Vertex operators in AdS |
and Koszul duality |
Andrei Mikhailov, IFT UNESP
Based on http://arxiv.org/abs/1207.2441
Passa Quatro, August 2012
1 Introduction
Theoretical physics has traditionally been very close to geometry.
But more recently it appears that our mathematical guide should be algebra, not geometry. It seems that geometrical constructions (such as smooth manifold in its traditional definition) are an artifact of taking the classical limit. Quantum mechanics, on the other hand, is based on algebra.
It would seem that some theoretical physics constructions could become more useful if translated into algebraic language.
Algebraically speaking, what is smooth manifold? Smooth manifold can be identified as a commutative algebra, namely the algebra of functions on this manifold. For exampleis identified with the algebra formed by two letters
and
with the relation:
Now, we can deform this algebra. For example:This is the noncommutative. We can ask ourselves how various geometrical constructions generalize to the noncommutative case.
In my talk I will discuss the technique which evolved around the noncommutative generalization of the de Rham differential. The commutative de Rham differencial is:
But what happens if we consider noncommutative? How to generalize
, and what would be its properties?
It turns out that this question is quite relevant to modern string worldsheet theory (Berkovits formalism).
2 Plan of the talk
Free algebras and algebras with quadratic relations
Koszul duality, Koszul differential
Relation to pure spinor formalism:
Pure spinors as Koszul algebra
BRST cohomology vs Lie algebra cohomology
Application to vertex operators in AdS and flat space
3 Free algebras
Algebra is a linear space
with a multiplication operation.
Example: tensor algebra, also called free algebra:
Take a linear space
and consider the space of all (finite) linear combinations of expressions of the form:
where
The notationimplies that
and
This is called tensor algebra, and denoted
4 Quadratic algebras
Quadratic algebras are tensor algebras with quadratic relations. This means that we consider the factorspace:
(quadratic relations)
What does this mean? Let us introduce the basisin
. Quadratic relations are defined by a set of constant matrices
:
That is, all the expressions of the form
are considered zero.
A standard notation for this algebra is:
5 Examples of quadratic algebras
Example 1: suppose thatrun over all antisymmetric matrices. In this case the resulting quadratic algebra
is just the algebra of polynomials of
variables, where
, or equivalently the algebra of symmetric tensors. Another name for it is “symmetric algebra”:
Example 2:runs over all symmetric matrices. In this case
is the exterior algebra
:
This is the algebra of antisymmetric tensors.
6 Dual algebra
We have just defined a quadratic algebra
.
Now we will define the dual algebra, which is denoted
.
Consider the linear spacewhich is dual to
:
As we introduced the basisin
, we can introduce the dual basis
in
. “Dual basis” means:
![]()
Remember thatwas defined using the set of constant matrices
, which were used to define the quadratic constraints (3). Consider the linear space of all the matrices
orthogonal to every
. “Orthogonal” means that
satisfies the equation:
Suppose that,
is a set of linearly independent solutions of this equation. Let us consider:
whereis generated by the tensors of the form:
Conclusion: We started from a quadratic algebra
and constructed another quadratic algebra
, which is called quadratic dual to
.
7 Example of mutually dual algebras
Example: We have already explained that the symmetric algebra
and exterior algebra
are particular cases of quadratic algebras. In fact, they are Koszul dual to each other:
Notice that the operation of “Koszul duality” applied twice brings us back to
:
In particular, we can also say:
8 Koszul Complex
Consider the following space:
- the space of linear maps fromto
.
It turns out that
naturally comes with some nilpotent operator
, which is a noncommutative generalization of the de Rham complex.
We will explain this using the particular example (8), (9) of dual algebras; in that particular case
is exactly the de Rham differential.
9 De Rham complex
As a partucular example, let us consider,
. In this case we get:
This is the de Rham complex for the flat space.
Explanation:
Ouris an
-dimensional vector space. Let us introduce a basis and call the basis elements
:
(this is just a notation, just use letterinstead of
). What is
? It is the space of polynomials of
, i.e.:
And what is the dual space? It is the space of “Taylor series”:
The dual spacehas the following basis:
Finally, what is? Here it is:
The De Rham differential acts as usual:
This operator is nilpotent. Moreover, it provides the resolution ofin the sense that the following complex:
is exact.Mathematically speaking, the de Rham complex provides:
An injective resolution of the
module
10 Koszul complex
The general Koszul complex is a more or less direct generalization of this construction.
generalizes to
Elements of
can be written as
. We have to explain how
acts on
. For that we need to remember that
and
Let us introduce a basisin
and a dual basis
in
; the “dual basis” means that:
We define:One can see that
11 Koszul complex
Notice that
and
are both graded algebras, therefore we can define
,
,
. Consider the complex:
It consists of injective modules, but is it exact?
The question about exactness is a nontrivial one. It depends on
.
Mathematicians have tools to address this question. (The theory of quadratic algebras.)
If (18) is exact, then the algebra
is called a “Koszul algebra”. If
is Koszul, then
is also Koszul, and vise versa. Duality of quadratic algebras is symmetric.
12 Koszul duality and string theory
We will now turn to string theory.
Unfortunately I do not have time to discuss the Berkovits formalism for the Type IIB superstring theory.
If you know this formalism, you will recognize the structures which I will discuss.
If not, I will try to make sure that at least the general physical meaning be understandable.
13 Pure spinors as a Koszul algebra
It turns out that the algebra of pure spinors in 10D:is Koszul.The dual algebra is the algebra of covariant derivatives of the 10-dimensional supersymmetric Yang-Mills theory:It is therefore also Koszul.We will denote:In this caseis the universal enveloping algebra:
whereis the Lie algebra formed by nested commutators of the letters
.
14 Koszul dual of a commutative algebra
General observation:
Consider the case when
is a commutative algebra. This means that
includes all antisymmetric tensors.
⇒
is a universal enveloping of a Lie algebra.
Example: If
is the algebra of polynomials of
variables
, then the dual Lie algebra is the algebra of antisymmetric polynomials
. This is the universal enveloping of the Lie superalgebra
.
15 BRST complex and Lie algebra cohomology
Main idea
The BRST complex, as it arizes in the pure spinor string theory, involves a representationof the algebra
(the algebra of covariant derivatives). The BRST operator
acts on the space of functions of
, taking values in
:
It acts onin the following way:
The exactness of the sequence (18) implies that the cohomology of (24) in fact coincides with the Lie algebra cohomology of the Lie algebra
(22):
16 Lie algebra cohomology
Lie algebra cohomology is well-known to physicists. Suppose thatis a representation of the Lie algebra
, with generators
. The Lie-algebraic (usually called Serre-Hochschild) cohomology
is the cohomology of the Faddeev-Popov operator:
17 Example: Maxwell theory
For example, consider the supersymmetric Maxwell theory in 10D in flat space. In the coset space approach, the flat space can be described as a group manifold of the Lie group
, which is generated by supersymmetries plus Poincare translations.
Solutions of the equations of motion correspond to the ghost number 1 vertex operators. The BRST complex is based on the functions ofand
:
![]()
The space of functions(or, rather, the Taylor series) is dual to the universal enveloping algebra
.
This is true for any group
with the Lie algebra
. The universal enveloping algebra can be identified with the left-invariant differential operators on
of finite order, i.e. expressions of the form:
![]()
where. Then given a function
on
, we can compute the derivative of
at the unit
:
![]()
This requires the knowledge of the coefficients of the Taylor series ofat
.
Therefore we observe the duality relation between the functions onand the elements of the universal enveloping algebra
.
Therefore in this case:Then Koszul duality (25) tells us that the space of vertex operators is:The relation betweenand
is the following:
whereis some ideal.
This is just to say that the
algebra “is a solution” to the Yang-Mills constraint:
(*)
in the following sense. If we put—
the generator of the supersymmetry transformation, then the constraint is satisfied. The definition (20), (22) of is such that
is the most general (“universal”) Lie algebra satisfying this constraint; therefore any algebra satisfying the Yang-Mills constraint should be a factoralgebra of
by some ideal.
The idealcan be described rather explicitly, in the following manner. Because of the quadratic relation (*),
is proportional to
:
(this is the definition of). It turns out, as a consequence of (*), that:
So defined
generates the ideal
![]()
Simply put, this ideal consists of the elements of the algebra
which vanish in the vacuum.
18 Using the Shapiro's lemma
One can show that:In particular, the vertex operators (corresponding to) are:
This formula may seem mysterious, but it has a transparent physical meaning. Indeed, it implies that the space of solutions of super-Maxwell equations is dual (as a linear space) to the space generated byand its derivatives
.
This is what we expect:
and its derivatives exhaust the gauge-invariant operators, i.e. gauge invariant linear functionals on the space of Maxwell solutions. Therefore:
This approach leads to the classification of gauge-invariant operators
Notice thatautomatically satisfies the Dirac equation:
19 Type IIB
We will now apply this method to Type IIB SUGRA.
We do not know what to do for a generic nonlinear SUGRA solution. This method (at least in its present form) can only be applied to the study of linearized fluctuations around the homogeneous space. But even this is technically nontrivial.
We will study the linearized excitations around a fixed background:
flat space
20 BRST complex
In Type IIB theory, there are two pure spinorsand
.
whereare the structure constants of
,
The BRST complex computing supergravity excitations on the backgroundis:
whereis the space of polynomials functions of the order
of two independent pure spinors
and
:
The space
is the space of functions of
and pure spinors
, more precisely: Taylor series in
and polynomials of the order
in
.
Theis given by:
Hereis the left multiplication by
.
21 BRST complex
We want to have some description similar to Eq. (32) for Maxwell.
In Type IIB the role ofis played by the Ramond-Ramond bispinor superfield
. In flat space one can think about
as the product of left and right
’s:
(but there are no such things asand
).
As we explained, in Maxwell theory the superfield
was interpreted as the generator of the ideal
, more precisely of its “abelianization”
. Is there a similar interpretation in Type IIB?
It turns out that in Type IIB the situation is more involved.
We will answer this question for Type IIB in
and in flat space. We will find an interpretation of the Type IIB gauge invariant operators in terms of an ideal in some infinite-dimensional Lie superalgebra which we will call
.
But:
will not be a quadratic Lie algebra
this interpretation will come with a puzzle
22 Our stratedy
Interpret the standard BRST complex as a Lie algebra cohomology complex for some infinite-dimensional Lie algebra; actually we will need the so-called relative Lie algebra cohomology
This relative cohomology can be interpreted as the cohomology (usual, not relative; and with trivial coefficients) of some ideal, which is defined similarly to the ideal
of the Maxwell theory
We can do this for bothand flat space.
23 Definition of the Lie algebra
We will start with
. We define the Lie algebra
as a sum of two copies of
, plus some finite-dimensional Lie algebra
(as linear spaces):
which are all “glued together” as follows:
whereis the structure constants of
—
the algebra of supersymmetries of . Comments:
Our
is not a quadratic algebra (because of Eqs. (A), (B) and (C))
Eq. (*) defines a quadratic subalgebra which we call
, and Eq. (**) defines
Each
and
is the same as the Yang-Mills (or Maxwell) algebra (20)
The rest of relations tell us how to “glue together”
and
Eq. (C) tells us that the generators
form
—
rotations around a point in
24 Relative cohomology
Consider any representation
of
. It is also a representation of
:
is a representation of
![]()
![]()
is a representation of
![]()
Indeed, let us remember the structure of the
. The generators are:
![]()
The subalgebrais generated by
. To have a representation
of
means to have operators
,
,
,
acting in the linear space
.
We then define the actionof
in
in terms of the action of
:
This is consistent. In fact, there is an ideal
such that
We definedso that
.
25 Relative cohomology
Let us consider the BRST cohomology:where
. The cohomology of this complex will be called:
On the other hand, consider the relative cohomology group:We claim that (42) coincides with (41):
In fact (41) is the usual (slightly generalized) BRST cohomology of the Berkovits formalism.
And (42) is a new description.
Let me explain why (41) is the usual BRST cohomology. Let us take as
the space dual to the universal enveloping
:
On the RHS, we haveacting on the space:
This is the space of sections (or rather, Taylor series of sections) of the pure spinor bundle over(was explained here). In other words, this is the “usual” BRST complex on the pure spinor worldsheet.
And now let us see what we have on the left hand side of (43).
26 Relation to the cohomology of the ideal
We now look at the relative Lie algebra cohomology:
There is an idealsuch that:
This ideal consists of those combinations of the covariant derivatives which vanish in the vacuum. It turns out that the relative cohomology (46) coincides with the cohomology of
:
The proof is in the paper.We will therefore proceed to study
.
27 Ghost number 1
The elements ofcorrespond to the global symmetry currents of the
-model. There are finitely many global symmetries. We have:
We will now show thatis a finite-dimensional representation of
, actually the adjoint representation of
. How do we see this?
As a representation of,
is generated by the following objects:
Remember
are defined from
and similarly
.
The bar over indexmeans:
We see that
is a finite-dimensional space.
Why there are no more elements? For example, let us consider
. Modulo
this is same as
, and using Eq. (39) this is proportional to
.
Similarly, all the other covariant derivatives of every element of (50) can be expressed as a linear sum of other elements. Therefore, they form a closed representation of
.
28 Ghost number 2
Ghost number 2 should correspond to the physical vertex operators:
We have only studied them in flat space.
As we will explain, there is a puzzle even in flat space.
It is more convenient to study the homology, instead of the cohomology. The relation between them is the Poincare duality:
The homology has a very straightforward elementary description. It consists of the elements of the form:
29 Ghost number 2
As in the case of Maxwell theory, elements of
should correspond to the gauge invariant operators in Type IIB theory.
Let us consider the case of flat space. The simplest thing we can reproduce is the Ramond-Ramond bispinor:
Here
is defined using the “left” part
of
—
see Eq. (38). It automatically satisfies the Dirac equation, because and
satisfy the Dirac equations (33).
30 Ghost number 2
The NSNS 3-form field strength should correspond to:It turns out that the linearized SUGRA equations of motion are not satisfied, because. However, the derivatives of
are all zero:
therefore this is a “zero mode effect”.This is a puzzle. Most likely, the zero modes are not correctly reproduced in the current version of the pure spinor formalism. The disagreement in the zero mode sector was discussed in arxiv:1005.0049 and arxiv:1203.0677. It would be useful to explicitly trace the connection between the nonzero
and the “non-physical” vertex found in those papers.
31 Conclusions
The pure spinor formalism is now well-developed. But the Koszul duality was not widely used, at least not in the string theory literature. We tried to study this duality by applying it to the Type IIB supergravity, more specifically to the classification of the linearized excitations of homogeneous spaces. We had to invent a modification of the duality, which involves the “left” and “right” quadratic algebras glued together in some special way. This gives an elegant description of the ghost number 1 operators, and a puzzle for the ghost number 2 operators.
More research is needed in this direction.