst interpreted physically by Gukov, Schwarz, andVafa). It is known that this more powerful theorycontains significantly more information about knots.

That would be interesting to discuss, but asphysicists, there is something else we shouldconsider.

Thus, Chern-Simons theory can be derived from amore powerful theory by taking an index. The morepowerful theory is known as Khovanov homology(developed in 2000 following ideas of I. Frenkel, andfirst interpreted physically by Gukov, Schwarz, andVafa). It is known that this more powerful theorycontains significantly more information about knots.That would be interesting to discuss, but asphysicists, there is something else we shouldconsider.

The D4-brane gauge theory isn’t ultravioletcomplete, but it has a well-known ultravioletcompletion in the M5-brane system, or more exactlyin the six-dimensional (0,2) superconformal fieldtheory. The whole construction can be usefullyexpressed in six-dimensional terms. The basic ideahere is that one just replaces the half-line R+ of theD4-brane worldvolume by a copy of R2 with a“cigar”-like metric:

The D4-brane gauge theory isn’t ultravioletcomplete, but it has a well-known ultravioletcompletion in the M5-brane system, or more exactlyin the six-dimensional (0,2) superconformal fieldtheory. The whole construction can be usefullyexpressed in six-dimensional terms. The basic ideahere is that one just replaces the half-line R+ of theD4-brane worldvolume by a copy of R2 with a“cigar”-like metric:

The cigar, which I will call D, is a cylinder ofrevolution. If one reduces the M5-brane theory onthe U(1) orbits, the M5-brane theory is replaced bya D4-brane theory, and D is replaced by

D/U(1) = R+.

This leads to the R+ factor in the D3-NS5, D3-D5,and D4-D6 descriptions.

The cigar, which I will call D, is a cylinder ofrevolution. If one reduces the M5-brane theory onthe U(1) orbits, the M5-brane theory is replaced bya D4-brane theory, and D is replaced by

D/U(1) = R+.

This leads to the R+ factor in the D3-NS5, D3-D5,and D4-D6 descriptions.

To a quantum field theorist, the M5-branedescription is the most perfect one, but the farthestfrom Chern-Simons theory.

To get back toChern-Simons theory, one reverses all the steps:Reduce on the U(1) orbits to replace D by R+ andM5-branes by D4-branes; then compactify on acircle to replace D4-branes by D3-branes (at thecost of losing some information); apply S-duality toget to a D3-NS5 system; and finally usesupersymmetric localization to get to a purelythree-dimensional description in Chern-Simonstheory. All steps are based on completely standardideas except the last.

To a quantum field theorist, the M5-branedescription is the most perfect one, but the farthestfrom Chern-Simons theory. To get back toChern-Simons theory, one reverses all the steps:Reduce on the U(1) orbits to replace D by R+ andM5-branes by D4-branes;

then compactify on acircle to replace D4-branes by D3-branes (at thecost of losing some information); apply S-duality toget to a D3-NS5 system; and finally usesupersymmetric localization to get to a purelythree-dimensional description in Chern-Simonstheory. All steps are based on completely standardideas except the last.

To a quantum field theorist, the M5-branedescription is the most perfect one, but the farthestfrom Chern-Simons theory. To get back toChern-Simons theory, one reverses all the steps:Reduce on the U(1) orbits to replace D by R+ andM5-branes by D4-branes; then compactify on acircle to replace D4-branes by D3-branes (at thecost of losing some information);

apply S-duality toget to a D3-NS5 system; and finally usesupersymmetric localization to get to a purelythree-dimensional description in Chern-Simonstheory. All steps are based on completely standardideas except the last.

To a quantum field theorist, the M5-branedescription is the most perfect one, but the farthestfrom Chern-Simons theory. To get back toChern-Simons theory, one reverses all the steps:Reduce on the U(1) orbits to replace D by R+ andM5-branes by D4-branes; then compactify on acircle to replace D4-branes by D3-branes (at thecost of losing some information); apply S-duality toget to a D3-NS5 system;

and finally usesupersymmetric localization to get to a purelythree-dimensional description in Chern-Simonstheory. All steps are based on completely standardideas except the last.

To a quantum field theorist, the M5-branedescription is the most perfect one, but the farthestfrom Chern-Simons theory. To get back toChern-Simons theory, one reverses all the steps:Reduce on the U(1) orbits to replace D by R+ andM5-branes by D4-branes; then compactify on acircle to replace D4-branes by D3-branes (at thecost of losing some information); apply S-duality toget to a D3-NS5 system; and finally usesupersymmetric localization to get to a purelythree-dimensional description in Chern-Simonstheory.

All steps are based on completely standardideas except the last.

To a quantum field theorist, the M5-branedescription is the most perfect one, but the farthestfrom Chern-Simons theory. To get back toChern-Simons theory, one reverses all the steps:Reduce on the U(1) orbits to replace D by R+ andM5-branes by D4-branes; then compactify on acircle to replace D4-branes by D3-branes (at thecost of losing some information); apply S-duality toget to a D3-NS5 system; and finally usesupersymmetric localization to get to a purelythree-dimensional description in Chern-Simonstheory. All steps are based on completely standardideas except the last.